{"commit":"715add8e9475f5193526bce0c26baef559a47b68","rows":[{"n":"Causalean.DAG","k":"structure","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"A Directed Acyclic Graph on a finite vertex type: a decidable edge relation together with the condition that no vertex is connected to itself by a directed path — the transitive closure of the edge relation is irreflexive. Irreflexivity of the transitive closure is exactly the statement that the graph has no directed cycle.","s":"(V : Type u_2) → [DecidableEq V] → [Fintype V] → Type u_2","l":51,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.BBDir","k":"inductive","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"The Bayes Ball arrival-direction type has exactly two cases: arrival at a vertex from one of its parents and arrival at a vertex from one of its children.","s":"Type","l":54,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.BBState","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a vertex population, a Bayes Ball state is a vertex together with an indication of whether the ball arrived from a parent or from a child.","s":"Type u_2 → Type (max u_2 0)","l":65,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.HasActivePath","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population, a source set, a target set, and a conditioning set, the active-path existence condition holds exactly when there is a vertex sequence with at least two vertices that is active relative to the conditioning set, whose first vertex belongs to the source set, and whose last vertex belongs to the target set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":94,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsActivePath","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population, a conditioning set, and a finite vertex sequence, the active-path condition holds exactly when each consecutive pair is joined by a directed edge in one direction or the other, and at every consecutive triple, its middle vertex is a collider that is either conditioned on or has a conditioned descendant, or else is a non-collider outside the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → List V → Prop","l":74,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsCollider","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population and three ordered vertices, the collider condition holds exactly when both outer vertices have directed edges into the middle vertex.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → Prop","l":67,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.IsCoveredEdge","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population and two vertices, the covered-edge condition holds exactly when there is a directed edge from the first vertex to the second and, for every other vertex, that vertex has an edge into the first if and only if it has an edge into the second.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":39,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.IsCoveredEdge.ne","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A covered edge is genuinely an edge a → b, so a ≠ b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V}, G.IsCoveredEdge a b → a ≠ b","l":65,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.IsImmorality","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population and three ordered vertices, the v-structure, or immorality, condition holds exactly when the first and third vertices are distinct, each has a directed edge into the second, and no directed edge joins the first and third in either direction.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → Prop","l":39,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.DAG.MoralAdj","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population, a ground set of vertices, and two vertices, moral adjacency holds exactly when the vertices are distinct members of the ground set and either a directed edge joins them in one direction or the other, or they have a common child in the ground set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → V → V → Prop","l":40,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralConn","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population, a ground set, a conditioning set, and two vertices, moral connectivity holds exactly when the first vertex reaches the second by a possibly empty sequence of moral-adjacent steps within the ground set, each of whose endpoints lies outside the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → V → V → Prop","l":50,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralSep","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population, a first vertex set, a second vertex set, and a conditioning set, moral separation holds exactly when no member of the first set is morally connected to any member of the second within the ancestral closure of the union of all three sets, while avoiding the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":54,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.MoralStep","k":"def","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population, a ground set, a conditioning set, and two vertices, a moral step holds exactly when the vertices are morally adjacent in the ground set and neither belongs to the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → V → V → Prop","l":46,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.OrderedLocalSG","k":"inductive","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":1,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, and an ambient set of random vertices, the ordered-local semi-graphoid closure contains precisely the triples of finite vertex sets that can be derived as follows: the empty first set is independent of any second and conditioning sets contained in the ambient set; a vertex in the ambient set is independent of a set of its non-descendants, excluding its ambient parents, conditional on those ambient parents, whenever that set contains those parents; and the relation is closed under symmetry, decomposition, weak union, and contraction.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Finset V → Prop","l":37,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.OrderedLocalSG.subset_random","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every triple appearing in an ordered-local derivation has all three sets contained in the ambient random set R. Used by the SCM interpretation to recover the ⊆ randomVars side-conditions of FullCondIndep.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {R X Y Z : Finset V}, G.OrderedLocalSG R X Y Z → X ⊆ R ∧ Y ⊆ R ∧ Z ⊆ R","l":63,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.UAdj","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population and two vertices, undirected adjacency holds exactly when a directed edge joins the two vertices in either direction.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":60,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.UAdj_symm","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Undirected adjacency is symmetric: if two vertices are adjacent, they remain adjacent in the opposite order.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.UAdj u v → G.UAdj v u","l":103,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.activePath_drop2","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Removing the first two vertices of an active path with at least three vertices leaves an active path.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b c : V} {r : List V}, G.IsActivePath Z (a :: b :: c :: r) → G.IsActivePath Z (c :: r)","l":116,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_forwardRun_absurd","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"An active path that starts with an arrow flowing away from an ancestor cannot end at a root when that ancestor has no directed route to any activated conditioning ancestor. The result rules out a forward run that must either enter a root or create an activated collider.","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {s : V}, (∀ k ∈ G.bbZAncestors C, ¬G.isAncestor s k) → ∀ (prev m : V) (rest : List V), G.IsActivePath C (prev :: m :: rest) → G.edge prev m → s = prev ∨ G.isAncestor s prev → (∀ (p : V), ¬G.edge p ((prev :: m :: rest).getLast (of_eq_true (Eq.trans (congrArg Not (eq_false' fun h => False.elim (noConfusion_of_Nat List.ctorIdx h))) not_false_eq_true)))) → False","l":357,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.DAG.activePath_head_triple","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"In an active path whose first three vertices are a, b, and c, the middle vertex obeys the active-path condition: a collider belongs to the ancestral closure of the conditioning set, and a non-collider is not conditioned on.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b c : V} {r : List V}, G.IsActivePath Z (a :: b :: c :: r) → if G.IsCollider a b c then b ∈ G.bbZAncestors Z else b ∉ Z","l":107,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_head_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"In an active path whose first two vertices are a and b, those vertices are adjacent in the underlying undirected graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b : V} {r : List V}, G.IsActivePath Z (a :: b :: r) → G.UAdj a b","l":100,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.activePath_join_at_collider","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Two active paths that meet at a conditioned collider can be joined into an active path when both incident edges point into the joining vertex.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {n x y : V} {pa pb : List V} (hpa_len : pa.length ≥ 2), pa.head? = some x → pa.getLast? = some n → G.IsActivePath Z pa → ∀ (hpb_len : pb.length ≥ 2), pb.head? = some n → pb.getLast? = some y → G.IsActivePath Z pb → G.edge (pa.get ⟨pa.length - 2, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_1 hpa_len a⟩) n → G.edge (pb.get ⟨1, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_2 hpb_len a⟩) n → n ∈ Z → have p := pa ++ pb.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":422,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.activePath_nodes_are_ancestors","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":1,"d":"Active-path nodes lie in the ancestral set. For finite vertex sets X, Y, Z, consider a path p that is active given Z running from a node x in X to a node y in Y. Then every vertex on p lies in the ancestral set of X ∪ Y ∪ Z. This is the main classical lemma used to justify ancestral reduction of d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} {x y : V} {p : List V}, x ∈ X → y ∈ Y → G.IsActivePath Z p → p.head? = some x → p.getLast? = some y → ∀ v ∈ p, v ∈ G.ancestralSet (X ∪ Y ∪ Z)","l":319,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.activePath_transfer_cond_to_source","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"An active path under an enlarged conditioning set can be replaced by an active path whose source may additionally come from the newly conditioned vertices.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Z S : Finset V} {p : List V} {x w : V}, x ∈ X → p.length ≥ 2 → G.IsActivePath (Z ∪ S) p → p.head? = some x → p.getLast? = some w → ∃ x' p', x' ∈ X ∪ S ∧ p'.length ≥ 2 ∧ G.IsActivePath Z p' ∧ p'.head? = some x' ∧ p'.getLast? = some w","l":216,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.activePath_witness_subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Active-path inner nodes lie in the ancestral set (informative form).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} {v : V}, v ∈ G.bbReachableVertices Z X → v ∈ Y → ∃ p, ∃ x ∈ X, p.length ≥ 2 ∧ G.IsActivePath Z p ∧ p.head? = some x ∧ p.getLast? = some v ∧ ∀ w ∈ p, w ∈ G.ancestralSet (X ∪ Y ∪ Z)","l":356,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.active_triple_swap_outer","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"The active-path condition at a three-vertex segment is unchanged when its two outer vertices are swapped.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u w z : V}, (if G.IsCollider u w z then w ∈ G.bbZAncestors Z else w ∉ Z) → if G.IsCollider z w u then w ∈ G.bbZAncestors Z else w ∉ Z","l":546,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.acyclic_of_topoOrder","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"Acyclicity from a topological ranking. Given an edge relation e on V and a ranking function τ into a type equipped with a transitive, irreflexive relation r, if τ strictly increases (with respect to r) along every edge of e, then e has no directed cycle: no vertex is reachable from itself via the transitive closure of e.","s":"∀ {V : Type u_2} {W : Type u_3} {r : W → W → Prop} [IsTrans W r] [Std.Irrefl r] {e : V → V → Prop} {τ : V → W}, (∀ (u v : V), e u v → r (τ u) (τ v)) → ∀ (v : V), ¬Relation.TransGen e v v","l":442,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.adjacent_iff_not_dSeparable","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Skeleton read-off. For two distinct vertices a and b, a and b are adjacent (in either direction) exactly when no conditioning set d-separates a from b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → (G.UAdj a b ↔ ¬∃ Z, G.dSep {a} {b} Z)","l":178,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.ancClosure","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the strict-ancestor set is obtained by starting with the vertex’s parents and applying backward ancestor expansion once for each vertex in the population.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":165,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancClosure_closed","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Every parent of a vertex in its computed ancestor set also belongs to that ancestor set.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V) {x : V}, x ∈ G.ancClosure v → G.parents x ⊆ G.ancClosure v","l":261,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancStep","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a finite vertex set, one backward ancestor-expansion step returns that set together with every parent of every member of the set.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":162,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorRank","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, its ancestor rank is the number of that vertex’s strict ancestors.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → ℕ","l":356,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorRank_lt_of_edge","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Along an edge the strict-ancestor count strictly increases.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, G.edge a b → G.ancestorRank a < G.ancestorRank b","l":359,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestors","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the ancestor set is the finite set of all vertices from which a directed path reaches that vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":304,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestorsSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a finite vertex set, the set of its strict ancestors contains exactly the vertices from which a directed path reaches at least one member of the given set.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":336,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestralSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a finite vertex set, its ancestral closure is that set together with every vertex from which a directed path reaches one of its members.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":340,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.ancestralSet_cases","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A vertex in the ancestral closure of a set either belongs to that set itself or is a strict ancestor of one of its elements.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {A : Finset V} {a : V}, a ∈ G.ancestralSet A → a ∈ A ∨ ∃ w ∈ A, G.isAncestor a w","l":328,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.ancestralSet_idem","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Taking the ancestors of an ancestral set gives back the same set.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V), G.ancestralSet (G.ancestralSet S) = G.ancestralSet S","l":34,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.DAG.ancestralSet_inter_subset_ancestralSet_of_dSep","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":1,"d":"Ancestral intersection from reachability separation. If X and Y are disjoint and no vertex of Y is Bayes-Ball-reachable from X given Z, then any vertex lying in both the ancestral closure of X and the ancestral closure of Y also lies in the ancestral closure of Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V}, Disjoint X Y → Disjoint (G.bbReachableVertices Z X) Y → G.ancestralSet X ∩ G.ancestralSet Y ⊆ G.ancestralSet Z","l":730,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.ancestralSet_mono","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"ancestralSet is monotone in its argument.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S T : Finset V}, S ⊆ T → G.ancestralSet S ⊆ G.ancestralSet T","l":62,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.ancestralSet_subset_of_subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"The ancestral closure of a subset of an ancestral closure remains inside the original ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Q Q' : Finset V}, Q' ⊆ G.ancestralSet Q → G.ancestralSet Q' ⊆ G.ancestralSet Q","l":119,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.asymm","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"If there is an edge from u to v, then there is no edge from v to u (a two-cycle is forbidden by acyclicity).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.edge u v → ¬G.edge v u","l":125,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.bbInit","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population and a source set, the initial Bayes Ball frontier contains, for every source vertex, each child paired with arrival from a parent and each parent paired with arrival from a child.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset (Causalean.DAG.BBState V)","l":253,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachAux","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"Iterative Bayes Ball with a step budget. Given a conditioning set, the current frontier of ball states, the states visited so far, and a budget of expansion rounds, the fuel-bounded reachable set is obtained by returning the visited states when the budget is exhausted, and otherwise expanding the frontier by one Bayes Ball step, stopping with the visited states if no new state appears and recursing on the new states with one less round otherwise. It is the reachable set only when the budget suffices for the traversal to close, as arranged by bbReachable.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset (Causalean.DAG.BBState V) → Finset (Causalean.DAG.BBState V) → ℕ → Finset (Causalean.DAG.BBState V)","l":112,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population, a conditioning set, and a source set, the Bayes Ball reachable-state set is the set returned by repeatedly applying the one-step transition to the initial frontier, with a limit of twice the number of graph vertices plus one rounds.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset (Causalean.DAG.BBState V)","l":265,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachableVertices","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population, a conditioning set, and a source set, the Bayes Ball reachable-vertex set is the set of vertices appearing in the reachable states, with their arrival directions discarded.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V","l":345,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachableVertices_iff_activePath","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"Bayes Ball correctness. For a source vertex set X and a conditioning vertex set Z and a vertex v, v lies in the breadth-first-search reachable set bbReachableVertices Z X if and only if there is an active path, given Z, from some vertex of X to v.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Z : Finset V) (v : V), v ∈ G.bbReachableVertices Z X ↔ ∃ x ∈ X, ∃ p, p.length ≥ 2 ∧ G.IsActivePath Z p ∧ p.head? = some x ∧ p.getLast? = some v","l":706,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.bbReachableVertices_mono_source","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"Bayes Ball reachable vertices are monotone in the source set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z X X' : Finset V}, X' ⊆ X → G.bbReachableVertices Z X' ⊆ G.bbReachableVertices Z X","l":364,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_bbStep_subset","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"bbReachable Z X is closed under bbStep Z. If s ∈ bbReachable Z X, then every state produced by bbStep Z s is also in bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V) {s : Causalean.DAG.BBState V}, s ∈ G.bbReachable Z X → G.bbStep Z s ⊆ G.bbReachable Z X","l":396,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_extend_directed_arm","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Extending an active path by a directed arm at a non-collider seam.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a c b : V} {pa q : List V}, pa.length ≥ 2 → pa.head? = some a → pa.getLast? = some c → G.IsActivePath Z pa → q.length ≥ 2 → q.head? = some c → q.getLast? = some b → (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.bbReachable_extend_directed_arm._proof_1 i hi a⟩) (q.get ⟨i + 1, hi⟩)) → (∀ (i : ℕ) (hi : i + 2 < q.length), q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.bbReachable_extend_directed_arm._proof_2 i hi a⟩ ∉ Z) → c ∉ Z → b ∈ G.bbReachableVertices Z {a}","l":1686,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.bbReachable_init_subset","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"The initial BFS frontier bbInit X is contained in bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V), G.bbInit X ⊆ G.bbReachable Z X","l":374,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_minimal","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"bbReachable Z X is the *least* superset of bbInit X closed under bbStep Z: for a conditioning set Z and source set X, if a candidate set S of Bayes-Ball states contains the initial frontier bbInit X and S is closed under the Bayes-Ball step relation bbStep Z, then S contains every state reachable via bbReachable Z X.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z X : Finset V) (S : Finset (Causalean.DAG.BBState V)), G.bbInit X ⊆ S → (∀ s ∈ S, G.bbStep Z s ⊆ S) → G.bbReachable Z X ⊆ S","l":380,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbReachable_mono_source","k":"theorem","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"Bayes Ball reachability is monotone in the source set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z X X' : Finset V}, X' ⊆ X → G.bbReachable Z X' ⊆ G.bbReachable Z X","l":354,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbStep","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population, a conditioning set, and a current Bayes Ball state, writing the state as its vertex and arrival direction, the one-step successor set sends an arrival from a child at an unconditioned vertex to its parents and children, and sends such an arrival at a conditioned vertex nowhere; it sends an arrival from a parent at an unconditioned vertex to its children and, when that vertex is an ancestor of a conditioned vertex, also to its parents, while an arrival from a parent at a conditioned vertex goes to its parents.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Causalean.DAG.BBState V → Finset (Causalean.DAG.BBState V)","l":76,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbZAncestors","k":"def","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph on a vertex population and a conditioning set, the activated-collider set consists of every conditioned vertex and every ancestor of a conditioned vertex.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":70,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.bbZAncestors_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same sets of vertices that are ancestors of the conditioning set and can activate colliders.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (Z : Finset V) (v : V), v ∈ G₁.bbZAncestors Z ↔ v ∈ G₂.bbZAncestors Z","l":239,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.bbZAncestors_flip_of_ne","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"If a vertex other than the reversed tail belongs to the Bayes-ball ancestor closure of a conditioning set before a covered reversal, it belongs to that closure after the reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b v : V} (hcov : G.IsCoveredEdge a b) (Z : Finset V), v ≠ a → v ∈ G.bbZAncestors Z → v ∈ (Causalean.DAG.flipEdge hcov).bbZAncestors Z","l":445,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.bbZAncestors_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"Adding directed edges can only enlarge the set of vertices that are ancestors of the conditioning set and can activate colliders.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ (Z : Finset V), G'.bbZAncestors Z ⊆ G.bbZAncestors Z","l":704,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.bbZAncestors_subset_ancestralSet_of_subset","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"bbZAncestors Z ⊆ ancestralSet S whenever Z ⊆ S. Used to lift a collider-activation witness m ∈ bbZAncestors Z into the larger ancestralSet (X ∪ Y ∪ Z).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z S : Finset V}, Z ⊆ S → G.bbZAncestors Z ⊆ G.ancestralSet S","l":82,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.bbZAncestors_union_eq","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"Ancestral-set distributes over union.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (Z S : Finset V), G.bbZAncestors (Z ∪ S) = G.bbZAncestors Z ∪ G.bbZAncestors S","l":87,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.branch_seed_eq","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"If a finite set contains a node, removing and then reinserting that node while taking unions recovers the same union as using the original set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {a b Zr c : Finset V} {n : V}, n ∈ Zr → a ∪ {n} ∪ (Zr.erase n ∪ b) ∪ c = a ∪ b ∪ Zr ∪ c","l":1266,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.chain_join_active","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Concatenating two active paths at a non-collider. If pa is active given Z ending at m, pb is active given Z starting at m, the first edge of pb points *out of* m (so m is a chain/fork point, not a collider), and m ∉ Z, then the glued path pa ++ pb.tail is active given Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {m x y : V} {pa pb : List V}, pa.length ≥ 2 → pa.head? = some x → pa.getLast? = some m → G.IsActivePath Z pa → ∀ (hpb_len : pb.length ≥ 2), pb.head? = some m → pb.getLast? = some y → G.IsActivePath Z pb → G.edge m (pb.get ⟨1, Decidable.byContradiction fun a => Causalean.DAG.activePath_join_at_collider._proof_2 hpb_len a⟩) → m ∉ Z → have p := pa ++ pb.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":618,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.children","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the child set is the finite set of all vertices to which that vertex has a directed edge.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":85,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.children_subset_descendants","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Children are a subset of descendants.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), G.children v ⊆ G.descendants v","l":326,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.dSep","k":"def","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"For a finite directed acyclic graph, a source set, a target set, and a conditioning set, the d-separation relation holds exactly when the source and target sets are disjoint, the source and conditioning sets are disjoint, the target and conditioning sets are disjoint, and no target vertex is Bayes-Ball reachable from the source set after conditioning on the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V → Finset V → Prop","l":55,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_iff_moralSep","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":1,"d":"The moralization criterion. For pairwise-disjoint X, Y, Z, X and Y are d-separated by Z exactly when they are moral-separated: no moral path inside the ancestral set An(X ∪ Y ∪ Z) connects them while avoiding Z. (Lauritzen–Dawid–Larsen–Speed.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V}, Disjoint X Y → Disjoint X Z → Disjoint Y Z → (G.dSep X Y Z ↔ G.MoralSep X Y Z)","l":502,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.dSep_mono_conditioningSet","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"d-separation transfers from a supergraph to a subgraph. For DAGs G and G' on the same vertex set, if every edge of G' is also an edge of G — G' is obtained from G by removing edges — and X and Y are d-separated by Z in G, then X and Y are also d-separated by Z in G'.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z : Finset V} (G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → G.dSep X Y Z → G'.dSep X Y Z","l":770,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_parents_of_topoOrder_lt","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"If one endpoint precedes a nonadjacent later endpoint topologically, conditioning on the later endpoint's parents d-separates the pair.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {x y : V}, G.topoOrder x < G.topoOrder y → ¬G.UAdj x y → G.dSep {x} {y} (G.parents y)","l":57,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.dSep_source_to_cond","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"Source-to-conditioning transfer for d-separation. If X and S are disjoint and X ∪ S and Y are d-separated by Z, then X and Y remain d-separated once S is moved into the conditioning set: G.dSep X Y (Z ∪ S). Equivalently: bbReachableVertices (Z ∪ S) X is a subset of bbReachableVertices Z (X ∪ S).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z S : Finset V}, Disjoint X S → G.dSep (X ∪ S) Y Z → G.dSep X Y (Z ∪ S)","l":657,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_subset_left","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"d-separation is monotone in X: smaller source sets preserve d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X X' Y Z : Finset V}, X' ⊆ X → G.dSep X Y Z → G.dSep X' Y Z","l":73,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_subset_right","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"d-separation is monotone in Y: shrinking the target set preserves d-separation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Y' Z : Finset V}, Y' ⊆ Y → G.dSep X Y Z → G.dSep X Y' Z","l":80,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_symm","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":1,"d":"d-separation is symmetric. For three finite vertex sets X, Y, Z, if X is d-separated from Y given Z then Y is d-separated from X given Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Y Z : Finset V), G.dSep X Y Z → G.dSep Y X Z","l":801,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.dSep_union_roots_right","k":"theorem","m":"Causalean.Graph.DSep.BackdoorBridges","a":"Graph/DSep","t":1,"d":"Adding root nodes (no incoming edges) to the conditioning set preserves d-separation. In a DAG G, suppose X and Y are d-separated by Z, every vertex of R has no incoming edge in G, i.e. R consists of root vertices, and R is disjoint from X and from Y. Then X and Y remain d-separated once the root vertices R are added to the conditioning set: G.dSep X Y (Z ∪ R).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Y Z R : Finset V}, G.dSep X Y Z → (∀ r ∈ R, ∀ (u : V), ¬G.edge u r) → Disjoint R X → Disjoint R Y → G.dSep X Y (Z ∪ R)","l":51,"f":"Causalean/Graph/DSep/BackdoorBridges.lean"},{"n":"Causalean.DAG.dSeparable_disjoint_of_not_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Disjoint skeleton read-off witness. Two distinct non-adjacent vertices are d-separated by a conditioning set disjoint from both endpoints (the parents of the topologically later one, which contain neither a nor b).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → ¬G.UAdj a b → ∃ Z, a ∉ Z ∧ b ∉ Z ∧ G.dSep {a} {b} Z","l":286,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.dSeparable_of_not_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Two distinct non-adjacent vertices can always be d-separated: conditioning on the parents of the topologically later vertex blocks every path between them.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, a ≠ b → ¬G.UAdj a b → ∃ Z, G.dSep {a} {b} Z","l":164,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.decDSep","k":"instance","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, a source set, target set, and conditioning set, the decision procedure for d-separation determines whether the source and target sets are d-separated conditional on the conditioning set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (X Y Z : Finset V) → Decidable (G.dSep X Y Z)","l":63,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.decIsAncestor","k":"instance","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"For a finite vertex population with decidable equality and a directed acyclic graph on that population, the decision procedure for the ancestor relation determines, for every ordered pair of vertices, whether the first is an ancestor of the second.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → DecidableRel G.isAncestor","l":298,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.decIsCollider","k":"instance","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, and three ordered vertices, the decision procedure for the collider condition determines whether each outer vertex has a directed edge into the middle vertex.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (l m r : V) → Decidable (G.IsCollider l m r)","l":70,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.decIsRoot","k":"instance","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, and a vertex, the decision procedure for the root condition determines whether no directed edge enters that vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (v : V) → Decidable (G.isRoot v)","l":427,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.decUAdj","k":"instance","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, and two vertices, the decision procedure for undirected adjacency determines whether a directed edge joins the vertices in either direction.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (u v : V) → Decidable (G.UAdj u v)","l":63,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.descendants","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the descendant set is the finite set of all vertices reachable from that vertex by a directed path.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":308,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.descendantsSet","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a finite vertex set, the set of its strict descendants contains exactly the vertices reachable by a directed path from at least one member of the given set.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Finset V","l":344,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.edge_to_later_of_nonCollider","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"On an unshielded two-edge triple, a non-collider middle vertex points into the topologically later endpoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {x y b : V}, G.topoOrder x ≤ G.topoOrder y → G.UAdj x b → G.UAdj y b → ¬G.IsCollider x b y → G.edge b y","l":190,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.exists_activePath_of_ancestor_avoiding","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"An ancestor outside the conditioning set's ancestral closure has an active path to its descendant.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u v : V}, G.isAncestor u v → u ∉ G.ancestralSet Z → ∃ p, p.length ≥ 2 ∧ p.head? = some u ∧ p.getLast? = some v ∧ G.IsActivePath Z p","l":404,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.exists_directedPath_avoiding","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If u is an ancestor of v and u ∉ ancestralSet Z, there is a list u = p₀, p₁, …, pₖ = v (k ≥ 1) of directed edges all avoiding Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → ∀ {Z : Finset V}, u ∉ G.ancestralSet Z → ∃ p, p.length ≥ 2 ∧ p.head? = some u ∧ p.getLast? = some v ∧ (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i, Causalean.DAG.exists_directedPath_avoiding._proof_2 p i hi⟩) (p.get ⟨i + 1, hi⟩)) ∧ ∀ x ∈ p, x ∉ Z","l":111,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.flipEdge","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex population, a directed acyclic graph, two vertices, and evidence that their directed edge is covered, the covered-edge reversal graph is the directed acyclic graph obtained by replacing the edge from the first vertex to the second with the edge from the second to the first.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → {a b : V} → G.IsCoveredEdge a b → Causalean.DAG V","l":124,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"In the graph obtained by reversing a covered edge, the edges are exactly the old edges except for deleting a → b and adding b → a.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) (u w : V), (Causalean.DAG.flipEdge hcov).edge u w ↔ G.edge u w ∧ ¬(u = a ∧ w = b) ∨ u = b ∧ w = a","l":128,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_edge_or_deleted","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Under a covered reversal of the edge from a to b, every directed edge of the original graph either remains an edge or is the deleted edge from a to b.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V} (hcov : G.IsCoveredEdge a b), G.edge u w → (Causalean.DAG.flipEdge hcov).edge u w ∨ u = a ∧ w = b","l":410,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_flipEdge_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Flipping the reversed covered edge restores the original edge relation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) (u w : V), (Causalean.DAG.flipEdge (Causalean.DAG.flipEdge_isCoveredEdge_back hcov)).edge u w ↔ G.edge u w","l":354,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_isCoveredEdge_back","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"In the flipped graph, the reversed edge b → a is covered.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), (Causalean.DAG.flipEdge hcov).IsCoveredEdge b a","l":322,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge preserves the immoralities. Because a and b share all other parents, no v-structure is created or destroyed by the single reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.SameImmoralities G (Causalean.DAG.flipEdge hcov)","l":154,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipEdge_sameSkeleton","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge preserves the skeleton. The undirected adjacency is unchanged: only the orientation of the single edge a — b flips.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.SameSkeleton G (Causalean.DAG.flipEdge hcov)","l":134,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipMinus","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population and two vertices, the edge relation with one deletion holds for two queried vertices exactly when the original graph has an edge from the first queried vertex to the second and the queried edge is not the edge from the first specified vertex to the second.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → V → Prop","l":45,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipMinus_le","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A flipMinus-edge is in particular a G-edge.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V}, G.flipMinus a b u w → G.edge u w","l":55,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipRel","k":"def","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"For a finite directed acyclic graph on a vertex population and two vertices, the edge relation with one reversal holds for two queried vertices exactly when the original graph has the queried edge other than the edge from the first specified vertex to the second, or the queried edge is the reversed edge from the second specified vertex to the first.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → V → V → Prop","l":49,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.flipRel_acyclic","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Reversing a covered edge keeps the graph acyclic. The transitive closure of the flipped relation is irreflexive. Key step (AMP Lemma 3.1): a directed G-path a ⇝ b of length ≥ 2 would end at a parent c ≠ a of b, hence (covered) a parent of a, closing a G-cycle; so no such detour exists and the single reversal introduces no cycle.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V}, G.IsCoveredEdge a b → ∀ (v : V), ¬Relation.TransGen (G.flipRel a b) v v","l":69,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.fork_isActivePath","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Two directed paths from one unconditioned vertex can be joined through that fork.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u x y : V} {xp yp : List V}, xp.length ≥ 2 → xp.head? = some u → xp.getLast? = some x → (∀ (i : ℕ) (hi : i + 1 < xp.length), G.edge (xp.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.walk_right_witness._proof_1✝ i hi a⟩) (xp.get ⟨i + 1, hi⟩)) → (∀ z ∈ xp, z ∉ Z) → yp.length ≥ 2 → yp.head? = some u → yp.getLast? = some y → (∀ (i : ℕ) (hi : i + 1 < yp.length), G.edge (yp.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.fork_isActivePath._proof_1 i hi a⟩) (yp.get ⟨i + 1, hi⟩)) → (∀ z ∈ yp, z ∉ Z) → have p := xp.reverse ++ yp.tail; p.length ≥ 2 ∧ p.head? = some x ∧ p.getLast? = some y ∧ G.IsActivePath Z p","l":420,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.hasActivePath_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Equal directed-edge relations preserve existence of active paths.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (X Y Z : Finset V), G₁.HasActivePath X Y Z ↔ G₂.HasActivePath X Y Z","l":312,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.hasActivePath_symm","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"HasActivePath is symmetric in X and Y.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (X Y Z : Finset V), G.HasActivePath X Y Z → G.HasActivePath Y X Z","l":380,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.immorality_iff_colliderSep","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Collider read-off. For vertices a, b, c forming an unshielded triple: a and b adjacent, c and b adjacent, and a, c non-adjacent and distinct, the middle vertex b is a collider a → b ← c — equivalently a → b ← c is a v-structure — exactly when b belongs to no conditioning set that d-separates a and c.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b c : V}, G.UAdj a b → G.UAdj c b → ¬G.UAdj a c → a ≠ c → (G.IsImmorality a b c ↔ ∀ (Z : Finset V), b ∈ Z → ¬G.dSep {a} {c} Z)","l":212,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.immorality_iff_colliderSep_disjoint","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Disjoint collider read-off. For an unshielded triple a — b — c, b is the collider a → b ← c iff every separator of a, c that contains b and excludes a, c fails to d-separate — equivalently, b lies in no such separator. (The endpoint-disjoint form used to transport immoralities across Markov-equivalent graphs.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b c : V}, G.UAdj a b → G.UAdj c b → ¬G.UAdj a c → a ≠ c → (G.IsImmorality a b c ↔ ∀ (Z : Finset V), b ∈ Z → a ∉ Z → c ∉ Z → ¬G.dSep {a} {c} Z)","l":301,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.instDecidableEqBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"","s":"DecidableEq Causalean.DAG.BBDir","l":58,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.instDecidableIsImmorality","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"For a finite vertex population with decidable equality, a directed acyclic graph on that population, and three ordered vertices, the decision procedure for the v-structure condition determines whether the first and third vertices are distinct non-adjacent parents of the second.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → (a b c : V) → Decidable (G.IsImmorality a b c)","l":45,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.DAG.instFintypeBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"The finite enumeration of Bayes Ball arrival directions lists the two arrival directions and certifies that every arrival direction is listed.","s":"Fintype Causalean.DAG.BBDir","l":60,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.instReprBBDir","k":"instance","m":"Causalean.Graph.DSep.BayesBall","a":"Graph/DSep","t":2,"d":"","s":"Repr Causalean.DAG.BBDir","l":58,"f":"Causalean/Graph/DSep/BayesBall.lean"},{"n":"Causalean.DAG.irrefl","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"No vertex has an edge to itself (a directed self-loop would be a length-one cycle).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), ¬G.edge v v","l":120,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isActivePath_Z_of_no_S_only_collider","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If q is active given Z ∪ S and every interior collider of q is Z-activated (not merely S-activated), then q is active given Z alone.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z S : Finset V} {q : List V}, G.IsActivePath (Z ∪ S) q → (∀ (i : ℕ) (hi : i + 2 < q.length), G.IsCollider (q.get ⟨i, Causalean.DAG.IsActivePath._proof_4 q i hi⟩) (q.get ⟨i + 1, Causalean.DAG.IsActivePath._proof_6 q i hi⟩) (q.get ⟨i + 2, hi⟩) → q.get ⟨i + 1, Causalean.DAG.IsActivePath._proof_6 q i hi⟩ ∈ G.bbZAncestors Z) → G.IsActivePath Z q","l":353,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_cons_of_active_triple","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Prepending an active triple to an active path keeps the path active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {z w u : V} {r : List V}, G.UAdj z w → (if G.IsCollider z w u then w ∈ G.bbZAncestors Z else w ∉ Z) → G.IsActivePath Z (w :: u :: r) → G.IsActivePath Z (z :: w :: u :: r)","l":522,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_cons_tail","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Removing the first vertex from an active path leaves an active path.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {u w : V} {r : List V}, G.IsActivePath Z (u :: w :: r) → G.IsActivePath Z (w :: r)","l":641,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_drop","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Removing any initial segment of an active graph path leaves a path that is still active under the same conditioning set.","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {p : List V} (j : ℕ), G.IsActivePath C p → G.IsActivePath C (List.drop j p)","l":409,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.DAG.isActivePath_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same active paths for every conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ (Z : Finset V) (p : List V), G₁.IsActivePath Z p ↔ G₂.IsActivePath Z p","l":268,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isActivePath_flip_of_not_mem","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A path that is active relative to a conditioning set and does not contain the tail of a covered reversed edge remains active relative to the same set after the reversal.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b) {Z : Finset V} {p : List V}, G.IsActivePath Z p → a ∉ p → (Causalean.DAG.flipEdge hcov).IsActivePath Z p","l":526,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isActivePath_insert_cond","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Adding a conditioning vertex preserves activity when that vertex appears internally only as a collider.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {D : Finset V} {p : List V} {n : V}, G.IsActivePath D p → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_1 i hi a⟩ = n → G.IsCollider (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_2 i hi a⟩) (p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_insert_cond._proof_1 i hi a⟩) (p.get ⟨i + 2, hi⟩)) → G.IsActivePath (insert n D) p","l":253,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.isActivePath_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"An active path in a graph with fewer edges remains active when those edges are restored, so path witnesses transfer across graph transformations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {Z : Finset V} {p : List V}, G'.IsActivePath Z p → G.IsActivePath Z p","l":742,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isActivePath_of_directed","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"A *forward*-directed path whose interior vertices avoid Z is an active path given Z. \"Forward-directed\" means each edge points from the *earlier* index to the *later* index. Every interior vertex is then a non-collider (incoming + outgoing), and by the avoidance hypothesis none is in Z; endpoints may lie in Z.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.walk_right_witness._proof_1✝ i hi a⟩) (p.get ⟨i + 1, hi⟩)) → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isActivePath_of_directed._proof_1 i hi a⟩ ∉ Z) → G.IsActivePath Z p","l":379,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.isActivePath_of_reversed_directed","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"A backward-directed path whose interior vertices avoid the conditioning set is active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, (∀ (i : ℕ) (hi : i + 1 < p.length), G.edge (p.get ⟨i + 1, hi⟩) (p.get ⟨i, Causalean.DAG.IsActivePath._proof_2 p i hi⟩)) → (∀ (i : ℕ) (hi : i + 2 < p.length), p.get ⟨i + 1, Causalean.DAG.IsActivePath._proof_6 p i hi⟩ ∉ Z) → G.IsActivePath Z p","l":172,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_pair","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Any two adjacent vertices form an active path of length one for every conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {a b : V}, G.UAdj a b → G.IsActivePath Z [a, b]","l":566,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_reverse","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Active paths are symmetric: reversing an active path is also active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V}, G.IsActivePath Z p → G.IsActivePath Z p.reverse","l":114,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.isActivePath_take","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every prefix of an active path is active.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Z : Finset V} {p : List V} {k : ℕ}, G.IsActivePath Z p → G.IsActivePath Z (List.take k p)","l":872,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.isAncestor","k":"inductive","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite vertex population with decidable equality and a directed acyclic graph on that population, the ancestor relation holds from one vertex to another exactly when there is a directed path from the former to the latter. It is established either by a directed edge from the former vertex to the latter or by an existing ancestor path followed by a directed edge.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":101,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestorAvoiding","k":"inductive","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"For a finite vertex set whose members can be compared for equality and a directed acyclic graph on it, a set of forbidden vertices, and two vertices, avoiding ancestry is the relation generated by a directed edge from the first vertex to the second and by extending an avoiding path with an edge from an intermediate vertex outside the forbidden set to its endpoint. Thus, it holds exactly when there is a directed path from the first vertex to the second whose strictly interior vertices avoid the forbidden set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → V → V → Prop","l":43,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.cons","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"An avoiding path can be extended backward by one edge when the new interior node avoids the forbidden set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u w v : V}, G.edge u w → w ∉ C → G.isAncestorAvoiding C w v → G.isAncestorAvoiding C u v","l":64,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.exists_path","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Avoiding ancestry can be materialized as a concrete directed path with all strictly interior nodes avoiding the forbidden set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u v : V}, G.isAncestorAvoiding C u v → ∃ q, q.length ≥ 2 ∧ q.head? = some u ∧ q.getLast? = some v ∧ (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.isAncestorAvoiding.exists_path._proof_1 q i hi a⟩) (q.get ⟨i + 1, hi⟩)) ∧ ∀ (i : ℕ) (hi : i + 2 < q.length), q.get ⟨i + 1, Decidable.byContradiction fun a => Causalean.DAG.isAncestorAvoiding.exists_path._proof_2 q i hi a⟩ ∉ C","l":76,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestorAvoiding.toIsAncestor","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Every avoiding ancestry relation is also ordinary directed ancestry.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {C : Finset V} {u v : V}, G.isAncestorAvoiding C u v → G.isAncestor u v","l":54,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.DAG.isAncestor_child","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"First-step decomposition: if u is an ancestor of v, then either edge u v or there exists a child c of u such that c is an ancestor of v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.edge u v ∨ ∃ c, G.edge u c ∧ G.isAncestor c v","l":144,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same ancestor relations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {u v : V}, G₁.isAncestor u v ↔ G₂.isAncestor u v","l":225,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isAncestor_has_parent","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Every vertex reached by a nonempty directed path has an incoming edge, namely the final edge of that path.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.parents v ≠ ∅","l":467,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_iff_transGen","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The inductive ancestor relation coincides with Relation.TransGen of the edge relation: both are the transitive closure of the edge relation.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v ↔ Relation.TransGen G.edge u v","l":106,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_irrefl","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestor relation is irreflexive: no vertex is its own ancestor (this is acyclicity, restated for the inductive ancestor relation).","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), ¬G.isAncestor v v","l":131,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_last","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"If one node is a strict ancestor of another in a directed acyclic graph, then it either has a direct edge to the latter or is a strict ancestor of a node that has a direct edge to the latter.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {H : Causalean.DAG V} {u v : V}, H.isAncestor u v → H.edge u v ∨ ∃ w, H.isAncestor u w ∧ H.edge w v","l":377,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isAncestor_mem_of_closed","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"A finite set that contains every parent of each of its vertices contains every ancestor of each vertex it contains.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {T : Finset V}, (∀ x ∈ T, G.parents x ⊆ T) → ∀ {u w : V}, G.isAncestor u w → w ∈ T → u ∈ T","l":270,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every directed edge of one graph is also an edge of another graph, every ancestor relation in the first graph also holds in the second graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {u v : V}, G'.isAncestor u v → G.isAncestor u v","l":695,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isAncestor_topoOrder_lt","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestors respect the topological order: if u is an ancestor of v then G.topoOrder u < G.topoOrder v, so ancestor pairs are strictly ordered by topoOrder.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, G.isAncestor u v → G.topoOrder u < G.topoOrder v","l":412,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isAncestor_trans","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Ancestor relation is transitive.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v w : V}, G.isAncestor u v → G.isAncestor v w → G.isAncestor u w","l":137,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isCollider_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same collider triples.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {l m r : V}, G₁.IsCollider l m r ↔ G₂.IsCollider l m r","l":261,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.isCollider_of_supergraph","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every edge of one graph is also an edge of another, a collider in the larger graph remains a collider in the smaller graph whenever its two adjacent pairs are present there.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {l m r : V}, G'.UAdj l m → G'.UAdj m r → G.IsCollider l m r → G'.IsCollider l m r","l":726,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.DAG.isDescendant","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and two vertices, the first and the second, the descendant relation holds precisely when there is a directed path from the second vertex to the first.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → V → Prop","l":155,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.isRoot","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the root condition holds exactly when no directed edge enters that vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Prop","l":424,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.lastIdxLt","k":"def","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":1,"d":"For a nonnegative integer bound and a predicate on nonnegative integers whose truth can be checked at every index, the optional last qualifying index is the largest index strictly below the bound that satisfies the predicate, if such an index exists, and is absent otherwise.","s":"ℕ → (P : ℕ → Prop) → [DecidablePred P] → Option ℕ","l":209,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.lastIdxLt_eq_none","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If there is no last index below a bound satisfying a predicate, then no index below the bound satisfies the predicate.","s":"∀ {n : ℕ} {P : ℕ → Prop} [inst : DecidablePred P], Causalean.DAG.lastIdxLt n P = none → ∀ i < n, ¬P i","l":237,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.lastIdxLt_eq_some","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"If the last index below a bound exists, it is below the bound, satisfies the predicate, and no larger index below the bound satisfies it.","s":"∀ {n : ℕ} {P : ℕ → Prop} [inst : DecidablePred P] {i : ℕ}, Causalean.DAG.lastIdxLt n P = some i → i < n ∧ P i ∧ ∀ (j : ℕ), i < j → j < n → ¬P j","l":218,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.last_edge_into_max","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"The final edge of a path to a topological maximum points into it. On an active path whose nodes all lie in an ancestral set with topological maximum n, if the path ends at n then its last edge points into n: a child of n on the path would be a strict descendant, hence a strictly larger node.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {D bigQ : Finset V} {p : List V} {n : V} (hp_len : p.length ≥ 2), G.IsActivePath D p → p.getLast? = some n → (∀ v ∈ p, v ∈ G.ancestralSet bigQ) → (∀ m ∈ G.ancestralSet bigQ, G.topoOrder m ≤ G.topoOrder n) → G.edge (p.get ⟨p.length - 2, Decidable.byContradiction fun a => Causalean.DAG.last_edge_into_max._proof_1 hp_len a⟩) n","l":392,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.le_card_iterate","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"If each of a specified number of successive applications of a function on finite sets strictly increases cardinality, the final set has grown by at least that number.","s":"∀ {V : Type u_2} (f : Finset V → Finset V) (S₀ : Finset V) (k : ℕ), (∀ j < k, (f^[j] S₀).card < (f^[j + 1] S₀).card) → (f^[0] S₀).card + k ≤ (f^[k] S₀).card","l":212,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.markovEquiv_flipEdge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":1,"d":"The analytic core (AMP, per-step invariance). In a DAG G, if the edge a → b is covered — every other parent of b is also a parent of a, and vice versa, then the DAG obtained by reversing that edge to b → a is Markov equivalent to G: the two graphs license exactly the same d-separation statements. This is the single-edge kernel used by the covered-reversal proof of the Verma--Pearl hard direction.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b : V} (hcov : G.IsCoveredEdge a b), Causalean.MarkovEquiv G (Causalean.DAG.flipEdge hcov)","l":1713,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.mem_ancClosure","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership in the backward-reachability fixpoint is exactly ancestry: a vertex lies in G.ancClosure v iff it is an ancestor of v. This makes the ancestor relation decidable using only the (decidable) edge relation, with no reference to any topological order.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V}, u ∈ G.ancClosure v ↔ G.isAncestor u v","l":279,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_ancestors","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for ancestors: u ∈ G.ancestors v ↔ G.isAncestor u v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v u : V}, u ∈ G.ancestors v ↔ G.isAncestor u v","l":312,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_ancestralSet_of_edge_to_mem","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"When a directed edge points to a vertex in the ancestral closure of a set, its source vertex is also in that ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u v : V} {S : Finset V}, G.edge u v → v ∈ G.ancestralSet S → u ∈ G.ancestralSet S","l":114,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.mem_ancestralSet_of_isAncestor","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"If u is an ancestor of some vertex w ∈ S, then u ∈ ancestralSet S.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {u w : V} {S : Finset V}, w ∈ S → G.isAncestor u w → u ∈ G.ancestralSet S","l":74,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.mem_children","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for children: w ∈ G.children v ↔ G.edge v w.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v w : V}, w ∈ G.children v ↔ G.edge v w","l":93,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_descendants","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for descendants: w ∈ G.descendants v ↔ G.isAncestor v w.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v w : V}, w ∈ G.descendants v ↔ G.isAncestor v w","l":316,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.mem_parents","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Membership characterization for parents: u ∈ G.parents v ↔ G.edge u v.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {v u : V}, u ∈ G.parents v ↔ G.edge u v","l":89,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.moralAdj_symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral adjacency is symmetric: an undirected moral edge from u to v is also one from v to u.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S : Finset V} {u v : V}, G.MoralAdj S u v → G.MoralAdj S v u","l":58,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_of_activePath","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"An active path inside a ground set, with endpoints outside the conditioning set, induces a connection in the corresponding moral graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} (n : ℕ) {x y : V} {p : List V}, p.length ≤ n → G.IsActivePath Z p → p.length ≥ 2 → (∀ v ∈ p, v ∈ S) → p.head? = some x → p.getLast? = some y → x ∉ Z → y ∉ Z → G.MoralConn S Z x y","l":126,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_of_step","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A single moral step yields moral connectivity.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v : V}, G.MoralStep S Z u v → G.MoralConn S Z u v","l":72,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralConn_trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral connectivity is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v w : V}, G.MoralConn S Z u v → G.MoralConn S Z v w → G.MoralConn S Z u w","l":77,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralStep_of_married","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A married pair (common child c ∈ S) of distinct non-Z vertices of S is a moral step.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v c : V}, u ≠ v → u ∈ S → v ∈ S → c ∈ S → G.edge u c → G.edge v c → u ∉ Z → v ∉ Z → G.MoralStep S Z u v","l":88,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.moralStep_of_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"A skeleton edge between two non-Z vertices of S is a moral step.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {S Z : Finset V} {u v : V}, u ≠ v → u ∈ S → v ∈ S → G.UAdj u v → u ∉ Z → v ∉ Z → G.MoralStep S Z u v","l":82,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.DAG.node_isAncestor_last_of_directed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"Every vertex on a forward directed path is either the path's endpoint or a strict ancestor of that endpoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {q : List V} {w : V}, q.length ≥ 2 → q.getLast? = some w → (∀ (i : ℕ) (hi : i + 1 < q.length), G.edge (q.get ⟨i, Decidable.byContradiction fun a => Causalean.DAG.node_isAncestor_last_of_directed._proof_1 i hi a⟩) (q.get ⟨i + 1, hi⟩)) → ∀ (j : ℕ) (hj : j < q.length), G.isAncestor (q.get ⟨j, hj⟩) w ∨ q.get ⟨j, hj⟩ = w","l":801,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.nonCollider_has_outgoing","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"A non-collider triple (l, m, r) has at least one outgoing edge from m.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {l m r : V}, G.UAdj l m → G.UAdj m r → ¬G.IsCollider l m r → G.edge m l ∨ G.edge m r","l":95,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.nonDescendants","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the non-descendant set contains exactly the vertices other than that vertex which cannot be reached from it by a directed path.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":348,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.not_dSep_iff_hasActivePath","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Bridge: failure of d-separation is exactly an active path. For pairwise-disjoint query sets, X and Y are *not* d-separated by Z iff there is an active path from X to Y given Z. Assembled from bbReachableVertices_iff_activePath. Reduces the covered-flip invariance to a pure active-path statement.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (H : Causalean.DAG V) (X Y Z : Finset V), Disjoint X Y → Disjoint X Z → Disjoint Y Z → (¬H.dSep X Y Z ↔ H.HasActivePath X Y Z)","l":198,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.not_dSeparable_of_uAdj","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"An edge makes its endpoints inseparable: if a and b are adjacent then no conditioning set d-separates them, because the single edge [a, b] is an active path regardless of the conditioning set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {a b : V}, G.UAdj a b → ∀ (Z : Finset V), ¬G.dSep {a} {b} Z","l":37,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.not_edge_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"No self-loops in a DAG.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (a : V), ¬G.edge a a","l":278,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.not_isAncestor_of_root'","k":"theorem","m":"Causalean.Graph.DSep.BackdoorBridges","a":"Graph/DSep","t":2,"d":"A vertex with no incoming edges has no proper ancestors.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {r : V}, (∀ (u : V), ¬G.edge u r) → ∀ (u : V), ¬G.isAncestor u r","l":39,"f":"Causalean/Graph/DSep/BackdoorBridges.lean"},{"n":"Causalean.DAG.not_uAdj_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"No self-adjacency in a DAG.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (a : V), ¬G.UAdj a a","l":282,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.DAG.ofAcyclic","k":"def","m":"Causalean.Graph.AcyclicConstruct","a":"Graph","t":1,"d":"Given an edge relation on a finite vertex set for which no vertex can return to itself by a nonempty directed path, the directed acyclic graph constructed from that relation has precisely that edge relation.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (e : V → V → Prop) → (∀ (v : V), ¬Relation.TransGen e v v) → Causalean.DAG V","l":32,"f":"Causalean/Graph/AcyclicConstruct.lean"},{"n":"Causalean.DAG.ofAcyclic_edge","k":"theorem","m":"Causalean.Graph.AcyclicConstruct","a":"Graph","t":1,"d":"The directed acyclic graph ofAcyclic e hac, built from an edge relation e together with a proof that e has no directed cycle, has exactly e as its edge relation.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (e : V → V → Prop) (hac : ∀ (v : V), ¬Relation.TransGen e v v), (Causalean.DAG.ofAcyclic e hac).edge = e","l":41,"f":"Causalean/Graph/AcyclicConstruct.lean"},{"n":"Causalean.DAG.orderedLocalSG_of_dSep_with_fixed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":1,"d":"d-separation yields an ordered-local derivation. In a DAG G, suppose every vertex of Zf has no parents, i.e. Zf consists of fixed roots, Zf is disjoint from R, and X, Y, and Zr are each contained in R. If X and Y are d-separated by Zr ∪ Zf, then the conditional-independence triple \"X ⊥ Y given Zr\" is derivable from the ordered-local basis on R via the semi-graphoid axioms.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (R X Y Zr Zf : Finset V), (∀ f ∈ Zf, G.parents f = ∅) → Disjoint Zf R → X ⊆ R → Y ⊆ R → Zr ⊆ R → G.dSep X Y (Zr ∪ Zf) → G.OrderedLocalSG R X Y Zr","l":1290,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.parents","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the parent set is the finite set of all vertices having a directed edge into that vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":81,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.parents_subset_ancestors","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Parents are a subset of ancestors.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (v : V), G.parents v ⊆ G.ancestors v","l":320,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.roots","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population, the root set is the finite set of all vertices with no incoming directed edge.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V","l":433,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.subset_ancestralSet","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"S ⊆ ancestralSet S: every member is in its own ancestral set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V), S ⊆ G.ancestralSet S","l":57,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.subset_ancestralSet_of_subset","k":"theorem","m":"Causalean.Graph.DSep.Ancestral","a":"Graph/DSep","t":2,"d":"Every element of a subset belongs to the ancestral closure of its ambient set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Y S : Finset V}, Y ⊆ S → Y ⊆ G.ancestralSet S","l":343,"f":"Causalean/Graph/DSep/Ancestral.lean"},{"n":"Causalean.DAG.subset_iterate_ancStep","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"Any finite set of graph nodes remains contained after applying the graph's ancestor-step operation any number of times.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (S : Finset V) (k : ℕ), S ⊆ G.ancStep^[k] S","l":194,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.take_suffix_at_last_S","k":"theorem","m":"Causalean.Graph.DSep.ActivePath","a":"Graph/DSep","t":2,"d":"Suffix-step for source-to-cond transfer.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {X Z S : Finset V} {p : List V} {x w : V}, x ∈ X → p.length ≥ 2 → G.IsActivePath (Z ∪ S) p → p.head? = some x → p.getLast? = some w → ∃ x' q, x' ∈ X ∪ S ∧ q.length ≥ 2 ∧ G.IsActivePath (Z ∪ S) q ∧ q.head? = some x' ∧ q.getLast? = some w ∧ ∀ (k : ℕ), 0 < k → ∀ (hk2 : k + 1 < q.length), q.get ⟨k, Causalean.DAG.take_suffix_at_last_S._proof_2 q k hk2⟩ ∉ S","l":250,"f":"Causalean/Graph/DSep/ActivePath.lean"},{"n":"Causalean.DAG.topoMax_mem_seed","k":"theorem","m":"Causalean.Graph.DSep.OrderedLocalSG","a":"Graph/DSep","t":2,"d":"A vertex with maximal topological order in the ancestral closure of a seed set must itself belong to that seed set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) {Q : Finset V} {n : V}, n ∈ G.ancestralSet Q → (∀ m ∈ G.ancestralSet Q, G.topoOrder m ≤ G.topoOrder n) → n ∈ Q","l":100,"f":"Causalean/Graph/DSep/OrderedLocalSG.lean"},{"n":"Causalean.DAG.topoOrder","k":"def","m":"Causalean.Graph.DAG","a":"Graph","t":1,"d":"For a finite directed acyclic graph on a vertex population and a vertex, the derived topological number is its number of strict ancestors times the population size, plus a fixed tie-breaking enumeration number. This number is injective across vertices and strictly increases along every directed edge.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → ℕ","l":372,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_injective","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The derived topological order is injective, so it provides a canonical total order on the finite vertex type.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V), Function.Injective G.topoOrder","l":383,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_lt","k":"theorem","m":"Causalean.Graph.DAG","a":"Graph","t":2,"d":"The derived topological order is edge-consistent: if there is an edge from u to v, then topoOrder u < topoOrder v. This witnesses acyclicity.","s":"∀ {V : Type u_2} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (u v : V), G.edge u v → G.topoOrder u < G.topoOrder v","l":396,"f":"Causalean/Graph/DAG.lean"},{"n":"Causalean.DAG.topoOrder_lt_of_flipMinus_transGen","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"A directed flipMinus-path strictly increases the topological order.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {a b u w : V}, Relation.TransGen (G.flipMinus a b) u w → G.topoOrder u < G.topoOrder w","l":58,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.uAdj_edge_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.CoveredReversal","a":"Graph/MarkovEquiv","t":2,"d":"Graphs with the same directed edges have exactly the same undirected adjacencies.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → ∀ {u v : V}, G₁.UAdj u v ↔ G₂.UAdj u v","l":254,"f":"Causalean/Graph/MarkovEquiv/CoveredReversal.lean"},{"n":"Causalean.DAG.uAdj_mono_edge","k":"theorem","m":"Causalean.Graph.DSep.Separation","a":"Graph/DSep","t":2,"d":"If every directed edge of one graph is also an edge of another graph, vertices adjacent in the first graph are also adjacent in the second graph.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G G' : Causalean.DAG V), (∀ (u v : V), G'.edge u v → G.edge u v) → ∀ {u v : V}, G'.UAdj u v → G.UAdj u v","l":717,"f":"Causalean/Graph/DSep/Separation.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily","k":"structure","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"An environment family for Invariant Causal Prediction (Peters, Bühlmann & Meinshausen 2016) bundles a finite, Fintype-indexed collection of structural causal models sharing a common target variable that every environment observes, a common observed-variable set, and a common latent-variable set. The environments never intervene on the target: they share the target's parent set, the target's structural mechanism, and the latent-noise law, while each environment carries the values assigned to its own intervened coordinates. It further packages the regularity needed to disintegrate the joint law into the target's conditional law given any predictor set — a standard Borel and nonempty target value space, and countable generation of the relevant kernels, and states the exogeneity assumption that in every environment the target's exogenous (latent) parents are independent of its observed parents under the joint law.","s":"(N : Type u_3) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_4) → [(n : N) → MeasurableSpace (Ω n)] → (ι : Type u_5) → [Fintype ι] → Type (max (max u_3 u_4) u_5)","l":40,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.Invariant","k":"def","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, and a finite predictor-node set that is observed in every environment, the assertion that this predictor set is invariant means that there exists a measure-valued rule for the target as a function of the predictor values and the values of fixed parents such that (1) after transporting fixed-parent values between environments, this law is the same in every pair of environments, and (2) in each environment, the observed conditional law of the target given the predictors agrees almost everywhere with that common law evaluated at that environment's fixed-parent values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (S : Finset (Causalean.SWIGNode N)) → (∀ (i : ι), S ⊆ (F.M i).observed) → Prop","l":44,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_eq_mechanismKernel_of_indep","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"Let $X$, $L$, and $Y$ be measurable random elements of a probability space, with $Y$'s value space standard Borel and nonempty, and let $Φ$ be a measurable map such that $Y$ equals $Φ(X,L)$ almost everywhere. If $L$ is independent of $X$, then the conditional distribution of $Y$ given $X$ agrees, for almost every pushed-forward value of $X$, with the mechanism kernel obtained by pushing the law of $L$ forward through $Φ$ paired with that value of $X$.","s":"∀ {α : Type u_4} {β : Type u_5} {γ : Type u_6} {δ : Type u_7} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : StandardBorelSpace δ] [inst_5 : Nonempty δ] (μ : MeasureTheory.Measure α) [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {X : α → β} {L : α → γ} {Y : α → δ} {Φ : β × γ → δ}, Measurable X → Measurable L → Measurable Y → Measurable Φ → ProbabilityTheory.IndepFun L X μ → (Y =ᵐ[μ] fun ω => Φ (X ω, L ω)) → (fun x => (ProbabilityTheory.condDistrib Y X μ) x) =ᵐ[MeasureTheory.Measure.map X μ] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map L μ) Φ)","l":325,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_eq_mechanismKernel_of_indep_of_pair_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Variant of condDistrib_eq_mechanismKernel_of_indep that takes equality of the joint (X,Y) push-forward measures directly. This avoids needing a MeasurableEq instance for the target when the equality is proved upstream by unfolding a map representation of μ.","s":"∀ {α : Type u_4} {β : Type u_5} {γ : Type u_6} {δ : Type u_7} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : StandardBorelSpace δ] [inst_5 : Nonempty δ] (μ : MeasureTheory.Measure α) [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {X : α → β} {L : α → γ} {Y : α → δ} {Φ : β × γ → δ}, Measurable X → Measurable L → Measurable Y → Measurable Φ → ProbabilityTheory.IndepFun L X μ → MeasureTheory.Measure.map (fun ω => (X ω, Y ω)) μ = MeasureTheory.Measure.map (fun ω => (X ω, Φ (X ω, L ω))) μ → (fun x => (ProbabilityTheory.condDistrib Y X μ) x) =ᵐ[MeasureTheory.Measure.map X μ] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map L μ) Φ)","l":382,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.condDistrib_target_eq_mechanismKernel","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"For an invariant-prediction environment family and a reference index i₀ together with an environment i, in environment i the conditional law of the target given its observed parents, under the joint kernel restricted to that environment, agrees with the mechanism kernel formed by pushing the latent-parent law of environment i forward through the target's structural mechanism function, paired with the parent value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), have hYrv := fun ⦃w⦄ hw => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hw) ▸ Finset.mem_union_left (F.M i).unobserved (F.hYobs i); have hPrv := Eq.mpr (id (congrArg (fun _a => _a ⊆ (F.M i).randomVars) (Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq F i₀ i))) (LE.le.trans Finset.inter_subset_right (id Finset.subset_union_left)); have hLrv := LE.le.trans Finset.inter_subset_right (id Finset.subset_union_right); (fun c => (ProbabilityTheory.condDistrib (Causalean.valuesProjection hYrv) (Causalean.valuesProjection hPrv) ((F.M i).jointKernel (F.s i))) c) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hPrv) ((F.M i).jointKernel (F.s i))] ⇑(Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i))) (F.mechanismFun i₀ i))","l":427,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.fixedParentVals","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, and an environment, the fixed-parent values are the values assigned by that environment's intervention, restricted to the target's fixed parents.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i : ι) → Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)","l":200,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.fixed_parent_mem_fixed_of_mem","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"A target parent that is fixed in one environment is fixed in every environment. The point is not that EnvFamily shares the whole fixed set, but that it shares the target parent set (E4), and a fixed-form SWIG node cannot be an observed or unobserved random-form node in the other environment, while a parent of Y is always classified as fixed/observed/unobserved (dag_edges_classified).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) {i j : ι} {d : Causalean.SWIGNode N}, d ∈ (F.M i).dag.parents F.yNode ∩ (F.M i).fixed → d ∈ (F.M j).fixed","l":149,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.icp_sound","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Soundness","a":"Discovery/InvariantPrediction","t":1,"d":"Theorem 1 (soundness). For an environment family and an index i₀ selecting the target's observed-parent set, the identified set is contained in the target's observed parents: every node selected by ICP is a genuine direct cause.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), F.idSet ⊆ ↑(F.paObs i₀)","l":34,"f":"Causalean/Discovery/InvariantPrediction/Soundness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.idSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, and an invariant-prediction environment family, the identified set is the intersection of all invariant predictor sets for that family.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Set (Causalean.SWIGNode N)","l":38,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.idSet_subset_of_mem","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":2,"d":"The identified set is contained in every invariant set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) {T : Set (Causalean.SWIGNode N)}, T ∈ F.invariantSets → F.idSet ⊆ T","l":46,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.invariantSets","k":"def","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, and an invariant-prediction environment family, the collection of invariant predictor sets consists exactly of node sets that can be represented by a finite predictor set observed in every environment and that satisfy invariance for that environment family.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Set (Set (Causalean.SWIGNode N))","l":29,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.jointKernel_map_paLat_eq_latentProduct_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Projecting the joint kernel to latent parents is the same as projecting the latent product directly to those latent coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i : ι) (hLrv : F.paLat i ⊆ (F.M i).randomVars) (hLun : F.paLat i ⊆ (F.M i).unobserved), MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i)) = MeasureTheory.Measure.map (Causalean.valuesProjection hLun) (F.M i).latentProduct","l":255,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.latentProduct_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment heterogeneous equality of latent product measures.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), (F.M i).latentProduct ≍ (F.M j).latentProduct","l":319,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.latentValues_measurableSpace_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment measurable-space transport for latent value spaces.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), inferInstance ≍ inferInstance","l":313,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.map_heq_transport","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Transport a push-forward μ.map g across a propositional equality of the source type together with HEq of the measurable structure, the measure, and the map. Used to identify latent-parent push-forwards across environments whose latent index sets coincide propositionally (via hUnobs) but whose latentDist families and value-space typing match only up to HEq.","s":"∀ {α₁ α₂ δ : Type u} [mα₁ : MeasurableSpace α₁] [mα₂ : MeasurableSpace α₂] [inst : MeasurableSpace δ] {μ₁ : MeasureTheory.Measure α₁} {μ₂ : MeasureTheory.Measure α₂} {g₁ : α₁ → δ} {g₂ : α₂ → δ}, α₁ = α₂ → mα₁ ≍ mα₂ → μ₁ ≍ μ₂ → g₁ ≍ g₂ → MeasureTheory.Measure.map g₁ μ₁ = MeasureTheory.Measure.map g₂ μ₂","l":277,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measurable_mechanismFun","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"mechanismFun is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), Measurable (F.mechanismFun i₀ i)","l":185,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measurable_mechanismFunCf","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"mechanismFunCf is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι) (cf : Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)), Measurable (F.mechanismFunCf i₀ i cf)","l":163,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.measure_pi_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Measure.pi transports heterogeneously across equality of the finite index sets and heterogeneous equality of the coordinate measure families.","s":"∀ {M : Type u_4} {Ω' : M → Type u_5} [inst : (n : M) → MeasurableSpace (Ω' n)] {I J : Finset M} {μI : (i : ↥I) → MeasureTheory.Measure (Ω' ↑i)} {μJ : (j : ↥J) → MeasureTheory.Measure (Ω' ↑j)}, I = J → μI ≍ μJ → MeasureTheory.Measure.pi μI ≍ MeasureTheory.Measure.pi μJ","l":299,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFun","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, a reference environment selecting the observed parents, and an environment supplying the structural mechanism and latent parents, the target mechanism function maps values of those observed parents and that environment's latent parents to the target's value, using the environment's intervention assignment for any fixed parents.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i₀ i : ι) → Causalean.ValuesOn (F.paObs i₀) (Causalean.swigΩ Ω) × Causalean.ValuesOn (F.paLat i) (Causalean.swigΩ Ω) → Causalean.ValuesOn {F.yNode} (Causalean.swigΩ Ω)","l":83,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFunCf","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, a reference environment selecting the observed parents, an environment supplying the structural mechanism and latent parents, and values for that environment's fixed target parents, the fixed-parent-parameterized target mechanism function maps values of the selected observed and latent parents to the target's value, using the supplied fixed-parent values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) → (i₀ i : ι) → Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω) → Causalean.ValuesOn (F.paObs i₀) (Causalean.swigΩ Ω) × Causalean.ValuesOn (F.paLat i) (Causalean.swigΩ Ω) → Causalean.ValuesOn {F.yNode} (Causalean.swigΩ Ω)","l":115,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismFunCf_fixedParentVals","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"At the environment's own fixed-parent values fixedParentVals i, the parameterized mechanism mechanismFunCf coincides with the version that reads fixed-parent values from the environment assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι), F.mechanismFunCf i₀ i (F.fixedParentVals i) = F.mechanismFun i₀ i","l":156,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanismKernel_cf_env_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"Fix an environment family, a base environment i₀, two environments i and j, and a fixed-parent value assignment cf for environment i. If the latent parents of the target in environment i are among its random variables and likewise for environment j, then the target-mechanism kernel built from environment i at cf equals the target-mechanism kernel built from environment j at the value obtained by transporting cf through the shared fixed-parent set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i j : ι) (cf : Causalean.ValuesOn (F.paFix i) (Causalean.swigΩ Ω)) (hLrv : F.paLat i ⊆ (F.M i).randomVars) (hLrv' : F.paLat j ⊆ (F.M j).randomVars), Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv) ((F.M i).jointKernel (F.s i))) (F.mechanismFunCf i₀ i cf) = Causalean.Mathlib.GraphMapProd.mechanismKernel (MeasureTheory.Measure.map (Causalean.valuesProjection hLrv') ((F.M j).jointKernel (F.s j))) (F.mechanismFunCf i₀ j (Causalean.valuesProjection (le_of_eq (Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_eq F j i)) cf))","l":568,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.mechanism_invariant","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"Mechanism invariance (the heart of soundness). For an environment family and a target index i₀, the target's observed parents paObs form an invariant set: in every environment the conditional law of the target given its observed parents equals one fixed structural factor, so ICP never rejects the parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), F.Invariant (F.paObs i₀) fun j => Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_subset_observed F i₀ j","l":189,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.obsKernel_map_valuesProjection_eq_jointKernel_map","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"The observed marginal of obsKernel is the corresponding random marginal of jointKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (S : Finset (Causalean.SWIGNode N)) (hSobs : S ⊆ M.observed) (hSrv : S ⊆ M.randomVars), MeasureTheory.Measure.map (Causalean.valuesProjection hSobs) (M.obsKernel s) = MeasureTheory.Measure.map (Causalean.valuesProjection hSrv) (M.jointKernel s)","l":69,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, and an environment, the fixed-parent set of the target is the set of target parents that are fixed by intervention in that environment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":137,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For any two environments i and j, the fixed parents of the target coincide between environment i and environment j.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paFix i = F.paFix j","l":176,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paFix_subset_fixed","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"The fixed parents are a subset of the fixed coordinates, so they can be read off the environment's intervention assignment s i.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i : ι), F.paFix i ⊆ (F.M i).fixed","l":195,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paLat","k":"def","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, and an environment, the latent-parent set of the target is the set of target parents that are unobserved in that environment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":47,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paLat_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"The latent-parent set is environment-independent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paLat i = F.paLat j","l":54,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, an invariant-prediction environment family, and an environment, the observed-parent set of the target is the set of target parents that are observed in that environment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → ι → Finset (Causalean.SWIGNode N)","l":121,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":2,"d":"paObs does not depend on the chosen environment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paObs i = F.paObs j","l":132,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_mem_invariantSets","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.IdentifiedSet","a":"Discovery/InvariantPrediction","t":1,"d":"For an invariant-prediction environment family and an index i₀ selecting the target's observed-parent set, the target's observed parents form an invariant predictor set — the membership form of mechanism_invariant.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ : ι), ↑(F.paObs i₀) ∈ F.invariantSets","l":51,"f":"Causalean/Discovery/InvariantPrediction/IdentifiedSet.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_subset_observed","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":2,"d":"The observed parents form a valid conditioning set in every environment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι), F.paObs i ⊆ (F.M j).observed","l":90,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.structFun_yNode_apply_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Cross-environment equality of the target structural function applied to coordinatewise-equal target-parent tuples.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i j : ι) {ξi : (w : ↥((F.M i).dag.parents F.yNode)) → Causalean.swigΩ Ω ↑w} {ξj : (w : ↥((F.M j).dag.parents F.yNode)) → Causalean.swigΩ Ω ↑w}, (∀ (d : Causalean.SWIGNode N) (hdi : d ∈ (F.M i).dag.parents F.yNode) (hdj : d ∈ (F.M j).dag.parents F.yNode), ξi ⟨d, hdi⟩ = ξj ⟨d, hdj⟩) → (F.M i).structFun ⟨F.yNode, F.hYobs i⟩ ξi = (F.M j).structFun ⟨F.yNode, F.hYobs j⟩ ξj","l":545,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.targetSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.Invariance","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, and an invariant-prediction environment family, the target set is the singleton containing that family's random-form target node.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Finset (Causalean.SWIGNode N)","l":39,"f":"Causalean/Discovery/InvariantPrediction/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.target_projection_evalMap_eq_mechanismFun","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Pointwise structural equation for the target, expressed through mechanismFun on observed and latent parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {ι : Type u_3} [inst_3 : Fintype ι] (F : Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι) (i₀ i : ι) (ℓ : (F.M i).LatentValues), Causalean.valuesProjection (have this := fun ⦃w⦄ hw => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hw) ▸ Finset.mem_union_left (F.M i).unobserved (F.hYobs i); this) ((F.M i).evalMap (F.s i) ℓ) = F.mechanismFun i₀ i (Causalean.valuesProjection (have this := Eq.mpr (id (congrArg (fun _a => _a ⊆ (F.M i).randomVars) (Causalean.Discovery.InvariantPrediction.EnvFamily.paObs_eq F i₀ i))) (LE.le.trans Finset.inter_subset_right (id Finset.subset_union_left)); this) ((F.M i).evalMap (F.s i) ℓ), Causalean.valuesProjection (have this := LE.le.trans Finset.inter_subset_right (id Finset.subset_union_right); this) ((F.M i).evalMap (F.s i) ℓ))","l":206,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.valuesOn_measurableSpace_heq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"ValuesOn measurable spaces transport heterogeneously across equality of the finite index sets.","s":"∀ {M : Type u_4} [Fintype M] {Ω' : M → Type u_5} [inst : (n : M) → MeasurableSpace (Ω' n)] {I J : Finset M}, I = J → inferInstance ≍ inferInstance","l":289,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.valuesProjection_randomToObserved_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.Helpers.MechanismFactor","a":"Discovery/InvariantPrediction/Helpers","t":2,"d":"Observed-coordinate projection after randomToObserved is the same as the corresponding random-coordinate projection.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)) (hSobs : S ⊆ M.observed) (hSrv : S ⊆ M.randomVars), Causalean.valuesProjection hSobs ∘ M.randomToObserved = Causalean.valuesProjection hSrv","l":59,"f":"Causalean/Discovery/InvariantPrediction/Helpers/MechanismFactor.lean"},{"n":"Causalean.Discovery.InvariantPrediction.EnvFamily.yNode","k":"def","m":"Causalean.Discovery.InvariantPrediction.Model","a":"Discovery/InvariantPrediction","t":1,"d":"For a finite node-label set, measurable coordinate outcome spaces, a finite environment index set, and an invariant-prediction environment family, the target node is the random-form node associated with the family's target variable.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {ι : Type u_3} → [inst_3 : Fintype ι] → Causalean.Discovery.InvariantPrediction.EnvFamily N Ω ι → Causalean.SWIGNode N","l":116,"f":"Causalean/Discovery/InvariantPrediction/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Env","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"A single do-intervention environment for the observational SEM M (sec:idfirst) bundles an intervention set that never includes the target node, the constant values assigned to each intervened coordinate, and measurable post-intervention coordinates that equal their assigned constant, almost surely, on the intervened set, satisfy the same structural equation as M — with M's coefficients and noises — on every coordinate left un-intervened, and for which the target's noise remains independent of each parent coordinate of the target.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Type u_1","l":165,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"An environment family for the linear-Gaussian ICP problem bundles an observational linear-Gaussian structural equation model together with a finite index set of interventional environments over that model.","s":"ℕ → Type (u_1 + 1)","l":212,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.HasShiftedSingleInterventions","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate and a linear-Gaussian environment family, the shifted single-intervention condition means that for every predictor coordinate there is an environment such that (1) that environment intervenes on that coordinate alone, and (2) its assigned value differs from the coordinate's observational mean.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Prop","l":84,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.InvarianceNull","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate, a linear-Gaussian environment family, and a predictor set, the regression-invariance null means that there exist a coefficient vector supported on that set and one residual measure such that (1) the coefficient vector is supported on the predictor set, (2) in the observational environment the residual is independent of each selected predictor, (3) its observational residual law is the common law, (4) in every interventional environment the residual is independent of each selected predictor, and (5) every interventional residual law is the same common law.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1)) → Prop","l":96,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.ObsIntegrable","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate and a linear-Gaussian environment family, the observational-integrability condition means that every observational coordinate is integrable under the observational probability law.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Prop","l":98,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.exists_youngest_nonzero","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Youngest-node selection (the \"youngest node X_{k₀}\" step). Given a nonempty set of coordinate indices $T$, there is an index $k_0 \\in T$ such that no other element of $T$ is a descendant of $k_0$ along the observational DAG's directed edges — i.e. $k_0$ is a sink of the subgraph induced by $T$. This is the \"youngest\" node with non-zero α of the paper's proof.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) (T : Finset (Fin (p + 1))), T.Nonempty → ∃ k₀ ∈ T, ∀ k ∈ T, k ≠ k₀ → ¬F.obs.dag.isAncestor k₀ k","l":145,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.icp_complete_linearGaussian","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Completeness for do-interventions — Theorem prop:1(i). For an observational linear-Gaussian SEM with integrable regressors such that every predictor $j$ receives at least one environment with a single do-intervention $A^e = \\{j\\}$ whose shifted value $a^e_j$ differs from its observational mean $E[X^1_j]$, the ICP identified set — the intersection of all invariant predictor sets — equals exactly the parent set of the target node, $S(E) = PA(Y)$.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → F.HasShiftedSingleInterventions → F.identifiedSet = F.paY","l":288,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.icp_sound_linearGaussian","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Soundness (propos:sem). Assuming the observational predictor coordinates are integrable, so their means E[X¹_j] are well defined, the identified set is contained in the target's parents, S(E) ⊆ PA(Y).","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → F.identifiedSet ⊆ F.paY","l":107,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.identifiedSet","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate and a linear-Gaussian environment family, the identified set is the intersection of all predictor sets in that family's collection of invariant predictor sets.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1))","l":150,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.instDecidablePredFinsetFinHAddNatOfNatInvarianceNull","k":"instance","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"For every number of predictor coordinates and linear-Gaussian environment family with those predictor coordinates and one target coordinate, a decision procedure for whether each predictor set satisfies the regression-invariance null is supplied by classical reasoning.","s":"{p : ℕ} → (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) → DecidablePred F.InvarianceNull","l":131,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.invariantSets","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate and a linear-Gaussian environment family, the collection of invariant predictor sets is the finite collection of predictor subsets whose regression-invariance null holds.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Finset (Fin (p + 1)))","l":141,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.mem_identifiedSet","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a predictor index k, k lies in the identified set S(E) exactly when k belongs to every predictor subset whose invariance null holds.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) {k : Fin (p + 1)}, k ∈ F.identifiedSet ↔ ∀ S ∈ F.invariantSets, k ∈ S","l":163,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.mem_invariantSets","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"Membership in invariantSets.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p) {S : Finset (Fin (p + 1))}, S ∈ F.invariantSets ↔ S ⊆ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p ∧ F.InvarianceNull S","l":170,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.obsMean","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate, a linear-Gaussian environment family, and a coordinate index, the observational mean of that coordinate is its expectation under the family's observational probability law.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Fin (p + 1) → ℝ","l":79,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.paY","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a family of intervention environments, the family's observational target-parent set is the target-parent set of its observational structural equation model.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p → Finset (Fin (p + 1))","l":236,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily.residual_mean_shift_of_doIntervention","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Completeness","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"Residual mean-shift (eq:help1/eq:help2). Fix an observational SEM with integrable regressors, a candidate coefficient vector γ, and an index k₀ such that the gap α_{k₀} = β₀,k₀ − γ_{k₀} between the causal coefficient and γ at k₀ is nonzero. Suppose k₀ is a youngest such index: every other index with a nonzero coefficient gap is not a descendant of k₀ in the observational DAG, and let i be an environment consisting of a single do-intervention pinning coordinate k₀ to a value different from its observational mean. Then the residual R^i = Y^i − Σ γ_k X_k^i computed in environment i and the observational residual R¹ computed the same way do not have the same distribution, since they have different means.","s":"∀ {p : ℕ} (F : Causalean.Discovery.InvariantPrediction.LinearGaussian.EnvFamily p), F.ObsIntegrable → ∀ (γ : Fin (p + 1) → ℝ) (k₀ : Fin (p + 1)), F.obs.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k₀ - γ k₀ ≠ 0 → (∀ (k : Fin (p + 1)), k ≠ k₀ → F.obs.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k - γ k ≠ 0 → ¬F.obs.dag.isAncestor k₀ k) → ∀ (i : F.ι), (F.env i).A = {k₀} → (F.env i).a k₀ ≠ F.obsMean k₀ → ¬ProbabilityTheory.IdentDistrib (Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual (F.env i) γ) (Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual F.obs γ) F.obs.P F.obs.P","l":163,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Completeness.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM","k":"structure","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"The observational linear-Gaussian structural equation model (eq:semmmmm, the e = 1 block): it bundles a sample space with its σ-algebra carrying a probability measure that is a genuine probability measure, together with structural coefficients β obeying Xⱼ = Σ_{k≠j} βⱼₖ Xₖ + εⱼ for measurable observed coordinates X, an acyclic graph whose edge k → j holds exactly when βⱼₖ ≠ 0, and the absence of self-loops in β. The noises ε are declared to be the structural residuals εⱼ = Xⱼ − Σ_{k≠j} βⱼₖ Xₖ, jointly independent, and each centered Gaussian with a positive variance σⱼ², and the target's noise is independent of every parent coordinate of the target.","s":"ℕ → Type (u_1 + 1)","l":77,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.mem_paY","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a predictor index k, k belongs to the target's parent set PA(Y) exactly when the target's structural coefficient on k is nonzero.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) {k : Fin (p + 1)}, k ∈ M.paY ↔ M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k ≠ 0","l":158,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.paY","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For an observational linear-Gaussian structural equation model, the target's parent set is the set of nodes with an arrow into the target in its acyclic graph.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Finset (Fin (p + 1))","l":144,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM.paY_subset_predictors","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"PA(Y) consists of predictors only (0 ∉ PA(Y)): the target is acyclic, so it is not its own parent.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p), M.paY ⊆ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p","l":150,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.SupportedOn","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate, a coefficient vector, and a set of coordinate indices, the assertion that the coefficient vector is supported on that set means that every nonzero coefficient has an index in the set.","s":"{p : ℕ} → (Fin (p + 1) → ℝ) → Finset (Fin (p + 1)) → Prop","l":61,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"For an observational linear-Gaussian structural equation model, the causal coefficient vector assigns to each variable its coefficient in the structural equation for the target outcome.","s":"{p : ℕ} → Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p → Fin (p + 1) → ℝ","l":31,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate, an observational linear structural-equation model, an interventional environment based on that model, a coefficient vector, and a realization of the model's exogenous variables, the interventional regression residual is the intervened target value minus the sum of each intervened coordinate value multiplied by its coefficient.","s":"{p : ℕ} → {M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p} → Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M → (Fin (p + 1) → ℝ) → M.Ω → ℝ","l":81,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual_eq_eps","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Interventional residual is the target noise. For an observational SEM and a do-intervention environment built on it — where the target is never itself intervened on, so it keeps its structural equation — the environment residual Yᵉ − Σ_k β₀ₖ Xₖᵉ, evaluated at the causal coefficient, equals the target's structural noise ε₀ almost everywhere.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (e : Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M), ∀ᵐ (ω : M.Ω) ∂M.P, Causalean.Discovery.InvariantPrediction.LinearGaussian.envResidual e (Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff M) ω = M.ε ω (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p)","l":52,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_aemeasurable","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"εⱼ is a.e.-measurable: by the structural-residual identity hε it agrees a.e. with the measurable map Xⱼ − Σ_{k≠j} βⱼₖ Xₖ.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), AEMeasurable (fun ω => M.ε ω j) M.P","l":27,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_integrable","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"The noise εⱼ is M.P-integrable (Gaussian marginal has a first moment).","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), MeasureTheory.Integrable (fun ω => M.ε ω j) M.P","l":37,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.eps_integral_zero","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Moments","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"For a linear-Gaussian observational SEM and coordinate index j, the structural noise εⱼ is centered: its expectation under M's probability measure is zero.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (j : Fin (p + 1)), ∫ (ω : M.Ω), M.ε ω j ∂M.P = 0","l":54,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Moments.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.mem_predictors","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":2,"d":"A node is a predictor exactly when it is not the target node.","s":"∀ {p : ℕ} {k : Fin (p + 1)}, k ∈ Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors p ↔ k ≠ 0","l":73,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.nonDescendant_invariance","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Invariance","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Non-descendant invariance. Let e be an environment of the observational SEM M in which the single intervened coordinate is k₀. Then every coordinate k that is neither k₀ nor a descendant of k₀ in the observational DAG keeps its observational value almost surely: Xₖᵉ = Xₖ¹. (The node k₀ itself is pinned to the assigned constant, hence excluded.)","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (e : Causalean.Discovery.InvariantPrediction.LinearGaussian.Env M) (k₀ : Fin (p + 1)), e.A = {k₀} → ∀ᵐ (ω : M.Ω) ∂M.P, ∀ (k : Fin (p + 1)), k ≠ k₀ → ¬M.dag.isAncestor k₀ k → e.X ω k = M.X ω k","l":36,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Invariance.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Regression","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with p predictor coordinates and one target coordinate, an observational linear structural-equation model, a coefficient vector, and a realization of that model's exogenous variables, the observational regression residual is the target value minus the sum of each coordinate value multiplied by its coefficient.","s":"{p : ℕ} → (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) → (Fin (p + 1) → ℝ) → M.Ω → ℝ","l":71,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Regression.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual_eq_eps","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":1,"d":"Observational residual is the target noise. For an observational SEM, evaluated at the causal coefficient γ* = β₀,·, the observational residual Y − Σ_k β₀ₖ X_k equals the target's structural noise ε₀ almost everywhere.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p), ∀ᵐ (ω : M.Ω) ∂M.P, Causalean.Discovery.InvariantPrediction.LinearGaussian.obsResidual M (Causalean.Discovery.InvariantPrediction.LinearGaussian.causalCoeff M) ω = M.ε ω (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p)","l":44,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.predictors","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with $p$ predictors, the predictor index set contains precisely the $p$ non-target variables among its $p+1$ variables.","s":"(p : ℕ) → Finset (Fin (p + 1))","l":69,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.sum_causalCoeff_eq","k":"theorem","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Helpers.Residual","a":"Discovery/InvariantPrediction/LinearGaussian/Helpers","t":2,"d":"With the causal coefficient, the full-sum Σ_k β₀ₖ X_k equals the structural-equation sum Σ_{k≠0} β₀ₖ X_k, since β₀₀ = 0.","s":"∀ {p : ℕ} (M : Causalean.Discovery.InvariantPrediction.LinearGaussian.ObsSEM p) (x : Fin (p + 1) → ℝ), ∑ k, M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k * x k = ∑ k ∈ Finset.univ.erase (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p), M.β (Causalean.Discovery.InvariantPrediction.LinearGaussian.target p) k * x k","l":36,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Helpers/Residual.lean"},{"n":"Causalean.Discovery.InvariantPrediction.LinearGaussian.target","k":"def","m":"Causalean.Discovery.InvariantPrediction.LinearGaussian.Model","a":"Discovery/InvariantPrediction/LinearGaussian","t":1,"d":"For a model with $p$ predictors, the target node is the first of its $p+1$ variables, represented by index zero.","s":"(p : ℕ) → Fin (p + 1)","l":65,"f":"Causalean/Discovery/InvariantPrediction/LinearGaussian/Model.lean"},{"n":"Causalean.Discovery.LiNGAM.colSupport_of_kurtosis","k":"theorem","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"Kurtosis-based column support. Let e be a family of real sources on a probability space such that each coordinate eⱼ is measurable, the coordinates are mutually independent, each has finite fourth moment, each is centered, and the fourth cumulant (excess kurtosis) of every coordinate is nonzero and of one common sign, all positive or all negative. For two distinct row indices i ≠ k of a mixing matrix W, if the linear forms Σⱼ Wᵢⱼ eⱼ and Σⱼ Wₖⱼ eⱼ are independent, then every column j satisfies Wᵢⱼ · Wₖⱼ = 0. This is the input required by genPerm_of_det_ne_zero_of_colSupport.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {e : Ω → Fin n → ℝ} {W : Matrix (Fin n) (Fin n) ℝ}, (∀ (j : Fin n), Measurable fun ω => e ω j) → ProbabilityTheory.iIndepFun (fun j ω => e ω j) P → (∀ (j : Fin n), MeasureTheory.MemLp (fun ω => e ω j) 4 P) → (∀ (j : Fin n), ∫ (ω : Ω), e ω j ∂P = 0) → ((∀ (j : Fin n), 0 < Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P) ∨ ∀ (j : Fin n), Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P < 0) → ∀ {i k : Fin n}, i ≠ k → ProbabilityTheory.IndepFun (fun ω => ∑ j, W i j * e ω j) (fun ω => ∑ j, W k j * e ω j) P → ∀ (j : Fin n), W i j * W k j = 0","l":614,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.cross_fourth_cumulant_eq_sum","k":"theorem","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"Fourth cross-cumulant identity (Isserlis / cumulant multilinearity). Let e be a family of real sources on a probability space such that each coordinate eⱼ is measurable, the coordinates are mutually independent, each has finite fourth moment, and each is centered. Then for any coefficient vectors a, b and the linear forms yₐ = Σⱼ aⱼ eⱼ, y_b = Σⱼ bⱼ eⱼ, the joint fourth cumulant cum(yₐ,yₐ,y_b,y_b) equals Σⱼ aⱼ² bⱼ² κ₄(eⱼ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {e : Ω → Fin n → ℝ} (a b : Fin n → ℝ), (∀ (j : Fin n), Measurable fun ω => e ω j) → ProbabilityTheory.iIndepFun (fun j ω => e ω j) P → (∀ (j : Fin n), MeasureTheory.MemLp (fun ω => e ω j) 4 P) → (∀ (j : Fin n), ∫ (ω : Ω), e ω j ∂P = 0) → ∫ (ω : Ω), (∑ j, a j * e ω j) ^ 2 * (∑ j, b j * e ω j) ^ 2 ∂P - (∫ (ω : Ω), (∑ j, a j * e ω j) ^ 2 ∂P) * ∫ (ω : Ω), (∑ j, b j * e ω j) ^ 2 ∂P - 2 * (∫ (ω : Ω), (∑ j, a j * e ω j) * ∑ j, b j * e ω j ∂P) ^ 2 = ∑ j, a j ^ 2 * b j ^ 2 * Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P","l":45,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.ica_genPerm_relation","k":"theorem","m":"Causalean.Discovery.LiNGAM.LiNGAMKurtosis","a":"Discovery/LiNGAM","t":1,"d":"Let A, A' be n × n real matrices such that A is invertible and A' is invertible. If the matrix A'⁻¹ A has at most one non-zero entry per column — the column-support fact — then there exist a permutation τ and a nowhere-zero scale vector d such that A'⁻¹ and A⁻¹ agree up to that generalized permutation: A'⁻¹ i j = d i · A⁻¹ (τ i) j for every i, j.","s":"∀ {n : ℕ} {A A' : Matrix (Fin n) (Fin n) ℝ}, IsUnit A.det → IsUnit A'.det → (∀ (j i k : Fin n), i ≠ k → (A'⁻¹ * A) i j = 0 ∨ (A'⁻¹ * A) k j = 0) → ∃ τ d, ∀ (i j : Fin n), A'⁻¹ i j = d i * A⁻¹ (τ i) j","l":35,"f":"Causalean/Discovery/LiNGAM/LiNGAMKurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.kurt","k":"def","m":"Causalean.Discovery.LiNGAM.Kurtosis","a":"Discovery/LiNGAM","t":1,"d":"For a measurable sample space, a real-valued random variable, and a measure on that sample space, its excess kurtosis is its fourth moment minus three times the square of its second moment.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Ω → ℝ) → MeasureTheory.Measure Ω → ℝ","l":35,"f":"Causalean/Discovery/LiNGAM/Kurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.lingam_identifiability_kurtosis","k":"theorem","m":"Causalean.Discovery.LiNGAM.LiNGAMKurtosis","a":"Discovery/LiNGAM","t":1,"d":"LiNGAM identification theorem (kurtosis route, Marcinkiewicz-free). Let A, A' be n × n real mixing matrices, both invertible, with coefficient matrices A⁻¹ = I − B, A'⁻¹ = I − B' both having unit diagonal, and with A⁻¹ acyclic with respect to a causal order σ: (A⁻¹) i j = 0 whenever σ i < σ j. Let e, e' be families of real disturbances on a probability space, each coordinate measurable, each family's coordinates mutually independent, each coordinate of e with finite fourth moment, each coordinate of e centered, and the fourth cumulant of every coordinate of e nonzero and of one common sign. If the structural equations A·e and A'·e' produce the same observed law, then the coefficient matrices coincide, A⁻¹ = A'⁻¹ (i.e. B = B').","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n : ℕ} {A A' : Matrix (Fin n) (Fin n) ℝ}, IsUnit A.det → IsUnit A'.det → (∀ (i : Fin n), A⁻¹ i i = 1) → (∀ (i : Fin n), A'⁻¹ i i = 1) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → A⁻¹ i j = 0) → ∀ {e e' : Ω → Fin n → ℝ}, (∀ (i : Fin n), Measurable fun ω => e ω i) → (∀ (i : Fin n), Measurable fun ω => e' ω i) → ProbabilityTheory.iIndepFun (fun i ω => e ω i) P → ProbabilityTheory.iIndepFun (fun i ω => e' ω i) P → (∀ (i : Fin n), MeasureTheory.MemLp (fun ω => e ω i) 4 P) → (∀ (i : Fin n), ∫ (ω : Ω), e ω i ∂P = 0) → ((∀ (j : Fin n), 0 < Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P) ∨ ∀ (j : Fin n), Causalean.Discovery.LiNGAM.kurt (fun ω => e ω j) P < 0) → MeasureTheory.Measure.map (fun ω => A.mulVec (e ω)) P = MeasureTheory.Measure.map (fun ω => A'.mulVec (e' ω)) P → A⁻¹ = A'⁻¹","l":73,"f":"Causalean/Discovery/LiNGAM/LiNGAMKurtosis.lean"},{"n":"Causalean.Discovery.LiNGAM.lingam_identifiable","k":"theorem","m":"Causalean.Discovery.LiNGAM.Identifiability","a":"Discovery/LiNGAM","t":1,"d":"LiNGAM coefficient identifiability (modulo ICA). Let B and B' be two n × n real coefficient matrices such that B has zero diagonal, B' has zero diagonal, B is acyclic with respect to a causal order σ: B i j = 0 whenever σ i < σ j, and the matrices I − B and I − B' are related by a generalized permutation — a permutation τ composed with a nonzero diagonal scale d, so (I − B') i j = d i · (I − B) (τ i) j (as delivered by ICA identifiability). Then B equals B'. This packages eq_of_genPerm_triangular_unitDiag for C = I − B.","s":"∀ {n : ℕ} {B B' : Matrix (Fin n) (Fin n) ℝ}, (∀ (i : Fin n), B i i = 0) → (∀ (i : Fin n), B' i i = 0) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → B i j = 0) → ∀ {τ : Equiv.Perm (Fin n)} {d : Fin n → ℝ}, (∀ (i j : Fin n), (1 - B') i j = d i * (1 - B) (τ i) j) → B = B'","l":36,"f":"Causalean/Discovery/LiNGAM/Identifiability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0H_gram_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"For every linear causal disentanglement solution, specified by natural numbers of latent variables, observed variables, and intervention contexts, an invertibility certificate for the Gram matrix of the product of its observational structural matrix and mixing pseudoinverse is constructed as a product of invertible matrices.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible (S.B0 * S.H * (S.B0 * S.H).transpose)","l":146,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_blockTriangular","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"B0 is upper triangular in the BlockTriangular id sense.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.B0.BlockTriangular id","l":63,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_det_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"det B0 = ∏ᵢ (B0)ᵢᵢ > 0.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), 0 < S.B0.det","l":82,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"For every linear causal disentanglement solution, specified by natural numbers of latent variables, observed variables, and intervention contexts, an invertibility certificate for its observational structural matrix is constructed from that matrix's nonzero determinant.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible S.B0","l":105,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.B0_source_row","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R3 ingredient) The iₖ-th row of B₀ at a source. If iₖ is a *source* node (no parents: (B₀)_{iₖ,j} = 0 for every j ≠ iₖ), then wₖ = B₀ᵀ *ᵥ e_{iₖ} = β e_{iₖ} with β = (B₀)_{iₖ,iₖ} > 0. This collapses the latent key identity (C) to the rank-one Δₖ = (λₖ² − β²) e_{iₖ} e_{iₖ}ᵀ, the base case of the monomial induction (R3): the transported equation then forces the i'ₖ-th row of M to be a multiple of e_{iₖ}.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), (∀ (j : Fin d), j ≠ i → S.B0 i j = 0) → S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = S.B0 i i • Causalean.Discovery.LinearDisentanglement.stdVec d i","l":422,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_blockTriangular","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Bint k is upper triangular: the perfect intervention only rewrites the target row (which keeps the diagonal at λₖ and zeroes the strictly-lower entries already zero in B0).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).BlockTriangular id","l":66,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_det_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"det (Bint k) = ∏ᵢ (Bint k)ᵢᵢ > 0.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), 0 < (S.Bint k).det","l":100,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_diag_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(Bint k)ᵢᵢ = (B0)ᵢᵢ off the target, and = λₖ on the target — in both cases strictly positive.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K) (i : Fin d), 0 < S.Bint k i i","l":87,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"For every linear causal disentanglement solution, specified by natural numbers of latent variables, observed variables, and intervention contexts and every intervention context in that solution, an invertibility certificate for the corresponding interventional structural matrix is constructed from that matrix's nonzero determinant.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → (k : Fin K) → Invertible (S.Bint k)","l":113,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_sub_B0","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Rank-one perturbation. Bₖ − B₀ = e_{iₖ} cₖᵀ (Matrix.vecMulVec): a perfect single-node intervention rewrites only the target row of B₀.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Bint k - S.B0 = Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.cvec S k)","l":271,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Bint_transpose_mulVec_target","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(B) Target row of Bₖ. (Bₖ)ᵀ *ᵥ e_{iₖ} = λₖ • e_{iₖ}: the iₖ-th row of a perfect-intervention matrix is λₖ e_{iₖ}ᵀ (the intervention zeroes the parent entries and sets the diagonal to λₖ).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) = S.lam k • Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)","l":290,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate","k":"structure","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"A concrete certificate that the selected two-coordinate shift cloud is contained in the affine line u x + v y = c, with a nonzero normal vector.","s":"{d : ℕ} → {E : Type u_1} → (E → Fin d → ℝ) → Fin d → Fin d → Type","l":28,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d matrix and two selected coordinates, pair-cycle admissibility holds exactly when the product of the $(i,j)$ and $(j,i)$ entries is not one; for distinct coordinates these are the two opposite off-diagonal entries of the selected pair.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → Prop","l":91,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.RepresentsCovarianceFamily","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a dimension, an environment collection, a covariance matrix for each environment, a diagonalizer and invariant matrix, and a diagonal shift vector for each environment, the covariance family is represented exactly when every environment's covariance matrix equals the covariance represented by those common matrices and that environment's shift vector. Invertibility of the diagonalizer is not part of this predicate; a singular diagonalizer represents only the all-zero family.","s":"{d : ℕ} → {E : Type u_1} → (E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → (E → Fin d → ℝ) → Prop","l":144,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.commonShiftCrossTerm","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference matrix, two selected coordinates, and a line normal, offset, and deformation magnitude, the common shift cross term is the line offset times the deformation magnitude, divided by both selected normalization denominators.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → ℝ → ℝ","l":97,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference diagonalizer, two selected coordinates, and a line normal and deformation magnitude, the deformed diagonalizer is the normalized two-coordinate deformation applied to the rows of the reference diagonalizer.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":84,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer_cycle_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the reference diagonal is unit-normalized, and both normalization denominators are nonzero, the selected cycle-product defect obeys the stated shear identity.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ∀ {u v t : ℝ}, Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t * Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t * (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t i j * Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t j i - 1) = (B i j * B j i - 1) * (1 - t ^ 2 * u * v)","l":260,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer_ne","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the reference diagonalizer is invertible, the line normal is nonzero, the deformation parameter is nonzero, and both normalization denominators are nonzero, the deformed diagonalizer differs from the reference.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → IsUnit (Matrix.det B) → ∀ {u v t : ℝ}, u ≠ 0 ∨ v ≠ 0 → t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t ≠ B","l":300,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer_pairCycleAdmissible","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the reference diagonal is unit-normalized, its selected two-cycle is admissible, both normalization denominators are nonzero, and the shear determinant is nonzero, the deformed selected two-cycle remains admissible.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B i j → ∀ {u v t : ℝ}, Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → 1 - t ^ 2 * u * v ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t) i j","l":282,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer_unitDiagonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the reference diagonal is unit-normalized, and both normalization denominators are nonzero, row normalization restores unit diagonal.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ∀ {u v t : ℝ}, Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t)","l":235,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"At zero deformation parameter, the deformed diagonalizer equals the reference diagonalizer.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d) (u v : ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v 0 = B","l":96,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference diagonalizer and invariant matrix, two selected coordinates, and a line normal, offset, and deformation magnitude, the deformed invariant matrix is obtained by first forming the normalized two-coordinate deformation, then taking its congruence transform of the invariant matrix and adding the symmetric common shift cross term.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":114,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant_isSymm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct and the original invariant is symmetric, the deformed invariant is symmetric.","s":"∀ {d : ℕ} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Matrix.IsSymm Ω → ∀ (u v c t : ℝ), Matrix.IsSymm (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j u v c t)","l":677,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"At zero deformation parameter, the transformed invariant is the original invariant matrix.","s":"∀ {d : ℕ} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d) (u v c : ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j u v c 0 = Ω","l":103,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference diagonalizer, two selected coordinates, a line normal and deformation magnitude, an original diagonal shift vector, and a coordinate, the deformed shift at that coordinate is the corresponding diagonal entry obtained after forming the normalized two-coordinate deformation and taking the congruence transform of the original diagonal shift matrix.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → (Fin d → ℝ) → Fin d → ℝ","l":125,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift_apply_first","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the first transformed shift is its stated nonnegative weighted sum.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (u v t : ℝ) (s : Fin d → ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v t s i = (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t)⁻¹ ^ 2 * (s i + t ^ 2 * v ^ 2 * s j)","l":349,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift_apply_of_ne","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the coordinate is neither selected coordinate, its transformed shift is unchanged.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j k : Fin d}, k ≠ i → k ≠ j → ∀ (u v t : ℝ) (s : Fin d → ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v t s k = s k","l":415,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift_apply_second","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the second transformed shift is its stated nonnegative weighted sum.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (u v t : ℝ) (s : Fin d → ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v t s j = (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t)⁻¹ ^ 2 * (t ^ 2 * u ^ 2 * s i + s j)","l":382,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift_nonnegative","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct and the original shift is coordinatewise nonnegative, the transformed shift is coordinatewise nonnegative.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (u v t : ℝ) (s : Fin d → ℝ), (∀ (k : Fin d), 0 ≤ s k) → ∀ (k : Fin d), 0 ≤ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v t s k","l":424,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"At zero deformation parameter, the transformed diagonal shift is the original shift.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d) (u v : ℝ) (s : Fin d → ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v 0 s = s","l":110,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.exists_algebraic_admissibility_radius","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.SmallParameter","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"Some positive radius makes both row-normalization denominators and the selected shear determinant nonzero.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d) (u v : ℝ), ∃ ρ > 0, ∀ (t : ℝ), |t| < ρ → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 ∧ 1 - t ^ 2 * u * v ≠ 0","l":172,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/SmallParameter.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.exists_collinear_simultaneous_congruence_ambiguity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Main","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"Given two distinct selected coordinates, a unit-diagonal reference diagonalizer, its invertibility, an admissible selected two-cycle, a positive-definite invariant, coordinatewise nonnegative shifts, a certificate that the two selected shift coordinates of every environment lie on one affine line with nonzero normal, and a positive parameter bound, there is a deformation parameter that is nonzero and strictly smaller than the bound in absolute value, whose deformed diagonalizer, invariant and shifts (given by the explicit two-coordinate deformation formulas) satisfy: the new diagonalizer is unit-diagonal, invertible, pair-cycle admissible and different from the reference one; the new invariant is symmetric and positive definite; the new shifts are nonnegative; every covariance represented by the reference data is positive definite; and every environment's represented covariance is the same under the new data as under the reference data. The environment index may be any type, finite or not.","s":"∀ {d : ℕ} {E : Type u_1} (B₀ Ω₀ : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) {i j : Fin d}, i ≠ j → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B₀ i j → Matrix.PosDef Ω₀ → (∀ (e : E) (k : Fin d), 0 ≤ s e k) → ∀ (cert : Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate s i j) {r : ℝ}, 0 < r → ∃ t B₁ Ω₁ s₁, 0 < |t| ∧ |t| < r ∧ B₁ = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B₀ i j cert.u cert.v t ∧ Ω₁ = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B₀ Ω₀ i j cert.u cert.v cert.c t ∧ (s₁ = fun e => Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B₀ i j cert.u cert.v t (s e)) ∧ Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₁ ∧ IsUnit (Matrix.det B₁) ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B₁ i j ∧ B₁ ≠ B₀ ∧ Matrix.IsSymm Ω₁ ∧ Matrix.PosDef Ω₁ ∧ (∀ (e : E) (k : Fin d), 0 ≤ s₁ e k) ∧ (∀ (e : E), Matrix.PosDef (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance B₀ Ω₀ (s e))) ∧ ∀ (e : E), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance B₁ Ω₁ (s₁ e) = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance B₀ Ω₀ (s e)","l":28,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Main.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.exists_deformedInvariant_posDef_radius","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.SmallParameter","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct and the original invariant is positive definite, some positive radius keeps every transformed invariant positive definite.","s":"∀ {d : ℕ} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Matrix.PosDef Ω → ∀ (u v c : ℝ), ∃ ρ > 0, ∀ (t : ℝ), |t| < ρ → Matrix.PosDef (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j u v c t)","l":155,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/SmallParameter.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.exists_interior_collinear_ambiguity_example","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Main","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"Given two distinct selected coordinates and coordinatewise nonnegative shifts, the prescribed affine-collinear shift family has a positive-definite interior covariance example with two distinct compatible normalized diagonalizers.","s":"∀ {d : ℕ} {E : Type u_1} (s : E → Fin d → ℝ) {i j : Fin d}, i ≠ j → (∀ (e : E) (k : Fin d), 0 ≤ s e k) → ∀ (cert : Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate s i j), ∃ Sigma B₀ B₁ Ω₀ Ω₁ s₁, B₀ = 1 ∧ Ω₀ = 1 ∧ Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ ∧ IsUnit (Matrix.det B₀) ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B₀ i j ∧ Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₁ ∧ IsUnit (Matrix.det B₁) ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B₁ i j ∧ B₁ ≠ B₀ ∧ Matrix.PosDef Ω₀ ∧ Matrix.IsSymm Ω₁ ∧ Matrix.PosDef Ω₁ ∧ (∀ (e : E) (k : Fin d), 0 ≤ s₁ e k) ∧ (∀ (e : E), Matrix.PosDef (Sigma e)) ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.RepresentsCovarianceFamily Sigma B₀ Ω₀ s ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.RepresentsCovarianceFamily Sigma B₁ Ω₁ s₁","l":79,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Main.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.exists_small_admissible_parameter","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.SmallParameter","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"Given two distinct selected coordinates, a unit-diagonal reference matrix, an invertible reference matrix, an admissible selected two-cycle, a positive-definite invariant, two line-normal coefficients that are not both zero, an affine offset entering the transformed invariant's cross term, coordinatewise nonnegative shifts, and a positive requested bound, there is a nonzero deformation parameter below that bound preserving all stated admissibility, positivity, and nonnegativity properties.","s":"∀ {d : ℕ} {E : Type u_1} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → IsUnit (Matrix.det B) → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible B i j → Matrix.PosDef Ω → ∀ (s : E → Fin d → ℝ) (u v c : ℝ), u ≠ 0 ∨ v ≠ 0 → (∀ (e : E) (k : Fin d), 0 ≤ s e k) → ∀ {r : ℝ}, 0 < r → ∃ t, 0 < |t| ∧ |t| < r ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 ∧ 1 - t ^ 2 * u * v ≠ 0 ∧ IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j u v t)) ∧ IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t)) ∧ Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t) ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.PairCycleAdmissible (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t) i j ∧ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j u v t ≠ B ∧ Matrix.PosDef (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j u v c t) ∧ ∀ (e : E) (k : Fin d), 0 ≤ Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j u v t (s e) k","l":204,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/SmallParameter.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference matrix, two selected coordinates, and a shear coefficient and magnitude, the first normalization denominator is $1+t v B_{ji}$.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ","l":51,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference matrix, two selected coordinates, and a line normal and deformation magnitude, the normalized two-coordinate deformation is the pair row normalizer multiplied by the elementary pair shear.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":77,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation_isUnit_det","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, both normalization denominators are nonzero, and the shear determinant is nonzero, the normalized pair deformation is invertible.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ {u v t : ℝ}, Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → 1 - t ^ 2 * u * v ≠ 0 → IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j u v t))","l":147,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"At zero deformation parameter, the normalized pair deformation is the identity matrix.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d) (u v : ℝ), Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j u v 0 = 1","l":82,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairRowNormalizer","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference matrix, two selected coordinates, and a line normal and deformation magnitude, the pair row normalizer is the diagonal matrix that rescales the first selected row by the reciprocal of the first normalization denominator, the second selected row (when it differs from the first) by the reciprocal of the second, and leaves all other rows unchanged; a zero denominator has reciprocal zero by convention.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":65,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairRowNormalizer_isUnit_det","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct and both row-normalization denominators are nonzero, the diagonal row normalizer is invertible.","s":"∀ {d : ℕ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ {u v t : ℝ}, Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.firstNormalizationDenom B i j v t ≠ 0 → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom B i j u t ≠ 0 → IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairRowNormalizer B i j u v t))","l":132,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairShear","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d$-dimensional coordinate system, two selected coordinates, and three real numbers specifying a line normal and deformation magnitude, the elementary two-row shear is the identity matrix plus $tv$ at position $(i,j)$ and $tu$ at position $(j,i)$. For distinct coordinates these are two off-diagonal entries; if the coordinates coincide, both increments land on the same diagonal entry.","s":"{d : ℕ} → Fin d → Fin d → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":43,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairShear_diagonal_congruence_decomposition","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the transformed diagonal shift matrix splits into the common symmetric off-diagonal term and its new diagonal shift.","s":"∀ {d : ℕ} {E : Type u_1} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (s : E → Fin d → ℝ) (cert : Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate s i j) (t : ℝ) (e : E), have T := Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j cert.u cert.v t; T * Matrix.diagonal (s e) * Matrix.transpose T = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairSymmetricOffDiagonal i j (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.commonShiftCrossTerm B i j cert.u cert.v cert.c t) + Matrix.diagonal (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j cert.u cert.v t (s e))","l":546,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairShear_isUnit_det","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct and the selected shear-block determinant is nonzero, the elementary pair shear is invertible.","s":"∀ {d : ℕ} {i j : Fin d}, i ≠ j → ∀ {u v t : ℝ}, 1 - t ^ 2 * u * v ≠ 0 → IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairShear i j u v t))","l":118,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.pairSymmetricOffDiagonal","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d$-dimensional coordinate system, two selected coordinates, and a real entry value, the symmetric selected off-diagonal matrix is the sum of the matrix with that value at position $(i,j)$ and the matrix with that value at position $(j,i)$. For distinct coordinates it has the value in both off-diagonal positions and zero elsewhere; if the coordinates coincide, it has twice the value on that diagonal entry.","s":"{d : ℕ} → Fin d → Fin d → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":106,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d diagonalizer and invariant matrix and a diagonal shift vector, the represented covariance matrix is the inverse congruence transform of the invariant matrix plus the diagonal shift matrix. This is meaningful for an invertible diagonalizer; for a singular one the matrix inverse is zero by convention, so the value is the zero matrix.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → (Fin d → ℝ) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":136,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance_deformation_eq","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"With two distinct selected coordinates, an invertible reference diagonalizer, and an invertible normalized deformation, the transformed representation gives exactly the same covariance matrix in each environment.","s":"∀ {d : ℕ} {E : Type u_1} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (s : E → Fin d → ℝ) (cert : Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate s i j) (t : ℝ) (e : E), IsUnit (Matrix.det B) → IsUnit (Matrix.det (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j cert.u cert.v t)) → Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedDiagonalizer B i j cert.u cert.v t) (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j cert.u cert.v cert.c t) (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j cert.u cert.v t (s e)) = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance B Ω (s e)","l":630,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance_posDef","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"Given an invertible diagonalizer, a positive-definite invariant, and a coordinatewise nonnegative shift, the represented covariance matrix is positive definite.","s":"∀ {d : ℕ} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : Fin d → ℝ), IsUnit (Matrix.det B) → Matrix.PosDef Ω → (∀ (k : Fin d), 0 ≤ s k) → Matrix.PosDef (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.representedCovariance B Ω s)","l":662,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.secondNormalizationDenom","k":"def","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Definitions","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":1,"d":"For a $d\\u2011by\\u2011$d reference matrix, two selected coordinates, and a shear coefficient and magnitude, the second normalization denominator is $1+t u B_{ij}$.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Fin d → Fin d → ℝ → ℝ → ℝ","l":58,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.total_congruence_decomposition","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.Algebra","a":"Discovery/LinearDisentanglement/CollinearAmbiguity","t":2,"d":"When the selected coordinates are distinct, the latent congruence is exactly the deformed invariant plus the transformed diagonal shift.","s":"∀ {d : ℕ} {E : Type u_1} (B Ω : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {i j : Fin d}, i ≠ j → ∀ (s : E → Fin d → ℝ) (cert : Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.AffineLineCertificate s i j) (t : ℝ) (e : E), have T := Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.normalizedPairDeformation B i j cert.u cert.v t; T * (Ω + Matrix.diagonal (s e)) * Matrix.transpose T = Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedInvariant B Ω i j cert.u cert.v cert.c t + Matrix.diagonal (Causalean.Discovery.LinearDisentanglement.CollinearAmbiguity.deformedShift B i j cert.u cert.v t (s e))","l":615,"f":"Causalean/Discovery/LinearDisentanglement/CollinearAmbiguity/Algebra.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.HHt_invertible","k":"instance","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"For every linear causal disentanglement solution, specified by natural numbers of latent variables, observed variables, and intervention contexts, an invertibility certificate for the Gram matrix of its mixing pseudoinverse is constructed from that Gram matrix's positive definiteness.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → Invertible (S.H * S.H.transpose)","l":135,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.HHt_posDef","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The latent Gram matrix H Hᵀ is positive definite.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), (S.H * S.H.transpose).PosDef","l":130,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.H_eq_recover_mul_Theta0","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The recovery identity H = W Θ₀. With C = B₀ H and Θ₀ = Cᵀ C, the matrix W = B₀⁻¹ (C Cᵀ)⁻¹ C satisfies W Θ₀ = H.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.B0⁻¹ * (S.B0 * S.H * (S.B0 * S.H).transpose)⁻¹ * (S.B0 * S.H) * S.Theta0 = S.H","l":176,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.H_mul_rightInv","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"R := Hᵀ (H Hᵀ)⁻¹ is a right inverse of H: H R = 1.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.H * (S.H.transpose * (S.H * S.H.transpose)⁻¹) = 1","l":158,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.IsPORQ","k":"structure","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Definition 1 (partial order RQ decomposition). A witness that a full-row-rank matrix H factors as H = R Q with R having a non-negative diagonal and support confined to the reflexive partial order — Rᵢⱼ = 0 unless i ⪯ j, and the rows of Q each of unit Euclidean norm and pairwise orthogonal along the strict order.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Matrix (Fin d) (Fin d) ℝ → Matrix (Fin d) (Fin p) ℝ → Prop","l":136,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.AffineMinorSeparated","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a finite environment collection, a diagonal shift vector for each environment, and a real margin, affine minor separation holds when there exist one base environment and one selected environment for each coordinate such that the absolute determinant of their shift-difference matrix is at least the margin.","s":"{d : ℕ} → {E : Type u_1} → [Fintype E] → (E → Fin d → ℝ) → ℝ → Prop","l":71,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.ApproximateCongruence","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a nonempty finite environment collection, an observed matrix for each environment, a prescribed diagonal vector for each environment, a candidate matrix, and a real tolerance, approximate congruence holds exactly when the simultaneous congruence residual is at most the tolerance.","s":"{d : ℕ} → {E : Type u_1} → [Fintype E] → [Nonempty E] → (E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) → (E → Fin d → ℝ) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → ℝ → Prop","l":134,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a finite environment collection, an observed matrix for each environment, a prescribed diagonal vector for each environment, and a reference matrix, exact congruence holds exactly when, in every environment, the reference matrix transforms the observed matrix into the diagonal matrix prescribed there.","s":"{d : ℕ} → {E : Type u_1} → [Fintype E] → (E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) → (E → Fin d → ℝ) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Prop","l":126,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairConditionBound","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a real condition bound, and a reference and candidate matrix, the pair condition bound holds exactly when the reference matrix is boundedly conditioned by that bound and the candidate matrix is boundedly conditioned by that bound.","s":"{d : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Prop","l":107,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairMatrixScaleBound","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a real scale bound, and a reference and candidate matrix, the pair matrix-scale bound holds exactly when the reference matrix's Euclidean operator norm is at most the scale bound and the candidate matrix's Euclidean operator norm is at most the scale bound.","s":"{d : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Prop","l":114,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.DetConditionEnvelope","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a condition-number envelope, and a square matrix, the determinant-condition envelope holds exactly when the matrix is nonsingular, its determinant has absolute value at most $p!$, and its Euclidean operator-norm condition number is at most the envelope.","s":"{p : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Prop","l":150,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InIdentityBranch","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a shift scale, a pairwise affine-separation margin, a matrix scale, and a reference and candidate matrix, membership in the identity branch holds exactly when the transition error has entrywise Euclidean size at most one and twice the pairwise aggregate factor times the shift scale times that size is at most one half.","s":"{p : ℕ} → ℝ → ℝ → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Prop","l":165,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InReferenceNeighborhood","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a real neighborhood radius, and a reference and candidate matrix, membership in the reference neighborhood holds exactly when their entrywise Euclidean distance is at most the radius.","s":"{p : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Prop","l":144,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a finite environment collection, an observed matrix for each environment, a candidate matrix, and a real tolerance, off-diagonal approximate congruence holds exactly when every off-diagonal entry of every transformed matrix has absolute value at most the tolerance.","s":"{p : ℕ} → {E : Type u_1} → [Fintype E] → (E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → ℝ → Prop","l":49,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairDetConditionEnvelope","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a condition-number envelope, and a reference and candidate matrix, the pair determinant-condition envelope holds exactly when the reference matrix satisfies that determinant-condition envelope and the candidate matrix satisfies that determinant-condition envelope.","s":"{p : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Prop","l":158,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairMatrixNormBound","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a real scale bound, and a reference and candidate matrix, the pair matrix-norm bound is the pair matrix-scale bound for the same data: both Euclidean operator norms are at most the scale bound. It is an abbreviation kept for compatibility; PairMatrixScaleBound is the canonical name.","s":"{p : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Prop","l":137,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a finite environment collection, a diagonal shift vector for each environment, and a real margin, pairwise affine separation holds exactly when every two distinct coordinates admit three, possibly pair-specific, environments whose affine determinant has absolute value at least the margin.","s":"{p : ℕ} → {E : Type u_1} → [Fintype E] → (E → Fin p → ℝ) → ℝ → Prop","l":41,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension and a condition-number envelope, the condition root is $((p!)\\kappa^{p-1})^{1/p}$ for positive dimension. At dimension zero the exponents are evaluated by convention (natural subtraction gives $p-1=0$ and $1/0=0$), so the value is one.","s":"ℕ → ℝ → ℝ","l":131,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.coordinateProductRemainder_le","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For a real square matrix, two rows, and two environments, if the shifts have scale at most M and M is nonnegative, then the weighted coordinate-product remainder is bounded by 2 M times the squared entrywise L² size.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] (s : E → Fin p → ℝ) (R : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M : ℝ}, 0 ≤ M → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → ∀ (e₁ e₀ : E) (i j : Fin p), |∑ k, R i k * R j k * (s e₁ k - s e₀ k)| ≤ 2 * M * Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R ^ 2","l":26,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.diagonalBranch_control","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For an invertible reference, unit-diagonal reference and candidate matrices, a nonnegative entry bound, and an off-diagonal transition-entry bound, each diagonal transition error is controlled by the same row's off-diagonal entries and the corresponding reference column.","s":"∀ {p : ℕ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {c : ℝ}, IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → 0 ≤ c → (∀ (i j : Fin p), i ≠ j → |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B i j| ≤ c) → ∀ (i : Fin p), |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B i i| ^ 2 ≤ ‖B₀‖ ^ 2 * ∑ k ∈ Finset.univ.erase i, |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B i k| ^ 2","l":268,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension and a square matrix, its entrywise Euclidean size is its Frobenius norm: the square root of the sum of squared absolute values of all its entries.","s":"{p : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → ℝ","l":64,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_eq_frobenius_norm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For a square matrix, its entrywise Euclidean size is, by definition, Mathlib's Frobenius norm of the matrix.","s":"∀ {p : ℕ} (R : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R = ‖R‖","l":73,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_eq_sqrt","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For a square matrix, its entrywise Euclidean size equals the square root of the sum of squared absolute values of its entries.","s":"∀ {p : ℕ} (R : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R = √(∑ i, ∑ j, |R i j| ^ 2)","l":78,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_le_dimension_mul_opNorm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension and a real square matrix, its entrywise-L² size is at most the dimension times its Euclidean operator norm.","s":"∀ {p : ℕ} (X : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), 0 < p → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 X ≤ ↑p * ‖X‖","l":32,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_mul_le","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For two real square matrices, the entrywise-L² size of their product is at most the product of their entrywise-L² sizes.","s":"∀ {p : ℕ} (X Y : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 (X * Y) ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 X * Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 Y","l":26,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_nonneg","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For a square matrix, its entrywise Euclidean size is nonnegative.","s":"∀ {p : ℕ} (R : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), 0 ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R","l":85,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2_transitionError_le","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension and nonnegative matrix and entry bounds, an invertible unit-diagonal reference and a unit-diagonal candidate, a reference norm bound, and a common off-diagonal transition-entry bound, the full transition error's entrywise L² size is bounded by the ordered-pair aggregation factor.","s":"∀ {p : ℕ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {L c : ℝ}, 0 < p → 0 ≤ L → 0 ≤ c → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ‖B₀‖ ≤ L → (∀ (i j : Fin p), i ≠ j → |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B i j| ≤ c) → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B) ≤ c * √(↑p * ↑(p - 1) * (1 + L ^ 2))","l":325,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.inIdentityBranch_of_small_residual","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"Small-residual local identity-branch selection. For positive dimension, shift scale, affine margin, matrix scale, and inverse envelope, bounded pairwise-separated shifts, an exact invertible unit-diagonal reference, a unit-diagonal candidate, a reference operator norm at most the matrix scale, an inverse bound, nonnegative approximate residual, the explicit residual smallness condition, and membership in the explicit ordinary reference neighborhood, the transition lies on the identity branch needed by the linear stability estimate.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M δ L J ε : ℝ}, 0 < p → 0 < M → 0 < δ → 1 ≤ L → 0 < J → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated s δ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ‖B₀‖ ≤ L → ‖B₀⁻¹‖ ≤ J → 0 ≤ ε → ε ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius p M δ L → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InReferenceNeighborhood (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius p M δ L J) B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InIdentityBranch M δ L B₀ B","l":92,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.invOpNorm_le_conditionEnvelope","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension, a unit-diagonal matrix, and its determinant/condition envelope, the reference inverse operator norm is at most the declared condition envelope.","s":"∀ {p : ℕ} {κ : ℝ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), 0 < p → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.DetConditionEnvelope κ B → ‖B⁻¹‖ ≤ κ","l":76,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.one_le_conditionRoot","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension and a condition envelope at least one, the determinant/condition root is at least one.","s":"∀ {p : ℕ} {κ : ℝ}, 0 < p → 1 ≤ κ → 1 ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ","l":315,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.opNorm_le_conditionRoot","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For positive dimension, condition envelope at least one, and a determinant/condition envelope for the matrix, the matrix operator norm is bounded by the determinant/condition root.","s":"∀ {p : ℕ} {κ : ℝ} (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), 0 < p → 1 ≤ κ → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.DetConditionEnvelope κ B → ‖B‖ ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ","l":327,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.opNorm_le_entryL2","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Stability","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For a real square matrix, its Euclidean operator norm is at most its entrywise L² size.","s":"∀ {p : ℕ} (R : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), ‖R‖ ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R","l":41,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Stability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.opNorm_sub_le_condition_specialization","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Stability","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"Under positive dimension, positive shift scale and pairwise margin, condition envelope at least one, bounded pairwise-separated shifts, exact and approximate unit-diagonal feasibility, a nonnegative residual tolerance, the determinant/condition envelope of the reference, and the explicit residual threshold and ordinary local reference neighborhood, the operator error is bounded by the requested displayed constant with L = ((p!) κ^(p-1))^(1/p).","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M γ κ ε : ℝ}, 0 < p → 0 < M → 0 < γ → 1 ≤ κ → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated s γ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.DetConditionEnvelope κ B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → 0 ≤ ε → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → ε ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius p M γ (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ) → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InReferenceNeighborhood (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius p M γ (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ) κ) B₀ B → ‖B - B₀‖ ≤ 16 * (6 * M / γ) * √(↑p * ↑(p - 1) * (1 + Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ ^ 2)) * Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ ^ 3 * ε","l":165,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Stability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.opNorm_sub_le_of_pairwise_affine","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Stability","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"Small-residual local pairwise-affine simultaneous-congruence stability. For positive dimension, shift scale, affine margin, matrix scale, and inverse envelope, bounded pairwise-separated shifts, an exact invertible unit-diagonal reference, a unit-diagonal candidate, a reference operator norm at most the matrix scale, a reference inverse bound, nonnegative off-diagonal residual below the explicit threshold, approximate feasibility, and membership in the explicit entrywise-L² neighborhood of the reference, the candidate-reference Euclidean operator distance is linear in the residual.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M δ L J ε : ℝ}, 0 < p → 0 < M → 0 < δ → 1 ≤ L → 0 < J → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated s δ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ‖B₀‖ ≤ L → ‖B₀⁻¹‖ ≤ J → 0 ≤ ε → ε ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius p M δ L → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InReferenceNeighborhood (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius p M δ L J) B₀ B → ‖B - B₀‖ ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseStabilityConstant p M δ L * ε","l":139,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Stability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.opNorm_sub_le_of_pairwise_affine_in_identity_branch","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Stability","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"Local pairwise-affine simultaneous-congruence stability. For positive dimension, shift scale, and affine margin, a matrix scale at least one, bounded pairwise-separated shifts, an exact invertible unit-diagonal reference, a unit-diagonal candidate, a reference operator norm at most the matrix scale, nonnegative off-diagonal residual tolerance, off-diagonal approximate feasibility, and the explicit local identity-branch condition, the candidate-reference Euclidean operator distance is linear in the residual.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M δ L ε : ℝ}, 0 < p → 0 < M → 0 < δ → 1 ≤ L → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated s δ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → ‖B₀‖ ≤ L → 0 ≤ ε → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InIdentityBranch M δ L B₀ B → ‖B - B₀‖ ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseStabilityConstant p M δ L * ε","l":60,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Stability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairAffineDet","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, an environment collection, a diagonal shift vector for each environment, two selected coordinates, and three environments, the pairwise affine determinant is the signed area determinant of the three corresponding two-coordinate shift points.","s":"{p : ℕ} → {E : Type u_1} → (E → Fin p → ℝ) → Fin p → Fin p → E → E → E → ℝ","l":32,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairMatrixNormBound_conditionRoot","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension, condition envelope at least one, and a common determinant/condition envelope, both matrix norms obey the common conditionRoot bound.","s":"∀ {p : ℕ} {κ : ℝ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), 0 < p → 1 ≤ κ → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairDetConditionEnvelope κ B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairMatrixNormBound (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.conditionRoot p κ) B₀ B","l":358,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pair_cramer_control","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"If the scale is nonnegative, the determinant margin is positive, and the residual bound is nonnegative, all four coefficients have scale at most 2M, the two-by-two coefficient determinant is separated, the two displayed equations hold, and both equation residuals are at most η, then both unknowns obey the conservative explicit Cramer bound.","s":"∀ {a b c d x y r₁ r₂ M δ η : ℝ}, 0 ≤ M → 0 < δ → 0 ≤ η → |a| ≤ 2 * M ∧ |b| ≤ 2 * M ∧ |c| ≤ 2 * M ∧ |d| ≤ 2 * M → δ ≤ |a * d - c * b| → a * x + b * y = r₁ → c * x + d * y = r₂ → |r₁| ≤ η → |r₂| ≤ η → |x| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor M δ * η ∧ |y| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor M δ * η","l":78,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseAggregateFactor","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a shift scale, a pairwise affine-separation margin, and a matrix scale, the pairwise aggregate factor is the pairwise solve factor times the square root of $p(p-1)(1+L^2)$.","s":"ℕ → ℝ → ℝ → ℝ → ℝ","l":98,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a shift scale, a pairwise affine-separation margin, a matrix scale, and an inverse-norm envelope, the pairwise local radius is the minimum of one and three divided by eight times the shift scale times the larger of one and the pairwise aggregate factor, all divided by the product of the dimension and the inverse-norm envelope.","s":"ℕ → ℝ → ℝ → ℝ → ℝ → ℝ","l":120,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension, shift scale, pairwise margin, and inverse envelope and a matrix scale at least one, the explicit local reference-neighborhood radius is strictly positive.","s":"∀ {p : ℕ} {M δ L J : ℝ}, 0 < p → 0 < M → 0 < δ → 1 ≤ L → 0 < J → 0 < Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseLocalRadius p M δ L J","l":183,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a shift scale, a pairwise affine-separation margin, and a matrix scale, the pairwise residual radius is the reciprocal of thirty-two times the squared maximum of one and the pairwise aggregate factor, times the shift scale.","s":"ℕ → ℝ → ℝ → ℝ → ℝ","l":111,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive shift scale and pairwise margin and a matrix scale at least one, the explicit residual threshold is strictly positive.","s":"∀ {p : ℕ} {M δ L : ℝ}, 0 < M → 0 < δ → 1 ≤ L → 0 < Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseResidualRadius p M δ L","l":175,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a shift scale and a pairwise affine-separation margin, the pairwise solve factor is $6M/\\delta$.","s":"ℝ → ℝ → ℝ","l":90,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseStabilityConstant","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension, a shift scale, a pairwise affine-separation margin, and a matrix scale, the pairwise stability constant is sixteen times the pairwise aggregate factor times $L^3$.","s":"ℕ → ℝ → ℝ → ℝ → ℝ","l":105,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwise_offDiagonal_control","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For nonnegative scale and residual parameters with a positive affine margin, pairwise affine-separated bounded shifts, an exact invertible reference, and an off-diagonally approximate candidate, every ordered off-diagonal transition error obeys the common pairwise estimate.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M δ ε : ℝ}, 0 ≤ M → 0 < δ → 0 ≤ ε → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseAffineSeparated s δ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → ∀ (i j : Fin p), i ≠ j → |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B i j| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor M δ * (2 * M * Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B) ^ 2 + 2 * ε)","l":249,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.selectedTriple_offDiagonal_control","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.PairwiseControl","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For distinct selected coordinates, nonnegative scale and residual parameters with a positive affine margin, an exact invertible reference, an off-diagonally approximate candidate, bounded shifts, and one selected triple whose affine determinant separates the coordinates, both off-diagonal transition errors for that coordinate pair are bounded by the selected triple's affine-system estimate.","s":"∀ {p : ℕ} {E : Type u_1} [inst : Fintype E] [Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) (s : E → Fin p → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {M δ ε : ℝ} (i j : Fin p), i ≠ j → ∀ (e₀ e₁ e₂ : E), 0 ≤ M → 0 < δ → 0 ≤ ε → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s M → δ ≤ |Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairAffineDet s i j e₀ e₁ e₂| → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.OffDiagonalApproximateCongruence A B ε → have R := Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B; have η := 2 * M * Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 R ^ 2 + 2 * ε; |R i j| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor M δ * η ∧ |R j i| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.pairwiseSolveFactor M δ * η","l":129,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/PairwiseControl.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.sub_eq_transitionError_mul","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Stability","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For an invertible reference, the candidate-reference difference factors through the transition error.","s":"∀ {p : ℕ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), IsUnit (Matrix.det B₀) → B - B₀ = Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B * B₀","l":52,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Stability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.Definitions","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":1,"d":"For a dimension and a reference and candidate matrix, the transition error is the transition matrix minus the identity matrix.","s":"{p : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p","l":59,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError_entryL2_le_of_neighborhood","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For positive dimension and inverse envelope, an invertible reference, a bound on its inverse operator norm, and a local entrywise-L² neighborhood, the transition error is bounded by dimension times inverse scale times neighborhood radius.","s":"∀ {p : ℕ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p) {J ρ : ℝ}, 0 < p → 0 ≤ J → IsUnit (Matrix.det B₀) → ‖B₀⁻¹‖ ≤ J → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.InReferenceNeighborhood ρ B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.entryL2 (Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B) ≤ ↑p * J * ρ","l":55,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError_eq_sub_mul_inv","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.LocalBranch","a":"Discovery/LinearDisentanglement/Quantitative/PairwiseAffine","t":2,"d":"For an invertible reference matrix, the transition error equals the candidate-reference difference multiplied by the reference inverse.","s":"∀ {p : ℕ} (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix p), IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.PairwiseAffine.transitionError B₀ B = (B - B₀) * B₀⁻¹","l":47,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/PairwiseAffine/LocalBranch.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a finite environment collection, a diagonal shift vector for each environment, and a real bound, the shift-scale bound holds exactly when every coordinate of every prescribed diagonal shift has absolute value at most the bound.","s":"{d : ℕ} → {E : Type u_1} → [Fintype E] → (E → Fin d → ℝ) → ℝ → Prop","l":80,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a nonnegative integer dimension, a square matrix is a real-valued matrix with that many rows and that many columns.","s":"ℕ → Type","l":27,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension and a square matrix, unit-diagonal normalization holds exactly when every diagonal entry of the matrix equals one.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Prop","l":88,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.WellConditioned","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a real condition bound, and a square matrix, bounded conditioning holds exactly when the matrix is nonsingular and its Euclidean operator-norm condition number is at most the bound.","s":"{d : ℕ} → ℝ → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Prop","l":101,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.abs_entry_le_opNorm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a real square matrix and a selected row and column, the absolute value of that entry is bounded by the Euclidean operator norm.","s":"∀ {d : ℕ} (M : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (i j : Fin d), |M i j| ≤ ‖M‖","l":143,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.admissibleRadius","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a shift scale, an affine-separation margin, a matrix scale, and a condition-number envelope, the admissible residual radius is the smaller of the two displayed reciprocal bounds determined by those quantities.","s":"ℕ → ℝ → ℝ → ℝ → ℝ → ℝ","l":159,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.admissibleRadius_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"When the matrix dimension is positive, the shift scale is positive, the affine separation margin is positive, and the matrix scale is positive, the declared small-residual radius is strictly positive.","s":"∀ {d : ℕ} {L δ R κ : ℝ}, 0 < d → 0 < L → 0 < δ → 0 < R → 0 < Causalean.Discovery.LinearDisentanglement.Quantitative.admissibleRadius d L δ R κ","l":176,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.affineSolveFactor","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a shift scale, and an affine-separation margin, the affine solve factor is $d(d-1)!(2L)^{d-1}/\\delta$.","s":"ℕ → ℝ → ℝ → ℝ","l":143,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.congruenceDefect","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, an observed matrix, a prescribed diagonal vector, and a candidate change-of-coordinates matrix, the congruence defect is $BAB^\\mathsf{T}$ minus the diagonal matrix formed from the prescribed vector.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → (Fin d → ℝ) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":31,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.continuous_simultaneousCongruenceResidual","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.CompactExclusion","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a finite real matrix family and its prescribed diagonal shifts, the simultaneous-congruence residual is continuous as a function of the candidate matrix.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ), Continuous (Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual A s)","l":30,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/CompactExclusion.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.coordinate_le_affineSolveFactor","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a real square coefficient matrix and a solution vector, if the determinant margin is positive, the scale bound is nonnegative, every coefficient is bounded, the determinant has the stated margin, and the linear-system residual is uniformly bounded, every solution coordinate is bounded by the explicit Cramer's-rule factor times the residual bound.","s":"∀ {d : ℕ} (V : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (x : Fin d → ℝ) {L δ η : ℝ}, 0 < δ → 0 ≤ L → (∀ (i j : Fin d), |V i j| ≤ 2 * L) → δ ≤ |Matrix.det V| → (∀ (i : Fin d), |Matrix.mulVec V x i| ≤ η) → ∀ (j : Fin d), |x j| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.affineSolveFactor d L δ * η","l":187,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.exists_simultaneousCongruence_exclusionRadius","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.CompactExclusion","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a finite nonempty real matrix family, its diagonal shifts, a candidate set of matrices, an open local chart, and a reference matrix, if the candidate set is compact, the chart is open, the reference is a candidate, the reference lies in the chart, the far candidates are nonempty, and only the reference has zero residual among candidates, then the far candidates have a strictly positive attained simultaneous-congruence residual minimum.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (K U : Set (Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d)) (B₀ : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), IsCompact K → IsOpen U → B₀ ∈ K → B₀ ∈ U → (K \\ U).Nonempty → (∀ B ∈ K, Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual A s B = 0 → B = B₀) → Nonempty (Causalean.Mathlib.Topology.CompactExclusion.PositiveExclusionRadius K U (Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual A s))","l":58,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/CompactExclusion.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.exists_sqMatrix_uniformExclusionTolerance","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.CompactExclusion","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"Given a parameter-matrix correspondence, a reference matrix section, a residual, and a radius function, if the correspondence is compact, the reference matrix section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference matrix has zero residual among feasible pairs, then one positive residual tolerance puts every feasible pair at or below it strictly inside its parameter-dependent Euclidean operator-norm reference ball.","s":"∀ {P : Type u_1} [inst : TopologicalSpace P] [T2Space P] {d : ℕ} (K : Set (P × Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d)) (B₀ : P → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (r : P × Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → ℝ) (ρ : P → ℝ), IsCompact K → Continuous B₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), (p, B) ∈ K → r (p, B) = 0 → B = B₀ p) → ∃ ε₀, 0 < ε₀ ∧ ∀ (p : P) (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), (p, B) ∈ K → r (p, B) ≤ ε₀ → ‖B - B₀ p‖ < ρ p","l":101,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/CompactExclusion.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.opNorm_le_dimension_mul_of_entry_bound","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a real square matrix, if the proposed entry bound is nonnegative and every entry obeys that bound, its Euclidean operator norm is at most the dimension times the bound.","s":"∀ {d : ℕ} (M : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {c : ℝ}, 0 ≤ c → (∀ (i j : Fin d), |M i j| ≤ c) → ‖M‖ ≤ ↑d * c","l":161,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.opNorm_sub_le_of_approximate_simultaneous_congruence","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"Explicit simultaneous-congruence stability. For positive matrix dimension, positive shift scale, positive affine separation, and positive matrix scale, if the diagonal shifts obey their scale bound, the shift family has a separated affine minor, the reference is unit-diagonal, the candidate is unit-diagonal, the pair obeys the matrix scale bound, the pair obeys the condition-number bound, the reference realizes every congruence exactly, the residual tolerance is nonnegative, the tolerance is admissibly small, and the candidate realizes the congruences up to that tolerance, then the candidate is within the explicit linear modulus times the tolerance of the reference in Euclidean operator norm.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {L δ R κ ε : ℝ}, 0 < d → 0 < L → 0 < δ → 0 < R → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s L → Causalean.Discovery.LinearDisentanglement.Quantitative.AffineMinorSeparated s δ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairMatrixScaleBound R B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairConditionBound κ B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → 0 ≤ ε → ε ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.admissibleRadius d L δ R κ → Causalean.Discovery.LinearDisentanglement.Quantitative.ApproximateCongruence A s B ε → ‖B - B₀‖ ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.stabilityConstant d L δ R κ * ε","l":458,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.operatorConditionNumber","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension and a square matrix, its Euclidean operator-norm condition number is the product of its Euclidean operator norm and the Euclidean operator norm of its inverse. This is the condition number for a nonsingular matrix; for a singular matrix the inverse is zero by convention, so the value is zero, which is why every bound on it in this library is paired with a nonsingularity requirement.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → ℝ","l":93,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.productControlFactor","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a shift scale, and an affine-separation margin, the product-control factor is twice the affine solve factor.","s":"ℕ → ℝ → ℝ → ℝ","l":148,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.productControlFactor_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"When the matrix dimension is positive, the shift scale is positive, and the affine separation margin is positive, the coordinate-product amplification factor is strictly positive.","s":"∀ {d : ℕ} {L δ : ℝ}, 0 < d → 0 < L → 0 < δ → 0 < Causalean.Discovery.LinearDisentanglement.Quantitative.productControlFactor d L δ","l":167,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a nonempty finite environment collection, an observed matrix for each environment, a prescribed diagonal vector for each environment, and a candidate change-of-coordinates matrix, the simultaneous congruence residual is the largest Euclidean operator norm of the individual congruence defects across environments.","s":"{d : ℕ} → {E : Type u_1} → [Fintype E] → [Nonempty E] → (E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) → (E → Fin d → ℝ) → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → ℝ","l":38,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual_le_iff","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a finite real matrix family, its prescribed diagonal shifts, a candidate change of coordinates, and a tolerance, the simultaneous residual is at most that tolerance exactly when every individual congruence defect is.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (ε : ℝ), Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual A s B ≤ ε ↔ ∀ (e : E), ‖Causalean.Discovery.LinearDisentanglement.Quantitative.congruenceDefect (A e) (s e) B‖ ≤ ε","l":59,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual_nonneg","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a finite real matrix family, its prescribed diagonal shifts, and a candidate change of coordinates, the simultaneous-congruence residual is nonnegative.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), 0 ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.simultaneousCongruenceResidual A s B","l":47,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.sqMatrix_uniformCompactCorrespondence_dichotomy","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.CompactExclusion","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"Given a parameter-matrix correspondence, a reference matrix section, a residual, and a radius function, if the correspondence is compact, the reference matrix section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference matrix has zero residual among feasible pairs, then either no operator-norm-far pair exists or one attains a strictly positive uniform residual minimum.","s":"∀ {P : Type u_1} [inst : TopologicalSpace P] [T2Space P] {d : ℕ} (K : Set (P × Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d)) (B₀ : P → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (r : P × Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → ℝ) (ρ : P → ℝ), IsCompact K → Continuous B₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), (p, B) ∈ K → r (p, B) = 0 → B = B₀ p) → Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K B₀ ρ = ∅ ∨ Nonempty (Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius K B₀ ρ r)","l":80,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/CompactExclusion.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.stabilityConstant","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension, a shift scale, an affine-separation margin, a matrix scale, and a condition-number envelope, the stability constant is $2dR\\max(1,\\kappa)$ times the product-control factor.","s":"ℕ → ℝ → ℝ → ℝ → ℝ → ℝ","l":153,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.transition","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Definitions","a":"Discovery/LinearDisentanglement/Quantitative","t":1,"d":"For a dimension and a reference and candidate matrix, the transition matrix is the candidate matrix multiplied by the inverse of the reference matrix.","s":"{d : ℕ} → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d","l":121,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Definitions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.transition_coordinate_product_control","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a finite real matrix family, its prescribed diagonal shifts, an exact reference, and a candidate, if the shift scale is nonnegative, the affine separation margin is positive, the shifts obey their scale bound, a separated affine minor exists, the reference is invertible, the reference realizes the congruences exactly, and the candidate has the stated approximate residual, then squared transition entries and cross-row products have the explicit coordinatewise residual bounds.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {L δ ε : ℝ}, 0 ≤ L → 0 < δ → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s L → Causalean.Discovery.LinearDisentanglement.Quantitative.AffineMinorSeparated s δ → IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.ApproximateCongruence A s B ε → (∀ (i a : Fin d), |Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B i a ^ 2 - if i = a then 1 else 0| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.productControlFactor d L δ * ε) ∧ ∀ (i j a : Fin d), i ≠ j → |Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B i a * Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B j a| ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.productControlFactor d L δ * ε","l":262,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.transition_diagonal_congruence","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For a real matrix family, its diagonal shifts, an exact reference matrix, a candidate matrix, if the reference is invertible and it exactly realizes every prescribed congruence, then the transition matrix conjugates each reference diagonal into the candidate congruence.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d), IsUnit (Matrix.det B₀) → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → ∀ (e : E), Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B * Matrix.diagonal (s e) * Matrix.transpose (Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B) = B * A e * Matrix.transpose B","l":240,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Quantitative.transition_diagonal_ge_half","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Quantitative.Quantitative","a":"Discovery/LinearDisentanglement/Quantitative","t":2,"d":"For positive matrix dimension, positive shift scale, positive affine separation, and positive matrix scale, if the shifts have the declared scale, their affine minor is separated, the reference is unit-diagonal, the candidate is unit-diagonal, both matrices obey the scale bound, both matrices obey the condition bound, the reference congruences are exact, the residual tolerance is nonnegative, it is below the admissible radius, and the candidate is approximately congruent, then every transition diagonal entry is at least one half.","s":"∀ {d : ℕ} {E : Type u_1} [inst : Fintype E] [inst_1 : Nonempty E] (A : E → Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) (s : E → Fin d → ℝ) (B₀ B : Causalean.Discovery.LinearDisentanglement.Quantitative.SqMatrix d) {L δ R κ ε : ℝ}, 0 < d → 0 < L → 0 < δ → 0 < R → Causalean.Discovery.LinearDisentanglement.Quantitative.ShiftScaleBound s L → Causalean.Discovery.LinearDisentanglement.Quantitative.AffineMinorSeparated s δ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B₀ → Causalean.Discovery.LinearDisentanglement.Quantitative.UnitDiagonal B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairMatrixScaleBound R B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.PairConditionBound κ B₀ B → Causalean.Discovery.LinearDisentanglement.Quantitative.ExactCongruence A s B₀ → 0 ≤ ε → ε ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.admissibleRadius d L δ R κ → Causalean.Discovery.LinearDisentanglement.Quantitative.ApproximateCongruence A s B ε → ∀ (i : Fin d), 1 / 2 ≤ Causalean.Discovery.LinearDisentanglement.Quantitative.transition B₀ B i i","l":358,"f":"Causalean/Discovery/LinearDisentanglement/Quantitative/Quantitative.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution","k":"structure","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"A linear causal disentanglement model with d latent variables, p observed variables, and K interventional contexts (Squires, Seigal, Bhate & Uhler 2023) bundles a full-row-rank mixing pseudoinverse from the observed to the latent space, a latent edge relation that respects the node order, and an observational structural matrix that is upper triangular, has a positive diagonal, and has off-diagonal support exactly equal to the edge set. For each interventional context it further carries a structural matrix, an intervention target node, and a positive perfect-intervention scaling, tied to the observational matrix by the perfect-single-node-intervention formula.","s":"ℕ → ℕ → ℕ → Type","l":62,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.An","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the inclusive ancestor set is the union of that node's ancestor set and the node itself.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":44,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.InSG","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a permutation of the latent-node indices, the order-preserving relabeling condition holds exactly when every directed edge from a parent to a child remains ordered with the child's relabeled index strictly smaller than the parent's.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Equiv.Perm (Fin d) → Prop","l":115,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Pa","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the inclusive parent set is the union of that node's parent set and the node itself.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":38,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Theta","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and an intervention context, the interventional precision matrix is the p-by-p real matrix obtained by sandwiching that context's structural-matrix Gram matrix between the mixing pseudoinverse and its transpose.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin K → Matrix (Fin p) (Fin p) ℝ","l":111,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.Theta0","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts, the observational precision matrix is the p-by-p real matrix obtained by sandwiching the observational structural matrix's Gram matrix between the mixing pseudoinverse and its transpose.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Matrix (Fin p) (Fin p) ℝ","l":107,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.anc","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the ancestor set is the set of latent nodes from which a nonempty directed path leads to that node.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":41,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.not_prec_self","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"≺ is irreflexive: a node is never its own strict ancestor.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), ¬S.prec i i","l":62,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.pa","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the parent set is the set of latent nodes having a directed edge into that node.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Set (Fin d)","l":35,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.permute","k":"def","m":"Causalean.Discovery.LinearDisentanglement.SigmaSolutions","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts, a permutation of its latent-node indices, and the condition that this permutation preserves the directed node order, the permuted solution is a linear causal disentanglement solution obtained by relabeling every latent coordinate with that permutation. Its mixing pseudoinverse is the row-permuted original, its observational structural matrix is the original conjugated by the permutation matrix, its interventional structural matrices are conjugated in the same way, and its intervention targets are relabeled by the permutation.","s":"{d p K : ℕ} → (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) → (σ : Equiv.Perm (Fin d)) → S.InSG σ → Causalean.Discovery.LinearDisentanglement.Solution d p K","l":84,"f":"Causalean/Discovery/LinearDisentanglement/SigmaSolutions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.prec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and two latent nodes, the strict causal precedence relation holds exactly when the second node is a strict ancestor of the first.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Fin d → Prop","l":47,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.prec_lt","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The strict partial order embeds into the Fin d linear order: if i ≺ j (j is a strict ancestor of i) then i < j. This is the acyclicity of 𝒢 lifted through the transitive closure, and the engine behind every \"process the nodes in a topological order\" argument below.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i j : Fin d}, S.prec i j → i < j","l":53,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.Solution.preceq","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and two latent nodes, the weak causal precedence relation holds exactly when the nodes coincide or the second is a strict ancestor of the first.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Fin d → Prop","l":50,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cancel_Ht_H","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Left-cancel Hᵀ and right-cancel H in Hᵀ X H = Hᵀ Y H. H has full row rank, so Hᵀ has a left inverse and H a right inverse.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {X Y : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * X * S.H = S.H.transpose * Y * S.H → X = Y","l":228,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.central_rank2_eq","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R1+C) The central rank-≤2 equation. Combining the Gram-difference transport Δₖ = Mᵀ Δ'ₖ M (gram_diff_transport) with the latent key identity (C) on both sides, Δₖ = λₖ²(eₖeₖᵀ) − wₖwₖᵀ and Δ'ₖ = λ'ₖ²(e'ₖe'ₖᵀ) − w'ₖw'ₖᵀ, gives the rank-≤2 identity in fully outer-product form, with the primed outer products transported by Mᵀ: λₖ²(eₖeₖᵀ) − wₖwₖᵀ = λ'ₖ²((Mᵀe'ₖ)(Mᵀe'ₖ)ᵀ) − (Mᵀw'ₖ)(Mᵀw'ₖ)ᵀ.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ (k : Fin K), S.lam k ^ 2 • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) - Matrix.vecMulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) = S'.lam k ^ 2 • Matrix.vecMulVec (M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))) (M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))) - Matrix.vecMulVec (M.transpose.mulVec (S'.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k)))) (M.transpose.mulVec (S'.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S'.target k))))","l":394,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.conj_vecMulVec_transpose","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Mᵀ (u vᵀ) M = (Mᵀ u)(Mᵀ v)ᵀ (conjugation of a rank-one matrix by Mᵀ · M).","s":"∀ {q : ℕ} (M : Matrix (Fin q) (Fin q) ℝ) (u v : Fin q → ℝ), M.transpose * Matrix.vecMulVec u v * M = Matrix.vecMulVec (M.transpose.mulVec u) (M.transpose.mulVec v)","l":388,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cvec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and an intervention context, the perfect-intervention perturbation row assigns to each latent node the intervention scaling times the standard-basis indicator of the target node, minus the corresponding entry of the observational structural matrix.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin K → Fin d → ℝ","l":267,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.cvec_ne_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(C, target-row form / sub-lemma (B)). The perturbation row is non-degenerate exactly when the intervention changes the precision matrix. Under Θₖ ≠ Θ₀, cₖ ≠ 0 (equivalently Bₖ ≠ B₀): if cₖ = 0 then Bₖ = B₀, hence Θₖ = Θ₀.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Theta k ≠ S.Theta0 → Causalean.Discovery.LinearDisentanglement.cvec S k ≠ 0","l":277,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.diag_permMat_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"A signed permutation matrix sends eᵢ to the signed basis vector ν (σ i) • e_{σ i}.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (ν : Fin d → ℝ) (i : Fin d), (Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ).mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = ν (σ i) • Causalean.Discovery.LinearDisentanglement.stdVec d (σ i)","l":456,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.disentanglement_identifiability","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Identifiability","a":"Discovery/LinearDisentanglement","t":1,"d":"Linear causal disentanglement identifiability (Theorem 2). Let S and S' be two solutions of the linear causal disentanglement model such that each solution's intervention-target map is a bijection onto the latent coordinates, i.e. one intervention per latent node and S's interventions are non-degenerate (the paper's genericity / Assumption 1(b)): every intervened precision matrix Θ_k differs from the observational precision matrix Θ_0. If S and S' share the same observational precision matrix and agree, context by context, on every interventional precision matrix, then S and S' are related by a single order-preserving relabeling σ of the latent coordinates, a nonzero scaling vector μ, and a ±1 sign vector ν: σ, μ, ν transport S's latent-direction matrix and structural coefficient matrices onto S''s, and σ carries S's intervention targets onto S''s.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ S'.H = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ * S.H ∧ S'.B0 * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":35,"f":"Causalean/Discovery/LinearDisentanglement/Identifiability.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.disentanglement_uniqueness","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":1,"d":"Uniqueness (⊆ direction of Theorem 2). Let S and S' be two solutions of the linear causal disentanglement model such that each solution's intervention-target map is a bijection onto the latent coordinates, i.e. one intervention per latent node and S's interventions are non-degenerate (the paper's genericity / Assumption 1(b)): every intervened precision matrix Θ_k differs from the observational precision matrix Θ_0. If S and S' share the same observational precision matrix and agree, context by context, on every interventional precision matrix, then S and S' are related by a single order-preserving relabeling σ of the latent coordinates, a nonzero scaling vector μ, and a ±1 sign vector ν, transporting S's latent-direction and structural coefficient matrices onto S''s, with σ carrying S's intervention targets onto S''s.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ S'.H = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ * S.H ∧ S'.B0 * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * (Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ) = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":1484,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.dotProduct_self_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The self dotProduct of a nonzero real vector is strictly positive (positive-definiteness of ⬝ᵥ). Equivalently, x ⬝ᵥ x = 0 ↔ x = 0; this is what makes \"normalize the projection residual\" well defined in the construction.","s":"∀ {p : ℕ} {x : Fin p → ℝ}, x ≠ 0 → 0 < x ⬝ᵥ x","l":87,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.eq_of_mul_eq_mul_row_indep","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Right cancellation by a full-row-rank Q. If the rows of Q are linearly independent and R Q = R' Q, then R = R'. This pins the R factor once Q is fixed, and is the algebraic half of the uniqueness proof.","s":"∀ {d p : ℕ} {R R' : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, LinearIndependent ℝ Q.row → R * Q = R' * Q → R = R'","l":100,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.exists_change_of_basis","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L1) The change-of-basis matrix. There is an invertible M with H' = M H. M is built from the recovery identity for H' (H' = W' Θ₀') by substituting Θ₀' = Θ₀ = Hᵀ B₀ᵀ B₀ H, giving H' = (W' Hᵀ B₀ᵀ B₀) H. Invertibility follows from the symmetric matrix N with H = N H' by right-cancelling the full-row-rank H/H'.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), S.Theta0 = S'.Theta0 → ∃ M, IsUnit M ∧ S'.H = M * S.H","l":188,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.exists_orderPerm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L4) Orthogonal correctness (full statement).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Bijective S.target → Function.Bijective S'.target → (∀ (k : Fin K), S.Theta k ≠ S.Theta0) → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∃ σ μ ν, S.InSG σ ∧ (∀ (i : Fin d), μ i ≠ 0) ∧ (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) ∧ M = Matrix.diagonal μ * Causalean.Discovery.LinearDisentanglement.permMat σ ∧ S'.B0 * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 ∧ (∀ (k : Fin K), S'.Bint k * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k) ∧ ∀ (k : Fin K), S'.target k = σ (S.target k)","l":1455,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.fact_transpose_mul","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.KeyIdentity","a":"Discovery/LinearDisentanglement","t":2,"d":"Fact (rank-one decomposition). For any square B, BᵀB = Σᵢ (Bᵀeᵢ)⊗², where v⊗² = v vᵀ (Matrix.vecMulVec v v).","s":"∀ {d : ℕ} (B : Matrix (Fin d) (Fin d) ℝ), B.transpose * B = ∑ i, Matrix.vecMulVec (B.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i)) (B.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i))","l":33,"f":"Causalean/Discovery/LinearDisentanglement/KeyIdentity.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_diff_transport","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(R1) Latent Gram-difference transport. Bₖᵀ Bₖ − B₀ᵀ B₀ = Mᵀ (B'ₖᵀ B'ₖ − B'₀ᵀ B'₀) M. Subtracting the observational conjugation identity from the k-th one.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → S.Theta0 = S'.Theta0 → (∀ (k : Fin K), S.Theta k = S'.Theta k) → ∀ (k : Fin K), (S.Bint k).transpose * S.Bint k - S.B0.transpose * S.B0 = M.transpose * ((S'.Bint k).transpose * S'.Bint k - S'.B0.transpose * S'.B0) * M","l":368,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L2) The per-context Gram identity. Given H' = M H and Θₖ = Θₖ' for the observational (B₀/B'₀) and interventional (Bₖ/B'ₖ) matrices, the d × d Gram identity BᵀB = (B' M)ᵀ (B' M) holds.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → B.transpose * B = (B' * M).transpose * (B' * M)","l":244,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_identity_conj","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"The per-context Gram identity in the symmetric form BᵀB = Mᵀ B'ᵀ B' M (the conjugation form of gram_identity).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ}, S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → B.transpose * B = M.transpose * (B'.transpose * B') * M","l":358,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.gram_to_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(L3) Gram ⟹ orthogonal. If XᵀX = YᵀY with X, Y invertible, then the transition matrix O = Y X⁻¹ is orthogonal: Oᵀ O = 1. This is the algebraic step that turns the equality of Gram matrices BᵀB = (B' M)ᵀ(B' M) into an orthogonality statement about O = B' M B⁻¹.","s":"∀ {q : ℕ} {X Y : Matrix (Fin q) (Fin q) ℝ} [Invertible X] [Invertible Y], X.transpose * X = Y.transpose * Y → (Y * X⁻¹).transpose * (Y * X⁻¹) = 1","l":47,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.hSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a set of latent-node indices, the selected mixing-row span is the real linear subspace generated by the mixing-pseudoinverse rows indexed by that set.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Set (Fin d) → Submodule ℝ (Fin p → ℝ)","l":40,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.inner_toLp_eq_dotProduct","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The EuclideanSpace ℝ (Fin p) inner product of toLp x and toLp y is exactly the dotProduct x ⬝ᵥ y. This is the bridge used to import all of Mathlib's inner-product-space machinery into the dotProduct world.","s":"∀ {p : ℕ} (x y : Fin p → ℝ), inner ℝ (WithLp.toLp 2 x) (WithLp.toLp 2 y) = x ⬝ᵥ y","l":75,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.key_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.KeyIdentity","a":"Discovery/LinearDisentanglement","t":1,"d":"Key identity (Proposition). For a linear causal disentanglement solution and an interventional context k with target iₖ, the difference of precision matrices Θ_k − Θ₀ equals exactly the difference between two outer products: one built from the target row of the interventional structural matrix, and one built from the target row of the observational structural matrix.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), S.Theta k - S.Theta0 = Matrix.vecMulVec (S.H.transpose.mulVec ((S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) (S.H.transpose.mulVec ((S.Bint k).transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) - Matrix.vecMulVec (S.H.transpose.mulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))) (S.H.transpose.mulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))))","l":42,"f":"Causalean/Discovery/LinearDisentanglement/KeyIdentity.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.latent_key_identity","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(C) Latent key identity. At the d × d (latent) level the difference of Gram matrices is a difference of two rank-one (outer-product) matrices: BₖᵀBₖ − B₀ᵀB₀ = λₖ² (e_{iₖ} e_{iₖ}ᵀ) − wₖ wₖᵀ where wₖ = B₀ᵀ *ᵥ e_{iₖ} is the iₖ-th row of B₀ (supported on Pa(iₖ)). This is the latent analogue of key_identity, and the structural engine of the orthogonal-correctness core.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K), (S.Bint k).transpose * S.Bint k - S.B0.transpose * S.B0 = S.lam k ^ 2 • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)) - Matrix.vecMulVec (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k))) (S.B0.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d (S.target k)))","l":306,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.leftInv_mul_Ht","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"L := (H Hᵀ)⁻¹ H is a left inverse of Hᵀ: L Hᵀ = 1.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), (S.H * S.H.transpose)⁻¹ * S.H * S.H.transpose = 1","l":163,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.orthogonal_of_diag_conj","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(S2) Diagonal conjugation ⟹ orthogonal rescaling. From Mᵀ diag(d') M = diag(d) with d, d' strictly positive, the matrix O = diag(fun i => √(d' i)) * M * diag(fun i => (√(d i))⁻¹) is orthogonal: Oᵀ O = 1.","s":"∀ {q : ℕ} {M : Matrix (Fin q) (Fin q) ℝ} {dv dv' : Fin q → ℝ}, (∀ (i : Fin q), 0 < dv i) → (∀ (i : Fin q), 0 < dv' i) → M.transpose * Matrix.diagonal dv' * M = Matrix.diagonal dv → ((Matrix.diagonal fun i => √(dv' i)) * M * Matrix.diagonal fun i => (√(dv i))⁻¹).transpose * ((Matrix.diagonal fun i => √(dv' i)) * M * Matrix.diagonal fun i => (√(dv i))⁻¹) = 1","l":669,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a nonnegative integer dimension and a permutation of its indices, the permutation matrix is the real square matrix whose entry in row i and column j is one exactly when i is the image of j under the permutation, and is zero otherwise.","s":"{d : ℕ} → Equiv.Perm (Fin d) → Matrix (Fin d) (Fin d) ℝ","l":58,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_apply_symm","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(permMat σ) a c = 1 iff c = σ.symm a, else 0 (column read-off of permMat).","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (a c : Fin d), Causalean.Discovery.LinearDisentanglement.permMat σ a c = if c = (Equiv.symm σ) a then 1 else 0","l":759,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_conj_apply","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Permutation conjugation entry. (permMat σ * X * (permMat σ)ᵀ) a b = X (σ.symm a) (σ.symm b): conjugating X by permMat σ relabels rows and columns by σ.symm.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (X : Matrix (Fin d) (Fin d) ℝ) (a b : Fin d), (Causalean.Discovery.LinearDisentanglement.permMat σ * X * (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose) a b = X ((Equiv.symm σ) a) ((Equiv.symm σ) b)","l":768,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"permMat σ *ᵥ eᵢ = e_{σ i}: the permutation matrix sends the i-th basis vector to the σ(i)-th.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)) (i : Fin d), (Causalean.Discovery.LinearDisentanglement.permMat σ).mulVec (Causalean.Discovery.LinearDisentanglement.stdVec d i) = Causalean.Discovery.LinearDisentanglement.stdVec d (σ i)","l":448,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_mul_transpose","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a dimension d and a permutation σ of Fin d, the permutation matrix permMat σ is orthogonal: its product with its own transpose is the identity matrix.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)), Causalean.Discovery.LinearDisentanglement.permMat σ * (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose = 1","l":123,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.permMat_transpose_mul","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":2,"d":"permMat σ is orthogonal: (permMat σ)ᵀ * permMat σ = 1.","s":"∀ {d : ℕ} (σ : Equiv.Perm (Fin d)), (Causalean.Discovery.LinearDisentanglement.permMat σ).transpose * Causalean.Discovery.LinearDisentanglement.permMat σ = 1","l":141,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqAncSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the strict-ancestor residual span is the real linear subspace generated by the partial-order RQ residuals of all strict ancestors of that node.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → Submodule ℝ (EuclideanSpace ℝ (Fin p))","l":226,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual","k":"def","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"For a linear causal disentanglement solution with d latent variables, p observed variables, and K intervention contexts and a latent node, the partial-order RQ residual is that node's mixing-pseudoinverse row, viewed as a Euclidean vector, minus its orthogonal projection onto the span of residuals of all strict ancestors. The ancestor-residual span used in this subtraction is generated by the residuals of precisely those strict ancestors.","s":"{d p K : ℕ} → Causalean.Discovery.LinearDisentanglement.Solution d p K → Fin d → EuclideanSpace ℝ (Fin p)","l":212,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_eq","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Unfolding equation for porqResidual: it is toLp hᵢ minus its orthogonal projection onto the ancestor span porqAncSpan i.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i = WithLp.toLp 2 (S.H i) - (Causalean.Discovery.LinearDisentanglement.porqAncSpan S i).starProjection (WithLp.toLp 2 (S.H i))","l":232,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_HspanLE","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The i-th residual lies in the span of {toLp hₘ : i ⪯ m} (the row itself and its strict ancestors). Proved by strong induction on the reversed Fin d order: the diagonal term is toLp hᵢ, and the projection lives in Wᵢ, whose generators porqResidual k (i ≺ k) lie, by induction, in span {toLp hₘ : k ⪯ m} ⊆ span {toLp hₘ : i ⪯ m} (by transitivity of ⪯).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ∈ Submodule.span ℝ {x | ∃ m, S.preceq i m ∧ x = WithLp.toLp 2 (S.H m)}","l":273,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_ancSpan","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Each strict-ancestor residual lies in the ancestor span porqAncSpan i.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i k : Fin d}, S.prec i k → Causalean.Discovery.LinearDisentanglement.porqResidual S k ∈ Causalean.Discovery.LinearDisentanglement.porqAncSpan S i","l":239,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_mem_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The i-th residual is orthogonal to its ancestor span: porqResidual i ∈ (Wᵢ)ᗮ.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ∈ (Causalean.Discovery.LinearDisentanglement.porqAncSpan S i)ᗮ","l":246,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_ne_zero","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Non-vanishing of the residual. porqResidual i ≠ 0: if it were zero then toLp hᵢ would lie in Wᵢ ⊆ span {toLp hₘ : i ≺ m}, contradicting the linear independence of {toLp hⱼ} (a member is never in the span of the strictly-others).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), Causalean.Discovery.LinearDisentanglement.porqResidual S i ≠ 0","l":303,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porqResidual_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Orthogonality along the order. The i-th residual is orthogonal to every strict-ancestor residual porqResidual k (i ≺ k), since the latter lies in Wᵢ and the former in (Wᵢ)ᗮ.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {i k : Fin d}, S.prec i k → inner ℝ (Causalean.Discovery.LinearDisentanglement.porqResidual S i) (Causalean.Discovery.LinearDisentanglement.porqResidual S k) = 0","l":252,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_diag_pos","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The diagonal of any PORQ factor is strictly positive.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (i : Fin d), 0 < R i i","l":154,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_exists","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Existence of the partial order RQ decomposition (Appendix B Proposition). For a linear causal disentanglement solution, there exist matrices R and Q realizing a partial order RQ decomposition of S's factor loading matrix H — that is, H = R·Q with R's diagonal entries nonnegative and R's support confined to the partial order on nodes, and with Q's rows unit-norm and mutually orthogonal along that order.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), ∃ R Q, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q","l":424,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_rowCoeffs","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Row coefficients of the existence factorization. Writing qₘ = ‖rₘ‖⁻¹ • rₘ for the normalized residuals, the row toLp hᵢ decomposes as ‖rᵢ‖ • qᵢ + ∑_{i ≺ k} (coefficient) • qₖ: the diagonal coefficient is the residual norm ‖rᵢ‖ ≥ 0, the off-diagonal coefficients are supported on the strict ancestors i ≺ k, and there are none off ⪯. This packages factor, diag_nonneg and supp into one existence statement (the i-th row of R).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (i : Fin d), ∃ c, (∀ (j : Fin d), ¬S.preceq i j → c j = 0) ∧ c i = ‖Causalean.Discovery.LinearDisentanglement.porqResidual S i‖ ∧ WithLp.toLp 2 (S.H i) = ∑ m, c m • ‖Causalean.Discovery.LinearDisentanglement.porqResidual S m‖⁻¹ • Causalean.Discovery.LinearDisentanglement.porqResidual S m","l":341,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_row_split","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"Row-split of a PORQ factorization. Splitting off the diagonal term, the i-th row of H is Rᵢᵢ • qᵢ plus a combination of the *other* rows qₖ (k ≠ i); by supp only the strict ancestors i ≺ k contribute, so this is exactly the triangular equation hᵢ = Rᵢᵢ qᵢ + ∑_{i ≺ k} Rᵢₖ qₖ. It is the membership half of the orthogonal-decomposition uniqueness argument.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (i : Fin d), S.H i = R i i • Q i + ∑ k ∈ Finset.univ.erase i, R i k • Q k","l":198,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.porq_unique","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":1,"d":"Uniqueness of the partial order RQ decomposition (Appendix B Proposition). Given a solution S, if both (R,Q) and (R',Q') are partial order RQ decompositions of S's latent-direction matrix H — factoring H as a lower- triangular-along-the-order matrix R times a row-orthonormal-along-the-order matrix Q, then the two decompositions coincide: R = R' and Q = Q'.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R R' : Matrix (Fin d) (Fin d) ℝ} {Q Q' : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → Causalean.Discovery.LinearDisentanglement.IsPORQ S R' Q' → R = R' ∧ Q = Q'","l":493,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.qSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"For a nonnegative latent dimension, a nonnegative observed dimension, a real matrix with d rows and p columns, and a set of latent-row indices, the selected RQ-row span is the real linear subspace generated by the rows indexed by that set.","s":"{d p : ℕ} → Matrix (Fin d) (Fin p) ℝ → Set (Fin d) → Submodule ℝ (Fin p → ℝ)","l":44,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowSpan","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"For a nonnegative integer observed dimension and a real square matrix of that dimension, the row span is the real linear subspace generated by all rows of the matrix.","s":"{p : ℕ} → Matrix (Fin p) (Fin p) ℝ → Submodule ℝ (Fin p → ℝ)","l":36,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.row_indep_of_mul_row_indep","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"A full-row-rank product forces a full-row-rank right factor. If the rows of R Q are linearly independent (d of them), then so are the rows of Q. The rows of R Q lie in the span of the rows of Q, which has at most d of them, so a dimension count forces the d rows of Q to be independent. Applied to H = R Q with H of full row rank, this shows the rows qᵢ of any PORQ factor are independent — the hypothesis eq_of_mul_eq_mul_row_indep needs.","s":"∀ {d p : ℕ} (R : Matrix (Fin d) (Fin d) ℝ) (Q : Matrix (Fin d) (Fin p) ℝ), LinearIndependent ℝ (R * Q).row → LinearIndependent ℝ Q.row","l":113,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_a","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(a). For a solution S, an intervention k, and a set of latent indices I, provided the intervention actually changes the observational precision matrix, Θₖ ≠ Θ₀ (the paper's genericity / Assumption (b)), the row span of the precision difference Θₖ − Θ₀ is contained in the span of the rows of the mixing pseudoinverse H indexed by I if and only if the parent set of the intervened target node is contained in I.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (k : Fin K) (I : Set (Fin d)), S.Theta k ≠ S.Theta0 → (Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.hSpan S I ↔ S.Pa (S.target k) ⊆ I)","l":205,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_b","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(b). For a solution S, an intervention k, and any partial order RQ decomposition of S's latent-direction matrix H, witnessed by R and Q, the row span of the precision difference Θₖ − Θ₀ is contained in the span of the rows of Q indexed by the ancestor set of the intervened target node.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (k : Fin K), Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.qSpan Q (S.An (S.target k))","l":344,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.rowspan_inclusion_c","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Rowspan","a":"Discovery/LinearDisentanglement","t":1,"d":"Lemma 1(c). For a solution S, an intervention k, and a set of latent indices I, given a partial order RQ decomposition of S's latent-direction matrix H, witnessed by R and Q, the non-degeneracy condition Θₖ ≠ Θ₀ (the paper's genericity / Assumption (b)), that I is ancestor-closed, and that the parent set of the intervened target node is not contained in I, then the row span of the precision difference Θₖ − Θ₀ is not contained in the span of the rows of Q indexed by I.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) {R : Matrix (Fin d) (Fin d) ℝ} {Q : Matrix (Fin d) (Fin p) ℝ}, Causalean.Discovery.LinearDisentanglement.IsPORQ S R Q → ∀ (k : Fin K) (I : Set (Fin d)), S.Theta k ≠ S.Theta0 → (∀ i ∈ I, S.An i ⊆ I) → ¬S.Pa (S.target k) ⊆ I → ¬Causalean.Discovery.LinearDisentanglement.rowSpan (S.Theta k - S.Theta0) ≤ Causalean.Discovery.LinearDisentanglement.qSpan Q I","l":522,"f":"Causalean/Discovery/LinearDisentanglement/Rowspan.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sigma_solutions","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.SigmaSolutions","a":"Discovery/LinearDisentanglement","t":1,"d":"For a solution S and a permutation σ of the latent coordinates such that σ is order-preserving, i.e. lies in the paper's group S(𝒢), the relabeled solution obtained by applying σ to S produces exactly the same observational precision matrix Θ₀, and for every intervention k the same interventional precision matrix Θₖ, as S itself.","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (σ : Equiv.Perm (Fin d)) (hσ : S.InSG σ), (S.permute σ hσ).Theta0 = S.Theta0 ∧ ∀ (k : Fin K), (S.permute σ hσ).Theta k = S.Theta k","l":140,"f":"Causalean/Discovery/LinearDisentanglement/SigmaSolutions.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.stdVec","k":"def","m":"Causalean.Discovery.LinearDisentanglement.Model","a":"Discovery/LinearDisentanglement","t":1,"d":"For a nonnegative integer dimension and an index in that dimension, the standard basis vector is the real vector that equals one at the specified index and zero at every other index.","s":"(d : ℕ) → Fin d → Fin d → ℝ","l":55,"f":"Causalean/Discovery/LinearDisentanglement/Model.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sum_latent_diff_unprimed","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(S1) The unprimed SUM TRICK. Let kof be a section of the surjective target map (S.target (kof n) = n). Summing the latent key identity over kof gives ∑ₙ (B_{kof n}ᵀ B_{kof n} − B₀ᵀB₀) = diagonal(d) − B₀ᵀB₀ with d n = (λ_{kof n})².","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K) (kof : Fin d → Fin K), (∀ (n : Fin d), S.target (kof n) = n) → ∑ n, ((S.Bint (kof n)).transpose * S.Bint (kof n) - S.B0.transpose * S.B0) = (Matrix.diagonal fun n => S.lam (kof n) ^ 2) - S.B0.transpose * S.B0","l":538,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.sum_smul_vecMulVec_stdVec_eq_diagonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"∑ n, (c n) • (e_n e_nᵀ) = diagonal c (the rank-one standard-basis outer products sum to a diagonal matrix).","s":"∀ {n : ℕ} (c : Fin n → ℝ), ∑ i, c i • Matrix.vecMulVec (Causalean.Discovery.LinearDisentanglement.stdVec n i) (Causalean.Discovery.LinearDisentanglement.stdVec n i) = Matrix.diagonal c","l":524,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.target_readoff","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Signed target read-off. Suppose the collapse holds for context k and the observational context: B'ₖ M = diagonal ν permMat σ Bₖ and B'₀ M = diagonal ν permMat σ B₀, with Θₖ ≠ Θ₀. Then the primed target equals the relabelled target: i'ₖ = σ(iₖ).","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ} {σ : Equiv.Perm (Fin d)} {ν : Fin d → ℝ} (k : Fin K), S.Theta k ≠ S.Theta0 → (∀ (i : Fin d), ν i = 1 ∨ ν i = -1) → S'.Bint k * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.Bint k → S'.B0 * M = Matrix.diagonal ν * Causalean.Discovery.LinearDisentanglement.permMat σ * S.B0 → S'.target k = σ (S.target k)","l":480,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.toLp_H_linearIndependent","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.PartialOrderRQ","a":"Discovery/LinearDisentanglement","t":2,"d":"The images of the rows of H under toLp are linearly independent (full row rank of H transported through the linear isomorphism toLp).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), LinearIndependent ℝ fun j => WithLp.toLp 2 (S.H j)","l":262,"f":"Causalean/Discovery/LinearDisentanglement/PartialOrderRQ.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.transition_orthogonal","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"(D) The transition factor is orthogonal. From the per-context Gram identity BₖᵀBₖ = (B'ₖ M)ᵀ(B'ₖ M) and invertibility, Oₖ = B'ₖ M Bₖ⁻¹ satisfies Oₖᵀ Oₖ = 1.","s":"∀ {d p K : ℕ} (S S' : Causalean.Discovery.LinearDisentanglement.Solution d p K) {M : Matrix (Fin d) (Fin d) ℝ}, S'.H = M * S.H → ∀ {B B' : Matrix (Fin d) (Fin d) ℝ} [Invertible B] [Invertible (B' * M)], S.H.transpose * B.transpose * B * S.H = S'.H.transpose * B'.transpose * B' * S'.H → (B' * M * B⁻¹).transpose * (B' * M * B⁻¹) = 1","l":341,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.transpose_mulVec_stdVec","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"Mᵀ *ᵥ eᵢ is the i-th row of M, viewed as a vector: (Mᵀ *ᵥ eᵢ) j = Mᵢⱼ. This is the bridge that turns the transported outer products of central_rank2_eq into statements about the rows of M.","s":"∀ {n : ℕ} (M : Matrix (Fin n) (Fin n) ℝ) (i : Fin n), M.transpose.mulVec (Causalean.Discovery.LinearDisentanglement.stdVec n i) = fun j => M i j","l":414,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.Discovery.LinearDisentanglement.vecMul_H_injective","k":"theorem","m":"Causalean.Discovery.LinearDisentanglement.Uniqueness","a":"Discovery/LinearDisentanglement","t":2,"d":"vecMul · H is injective (full row rank of H).","s":"∀ {d p K : ℕ} (S : Causalean.Discovery.LinearDisentanglement.Solution d p K), Function.Injective fun v => Matrix.vecMul v S.H","l":125,"f":"Causalean/Discovery/LinearDisentanglement/Uniqueness.lean"},{"n":"Causalean.EdgeType","k":"inductive","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"An edge-type classification is either nonparametric, monotonic with a specified monotonicity classification, linear, or parametric.","s":"Type","l":45,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.decRefines","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"For each first edge-type assumption and each second edge-type assumption, a decision procedure for whether the first refines the second is provided.","s":"(e₁ e₂ : Causalean.EdgeType) → Decidable (e₁.refines e₂)","l":93,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"For two edge-type assumptions, the refinement relation holds exactly when their Boolean refinement indicator is true; thus the first assumption is at least as restrictive as the second.","s":"Causalean.EdgeType → Causalean.EdgeType → Prop","l":86,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refinesBool","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"For two edge-type assumptions, the Boolean refinement indicator is true exactly when the first assumption is at least as restrictive as the second: every assumption refines the nonparametric class, matching monotonicity kinds refine one another, and linear or parametric classes refine only their respective matching classes.","s":"Causalean.EdgeType → Causalean.EdgeType → Bool","l":69,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines_nonparametric","k":"theorem","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"Every edge-type functional-form assumption e refines the nonparametric assumption: nonparametric is the weakest assumption in the refinement order, so every other assumption is at least as specific as it.","s":"∀ (e : Causalean.EdgeType), e.refines Causalean.EdgeType.nonparametric","l":104,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeType.refines_refl","k":"theorem","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"Every edge-type assumption refines itself.","s":"∀ (e : Causalean.EdgeType), e.refines e","l":97,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment","k":"structure","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"An edge type assignment attaches a functional-assumption label to each directed edge of a graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Type u_1","l":114,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.allNonparametric","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"For a directed acyclic graph, the all-nonparametric edge-type assignment labels every ordered pair of its vertices as nonparametric.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → Causalean.EdgeTypeAssignment G","l":133,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.decIsFullyNonparametric","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"For a finite vertex set whose members can be compared for equality and a directed acyclic graph on it and an edge-type assignment on that graph, a decision procedure for whether every directed edge has the nonparametric label is provided.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → (a : Causalean.EdgeTypeAssignment G) → Decidable a.isFullyNonparametric","l":148,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.incomingTypes","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"For an edge-type assignment and a vertex in its graph, the incoming edge-type set is the finite set of labels assigned to all parents of that vertex.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → Causalean.EdgeTypeAssignment G → V → Finset Causalean.EdgeType","l":138,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.EdgeTypeAssignment.isFullyNonparametric","k":"def","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"For an edge-type assignment, full nonparametricity holds exactly when, for every ordered pair of vertices joined by a directed edge, the assigned label is nonparametric.","s":"{V : Type u_2} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → Causalean.EdgeTypeAssignment G → Prop","l":143,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem","k":"structure","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"A back-door estimation system extends a potential-outcome back-door system with value-space representatives of the nuisance parameters used by AIPW-style estimators: an outcome regression μ(a,x) and a propensity score e(x) that is bounded away from 0 and away from 1, together with the outcome regression's agreement, almost everywhere, with the σ(X)-measurable observable regression adjustedCE, composed with the factual covariate and the analogous agreement of the propensity score with the observable propensity propScore.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":52,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a measurable covariate space and a real overlap level, the pointwise overlap-bounded nuisance class is the set of all nuisance vectors whose propensity score lies between $ε$ and $1-ε$ at every covariate value.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → ℝ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":294,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door estimation system, and a real overlap level, the almost-everywhere $L^2$ AIPW nuisance class is the set of nuisance vectors whose propensity score lies between $ε$ and $1-ε$ almost everywhere under the covariate law, whose two outcome regressions are square-integrable under that law, and whose propensity score is essentially bounded under that law.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":301,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_e_memLp_top","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The propensity component of an H_ε_aeL2 nuisance is in L∞(P_X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → MeasureTheory.MemLp η.e_fn ⊤ S.P_X","l":351,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_mu_memLp","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For an estimation system S, overlap level ε, and treatment arm d, if η is a member of the ε-overlap L² nuisance class H_ε_aeL2, then η's outcome-regression component at arm d lies in L²(P_X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ (d : Bool), MeasureTheory.MemLp (η.μ_fn d) 2 S.P_X","l":342,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_overlap_P_Z","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Membership in H_ε_aeL2 transports its a.e. overlap condition from the covariate law to the observed-data law along the covariate projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, ε ≤ η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ∧ η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ≤ 1 - ε","l":327,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε_aeL2_overlap_factualX","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Membership in H_ε_aeL2 transports its a.e. overlap condition from the covariate law to the original probability space along the observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATE.NuisanceVec γ}, η ∈ S.H_ε_aeL2 ε → ∀ᵐ (ω : P.Ω) ∂P.μ, ε ≤ η.e_fn (S.factualX ω) ∧ η.e_fn (S.factualX ω) ≤ 1 - ε","l":312,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.K_AIPW","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"For a real number, the AIPW Lipschitz constant is $1+2/ε+2/ε^2$.","s":"ℝ → ℝ","l":72,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_X","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the covariate marginal law is the image of the population measure under the factual covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → MeasureTheory.Measure γ","l":125,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_X_eq","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The covariate marginal is the image of the population measure under the factual covariate map.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.P_X = MeasureTheory.Measure.map S.factualX P.μ","l":129,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the observed-data law is the image of the population measure under the factual covariate--treatment--outcome map.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → MeasureTheory.Measure (γ × Bool × ℝ)","l":151,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_Z_eq","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The joint data law is the image of the population measure under the map recording the observed covariate, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.P_Z = MeasureTheory.Measure.map S.factualZ P.μ","l":156,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.P_Z_map_projX_eq_P_X","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The covariate marginal P_X is the pushforward of P_Z along the projection (x, a, y) ↦ x. Used to bridge integrals/eLpNorm between P_X (covariates only) and P_Z (full data triple).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), MeasureTheory.Measure.map (fun z => z.1) S.P_Z = S.P_X","l":163,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.StrictOverlap","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door estimation system, and a real overlap level, the strict-overlap condition holds precisely when $ε>0$, $ε\\leq 1/2$, and almost surely under the population measure, the probability of treatment is between $ε$ and $1-ε$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ → Prop","l":115,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.Tfull","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Given a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the full mean-zero tangent space is the orthogonal complement of the constant functions in the observed-data $L^2$ space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":455,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.YMuVal_residual_sq","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the squared residual-sum integrand maps each observed triple to the square of the sum of the absolute residuals from its two true outcome regressions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":351,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwLp","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Given a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door estimation system, a real overlap level satisfying strict overlap, the back-door identification assumptions, an integrable squared factual outcome, and integrable squared potential outcomes under each treatment, the square-integrable AIPW influence function is the equivalence class of that influence function in the observed-data $L^2$ space.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → {ε : ℝ} → S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":440,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwLp_mem_tangent","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The AIPW influence function lies in the mean-zero tangent space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), S.aipwLp h_overlap hA h_y2 h_yd2 ∈ S.Tfull","l":483,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMoment","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a covariate space, an observed triple, two treatment-specific outcome-regression functions, a propensity-score function, and a real target value, the augmented inverse-probability-weighting moment is the outcome-regression contrast plus the treated inverse-propensity-weighted residual minus the control inverse-propensity-weighted residual, less the target value.","s":"{γ : Type u_1} → γ × Bool × ℝ → (Bool → γ → ℝ) → (γ → ℝ) → ℝ → ℝ","l":74,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a measurable covariate space, the AIPW moment functional maps a nuisance vector, an observed covariate--treatment--outcome triple, and a real target value to the corresponding AIPW moment.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → γ × Bool × ℝ → ℝ → ℝ","l":358,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMoment_eq","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The augmented inverse-probability-weighting moment at a data point, a nuisance pair, and a parameter value is the contrast of the two outcome regressions at the covariate, plus the treated inverse-propensity-weighted outcome residual, minus the control inverse-propensity-weighted outcome residual, minus the parameter value.","s":"∀ {γ : Type u_1} [MeasurableSpace γ] (z : γ × Bool × ℝ) (μ_fn : Bool → γ → ℝ) (e_fn : γ → ℝ) (θ : ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMoment z μ_fn e_fn θ = μ_fn true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - μ_fn false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) + Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - μ_fn true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) - (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - μ_fn false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) - θ","l":86,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_finite_var","k":"theorem","m":"Causalean.Estimation.ATE.Score.FiniteVar","a":"Estimation/ATE/Score","t":1,"d":"Finite variance of ψ_AIPW — lem:est-aipw-finite-var. For an estimation system S, if the true propensity satisfies strict overlap at some level in (0, 1/2], the observed outcome has finite second moment, and each treatment arm's outcome regression, evaluated at the covariate, is square-integrable, then the AIPW influence function is square-integrable against the observed-data law P_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.MemLp (fun ω => S.μ_val d (S.factualX ω)) 2 P.μ) → MeasureTheory.Integrable (fun z => S.ψ_AIPW z ^ 2) S.P_Z","l":39,"f":"Causalean/Estimation/ATE/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_finite_var_of_counterfactual_sq","k":"theorem","m":"Causalean.Estimation.ATE.Score.FiniteVar","a":"Estimation/ATE/Score","t":2,"d":"Counterfactual second moments are a stronger sufficient condition for aipw_finite_var.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.Integrable (fun z => S.ψ_AIPW z ^ 2) S.P_Z","l":174,"f":"Causalean/Estimation/ATE/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_isPathwiseGradient_ATE","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"Hahn's pathwise-derivative identity makes the AIPW score a pathwise gradient of the backdoor ATE functional along every regular submodel.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : MeasureTheory.Measure (γ × Bool × ℝ) → ℝ), (∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), HasDerivAt (fun t => ψ (m.path t)) (∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑m.score z ∂S.P_Z) 0) → Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ (S.aipwLp h_overlap hA h_y2 h_yd2)","l":82,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_is_efficientInfluenceFunction","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":1,"d":"AIPW is the efficient influence function for the backdoor ATE. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Suppose a functional ψ of the observed-data law satisfies Hahn's pathwise-derivative identity — along every regular submodel, ψ's derivative at the true law equals the covariance of the AIPW score with the submodel's score, and that the full mean-zero space is a genuine nonparametric tangent space, i.e. it is contained in the closed span of regular-submodel scores. Then the AIPW influence function is a pathwise gradient of ψ, lies in the full mean-zero tangent space, and is the canonical efficient influence function: for every other pathwise gradient g of ψ, projecting g onto the tangent space returns the AIPW influence function.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : MeasureTheory.Measure (γ × Bool × ℝ) → ℝ), (∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), HasDerivAt (fun t => ψ (m.path t)) (∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑m.score z ∂S.P_Z) 0) → S.Tfull ≤ Causalean.Estimation.Efficiency.tangentSpace S.oneLp S.P_Z → Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ (S.aipwLp h_overlap hA h_y2 h_yd2) ∧ S.aipwLp h_overlap hA h_y2 h_yd2 ∈ S.Tfull ∧ ∀ (g : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), Causalean.Estimation.Efficiency.IsPathwiseGradient S.oneLp S.P_Z ψ g → Causalean.Estimation.Efficiency.efficientIF S.Tfull g = S.aipwLp h_overlap hA h_y2 h_yd2","l":104,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_mean_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":1,"d":"Mean zero of the AIPW influence function. Under the back-door identification assumptions, if the inverse-propensity-weighted residual correction on the treated arm, 1{D=1}/e(X) · (Y − μ(1,X)), is integrable and the analogous correction on the control arm, 1{D=0}/(1 − e(X)) · (Y − μ(0,X)), is integrable, then the AIPW influence function ψ_AIPW has mean zero under the joint law of the covariates, treatment indicator, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → MeasureTheory.Integrable (fun ω => 1 / S.e_val (S.factualX ω) * (S.dVar.indicator true ω * (S.factualY ω - S.μ_val true (S.factualX ω)))) P.μ → MeasureTheory.Integrable (fun ω => 1 / (1 - S.e_val (S.factualX ω)) * (S.dVar.indicator false ω * (S.factualY ω - S.μ_val false (S.factualX ω)))) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ∂S.P_Z = 0","l":615,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_mean_zero_of_square_integrable","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"A stronger sufficient-condition corollary for aipw_mean_zero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ∂S.P_Z = 0","l":643,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_memLp","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The AIPW influence function is square-integrable under the observed data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.MemLp S.ψ_AIPW 2 S.P_Z","l":429,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const","k":"def","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":1,"d":"For a real overlap level, the AIPW remainder constant is $2/[ε(1-ε)]$, the uniform strict-overlap weight bound used in the second-order remainder estimate.","s":"ℝ → ℝ","l":94,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_bound","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Bound","a":"Estimation/ATE/Remainder","t":1,"d":"Fix strict overlap for the true propensity at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a candidate nuisance vector η such that η lies in the ε-overlap L² nuisance class H_ε_aeL2, with each treatment-arm outcome-regression error in L²(P_X) and the propensity error in L²(P_X), the population AIPW moment functional at η and the true ATE θ₀ is bounded in absolute value by an overlap-dependent constant times the sum, over treatment arms, of the product of the outcome-regression and propensity L² errors.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ η ∈ S.H_ε_aeL2 ε, (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → |∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const ε * ∑ a, (MeasureTheory.eLpNorm (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":45,"f":"Causalean/Estimation/ATE/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_identity","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":1,"d":"Integrated AIPW remainder identity. Fix strict overlap for the true propensity at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a nuisance vector η such that η lies in the ε-overlap L² nuisance class H_ε_aeL2 with each treatment-arm outcome-regression error in L²(P_X), the population AIPW moment functional at η and the true ATE θ₀, integrated over the observed-data law P_Z, equals the covariate-law integral of η's propensity error (its propensity estimate minus the truth) times the sum of each treatment-arm outcome-regression error divided by the corresponding true or complementary propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ η ∈ S.H_ε_aeL2 ε, (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ ∂S.P_Z = ∫ (x : γ), (η.e_fn x - S.e_val x) * ((η.μ_fn true x - S.μ_val true x) / η.e_fn x + (η.μ_fn false x - S.μ_val false x) / (1 - η.e_fn x)) ∂S.P_X","l":121,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_remainder_op","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Bound","a":"Estimation/ATE/Remainder","t":1,"d":"AIPW remainder is o_p(n^{-1/2}) under the product rate. Fix strict overlap at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a sequence of nuisance estimators η̂ such that every realization η̂(n,ω) lies in the ε-overlap L² nuisance class, with outcome-regression errors in L²(P_X) at every horizon and realization and propensity errors in L²(P_X) at every horizon and realization, and whose L² outcome-regression and propensity errors have product rate o_p(n^{-1/2}) for each treatment arm, the population AIPW moment functional at the random nuisance η̂(n) and θ₀ is o_p(n^{-1/2}) under μ. Direct consequence of aipw_remainder_bound plus closure of IsLittleOp under finite sums and constant scaling.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":230,"f":"Causalean/Estimation/ATE/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":1,"d":"Fix strict overlap at level ε, the back-door identification assumptions, and finite second moments of the observed and potential outcomes. For a sequence of nuisance estimators η̂ such that every realization lies in the ε-overlap L² nuisance class, with outcome-regression errors in L²(P_X) at every horizon and realization and propensity errors in L²(P_X) at every horizon and realization, if the outcome-regression error converges to zero in L²(P_X) in probability and the propensity error converges to zero in L²(P_X) in probability, then the L²(P_Z) norm of the AIPW score difference between the estimated and true nuisance converges to zero in probability.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":941,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_diff_pointwise_bound","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":2,"d":"AIPW score Lipschitz bound on H_ε, P_Z-a.e.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → ∀ η ∈ S.H_ε_aeL2 ε, ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, |Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z S.θ₀ - Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.K_AIPW ε * (|η.μ_fn true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + |η.μ_fn false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + (|Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + |Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|) * |η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|)","l":270,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_score_meanZero_projection_eq","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The square-integrable AIPW score is already mean-zero, so projecting it onto the full mean-zero tangent space leaves it unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.efficientIF S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = S.aipwLp h_overlap hA h_y2 h_yd2","l":503,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_variance_hahn_decomposition","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEVariance","a":"Estimation/Efficiency","t":1,"d":"Hahn (1998) variance decomposition of the AIPW influence function. Assume the back-door identifying assumptions and strict overlap of the true propensity score with margin ε, and suppose the factual outcome is square-integrable and every potential outcome Y(d) is square-integrable. Then the variance of the augmented inverse-propensity-weighted (AIPW) influence function decomposes as the variance of the conditional treatment-effect function μ₁ − μ₀ − θ₀ plus two inverse-propensity-weighted conditional-outcome-variance terms — the Hahn semiparametric efficiency bound V_H.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ^ 2 ∂S.P_Z = ∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z","l":153,"f":"Causalean/Estimation/Efficiency/ATEVariance.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.cond_exp_residual_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"Conditional expectation of the residual ind_d · (Y − μ_val d X) given σ(X) is zero a.s. — the σ(X)-cleared form of lem:est-aipw-mean-zero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), P.μ[fun ω => S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":125,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.e_val_label","k":"def","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door estimation system, a treatment label, and a covariate value, the label-specific value-space propensity equals the propensity score for the treated label and one minus that score for the control label.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → Bool → γ → ℝ","l":40,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_hahn","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Semiparametric efficiency bound equals Hahn's three-term variance formula. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Then the semiparametric efficiency bound for the backdoor ATE equals the sum of the between-arms regression-contrast variance ∫(μ(1,X)-μ(0,X)-θ₀)²dP_X, the treated-arm weighted residual variance ∫(A/e(X)²)(Y-μ(1,X))²dP_Z, and the control-arm weighted residual variance ∫((1-A)/(1-e(X))²)(Y-μ(0,X))²dP_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = ∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z","l":546,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_of_smaller_tangent_containing_aipw","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"For any supplied smaller tangent space that is contained in the mean-zero tangent space and still contains the AIPW score, the abstract squared-norm efficiency bound is unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (T' : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 S.P_Z)) [inst_3 : T'.HasOrthogonalProjection], T' ≤ S.Tfull → S.aipwLp h_overlap hA h_y2 h_yd2 ∈ T' → Causalean.Estimation.Efficiency.effBound T' (S.aipwLp h_overlap hA h_y2 h_yd2) = Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2)","l":601,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.effBound_eq_variance","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Semiparametric efficiency bound equals the AIPW variance. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. Then the semiparametric efficiency bound for the backdoor ATE, computed against the full mean-zero tangent space, equals the second moment of the AIPW influence function ψ_AIPW under the observed-data law P_Z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ), Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) = ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z ^ 2 ∂S.P_Z","l":521,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.efficiency_bound_optimal","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"The efficiency bound lower-bounds every gradient's squared norm. Let S be a backdoor average-treatment-effect estimation system with strict overlap at level ε, satisfying the system's core identification assumptions, in which the observed outcome has finite second moment and each potential outcome under treatment level d has finite second moment. If ψ is a square-integrable element of L²(P_Z) that is a gradient for the AIPW influence function relative to the full mean-zero tangent space, then the semiparametric efficiency bound is at most the squared L² norm of ψ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (ψ : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), Causalean.Estimation.Efficiency.IsGradient S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) ψ → Causalean.Estimation.Efficiency.effBound S.Tfull (S.aipwLp h_overlap hA h_y2 h_yd2) ≤ ‖ψ‖ ^ 2","l":576,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the factual data map sends every sample point to its observed covariate, binary treatment, and outcome triple.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → P.Ω → γ × Bool × ℝ","l":136,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.indA","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a covariate space and an observed covariate--treatment--outcome triple, the real-valued treatment indicator equals one when the treatment is true and zero when it is false.","s":"{γ : Type u_1} → γ × Bool × ℝ → ℝ","l":70,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.indicator_to_propScore_integral","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"Propensity-score pull-out for the treatment indicator. Fix a treatment label $d$, under the back-door identification assumptions. If f : γ → ℝ is measurable and the product f(X) · 1{D=d} is integrable, then replacing the treatment indicator 1{D=d} by the value-space propensity e_val_label d inside the integral leaves the integral unchanged: ∫ f(X) · 1{D=d} dμ = ∫ f(X) · e_val_label d(X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool) (f : γ → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualX ω) * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), f (S.factualX ω) * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualX ω) * S.e_val_label d (S.factualX ω) ∂P.μ","l":225,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.inner_aipwLp_eq_integral","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"The covariance pairing between the AIPW influence function and any square-integrable score equals the Hilbert-space inner product against the AIPW element in the observed-data square-integrable space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (h_overlap : S.StrictOverlap ε) (hA : S.Assumptions) (h_y2 : MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ) (h_yd2 : ∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) (f : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), inner ℝ (S.aipwLp h_overlap hA h_y2 h_yd2) f = ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * ↑↑f z ∂S.P_Z","l":35,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.inner_oneLp","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"Inner product against the constant-one function equals integration under the observed data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (f : ↥(MeasureTheory.Lp ℝ 2 S.P_Z)), inner ℝ f S.oneLp = ∫ (z : γ × Bool × ℝ), ↑↑f z ∂S.P_Z","l":474,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instHasOrthogonalProjection_Tfull","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"For a potential-outcome system with a standard-Borel sample space and finite population measure, a measurable covariate space, and a back-door estimation system, the full mean-zero tangent space admits an orthogonal projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Tfull.HasOrthogonalProjection","l":467,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instHasOrthogonalProjection_span_oneLp","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"For a potential-outcome system with a standard-Borel sample space and finite population measure, a measurable covariate space, and a back-door estimation system, the one-dimensional subspace spanned by the constant-one observed-data function admits an orthogonal projection.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), (ℝ ∙ S.oneLp).HasOrthogonalProjection","l":460,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.instIsProbabilityMeasure_P_Z","k":"instance","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"For a potential-outcome system with a standard-Borel sample space and finite population measure, a measurable covariate space, and a back-door estimation system, the system's observed-data law is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), MeasureTheory.IsProbabilityMeasure S.P_Z","l":422,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.isTangentSpace_Tfull","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"The full mean-zero space is a genuine tangent space once the supplied nonparametric-model hypothesis says it is contained in the closed span of regular-submodel scores.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Tfull ≤ Causalean.Estimation.Efficiency.tangentSpace S.oneLp S.P_Z → Causalean.Estimation.Efficiency.IsTangentSpace S.oneLp S.P_Z S.Tfull","l":70,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_aipwMomentFunctional","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":2,"d":"The AIPW moment functional is measurable in the observed data triple for any fixed nuisance vector and target value.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATE.NuisanceVec γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional η z θ","l":97,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_e_val_label","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"The value-space propensity for any treatment label is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (d : Bool), Measurable (S.e_val_label d)","l":45,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"The observed covariate, treatment, and outcome triple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), Measurable S.factualZ","l":143,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.measurable_ψ_AIPW","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The AIPW influence function is measurable as a function of the observed data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), Measurable S.ψ_AIPW","l":48,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.oneLp","k":"def","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":1,"d":"Given a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the constant-one element of the observed-data $L^2$ space is the equivalence class of the function that equals one everywhere.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → ↥(MeasureTheory.Lp ℝ 2 S.P_Z)","l":450,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.plugin_bias_le_eLpNorm","k":"theorem","m":"Causalean.Estimation.ATE.Remainder.Identity","a":"Estimation/ATE/Remainder","t":2,"d":"Plug-in bias bound. The integrated plug-in bias in component a satisfies |∫ (μ̂(a,·) − μ_val(a,·)) dP_X| ≤ ‖μ̂(a,·) − μ_val(a,·)‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) [MeasureTheory.IsProbabilityMeasure P.μ] {μ_fn : Bool → γ → ℝ}, (∀ (a : Bool), Measurable (μ_fn a)) → (∀ (a : Bool), MeasureTheory.MemLp (fun x => μ_fn a x - S.μ_val a x) 2 S.P_X) → ∀ (a : Bool), |∫ (x : γ), μ_fn a x - S.μ_val a x ∂S.P_X| ≤ (MeasureTheory.eLpNorm (fun x => μ_fn a x - S.μ_val a x) 2 S.P_X).toReal","l":60,"f":"Causalean/Estimation/ATE/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projA","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a covariate space, the treatment projection maps every observed triple $(x,a,y)$ to its binary treatment component $a$.","s":"{γ : Type u_1} → γ × Bool × ℝ → Bool","l":52,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projA_apply","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The treatment projection of an observed data triple returns its treatment component.","s":"∀ {γ : Type u_1} [MeasurableSpace γ] (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.projA z = z.2.1","l":62,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projX","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a covariate space, the covariate projection maps every observed triple $(x,a,y)$ to its covariate component $x$.","s":"{γ : Type u_1} → γ × Bool × ℝ → γ","l":49,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projX_apply","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The covariate projection of an observed data triple returns its covariate component.","s":"∀ {γ : Type u_1} [MeasurableSpace γ] (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z = z.1","l":58,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projY","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a covariate space, the outcome projection maps every observed triple $(x,a,y)$ to its real-valued outcome component $y$.","s":"{γ : Type u_1} → γ × Bool × ℝ → ℝ","l":55,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.projY_apply","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"The outcome projection of an observed data triple returns its outcome component.","s":"∀ {γ : Type u_1} [MeasurableSpace γ] (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z = z.2.2","l":66,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_eq_e_val_label_ae","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"The σ(X)-conditional expectation of 1_{D=d} is e_val_label d (X) a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), S.propScore d =ᵐ[P.μ] fun ω => S.e_val_label d (S.factualX ω)","l":99,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_false_ae","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":2,"d":"propScore false =ᵐ 1 - propScore true under back-door assumptions. The indicator-pair sums to one pointwise, conditional expectation is linear, and preserves constants.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.propScore true ω","l":53,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.propScore_ne_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The conditional treatment probability for either treatment label is nonzero almost surely under the back-door assumptions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0","l":72,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.score_mem_Tfull","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEEfficientIF","a":"Estimation/Efficiency","t":2,"d":"Every regular submodel score is mean-zero, so every such score belongs to the full nonparametric tangent space for the observed-data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (m : Causalean.Estimation.Efficiency.RegularSubmodel S.oneLp S.P_Z), m.score ∈ S.Tfull","l":59,"f":"Causalean/Estimation/Efficiency/ATEEfficientIF.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.theta_zero_factualX_integral","k":"theorem","m":"Causalean.Estimation.ATE.Score.MeanZero","a":"Estimation/ATE/Score","t":2,"d":"The estimand θ₀ = ∫ x, (μ_val(1,x) − μ_val(0,x)) ∂P_X lifts back to the ambient measure via factualX.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.θ₀ = ∫ (ω : P.Ω), S.μ_val true (S.factualX ω) - S.μ_val false (S.factualX ω) ∂P.μ","l":261,"f":"Causalean/Estimation/ATE/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.weighted_residual_integral_zero","k":"theorem","m":"Causalean.Estimation.ATE.Score.ScorePullout","a":"Estimation/ATE/Score","t":1,"d":"Weighted-residual mean-zero identity (pull-out lemma). Fix a treatment label $d$ and a measurable weight function g : γ → ℝ on the covariates, under the back-door identification assumptions. If the product g(X) · 1{D=d} · (Y − μ(d,X)) is integrable and the σ(X)-conditional expectation of the treatment-d residual 1{D=d} · (Y − μ(d,X)) vanishes almost surely, then the integral of the weighted residual against the observed-data law vanishes: ∫ g(X) · 1{D=d} · (Y − μ(d,X)) dμ = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool) (g : γ → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualX ω) * (S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)))) P.μ → (P.μ[fun ω => S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0) → ∫ (ω : P.Ω), g (S.factualX ω) * (S.dVar.indicator d ω * (S.factualY ω - S.μ_val d (S.factualX ω))) ∂P.μ = 0","l":114,"f":"Causalean/Estimation/ATE/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.yMuVal_residual_sq_integrable","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWScoreL2","a":"Estimation/ATE/Score","t":2,"d":"The cross-term integrand is P_Z-integrable, with the bound coming from (a+b)² ≤ 2(a² + b²) and Y² ∈ L¹(P_Z), μ_val(d, X)² ∈ L¹(P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → MeasureTheory.Integrable S.YMuVal_residual_sq S.P_Z","l":376,"f":"Causalean/Estimation/ATE/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.η₀","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the true nuisance vector consists of the system's treatment-specific outcome regressions and propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → Causalean.Estimation.ATE.NuisanceVec γ","l":290,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, and a back-door estimation system, the value-space average treatment effect is the covariate-law integral of the true treated outcome regression minus the true control outcome regression.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → ℝ","l":175,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.θ₀_eq_ATE","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Value-space estimand equals the potential-outcome ATE. Under the back-door identification assumptions, the value-space estimand θ₀ = ∫ (μ(1,x) − μ(0,x)) dP_X, built from the outcome-regression nuisance, coincides with the average treatment effect defined on potential outcomes.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → S.θ₀ = S.ATE","l":179,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.μ_compat","k":"theorem","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":2,"d":"Counterfactual compatibility (derived, under identification). Recovers the former μ_compat field — the *counterfactual* reading μ[Y(d) | σ(X)] =ᵐ μ_val d ∘ factualX — now as a theorem rather than an assumption. It is the observable μ_reg_compat (μ_val d ∘ factualX =ᵐ adjustedCE d) composed with back-door identification (cate_backdoor : μ[Y(d)|σX] =ᵐ adjustedCE d), so the counterfactual binding is NOT part of the estimation system's data: it holds only under Assumptions. Every downstream proof that used the old field calls this with the ambient hA.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ (d : Bool), P.μ[S.YofD d | S.sigmaX] =ᵐ[P.μ] fun ω => S.μ_val d (S.factualX ω)","l":101,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.Estimation.ATE.BackdoorEstimationSystem.ψ_AIPW","k":"def","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door estimation system, and an observed triple, the AIPW influence function at the system's true nuisance functions and average treatment effect is its AIPW moment evaluated at that triple.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":100,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec","k":"structure","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":1,"d":"A pair of value-space AIPW nuisance functions, used as the abstract nuisance space for the AIPW moment functional: a treatment-arm-indexed outcome regression and a propensity score, together with their measurability.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":114,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.ext","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"Two AIPW nuisance vectors are equal when their outcome regressions and propensity functions agree everywhere.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATE.NuisanceVec γ}, (∀ (b : Bool) (x : γ), η.μ_fn b x = η'.μ_fn b x) → (∀ (x : γ), η.e_fn x = η'.e_fn x) → η = η'","l":168,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.ext_iff","k":"theorem","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATE.NuisanceVec γ}, η = η' ↔ (∀ (b : Bool) (x : γ), η.μ_fn b x = η'.μ_fn b x) ∧ ∀ (x : γ), η.e_fn x = η'.e_fn x","l":170,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instAdd","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the addition operation on AIPW nuisance vectors is componentwise addition of the treatment-specific outcome regressions and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.ATE.NuisanceVec γ)","l":135,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instAddCommGroup","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the additive commutative group structure on AIPW nuisance vectors uses the zero vector, componentwise addition, componentwise negation, componentwise subtraction, natural-number scalar multiplication, and integer scalar multiplication, and satisfies the natural-zero rule, the natural-successor rule, the integer-zero rule, the positive-integer-successor rule, the negative-integer-successor rule, subtraction as addition of an inverse, associativity, the left-zero law, the right-zero law, inverse cancellation, and commutativity.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.ATE.NuisanceVec γ)","l":183,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instModuleReal","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the real vector-space structure on AIPW nuisance vectors uses componentwise scalar multiplication and satisfies multiplication by one, compatibility of successive scalar multiplications, multiplication of zero vectors, distribution over vector addition, distribution over scalar addition, and multiplication by the zero scalar.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.ATE.NuisanceVec γ)","l":235,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instNeg","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the negation operation on AIPW nuisance vectors negates each treatment-specific outcome regression and the propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.ATE.NuisanceVec γ)","l":144,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instSMulReal","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the real scalar-multiplication operation on AIPW nuisance vectors scales each treatment-specific outcome regression and the propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.ATE.NuisanceVec γ)","l":160,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instSub","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the subtraction operation on AIPW nuisance vectors is componentwise subtraction of the treatment-specific outcome regressions and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.ATE.NuisanceVec γ)","l":151,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.NuisanceVec.instZero","k":"instance","m":"Causalean.Estimation.ATE.Score.AIPWMoment","a":"Estimation/ATE/Score","t":2,"d":"For a measurable covariate space, the zero operation on AIPW nuisance vectors sets both treatment-specific outcome regressions and the propensity score to zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.ATE.NuisanceVec γ)","l":129,"f":"Causalean/Estimation/ATE/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATE.aipwGeneralMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a potential-outcome system with a measurable covariate space, a back-door estimation system, and a real radius for which the system's true nuisance functions belong to its almost-everywhere $L^2$ nuisance class, the AIPW general-moment specification is the abstract moment model whose data are the observed covariate, treatment, and outcome and whose target is the back-door average treatment effect.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → {ε : ℝ} → S.η₀ ∈ S.H_ε_aeL2 ε → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × Bool × ℝ) S.P_Z (Causalean.Estimation.ATE.NuisanceVec γ)","l":56,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":2,"d":"AIPW satisfies BilinearRemainder with constant aipw_rem_const ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → (∀ η ∈ S.H_ε_aeL2 ε, (∀ (d : Bool), MeasureTheory.MemLp (fun x => η.μ_fn d x - S.μ_val d x) 2 S.P_X) ∧ MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem) C","l":102,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":1,"d":"Headline AIPW DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. For the back-door AIPW estimator with true nuisance η₀ known to lie in the ε-ball H_ε_aeL2 S ε, assume the propensity score has ε-strict overlap, that the identification assumptions of the back-door system hold, and that the factual and potential outcomes are square-integrable. Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂ that stay in the ε-ball at every fold and sample point with outcome-regression and propensity-score differences from the truth square-integrable in S.P_X. Assume the technical regularity package that the AIPW moment functional at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable under S.P_Z. Finally suppose the two nuisance-error rates are individually negligible ρ₁(η̂, η₀) = o_P(1), ρ₂(η̂, η₀) = o_P(1), and their product decays at the parametric rate ρ₁(η̂, η₀) · ρ₂(η̂, η₀) = o_P(n^{-1/2}). Then the Chernozhukov one-step AIPW-DML estimator is asymptotically linear at the true parameter S.θ₀ with the standard AIPW influence function ψ(z) = −J₀⁻¹ · ψ_AIPW(η₀, z), indexed over the fold-B subsample.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε_aeL2 ε) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (η_hat n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem).J₀_inv * Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":138,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.aipw_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AIPWInstance","a":"Estimation/OrthogonalMoments","t":2,"d":"AIPW satisfies MeanZero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε_aeL2 ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.ATE.aipwGeneralMoment S hη₀_mem)","l":89,"f":"Causalean/Estimation/OrthogonalMoments/AIPWInstance.lean"},{"n":"Causalean.Estimation.ATE.dmlEstimator","k":"def","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a measurable covariate space, a back-door estimation system, an independent and identically distributed sample of observed covariate, treatment, and outcome triples from its observable data law, a one-shot split of that sample, an outcome-regression learner indexed by sample size and population realization, a propensity-score learner indexed in the same way, and a nonnegative integer sample-size index, the one-shot DML/AIPW estimator of the back-door average treatment effect assigns to each population realization the average AIPW moment over that index's estimation fold.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Bool → γ → ℝ) → (ℕ → P.Ω → γ → ℝ) → ℕ → P.Ω → ℝ","l":62,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_attains_hahn_bound","k":"theorem","m":"Causalean.Estimation.Efficiency.ATEVariance","a":"Estimation/Efficiency","t":1,"d":"The one-shot DML ATE attains the Hahn efficiency bound. Assume the back-door identifying assumptions and strict overlap of the true propensity score with margin ε, with the factual outcome and every potential outcome Y(d) square-integrable. Let sample be an i.i.d. draw of the observed data and split a one-shot fold split whose training-fold share converges to a limit c strictly between 0 and 1. For the nuisance estimators μ_hat, e_hat fit on the training fold, assume each is jointly measurable in the training data and the evaluation point and square-integrable against P_X, and each is a measurable function of the training fold alone, that the estimated propensity score also satisfies overlap with margin ε, that each nuisance estimator is L²(P_X)-consistent, and that the product of the two L² estimation errors is o_P(n^{-1/2}) — the Neyman-orthogonality rate condition. Assume finally the AIPW influence function, the rescaled estimator, and the normalized influence-function sum are all measurable. Then the rescaled one-shot DML ATE estimator converges in distribution to the mean-zero Gaussian law whose variance is the Hahn semiparametric efficiency bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Measurable S.ψ_AIPW → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample S.ψ_AIPW split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ), (S.μ_val true x - S.μ_val false x - S.θ₀) ^ 2 ∂S.P_X + ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z / S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val true (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z + ∫ (z : γ × Bool × ℝ), (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) / (1 - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ_val false (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":426,"f":"Causalean/Estimation/Efficiency/ATEVariance.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"Asymptotic linearity of the one-shot DML ATE — thm:est-dml-ate-al. Fix the back-door identification assumptions for the estimation system S, with strict overlap $\\varepsilon \\le e(X) \\le 1-\\varepsilon$ for the true propensity and a.e. overlap at the same $\\varepsilon$ for every learner realization $\\hat e(n,\\omega)$, a finite second moment for the observed outcome, and a finite second moment for each potential outcome. Take a one-shot sample split whose training-fold size fraction converges to a limit $c$ with $0 < c < 1$. Suppose the outcome-regression and propensity learners $\\hat\\mu, \\hat e$ are measurable, lie in $L^2(P_X)$ at every realization, depend only on the nuisance-training fold $A(n)$, both as functions of that fold alone and jointly with the covariate, and converge individually to the truth in $L^2(P_X)$ at rate $o_p(1)$ with a product rate of $o_p(n^{-1/2})$. Then the one-shot DML/AIPW estimator of the back-door ATE is asymptotically linear at the true ATE $\\theta_0$ with influence function $\\psi_{AIPW}$ along the training folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ S.ψ_AIPW sample split.foldB","l":101,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.dml_ATE_tendstoNormal","k":"theorem","m":"Causalean.Estimation.ATE.DML","a":"Estimation/ATE","t":1,"d":"Asymptotic normality of the one-shot DML ATE (thm:est-dml-ate-al, \"In particular ...\" clause). Under the back-door identification assumptions for S, with strict overlap for the true propensity and a.e. overlap for every learner realization, finite second moments of the observed and potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1: suppose the learners μ̂, ê are measurable, in L²(P_X) at every realization, depend only on the nuisance-training fold, marginally and jointly with the covariate, and converge individually at rate o_p(1) with product rate o_p(n^{-1/2}) — the same hypotheses as dml_ATE_isAsymLinear. Given in addition measurability of the AIPW influence function, a.e. measurability of the rescaled estimator at every horizon, and a.e. measurability of the normalized influence-sum at every horizon, then the rescaled estimator √|B(n)| (θ̂ⁿ − θ₀) converges in distribution to N(0, ∫ ψ_AIPW² dP_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ) (e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, ε ≤ e_hat n ω x ∧ e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal) (fun x => 1) P.μ) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ) → Measurable S.ψ_AIPW → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample S.ψ_AIPW split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.dmlEstimator S sample split μ_hat e_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ × Bool × ℝ), S.ψ_AIPW x ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":936,"f":"Causalean/Estimation/ATE/DML.lean"},{"n":"Causalean.Estimation.ATE.plugInEstimator","k":"def","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a measurable covariate space, a back-door estimation system, an independent and identically distributed sample of observed covariate, treatment, and outcome triples from its observable data law, a one-shot split of that sample, an outcome-regression learner indexed by sample size and population realization, and a nonnegative integer sample-size index, the sample-split plug-in estimator of the back-door average treatment effect assigns to each population realization the estimation-fold average of the estimated treated-minus-control outcome regression at the observed covariates.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Bool → γ → ℝ) → ℕ → P.Ω → ℝ","l":64,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.plugIn_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Asymptotic linearity of the plug-in ATE — thm:est-plug-in-ate-al. Fix the back-door identification assumptions for S, square-integrability of both potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1. Suppose the outcome-regression learner μ̂ is measurable, lies in L²(P_X) at every realization, and depends only on the nuisance-training fold, marginally and jointly with the covariate, with joint L²(P_X) estimation error at rate o_p(n^{-1/2}). Then the sample-split plug-in estimator of the back-door ATE is asymptotically linear at the true ATE θ₀ with influence function ψ_plugin along the training folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → Causalean.Stat.IsLittleOp (fun n ω => √(∑ a, (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal ^ 2)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ (Causalean.Estimation.ATE.ψ_plugin S) sample split.foldB","l":109,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.plugIn_tendstoNormal","k":"theorem","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"Asymptotic normality of the plug-in ATE (thm:est-plug-in-ate-al, \"In particular ...\" clause). Under the back-door identification assumptions for S, square-integrability of both potential outcomes, and a one-shot sample split whose training-fold fraction converges to some c with 0 < c < 1: suppose the learner μ̂ is measurable, in L²(P_X) at every realization, depends only on the nuisance-training fold, marginally and jointly with the covariate, with joint L²(P_X) estimation error at rate o_p(n^{-1/2}) — the same hypotheses as plugIn_isAsymLinear. Given in addition a.e. measurability of the rescaled estimator at every horizon and a.e. measurability of the normalized influence-sum at every horizon, then the rescaled estimator √|B(n)| (θ̂ⁿ − θ₀) converges in distribution to N(0, ∫ ψ_plugin² dP_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ_hat : ℕ → P.Ω → Bool → γ → ℝ), (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => μ_hat n ω a x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ_hat n)) → (∀ (n : ℕ) (a : Bool), Measurable fun p => μ_hat n p.1 a p.2) → Causalean.Stat.IsLittleOp (fun n ω => √(∑ a, (MeasureTheory.eLpNorm (fun x => μ_hat n ω a x - S.μ_val a x) 2 S.P_X).toReal ^ 2)) (fun n => ↑n ^ (-(1 / 2))) P.μ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.ATE.ψ_plugin S) split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.ATE.plugInEstimator S sample split μ_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : γ × Bool × ℝ), Causalean.Estimation.ATE.ψ_plugin S x ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":781,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATE.ψ_plugin","k":"def","m":"Causalean.Estimation.ATE.PlugIn","a":"Estimation/ATE","t":1,"d":"For a potential-outcome system with a measurable covariate space, a back-door estimation system, and an observed covariate, treatment, and outcome triple, the plug-in influence-function value is the true treated-minus-control outcome regression at that triple's covariate, centered at the system's true average treatment effect.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":98,"f":"Causalean/Estimation/ATE/PlugIn.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem","k":"structure","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"A treated estimation system extends a potential-outcome back-door system with the value-space nuisance representatives needed for ATT AIPW estimation: the control-arm outcome regression μ₀(x) and a propensity score e(x) that is bounded away from 1, together with the control regression's agreement, almost everywhere, with the σ(X)-measurable observable control regression adjustedCE false, composed with the factual covariate and the analogous agreement of the propensity score with the observable propensity propScore.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":57,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"For a potential-outcome system with a measurable covariate space, a treated estimation system, and a real overlap margin, the ATT nuisance class consists exactly of nuisance pairs for which the propensity score is at most $1-ε$ for almost every covariate, the control-arm outcome regression is square-integrable under the covariate distribution, and the propensity score is essentially bounded under that distribution.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ → Set (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":287,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε_overlap_P_Z","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"Overlap transports to the joint data law. For a candidate nuisance pair lying in the ATT overlap-bounded realization set H_ε, in particular satisfying the one-sided upper overlap bound η.e_fn ≤ 1 − ε almost everywhere on the covariates, the same bound holds almost everywhere under the joint law of the covariates, treatment, and outcome, once the candidate propensity is evaluated at the covariate projection of the data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η ∈ S.H_ε ε → ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) ≤ 1 - ε","l":319,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.H_ε_overlap_factualX","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Membership in the ATT H_ε transports the one-sided a.e. overlap bound to the original probability space along the observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} {η : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η ∈ S.H_ε ε → ∀ᵐ (ω : P.Ω) ∂P.μ, η.e_fn (S.factualX ω) ≤ 1 - ε","l":306,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"For every real overlap margin, the ATT AIPW Lipschitz constant is $1 + 2/ε + 2/ε^2$.","s":"ℝ → ℝ","l":76,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.OneSidedOverlap","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space, a treated estimation system, and a real margin, the one-sided overlap condition holds exactly when $0<ε≤1/2$ and the conditional probability of treatment given the covariates is at most $1-ε$ almost surely under the population measure.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ → Prop","l":146,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_X","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the covariate distribution is the image of the population measure under the factual covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → MeasureTheory.Measure γ","l":161,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_X_eq","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"The covariate marginal is the image of the population measure under the factual covariate map.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.P_X = MeasureTheory.Measure.map S.factualX P.μ","l":167,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the joint observable-data distribution is the image of the population measure under the factual map recording covariate, treatment, and outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → MeasureTheory.Measure (γ × Bool × ℝ)","l":191,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_Z_eq","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"The joint data law is the image of the population measure under the map recording the observed covariate, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.P_Z = MeasureTheory.Measure.map S.factualZ P.μ","l":198,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.P_Z_map_projX_eq_P_X","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"The covariate marginal P_X is the pushforward of P_Z along the projection (x, a, y) ↦ x. Used to bridge integrals/eLpNorm between P_X (covariates only) and P_Z (full data triple).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), MeasureTheory.Measure.map (fun z => z.1) S.P_Z = S.P_X","l":205,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.YMuVal_residual_sq_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"For a potential-outcome system with a measurable covariate space, a treated estimation system, and an observed covariate, treatment, and outcome triple, the squared ATT control-regression residual is the square of the observed outcome minus the true control-arm outcome regression evaluated at the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":284,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"Given a measurable covariate space, an observed covariate, treatment, and outcome triple, a control-arm outcome-regression function, a propensity-score function, and a real candidate effect on the treated, the unnormalized augmented inverse-probability-weighting moment for the average treatment effect on the treated is the treated residual minus the odds-weighted control residual minus the treated indicator times the candidate effect.","s":"{γ : Type u_1} → γ × Bool × ℝ → (γ → ℝ) → (γ → ℝ) → ℝ → ℝ","l":53,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"Given a measurable covariate space, the ATT AIPW moment functional maps a nuisance pair, an observed covariate-treatment-outcome triple, and a candidate effect on the treated to the corresponding unnormalized ATT AIPW moment.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATT.TreatedNuisanceVec γ → γ × Bool × ℝ → ℝ → ℝ","l":339,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT_eq","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"The augmented inverse-probability-weighting moment for the effect on the treated, at a data point, a nuisance pair, and a parameter value, is the treated outcome residual against the control regression, minus the odds-weighted control outcome residual, minus the parameter value on the treated.","s":"∀ {γ : Type u_1} [MeasurableSpace γ] (z : γ × Bool × ℝ) (μ₀_fn e_fn : γ → ℝ) (θ : ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z μ₀_fn e_fn θ = Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) - (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)) - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z * θ","l":76,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_finite_var_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":1,"d":"Finite variance of the ATT AIPW moment. Under one-sided overlap: the true treatment propensity is bounded above by 1 − ε for some ε ∈ (0, 1/2] almost surely, the one-sided back-door ATT assumptions, and square-integrability of the factual outcome together with square-integrability of the untreated potential outcome Y(0), the ATT AIPW moment evaluated at the truth nuisance is square-integrable under the joint law of the covariates, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀ ^ 2) S.P_Z","l":49,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_mean_zero_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":1,"d":"Mean zero of the ATT AIPW moment. Under the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, and integrability of the control-arm inverse-propensity-weighted residual correction (1 − 1{D=1}) · (e(X)/(1 − e(X))) · (Y − μ₀(X)), the ATT AIPW moment evaluated at the truth has mean zero under the joint law of the covariates, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀ ∂S.P_Z = 0","l":112,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_rem_const_ATT","k":"def","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"For a real number $\\varepsilon$, the ATT augmented inverse-probability-weighting remainder constant is $1/\\varepsilon$.","s":"ℝ → ℝ","l":50,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_bound_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder bound (ATT). Fix a candidate nuisance pair η. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, and square-integrability of the untreated potential outcome Y(0): if η lies in the overlap-bounded candidate realization set H_ε, its control-regression error μ̂₀ − μ₀ is square-integrable against the covariate law, its propensity error ê − e is square-integrable against the covariate law, and its IPW correction is integrable against the observed data law, then the absolute value of the population AIPW moment at η is bounded by (1/ε) · ‖μ̂₀ − μ₀‖_{L²(P_X)} · ‖ê − e‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z → |∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_rem_const_ATT ε * (MeasureTheory.eLpNorm (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":59,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_identity_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Identity","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder identity (ATT). Fix a candidate nuisance pair η. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, square-integrability of the untreated potential outcome Y(0): if η lies in the overlap-bounded candidate realization set H_ε, its control-regression error μ̂₀ − μ₀ is square-integrable against the covariate law, its propensity error ê − e is square-integrable against the covariate law, and its IPW correction is integrable against the observed data law, then the population AIPW moment at η equals the single cross-product ∫ ((ê(x) − e(x))/(1 − ê(x))) · (μ̂₀(x) − μ₀(x)) dP_X.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ ∂S.P_Z = ∫ (x : γ), (η.e_fn x - S.e_val x) / (1 - η.e_fn x) * (η.μ₀_fn x - S.μ₀_val x) ∂S.P_X","l":67,"f":"Causalean/Estimation/ATT/Remainder/Identity.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_remainder_op_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Remainder.Bound","a":"Estimation/ATT/Remainder","t":1,"d":"AIPW remainder is o_p(n^{-1/2}) under the ATT product rate. Fix a sequence of random candidate nuisance pairs indexed by sample size. Under one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome, and square-integrability of the untreated potential outcome Y(0): if every draw of the candidate lies in the overlap-bounded realization set H_ε, each control-regression error is square-integrable against the covariate law, each propensity error is square-integrable against the covariate law, each candidate IPW correction is integrable against the observed data law, and the product of the two L²(P_X) error norms is o_p(n^{-1/2}), then the population AIPW moment evaluated at the random candidate nuisance is itself o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * ((η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - (η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - (η_hat n ω).μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":188,"f":"Causalean/Estimation/ATT/Remainder/Bound.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_score_diff_isLittleOp_one_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":1,"d":"Headline L²(P_Z) o_p(1) continuity bound for the ATT AIPW score. Fix a sequence of random candidate nuisance pairs indexed by sample size. Under one-sided overlap ε on the true propensity, membership of the truth nuisance in the overlap-bounded realization set H_ε, nonnegativity of the true propensity, the one-sided back-door ATT assumptions, square-integrability of the factual outcome and of the untreated potential outcome Y(0): if every draw of the candidate lies in the overlap-bounded realization set H_ε, every candidate propensity is nonnegative, each candidate control-regression error admits an L²(P_X) witness, each candidate propensity error admits an L²(P_X) witness, the control-regression error rate is o_p(1) in L²(P_X), and the propensity error rate is o_p(1) in L²(P_X), then the L²(P_Z) distance between the ATT AIPW score evaluated at the random candidate nuisance and at the truth nuisance is o_p(1).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.η₀ ∈ S.H_ε ε → (∀ (x : γ), 0 ≤ S.e_val x) → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ (η_hat n ω).e_fn x) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":604,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.aipw_score_lipschitz_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWScoreL2","a":"Estimation/ATT/Score","t":2,"d":"ATT AIPW score Lipschitz bound on the a.e. one-sided-overlap set H_ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.η₀ ∈ S.H_ε ε → ∀ η ∈ S.H_ε ε, ∀ᵐ (z : γ × Bool × ℝ) ∂S.P_Z, |Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z S.θ₀ - Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT ε * |η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.μ₀_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| + Causalean.Estimation.ATT.TreatedEstimationSystem.K_AIPW_ATT ε * |Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - S.μ₀_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)| * |η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z)|","l":240,"f":"Causalean/Estimation/ATT/Score/AIPWScoreL2.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.control_cate_backdoor","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Control-arm backdoor CATE under ATT assumptions. The conditional mean of the untreated potential outcome given the covariates equals the observable control regression when consistency, conditional ignorability, integrability, and one-sided control overlap hold. No treated-arm overlap is used.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.CATE false =ᵐ[P.μ] S.adjustedCE false","l":110,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the factual data-recording map sends every population unit to its observed covariate, observed treatment, and observed outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → P.Ω → γ × Bool × ℝ","l":174,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.indicator_to_propScore_integral","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"Propensity-score pull-out for the treatment indicator (ATT). Fix a treatment label d, under the one-sided back-door ATT assumptions. If f : γ → ℝ is measurable and the product f(X) · 1{D=d} is integrable, then replacing the treatment indicator 1{D=d} by the value-space propensity — e_val when d is true, 1 − e_val when d is false — inside the integral leaves the integral unchanged.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → ∀ (d : Bool) (f : γ → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualX ω) * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), f (S.factualX ω) * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualX ω) * if d = true then S.e_val (S.factualX ω) else 1 - S.e_val (S.factualX ω) ∂P.μ","l":341,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipwWeight_false","k":"def","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"For a potential-outcomes system with a standard Borel sample space and finite probability measure, a measurable covariate space, a treated estimation system based on that system and covariate space, and a covariate value, the control-arm inverse-probability weight used in the ATT correction is the propensity score at that covariate value divided by one minus that propensity score. Equivalently, it is $e(x)/(1-e(x))$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ → ℝ","l":45,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipw_estimated_integrable","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":2,"d":"Integrability of an estimated-nuisance ATT IPW correction (over P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, 0 < ε → ∀ (η : Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ᵐ (x : γ) ∂S.P_X, 0 ≤ η.e_fn x) → (∀ᵐ (x : γ) ∂S.P_X, η.e_fn x ≤ 1 - ε) → MeasureTheory.MemLp η.μ₀_fn 2 S.P_X → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z","l":229,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ipw_truth_integrable","k":"theorem","m":"Causalean.Estimation.ATT.Score.FiniteVar","a":"Estimation/ATT/Score","t":2,"d":"Integrability of the truth-nuisance ATT IPW correction (over P.μ).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ","l":311,"f":"Causalean/Estimation/ATT/Score/FiniteVar.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_aipwMomentATTFunctional","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the ATT AIPW moment functional in the data variable z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATT.TreatedNuisanceVec γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional η z θ","l":349,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_aipwMomentATT_at_θ₀","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":2,"d":"Measurability of aipwMomentATT z η₀ θ₀ as a function of z.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z S.μ₀_val S.e_val S.θ₀","l":71,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Measurability of the data triple.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.factualZ","l":183,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_ipwWeight_false","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the value-space IPW weight e/(1−e).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.ipwWeight_false","l":50,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.measurable_ψ_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Score.MeanZero","a":"Estimation/ATT/Score","t":2,"d":"Measurability of the ATT influence function ψ_ATT.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), Measurable S.ψ_ATT","l":59,"f":"Causalean/Estimation/ATT/Score/MeanZero.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_eq_e_val_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"The treated-arm propensity propScore true factors through factualX via the value-space e_val. Direct restatement of S.e_compat.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore true =ᵐ[P.μ] fun ω => S.e_val (S.factualX ω)","l":104,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_false_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"propScore false =ᵐ 1 − propScore true. The indicator pair sums to one pointwise, conditional expectation is linear and preserves constants.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.propScore true ω","l":56,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_false_eq_one_minus_e_val_ae","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"Control-arm propensity factors through factualX as 1 − e_val.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.propScore false =ᵐ[P.μ] fun ω => 1 - S.e_val (S.factualX ω)","l":113,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.propScore_true_nonneg_ae","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Treated propensity nonnegativity. The conditional treatment probability P[D=1 | X] is nonnegative almost surely because it is the conditional expectation of a nonnegative treatment indicator.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), ∀ᵐ (ω : P.Ω) ∂P.μ, 0 ≤ S.propScore true ω","l":122,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.residual_false_condExp_zero","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":2,"d":"The σ(X)-conditional expectation of 1_{D=false}·(Y − μ₀(X)) vanishes a.s. The witness Y(false) =ᵐ μ₀_val ∘ factualX comes from μ₀_compat together with consistency on {D = false}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → P.μ[fun ω => S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω)) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":130,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.weighted_residual_false_integral_zero","k":"theorem","m":"Causalean.Estimation.ATT.Score.ScorePullout","a":"Estimation/ATT/Score","t":1,"d":"Weighted-residual mean-zero identity, control arm (ATT). Under the one-sided back-door ATT assumptions, if g : γ → ℝ is measurable and the product g(X) · 1{D=false} · (Y − μ₀(X)) is integrable, then the integral of the weighted control-arm residual against the observed-data law vanishes: ∫ g(X) · 1{D=false} · (Y − μ₀(X)) dμ = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → ∀ (g : γ → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω)))) P.μ → ∫ (ω : P.Ω), g (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ₀_val (S.factualX ω))) ∂P.μ = 0","l":231,"f":"Causalean/Estimation/ATT/Score/ScorePullout.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.η₀","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the true ATT nuisance pair consists of that system's control-arm outcome regression and propensity-score functions.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → Causalean.Estimation.ATT.TreatedNuisanceVec γ","l":281,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the value-space average treatment effect on the treated is the system's adjusted control-regression functional.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ","l":226,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.θ₀_eq_ATT","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Value-space estimand equals the potential-outcome ATT. Under the one-sided back-door ATT assumptions, the value-space ATT estimand θ₀ (the adjusted control-regression functional) coincides with the average treatment effect on the treated defined on potential outcomes.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → S.θ₀ = S.ATT","l":235,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.μ₀_compat","k":"theorem","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":2,"d":"Counterfactual compatibility (derived, under ATT identification). The control-arm outcome-regression representative equals the conditional mean of the untreated potential outcome given the covariates, almost surely.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ), S.ATTAssumptions → P.μ[S.YofD false | S.sigmaX] =ᵐ[P.μ] fun ω => S.μ₀_val (S.factualX ω)","l":132,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.π_val","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"For a potential-outcome system with a measurable covariate space and a treated estimation system, the marginal treatment probability is the population probability that the factual treatment equals one.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → ℝ","l":217,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.Estimation.ATT.TreatedEstimationSystem.ψ_ATT","k":"def","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"For a potential-outcome system with a measurable covariate space, a treated estimation system, and an observed covariate, treatment, and outcome triple, the ATT AIPW influence-function value is the unnormalized true-nuisance moment at zero, divided by the population treatment probability and centered at the true average treatment effect on the treated.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.ATT.TreatedEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":90,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec","k":"structure","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":1,"d":"A pair of value-space ATT nuisance functions, used as the abstract nuisance space for the ATT AIPW moment functional: the control-arm outcome regression and a propensity score, together with their measurability.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":114,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.ext","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"Two treated-nuisance vectors are equal when their control regressions and propensity functions agree everywhere.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, (∀ (x : γ), η.μ₀_fn x = η'.μ₀_fn x) → (∀ (x : γ), η.e_fn x = η'.e_fn x) → η = η'","l":167,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.ext_iff","k":"theorem","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.ATT.TreatedNuisanceVec γ}, η = η' ↔ (∀ (x : γ), η.μ₀_fn x = η'.μ₀_fn x) ∧ ∀ (x : γ), η.e_fn x = η'.e_fn x","l":168,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instAdd","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the addition operation on ATT nuisance vectors is componentwise addition of the control-arm outcome regression and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":134,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instAddCommGroup","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the additive commutative group structure on ATT nuisance vectors uses the zero vector, componentwise addition, componentwise negation, componentwise subtraction, natural-number scalar multiplication, and integer scalar multiplication, and satisfies the natural-zero rule, the natural-successor rule, the integer-zero rule, the positive-integer-successor rule, the negative-integer-successor rule, subtraction as addition of an inverse, associativity, the left-zero law, the right-zero law, inverse cancellation, and commutativity.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":181,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instModuleReal","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the real vector-space structure on ATT nuisance vectors uses componentwise scalar multiplication and satisfies multiplication by one, compatibility of successive scalar multiplications, multiplication of zero vectors, distribution over vector addition, distribution over scalar addition, and multiplication by the zero scalar.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":221,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instNeg","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the negation operation on ATT nuisance vectors negates the control-arm outcome regression and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":143,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instSMulReal","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the real scalar-multiplication operation on ATT nuisance vectors scales the control-arm outcome regression and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":159,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instSub","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the subtraction operation on ATT nuisance vectors is componentwise subtraction of the control-arm outcome regression and propensity score.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":150,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.TreatedNuisanceVec.instZero","k":"instance","m":"Causalean.Estimation.ATT.Score.AIPWMoment","a":"Estimation/ATT/Score","t":2,"d":"For a measurable covariate space, the zero operation on ATT nuisance vectors sets the control-arm outcome regression and the propensity score to zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":129,"f":"Causalean/Estimation/ATT/Score/AIPWMoment.lean"},{"n":"Causalean.Estimation.ATT.attGeneralMoment","k":"def","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":1,"d":"For a potential-outcomes system with a standard Borel sample space and finite probability measure, a measurable covariate space, a treated estimation system, a real overlap radius, the condition that the system's true nuisance vector belongs to its overlap-bounded candidate set at that radius, and the condition that its marginal treatment probability is strictly positive, the ATT augmented inverse-probability-weighted general moment is the general moment whose score is the ATT AIPW score, whose target nuisance vector and target parameter are the system's true nuisance vector and ATT, whose candidate set is that overlap-bounded set, whose two seminorms are the $L^2$ distances between control-outcome regressions and between propensity scores, and whose Jacobian is the negative marginal treatment probability.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) → {ε : ℝ} → S.η₀ ∈ S.H_ε ε → 0 < S.π_val → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × Bool × ℝ) S.P_Z (Causalean.Estimation.ATT.TreatedNuisanceVec γ)","l":73,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_bilinearRem","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":2,"d":"ATT AIPW satisfies BilinearRemainder with constant aipw_rem_const_ATT ε.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → (∀ η ∈ S.H_ε ε, MeasureTheory.MemLp (fun x => η.μ₀_fn x - S.μ₀_val x) 2 S.P_X ∧ MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X) → (∀ η ∈ S.H_ε ε, MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * (η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - η.e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - η.μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos) C","l":130,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":1,"d":"Headline ATT AIPW DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. Fix an estimated-nuisance sequence η_hat, an i.i.d. sample of the data triple, and a one-shot cross-fitting split of that sample. Under membership of the truth nuisance in the overlap-bounded realization set H_ε, nonnegativity of the true propensity, one-sided overlap ε on the true propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome and of the untreated potential outcome Y(0), integrability of the truth-side control-arm IPW correction, and a limiting fold-size fraction c strictly between 0 and 1 with the treated-fold cardinality fraction converging to c: if every candidate draw η_hat n ω lies in the overlap-bounded realization set H_ε, every candidate propensity is nonnegative, each candidate control-regression and propensity error admits an L²(P_X) witness, each candidate IPW correction is integrable, the AIPW moment functional is measurable jointly in the probability-space and data arguments, and on each cross-fitting fold, both singly and jointly, the moment at every candidate nuisance is integrable and square-integrable against the observed data law, the control-regression and propensity error rates are individually o_p(1) in L²(P_X), and their product is o_p(n^{-1/2}), then the Chernozhukov one-step DML estimator built from the ATT AIPW moment, the sample, the split, and the candidate nuisance sequence is asymptotically linear at the true ATT θ₀, with influence function ψ(z) = (1/π_T) · aipwMomentATTFunctional η₀ z θ₀.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), (∀ (x : γ), 0 ≤ S.e_val x) → S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.ATT.TreatedNuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ S.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ (η_hat n ω).e_fn x) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).μ₀_fn x - S.μ₀_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => (1 - Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z) * ((η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) / (1 - (η_hat n ω).e_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.projY z - (η_hat n ω).μ₀_fn (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) S.P_Z) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos).J₀_inv * Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATTFunctional S.η₀ z S.θ₀) sample split.foldB","l":168,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.att_meanZero","k":"theorem","m":"Causalean.Estimation.ATT.ATTInstance","a":"Estimation/ATT","t":2,"d":"ATT AIPW satisfies MeanZero. Direct repackaging of aipw_mean_zero_ATT from Estimation/ATT/Score/MeanZero.lean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ} (hη₀_mem : S.η₀ ∈ S.H_ε ε), S.OneSidedOverlap ε → S.ATTAssumptions → ∀ (hπ_pos : 0 < S.π_val), MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.ATT.attGeneralMoment S hη₀_mem hπ_pos)","l":110,"f":"Causalean/Estimation/ATT/ATTInstance.lean"},{"n":"Causalean.Estimation.ATT.dmlEstimator_ATT","k":"def","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":1,"d":"For a potential-outcomes system with a standard Borel sample space and finite probability measure, a measurable covariate space, a treated estimation system, an independent and identically distributed sample of observed covariate, treatment, and outcome triples from that system's observed-data distribution, a one-shot split of that sample, a sequence of control-arm outcome-regression estimators, a sequence of propensity-score estimators, and a sample-size index, the one-shot double-machine-learning estimator of the back-door average treatment effect on the treated is the marginal-treatment-probability-normalized mean, over the split's evaluation fold at that index, of the ATT AIPW score evaluated at zero using the supplied nuisance functions at that index.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → γ → ℝ) → (ℕ → P.Ω → γ → ℝ) → ℕ → P.Ω → ℝ","l":82,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.dml_ATT_isAsymLinear","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":1,"d":"Asymptotic linearity of the one-shot DML ATT — thm:est-dml-att-al. Fix candidate control-regression and propensity estimator sequences μ₀_hat and e_hat, an i.i.d. sample of the data triple, and a one-shot cross-fitting split of that sample. Under the true propensity bounded above by 1 − ε almost everywhere, nonnegativity of the true propensity, one-sided overlap ε on the treated-arm propensity, the one-sided back-door ATT assumptions, a strictly positive marginal treatment probability, square-integrability of the factual outcome and of the untreated potential outcome Y(0), and a limiting fold-size fraction c strictly between 0 and 1 with the treated-fold cardinality fraction converging to c: if the candidate propensity is bounded above by 1 − ε almost everywhere, for every n, ω, the candidate propensity is nonnegative everywhere, the candidate regressions are jointly measurable in the probability-space and covariate arguments, each candidate regression, at every n, ω, is square-integrable against the covariate law, each candidate regression depends only on its own cross-fitting fold, singly and jointly with the covariate, the ATT AIPW moment at every candidate regression pair is integrable and square-integrable against the observed data law, the control-regression and propensity error rates are individually o_p(1) in L²(P_X), and their product is o_p(n^{-1/2}), then the population-π one-shot DML/AIPW ATT estimator is asymptotically linear at the true ATT θ₀, with influence function ψ_ATT, along the sample and the cross-fitting folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, (∀ᵐ (x : γ) ∂S.P_X, S.e_val x ≤ 1 - ε) → (∀ (x : γ), 0 ≤ S.e_val x) → S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ₀_hat e_hat : ℕ → P.Ω → γ → ℝ), (∀ (n : ℕ) (ω : P.Ω), ∀ᵐ (x : γ) ∂S.P_X, e_hat n ω x ≤ 1 - ε) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), 0 ≤ e_hat n ω x) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => μ₀_hat n ω x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => e_hat n ω x) 2 S.P_X) → (∀ (n : ℕ), Measurable (μ₀_hat n)) → (∀ (n : ℕ), Measurable (e_hat n)) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e_hat n p.1 p.2) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z (μ₀_hat n ω) (e_hat n ω) S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => Causalean.Estimation.ATT.TreatedEstimationSystem.aipwMomentATT z (μ₀_hat n ω) (e_hat n ω) S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ₀_hat n ω x - S.μ₀_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => μ₀_hat n ω x - S.μ₀_val x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => e_hat n ω x - S.e_val x) 2 S.P_X).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.ATT.dmlEstimator_ATT S sample split μ₀_hat e_hat) S.θ₀ S.ψ_ATT sample split.foldB","l":239,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.ψ_ATT_finite_var","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":2,"d":"Finite variance of the centered population-π ATT influence function. Integrable ψ_ATT² against P_Z. Derived (not assumed) from aipw_finite_var_ATT (square-integrability of the moment at θ₀) and boundedness of the treatment indicator, via ψ_ATT = (1/π_T)·(m(η₀,·,θ₀) + θ₀·A) − θ₀ ∈ L²(P_Z).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun z => S.ψ_ATT z ^ 2) S.P_Z","l":191,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.ATT.ψ_ATT_integral_zero","k":"theorem","m":"Causalean.Estimation.ATT.DML","a":"Estimation/ATT","t":2,"d":"Mean zero of the centered population-π ATT influence function ψ_ATT.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATT.TreatedEstimationSystem P γ) {ε : ℝ}, S.OneSidedOverlap ε → S.ATTAssumptions → 0 < S.π_val → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => S.YofD false ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_ATT z ∂S.P_Z = 0","l":113,"f":"Causalean/Estimation/ATT/DML.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem","k":"structure","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"Wrapper around BackdoorEstimationSystem whose semantic role is to expose the CATE target τ_val x = μ_val 1 x − μ_val 0 x as a derived function-valued estimand. No new fields are introduced; the underlying BackdoorEstimationSystem carries consistency, conditional exchangeability, overlap, integrability, the value-space μ_val / e_val, observable outcome-regression compatibility μ_reg_compat, and propensity compatibility e_compat. Counterfactual outcome-regression compatibility is derived later from the back-door causal assumptions, not inherited as a field.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max u_1 u_2) u_3)","l":48,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.measurable_τ_val","k":"theorem","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":2,"d":"The value-space CATE is measurable, since both arms of μ_val are.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), Measurable S.τ_val","l":71,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.phi_eta_cond_exp_eq_tau","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcomeMean","a":"Estimation/CATE/Core","t":1,"d":"Value-space form of the DR pseudo-outcome mean-CATE identity. Under the back-door identification assumptions, two-sided strict overlap: the treatment propensity lies in [ε, 1 − ε] for some ε ∈ (0, 1/2] almost surely, and integrability of the true DR pseudo-outcome φ₀ evaluated at the factual data triple, for covariate-law-almost-every x, the mean of φ₀ under the regular conditional distribution of the data triple given the covariate value x equals the value-space CATE τ_val x.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → ∀ᵐ (x : γ) ∂S.P_X, ∫ (z : γ × Bool × ℝ), Causalean.Estimation.CATE.phi₀ S z ∂(ProbabilityTheory.condDistrib S.factualZ S.factualX P.μ) x = S.τ_val x","l":341,"f":"Causalean/Estimation/CATE/Core/PseudoOutcomeMean.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.phi₀_factualZ_cond_exp","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcomeMean","a":"Estimation/CATE/Core","t":1,"d":"σ(X)-conditional mean of the true DR pseudo-outcome equals the CATE. Under the back-door identification assumptions and two-sided strict overlap: the treatment propensity lies in [ε, 1 − ε] for some ε ∈ (0, 1/2] almost surely, the σ(X)-conditional expectation of the true DR pseudo-outcome φ₀, evaluated at the factual data triple, equals the value-space CATE τ_val pulled back along the covariate, almost surely.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → P.μ[fun ω => Causalean.Estimation.CATE.phi₀ S (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun ω => S.τ_val (S.factualX ω)","l":134,"f":"Causalean/Estimation/CATE/Core/PseudoOutcomeMean.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.tau_val_eq_CATE","k":"theorem","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"Value-space CATE recovers the conditional average treatment effect. Under the back-door causal assumptions — consistency of observed and potential outcomes, treatment ignorability given the covariates, two-sided overlap, and integrability of both potential outcomes, the value-space CATE, evaluated at the observed covariate and viewed as a random variable on the sample space, agrees almost surely with the σ(X)-conditional expectation of the potential-outcome contrast Y(1) − Y(0). This is the last sentence of def:est-cate-causal-assumptions:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → (fun ω => S.τ_val (S.factualX ω)) =ᵐ[P.μ] P.μ[fun ω => S.YofD true ω - S.YofD false ω | S.sigmaX]","l":77,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.CATEEstimationSystem.τ_val","k":"def","m":"Causalean.Estimation.CATE.Setup","a":"Estimation/CATE","t":1,"d":"For a potential-outcomes system with a standard Borel sample space and finite probability measure, a measurable covariate space, a CATE estimation system, and a covariate value, the value-space conditional average treatment effect is the conditional mean outcome under treatment at that covariate value minus the conditional mean outcome under control at that covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → γ → ℝ","l":65,"f":"Causalean/Estimation/CATE/Setup.lean"},{"n":"Causalean.Estimation.CATE.abs_integral_phiDiff_mul_le_product","k":"theorem","m":"Causalean.Estimation.CATE.Core.SecondOrderBias","a":"Estimation/CATE/Core","t":1,"d":"Second-order product bias bound for the DR-Learner. Fix a candidate nuisance pair η and a test function w. Under ε strictly positive, the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], each candidate outcome-regression arm, composed with the covariate, is integrable, w is measurable, w is dominated by a nonnegative bound B, the w-weighted pseudo-outcome bias is integrable, each candidate outcome-regression error is square-integrable against the covariate law, and the propensity error is square-integrable against the covariate law, then the absolute value of the population integral of the w-weighted pseudo-outcome bias (phi_eta η − φ₀) · w is bounded by B/ε times the sum, over the two treatment arms, of ‖η.μ_fn a − μ_val a‖_{L²(P_X)} · ‖η.e_fn − e_val‖_{L²(P_X)}.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) {ε : ℝ}, 0 < ε → S.Assumptions → ∀ η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε, S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → ∀ (w : γ → ℝ), Measurable w → ∀ {B : ℝ}, 0 ≤ B → (∀ (x : γ), |w x| ≤ B) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * w (S.factualX ω)) P.μ → (∀ (a : Bool), MeasureTheory.MemLp (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => η.e_fn x - S.e_val x) 2 S.P_X → |∫ (z : γ × Bool × ℝ), (Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z) * w z.1 ∂S.P_Z| ≤ B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => η.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => η.e_fn x - S.e_val x) 2 S.P_X).toReal","l":147,"f":"Causalean/Estimation/CATE/Core/SecondOrderBias.lean"},{"n":"Causalean.Estimation.CATE.cate_dr_oracle_efficient_linear","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.LinearSmootherCATE","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle efficiency for the DR-Learner with a linear-smoother second stage. Fix a CATE estimation system, a linear-smoother second-stage operator op, an estimated nuisance sequence η_hat, an evaluation point x, a centering-rate sequence d_n, and a bias-identity relation BiasIdent. Under the back-door identification assumptions and two-sided strict overlap for the truth, if the smoothed oracle estimator is stable at (τ_val, d_n, x) relative to BiasIdent, the centering sequence d_n converges to 0 in probability, the pseudo-outcome bias, the true pseudo-outcome, and the smoothed conditional bias jointly satisfy the identity relation BiasIdent, and the smoothed conditional-bias evaluation is o_p of the oracle risk scale, then the DR-Learner CATE estimator and the oracle estimator, both built from the linear-smoother second-stage operator, differ by o_p of the oracle risk scale.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → ∀ (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op.toSecondStageOperator S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op.toSecondStageOperator x n) P.μ → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op.toSecondStageOperator η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op.toSecondStageOperator n ω x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op.toSecondStageOperator x n) P.μ","l":227,"f":"Causalean/Estimation/CATE/Kennedy/LinearSmootherCATE.lean"},{"n":"Causalean.Estimation.CATE.cate_linear_smoother_bias_bound","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.LinearSmootherCATE","a":"Estimation/CATE/Kennedy","t":1,"d":"Linear-smoother bias bound for the DR-Learner CATE estimator. Fix a candidate nuisance sequence η_hat, a linear-smoother operator op, a sample index n, a realization ω, an evaluation point x, a data enumeration xs over an index set B with weights w, and constants c_n, p, q. Under two-sided strict overlap for the truth, if the estimated nuisance η_hat n ω has propensity uniformly bounded in [ε, 1 − ε], op realizes the linear smoother Σ_i w_i · f(xs i) at (n, ω, x) over B, the weights satisfy the absolute-value envelope Σ |w_i| ≤ c_n, and p and q are Hölder-conjugate exponents, then the smoothed conditional-bias evaluation op.evalAt n ω (condBias η_hat η₀ ∘ proj₁) x is bounded in absolute value by aipw_rem_const ε · c_n times the weighted-p-norm of the propensity error Δπ times the sum over treatment arms of the weighted-q-norm of the outcome-regression error Δμ_a.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → ∀ (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (n : ℕ) (ω : P.Ω) (x : γ) {ι : Type u_2} (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n p q : ℝ), η_hat n ω ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → p.HolderConjugate q → |op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x| ≤ Causalean.Estimation.ATE.BackdoorEstimationSystem.aipw_rem_const ε * c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => (η_hat n ω).e_fn (xs i).1 - S.e_val (xs i).1) p * ∑ a, Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => (η_hat n ω).μ_fn a (xs i).1 - S.μ_val a (xs i).1) q","l":46,"f":"Causalean/Estimation/CATE/Kennedy/LinearSmootherCATE.lean"},{"n":"Causalean.Estimation.CATE.condBias","k":"def","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"For a measurable covariate space, a candidate pair of outcome-regression and propensity-score nuisance functions, a reference pair of such nuisance functions, and a covariate value, the closed-form conditional bias of the doubly robust pseudo-outcome contrast is the sum over the two treatment arms of the product of the candidate-minus-reference propensity error and the corresponding outcome-regression error, divided by the candidate probability of that arm at the covariate value.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Causalean.Estimation.ATE.NuisanceVec γ → γ → ℝ","l":65,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.condBias_zero_of_outcome_match","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"If both outcome arms match the truth at x, then condBias η η₀ x = 0.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ) (x : γ), (∀ (a : Bool), η.μ_fn a x = η₀.μ_fn a x) → Causalean.Estimation.CATE.condBias η η₀ x = 0","l":742,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.condBias_zero_of_propensity_match","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"If the propensity matches the truth at x, then condBias η η₀ x = 0.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ) (x : γ), η.e_fn x = η₀.e_fn x → Causalean.Estimation.CATE.condBias η η₀ x = 0","l":728,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.cond_exp_phi_eta_dir_deriv_at_truth_zero","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"The σ(X)-conditional expectation of the directional derivative phi_eta_dir_deriv at the truth g₀ = η₀ is zero a.e.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (v : Causalean.Estimation.ATE.NuisanceVec γ), (∃ Cμ, ∀ (b : Bool) (x : γ), |v.μ_fn b x| ≤ Cμ) → (∃ Ce, ∀ (x : γ), |v.e_fn x| ≤ Ce) → P.μ[fun ω => Causalean.Estimation.CATE.phi_eta_dir_deriv S.η₀ v (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun x => 0","l":760,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.drLearnerEstimator","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"For a population outcome system and a covariate space, given a CATE estimation system, a second-stage regression operator, a sequence of estimated nuisance vectors indexed by sample size and sample realization, a sample size, a sample realization, and a covariate query point, the DR-Learner CATE estimator is that operator, at the stated sample size, realization, and query point, applied to the uncentered augmented inverse-probability-weighted pseudo-outcome formed with the corresponding estimated nuisance vector.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → (ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) → ℕ → P.Ω → γ → ℝ","l":47,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleEstimator","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"For a population outcome system and a covariate space, given a CATE estimation system, a second-stage regression operator, a sample size, a sample realization, and a covariate query point, the oracle DR-Learner estimator is the operator at that sample size, realization, and query point applied to the true augmented inverse-probability-weighted pseudo-outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → ℕ → P.Ω → γ → ℝ","l":66,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleEstimator_eq_oracleEstimator","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"For a CATE estimation system and a second-stage operator, the oracle DR-learner estimator built from S and op equals the abstract oracle estimator of op applied to the true pseudo-outcome φ_0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ), Causalean.Estimation.CATE.drOracleEstimator S op = op.oracleEstimator fun z => Causalean.Estimation.CATE.phi₀ S z","l":99,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.drOracleRiskScale","k":"def","m":"Causalean.Estimation.CATE.Kennedy.DRLearner","a":"Estimation/CATE/Kennedy","t":1,"d":"For a population outcome system and a covariate space, given a CATE estimation system, a second-stage regression operator, a covariate query point, and a sample size, the oracle pointwise risk scale is the square root of the population expectation of the squared difference between the oracle DR-Learner at that point and the system's conditional average treatment-effect target.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ → γ → ℕ → ℝ","l":81,"f":"Causalean/Estimation/CATE/Kennedy/DRLearner.lean"},{"n":"Causalean.Estimation.CATE.dr_oracle_efficient","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.OracleExpansion","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle efficiency for the DR-Learner CATE estimator, at a query point x — corollary of dr_oracle_expansion. Under the back-door causal assumptions and the same operator-stability, consistency, and bias-identification hypotheses as dr_oracle_expansion — op is stable at x w.r.t. a distance d_n, via the bias-identification predicate BiasIdent, d_n converges to zero in probability, and the AIPW pseudo-outcome contrast is identified with the closed-form conditional bias condBias(η_hat, η₀) — if in addition that smoothed conditional-bias term, the operator applied to condBias(η_hat, η₀) at x, is itself o_p(R*_n(x)), then the DR-Learner CATE estimator is oracle-efficient at x: it differs from its oracle counterpart by o_p(R*_n(x)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op n ω x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ","l":98,"f":"Causalean/Estimation/CATE/Kennedy/OracleExpansion.lean"},{"n":"Causalean.Estimation.CATE.dr_oracle_expansion","k":"theorem","m":"Causalean.Estimation.CATE.Kennedy.OracleExpansion","a":"Estimation/CATE/Kennedy","t":1,"d":"Oracle expansion for the DR-Learner CATE estimator. Fix a CATE estimation system under the back-door causal assumptions, a query point x, and a sequence of estimated nuisance vectors η_hat. Suppose the abstract second-stage regression operator op is stable at x with respect to a distance d_n, meaning the caller-supplied bias-identification predicate BiasIdent correctly separates the operator-level discrepancy between the estimated-nuisance and true pseudo-outcomes into a bias term plus a negligible remainder; d_n converges to zero in probability; and the contrast between the AIPW pseudo-outcome built from η_hat and the true pseudo-outcome is identified, through BiasIdent, with the closed-form conditional bias condBias(η_hat, η₀). Then the DR-Learner CATE estimator, minus its oracle counterpart, minus the operator applied to that closed-form bias, equals o_p(R*_n(x)), where R*_n(x) is the oracle risk scale at x.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator P.Ω P.μ γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ) (x : γ) (d_n : ℕ → P.Ω → ℝ) (BiasIdent : (ℕ → P.Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → P.Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op S.τ_val d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) P.μ → (BiasIdent (fun n ω z => Causalean.Estimation.CATE.phi_eta z (η_hat n ω)) (fun z => Causalean.Estimation.CATE.phi₀ S z) fun n ω u => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ u) → Causalean.Stat.IsLittleOp (fun n ω => Causalean.Estimation.CATE.drLearnerEstimator S op η_hat n ω x - Causalean.Estimation.CATE.drOracleEstimator S op n ω x - op.evalAt n ω (fun z => Causalean.Estimation.CATE.condBias (η_hat n ω) S.η₀ z.1) x) (fun n => Causalean.Estimation.CATE.drOracleRiskScale S op x n) P.μ","l":49,"f":"Causalean/Estimation/CATE/Kennedy/OracleExpansion.lean"},{"n":"Causalean.Estimation.CATE.integral_phiDiff_mul_eq_condBias","k":"theorem","m":"Causalean.Estimation.CATE.Core.SecondOrderBias","a":"Estimation/CATE/Core","t":2,"d":"Conditioning identity for the weighted pseudo-outcome bias.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → ∀ (w : γ → ℝ), Measurable w → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * w (S.factualX ω)) P.μ → ∫ (z : γ × Bool × ℝ), (Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z) * w z.1 ∂S.P_Z = ∫ (x : γ), Causalean.Estimation.CATE.condBias η S.η₀ x * w x ∂S.P_X","l":66,"f":"Causalean/Estimation/CATE/Core/SecondOrderBias.lean"},{"n":"Causalean.Estimation.CATE.measurable_condBias","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":2,"d":"condBias η η₀ is measurable in x.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η η₀ : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun x => Causalean.Estimation.CATE.condBias η η₀ x","l":83,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi_eta","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":2,"d":"The uncentered AIPW pseudo-outcome is measurable in the data argument.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun z => Causalean.Estimation.CATE.phi_eta z η","l":72,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi_eta_dir_deriv","k":"theorem","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":2,"d":"phi_eta_dir_deriv g₀ v is measurable in z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (g₀ v : Causalean.Estimation.ATE.NuisanceVec γ), Measurable fun z => Causalean.Estimation.CATE.phi_eta_dir_deriv g₀ v z","l":97,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.measurable_phi₀","k":"theorem","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"For a CATE estimation system, the true doubly-robust pseudo-outcome φ_0 is measurable as a function of the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), Measurable fun z => Causalean.Estimation.CATE.phi₀ S z","l":102,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.phi_eta","k":"def","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"For a covariate space, given an observed covariate, binary treatment, and outcome triple and a nuisance vector consisting of outcome regressions and a treatment propensity, the uncentered augmented inverse-probability-weighted pseudo-outcome is the difference between the two outcome regressions plus the treated and control inverse-propensity-weighted residual corrections.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → γ × Bool × ℝ → Causalean.Estimation.ATE.NuisanceVec γ → ℝ","l":44,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_dir_deriv","k":"def","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":1,"d":"For a covariate space, given a baseline nuisance vector, a nuisance-vector direction, and an observed covariate, binary treatment, and outcome triple, the directional derivative of the uncentered augmented inverse-probability-weighted pseudo-outcome is the displayed closed-form first-order change at the baseline nuisance vector in the stated direction.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Causalean.Estimation.ATE.NuisanceVec γ → γ × Bool × ℝ → ℝ","l":76,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_dir_deriv_tendsto","k":"theorem","m":"Causalean.Estimation.CATE.Core.PhiEtaDeriv","a":"Estimation/CATE/Core","t":1,"d":"Pointwise convergence of the DR pseudo-outcome difference quotient. Fix an anchor nuisance pair g₀, a perturbation direction v, and a data point z. If ε is strictly positive and the anchor g₀ has propensity uniformly bounded in [ε, 1 − ε] for every covariate value, then as the step size t tends to 0 along nonzero values, the difference quotient (phi_eta z (g₀ + t•v) − phi_eta z g₀) / t converges to the closed-form directional derivative phi_eta_dir_deriv g₀ v z.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (g₀ v : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, 0 < ε → g₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (z : γ × Bool × ℝ), Filter.Tendsto (fun t => (Causalean.Estimation.CATE.phi_eta z (g₀ + t • v) - Causalean.Estimation.CATE.phi_eta z g₀) / t) (nhdsWithin 0 {0}ᶜ) (nhds (Causalean.Estimation.CATE.phi_eta_dir_deriv g₀ v z))","l":219,"f":"Causalean/Estimation/CATE/Core/PhiEtaDeriv.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_minus_phi₀_at_x","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"Value-space form of the DR pseudo-outcome bias identity. Fix a candidate nuisance pair η. Under the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], ε is strictly positive, each candidate outcome-regression arm, composed with the covariate, is integrable, and the DR pseudo-outcome contrast φ_η − φ_0 is integrable, then for covariate-law-almost-every x, the mean of φ_η − φ_0 under the regular conditional distribution of the data triple given the covariate value x equals the closed-form cross-product remainder condBias η η₀ x.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → 0 < ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → ∀ᵐ (x : γ) ∂S.P_X, ∫ (z : γ × Bool × ℝ), Causalean.Estimation.CATE.phi_eta z η - Causalean.Estimation.CATE.phi₀ S z ∂(ProbabilityTheory.condDistrib S.factualZ S.factualX P.μ) x = Causalean.Estimation.CATE.condBias η S.η₀ x","l":618,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.phi_eta_minus_phi₀_cond_exp","k":"theorem","m":"Causalean.Estimation.CATE.Core.ConditionalBias","a":"Estimation/CATE/Core","t":1,"d":"σ(X)-conditional bias identity for the DR pseudo-outcome (Kennedy / DR-Learner Proposition 2). Fix a candidate nuisance pair η. Under the back-door identification assumptions, if η's propensity is uniformly bounded in [ε, 1 − ε] for every covariate value, the truth nuisance likewise has propensity uniformly bounded in [ε, 1 − ε], ε is strictly positive, and each candidate outcome-regression arm, composed with the covariate, is integrable, then the σ(X)-conditional expectation of the DR pseudo-outcome contrast φ_η − φ_0 equals the closed-form cross-product remainder condBias η η₀, evaluated at the covariate, almost surely.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {P : Causalean.PO.POSystem} [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ) {ε : ℝ}, η ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → 0 < ε → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => η.μ_fn a (S.factualX ω)) P.μ) → P.μ[fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) η - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω) | S.sigmaX] =ᵐ[P.μ] fun ω => Causalean.Estimation.CATE.condBias η S.η₀ (S.factualX ω)","l":219,"f":"Causalean/Estimation/CATE/Core/ConditionalBias.lean"},{"n":"Causalean.Estimation.CATE.phi₀","k":"def","m":"Causalean.Estimation.CATE.Core.PseudoOutcome","a":"Estimation/CATE/Core","t":1,"d":"For a covariate space and a population outcome system, given a CATE estimation system and an observed covariate, binary treatment, and outcome triple, the true doubly robust pseudo-outcome is the uncentered augmented inverse-probability-weighted pseudo-outcome evaluated with that system's true nuisance vector.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → {P : Causalean.PO.POSystem} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → γ × Bool × ℝ → ℝ","l":61,"f":"Causalean/Estimation/CATE/Core/PseudoOutcome.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem","k":"structure","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"This structure extends a two-stage potential-outcome dynamic-treatment-regime system with, at a fixed target regime, measurable value-space representatives of the stage-0 and stage-1 outcome regressions and of the propensities at both stages, bounded away from zero and one, each required to agree almost surely with the corresponding observable conditional regression or propensity built from the factual treatment and covariate history.","s":"(P : Causalean.PO.POSystem) → (δ : Type) → (γ : Fin 2 → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin 2) → MeasurableSpace (γ k)] → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max u_1 u_2)","l":62,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.H_ε","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, given a real number, the overlap-bounded nuisance set consists exactly of stagewise nuisance vectors for which, at every first-period state and every second-period history, each respective treatment propensity lies between ε and $1-ε$, inclusively.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → ℝ → Set (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":400,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR","k":"def","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"Given a real overlap level, the sequential doubly robust Lipschitz constant is $1 + 2/ε + 4/ε^2 + 4/ε^3$.","s":"ℝ → ℝ","l":51,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_inv_cubed_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the cubed inverse-overlap bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε ^ 3 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":102,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_inv_sq_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the squared inverse-overlap bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε ^ 2 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":95,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_mu0_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the stage-zero regression coefficient bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 + 1 / ε ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":81,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_mu1_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates the stage-one regression coefficient bound.","s":"∀ {ε : ℝ}, 0 < ε → 1 / ε + 1 / ε ^ 2 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":88,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR_one_le","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The sequential doubly robust Lipschitz constant dominates one under positive overlap.","s":"∀ {ε : ℝ}, 0 < ε → 1 ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε","l":74,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_H₀","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the marginal law of the first-period state is the population distribution induced by that system's factual first-period state.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 0)","l":465,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_H₁","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the marginal law of the second-period history is the population distribution of the second-period state, first-period treatment, and first-period state, in that order.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 1 × δ × γ 0)","l":470,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the joint law of the factual two-stage data tuple is the population distribution induced by its factual two-stage data map.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → MeasureTheory.Measure (γ 0 × δ × γ 1 × δ × ℝ)","l":502,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The joint two-stage data law is the image of the population measure under the map recording the full observed two-stage data tuple.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.P_Z = MeasureTheory.Measure.map S.factualZ P.μ","l":509,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z_map_histH₁_eq_P_H₁","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a dynamic-treatment-regime estimation system, pushing the full observed-data law forward through the stage-1 history projection yields exactly the stage-1 history marginal law.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.Measure.map (fun z => Causalean.Estimation.DTR.histH₁ z) S.P_Z = S.P_H₁","l":502,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.P_Z_map_projS₀_eq_P_H₀","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a dynamic-treatment-regime estimation system, pushing the full observed-data law forward through the initial-state projection yields exactly the stage-0 history marginal law.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.Measure.map (fun z => Causalean.Estimation.DTR.projS₀ z) S.P_Z = S.P_H₀","l":494,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.StrictOverlap","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system and a real number, the strict-overlap condition holds exactly when $0<ε≤1/2$ and, almost surely under the population measure, both conditional probabilities of the system's target treatment at their respective stages lie between ε and $1-ε$, inclusively.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → ℝ → Prop","l":389,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.cond_exp_residual_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-0 residual conditional expectation is zero a.s.: under DTR assumptions, μ[1{D₀=dbar 0}·(μ₁_val(history₁) − μ₀_val(S₀)) | σ(historyBundle 0)] =ᵐ 0.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ((S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) =ᵐ[P.μ] fun x => 0","l":92,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.cond_exp_residual_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-1 residual conditional expectation is zero a.s.: under DTR assumptions, the σ(historyBundle 1)-conditional expectation of 1{D₀ = dbar 0} · 1{D₁ = dbar 1} · (factualY − μ₁_val(S₁,D₀,S₀)) is zero a.s.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ((S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) =ᵐ[P.μ] fun x => 0","l":316,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eLpNorm_comp_histH₁_eq","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Pulling a stage-one history function back along the full DTR data law preserves its L² norm.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {f : γ 1 × δ × γ 0 → ℝ}, MeasureTheory.AEStronglyMeasurable f S.P_H₁ → MeasureTheory.eLpNorm (fun z => f (Causalean.Estimation.DTR.histH₁ z)) 2 S.P_Z = MeasureTheory.eLpNorm f 2 S.P_H₁","l":124,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eLpNorm_comp_projS₀_eq","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Pulling a stage-zero function back along the full DTR data law preserves its L² norm.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {f : γ 0 → ℝ}, MeasureTheory.AEStronglyMeasurable f S.P_H₀ → MeasureTheory.eLpNorm (fun z => f (Causalean.Estimation.DTR.projS₀ z)) 2 S.P_Z = MeasureTheory.eLpNorm f 2 S.P_H₀","l":109,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eta_e0_pos_of_mem_Hε","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"Any nuisance vector in the overlap-bounded set has a positive stage-zero propensity.","s":"∀ {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ} {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε → ∀ (s₀ : γ 0), 0 < η.e₀_fn s₀","l":97,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.eta_e1_pos_of_mem_Hε","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"Any nuisance vector in the overlap-bounded set has a positive stage-one propensity.","s":"∀ {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ} {ε : ℝ}, 0 < ε → η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε → ∀ (h : γ 1 × δ × γ 0), 0 < η.e₁_fn h","l":105,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.factualZ","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the factual two-stage data map sends each sample realization to its factual first-period state, first-period treatment, second-period state, second-period treatment, and outcome, in that order.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → P.Ω → γ 0 × δ × γ 1 × δ × ℝ","l":480,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indD_mul_μ₁_val_comp_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"Same as μ₁_val_comp_mul_indD_eq, with the partial regime indicator written on the left.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.indD S.dbar 1 ω * S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":371,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_factualD0_eq_indicator","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero equality indicator agrees with the stage-zero treatment indicator for the target regime.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (ω : P.Ω), Causalean.Estimation.DTR.indEq (S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) = (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":41,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_factualD1_eq_indicator","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-one equality indicator agrees with the stage-one treatment indicator for the target regime.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (ω : P.Ω), Causalean.Estimation.DTR.indEq (S.factualD ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω) (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) = (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":69,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_le_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The real-valued equality indicator is always bounded above by one.","s":"∀ {δ : Type} (d d' : δ), Causalean.Estimation.DTR.indEq d d' ≤ 1","l":69,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indEq_nonneg","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The real-valued equality indicator is always nonnegative.","s":"∀ {δ : Type} (d d' : δ), 0 ≤ Causalean.Estimation.DTR.indEq d d'","l":63,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_to_propScore_integral_stage0","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-0 indicator-to-propensity rewrite. Let S be a two-stage dynamic-treatment-regime estimation system and let f be a real-valued function of the stage-0 history that is measurable and for which the product f(S₀) · 1{D₀ = dbar 0} is integrable. Then the expectation of f(S₀) times the indicator of following the target stage-0 treatment dbar 0 equals the expectation of f(S₀) times the true stage-0 propensity score e₀_val(S₀).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f : γ 0 → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω) P.μ → ∫ (ω : P.Ω), f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.e₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":189,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_to_propScore_integral_stage1","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-1 indicator-to-propensity rewrite. Let S be a two-stage dynamic-treatment-regime estimation system and let f be a real-valued function of the stage-1 history (S₁,D₀,S₀) that is measurable and for which the product f(S₁,D₀,S₀) · 1{D₁ = dbar 1} is integrable. Then the expectation of f(S₁,D₀,S₀) times the indicator of following the target stage-1 treatment dbar 1 equals the expectation of f(S₁,D₀,S₀) times the true stage-1 propensity score e₁_val(S₁,D₀,S₀).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f : γ 1 × δ × γ 0 → ℝ), Measurable f → MeasureTheory.Integrable (fun ω => f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω) P.μ → ∫ (ω : P.Ω), f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω ∂P.μ = ∫ (ω : P.Ω), f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.e₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":406,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu0_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero indicator-weighted stage-zero outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) * (η.μ₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":113,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu1_stage0_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The stage-zero indicator-weighted stage-one outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) * (η.μ₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":174,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.indicator_weighted_delta_mu1_stage1_integrable","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":2,"d":"The double-indicator-weighted stage-one outcome-regression error is integrable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, 0 < ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.Integrable (fun ω => (S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (1 / (η.e₀_fn (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * η.e₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) * (η.μ₁_fn (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) P.μ","l":246,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The full observed two-stage data tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable S.factualZ","l":492,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_histH₁","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The cons-ordered stage-1 history projection is measurable.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] {γ : Fin 2 → Type} [inst_1 : (k : Fin 2) → MeasurableSpace (γ k)], Measurable fun z => Causalean.Estimation.DTR.histH₁ z","l":488,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_projS₀","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The initial-state projection is measurable.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] {γ : Fin 2 → Type} [inst_1 : (k : Fin 2) → MeasurableSpace (γ k)], Measurable fun z => Causalean.Estimation.DTR.projS₀ z","l":481,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_seqDRMomentFunctional","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"The sequential doubly robust moment functional is measurable as a function of the observed data tuple.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {P : Causalean.PO.POSystem} [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (η : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ) (θ : ℝ), Measurable fun z => S.seqDRMomentFunctional η z θ","l":418,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_ψ_seqDR","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Measurability of the sequential DR influence function on the data tuple (s₀, d₀, s₁, d₁, y) : γ 0 × δ × γ 1 × δ × ℝ. Decomposes into Measurable.add/Measurable.mul/Measurable.div chained against the projections, the indicator functions indEq, and the value-space nuisance functions stored in S.η₀.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable S.ψ_seqDR","l":49,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.measurable_ψ_seqDR_squared","k":"theorem","m":"Causalean.Estimation.DTR.FiniteVar","a":"Estimation/DTR","t":2,"d":"Measurability helper: the squared sequential DR influence function on the data tuple is measurable. Used in the L² bookkeeping for seqDR_finite_var.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), Measurable fun z => S.ψ_seqDR z ^ 2","l":42,"f":"Causalean/Estimation/DTR/FiniteVar.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.propScore_ne_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-0 propensity is a.e. nonzero under the DTR backdoor assumptions. The conditional indicator μ[1{D₀ = dbar 0} | σ(historyBundle 0)] is identified via e₀_compat with e₀_val ∘ factualS 0, and e₀_val > 0 pointwise on γ 0.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ≠ 0","l":61,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.propScore_ne_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"Stage-1 propensity is a.e. nonzero under the DTR backdoor assumptions. Analogous to propScore_ne_zero_stage0 via e₁_compat and e₁_pos.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ≠ 0","l":74,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.residual_mul_error_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"Multiplying an L²-convergent error by a fixed square-integrable residual still gives a stochastic little-o L² norm.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {ν : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure ν] {R : α → ℝ}, Measurable R → (∀ (z : α), 0 ≤ R z) → MeasureTheory.MemLp R 2 ν → ∀ {deZ : ℕ → P.Ω → α → ℝ}, (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (deZ n ω) 2 ν) → (∀ (n : ℕ) (ω : P.Ω), Measurable (deZ n ω)) → (∀ (n : ℕ) (ω : P.Ω) (z : α), |deZ n ω z| ≤ 1) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (deZ n ω) 2 ν).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => R z * |deZ n ω z|) 2 ν).toReal) (fun x => 1) P.μ","l":142,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDRMoment","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, an observed two-stage data tuple, a stagewise nuisance vector, and a candidate regime mean, the system-specific sequential doubly robust moment is the sequential doubly robust moment for the treatment regime selected by the system.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → γ 0 × δ × γ 1 × δ × ℝ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → ℝ → ℝ","l":381,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDRMomentFunctional","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the sequential doubly robust moment functional maps every stagewise nuisance vector, observed two-stage data tuple, and candidate regime mean to the sequential doubly robust moment for the system's target regime.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → γ 0 × δ × γ 1 × δ × ℝ → ℝ → ℝ","l":409,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_finite_var","k":"theorem","m":"Causalean.Estimation.DTR.FiniteVar","a":"Estimation/DTR","t":1,"d":"Finite variance of the sequential doubly robust score — sequential DR (DTR) analogue of aipw_finite_var. Under the two-stage DTR backdoor assumptions — sequential exchangeability, consistency, stagewise positivity, and integrability of every counterfactual outcome, uniform two-stage strict overlap: the target-regime propensity at each stage lies almost surely in [ε, 1-ε] for some ε in (0, 1/2], a finite second moment for the observed factual outcome, and a finite second moment for every counterfactual outcome under a treatment sequence, the sequential doubly robust influence function ψ_seqDR is square-integrable against the joint law of the observed two-stage data tuple.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → MeasureTheory.Integrable (fun z => S.ψ_seqDR z ^ 2) S.P_Z","l":56,"f":"Causalean/Estimation/DTR/FiniteVar.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_mean_zero","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":1,"d":"Mean zero of the sequential doubly robust score. Under the two-stage DTR backdoor assumptions — sequential exchangeability, consistency, stagewise positivity, and integrability of every counterfactual outcome, uniform two-stage strict overlap: the target-regime propensity at each stage lies almost surely in [ε, 1-ε] for some ε in (0, 1/2], and a finite second moment for the observed factual outcome, then the sequential doubly robust influence function ψ_seqDR has expectation zero under the observed two-stage data law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.ψ_seqDR z ∂S.P_Z = 0","l":888,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_real_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"Pointwise Lipschitz bound for the two-stage sequential DR score (real-valued form). Consider a stage-0 propensity e0 and its estimate ê0, a stage-1 propensity e1 and its estimate ê1, true and estimated stage-0/stage-1 outcome-regression values μ0, μ0h, μ1, μ1h, an outcome value y, a target parameter θ, and treatment indicators I0, I1, where ε is strictly positive, both the true and estimated stage-0 propensities are at least ε, both the true and estimated stage-1 propensities are at least ε, and both treatment indicators have absolute value at most 1. Then the absolute difference between the sequential doubly-robust moment built from the estimated nuisances (ê0, ê1, μ0h, μ1h) and from the true nuisances (e0, e1, μ0, μ1) is bounded by the Lipschitz constant K_seqDR ε times the sum of the stage-0 regression error, the stage-1 regression error, and cross terms in which the stage-0 and stage-1 propensity errors are weighted by outcome/regression residuals.","s":"∀ {ε e0 ê0 e1 ê1 μ0 μ0h μ1 μ1h y θ I0 I1 : ℝ}, 0 < ε → ε ≤ e0 → ε ≤ ê0 → ε ≤ e1 → ε ≤ ê1 → |I0| ≤ 1 → |I1| ≤ 1 → |μ0h + I0 / ê0 * (μ1h - μ0h) + I0 * I1 / (ê0 * ê1) * (y - μ1h) - θ - (μ0 + I0 / e0 * (μ1 - μ0) + I0 * I1 / (e0 * e1) * (y - μ1) - θ)| ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε * (|μ0h - μ0| + |μ1h - μ1| + (|μ1 - μ0| + |y - μ1|) * |ê0 - e0| + |y - μ1| * |ê1 - e1|)","l":415,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_rem_const","k":"def","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"For a real overlap margin, the sequential doubly robust remainder constant is $2/[\\varepsilon^2(1-\\varepsilon)]$.","s":"ℝ → ℝ","l":71,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_bound","k":"theorem","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"Sequential DR (DTR, n = 2) remainder bound. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions, for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. For any candidate nuisance vector η whose propensities likewise lie in this strict-overlap band, and whose stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error are each square-integrable against the corresponding stage's history law, the absolute value of the population sequential doubly robust moment at η and the true target θ₀ is at most an explicit O(ε⁻²) constant times the sum of the two stagewise outcome-regression L² errors, times the sum of the two stagewise propensity L² errors.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁ → |∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ Causalean.Estimation.DTR.DTREstimationSystem.seqDR_rem_const ε * ((MeasureTheory.eLpNorm (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal + (MeasureTheory.eLpNorm (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal) * ((MeasureTheory.eLpNorm (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal + (MeasureTheory.eLpNorm (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal)","l":81,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_identity","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity","a":"Estimation/DTR/RemainderIdentity","t":1,"d":"Sequential DR (DTR, n = 2) remainder identity. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions (consistency and sequential ignorability), for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. For any candidate nuisance vector η whose stage-0 and stage-1 propensities likewise lie in this strict-overlap band, and whose stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error are each square-integrable against the corresponding stage's history law, the population sequential doubly robust moment at η and the true target θ₀ equals the sum of two stagewise cross-product integrals — propensity error times inverse-propensity weight times outcome-regression error, at stage 0 against the stage-0 history law and at stage 1 against the stage-1 history law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ → MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ → MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional η z S.θ₀ ∂S.P_Z = ∫ (s₀ : γ 0), (η.e₀_fn s₀ - S.e₀_val s₀) * (1 / η.e₀_fn s₀) * (η.μ₀_fn s₀ - S.μ₀_val s₀) ∂S.P_H₀ + ∫ (h : γ 1 × δ × γ 0), Causalean.Estimation.DTR.indEq h.2.1 (S.dbar 0) * (η.e₁_fn h - S.e₁_val h) * (1 / (η.e₀_fn h.2.2 * η.e₁_fn h)) * (η.μ₁_fn h - S.μ₁_val h) ∂S.P_H₁","l":96,"f":"Causalean/Estimation/DTR/RemainderIdentity.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_remainder_op","k":"theorem","m":"Causalean.Estimation.DTR.RemainderBound","a":"Estimation/DTR","t":1,"d":"Sequential DR remainder is o_p(n^{-1/2}) under the two-stage product rate. Consider a two-stage dynamic treatment-regime estimation system satisfying the sequential causal assumptions, for which the stage-0 and stage-1 propensity scores are bounded within a margin ε of 0 and 1 (strict overlap), and where the factual outcome and the potential outcome under every fixed treatment regime each have finite second moment. Let η̂ₙ be a sequence of sample-size-indexed, possibly random, candidate nuisance vectors that always land in the strict-overlap band, for every sample size and every outcome of the underlying randomness, with stage-0 outcome-regression error, stage-1 outcome-regression error, stage-0 propensity error, and stage-1 propensity error each square-integrable against the corresponding stage's history law at every sample size and outcome. If the four stagewise L² products of outcome-regression and propensity error — own-stage at stage 0, own-stage at stage 1, stage-0 outcome-regression with stage-1 propensity, and stage-1 outcome-regression with stage-0 propensity — are each o_p(n^{-1/2}), then the population sequential doubly robust moment evaluated at the random nuisance η̂ₙ is itself o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ ∂S.P_Z) (fun n => ↑n ^ (-(1 / 2))) P.μ","l":310,"f":"Causalean/Estimation/DTR/RemainderBound.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2","a":"Estimation/DTR/ScoreL2","t":1,"d":"Sequential DR score L²(P_Z) continuity. Consider a two-stage dynamic-treatment-regime estimation system in which the estimated propensity scores stay strictly between ε and 1-ε, the system's identification assumptions (consistency, sequential exchangeability, positivity) hold, the observed outcome has finite second moment, and every potential outcome under a fixed two-stage treatment regime has finite second moment. Given a sequence of nuisance-estimator draws η̂(n) that always land inside the ε-overlap-bounded nuisance set, whose stage-0 outcome-regression, stage-1 outcome-regression, stage-0 propensity-score, and stage-1 propensity-score estimation errors are each square-integrable against the corresponding history law, for every draw and every sample point, and whose four stagewise L² estimation-error rates each vanish in probability (are $o_p(1)$), then the L²(P_Z) distance between the sequential doubly-robust score evaluated at η̂(n) and at the true nuisance η₀ is itself $o_p(1)$.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ - S.seqDRMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":72,"f":"Causalean/Estimation/DTR/ScoreL2.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_score_diff_pointwise_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":1,"d":"Pointwise Lipschitz bound for the sequential DR score difference. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε (the true propensity scores lie in [ε, 1-ε]), and let η be a candidate nuisance vector whose estimated propensity components are likewise confined to [ε, 1-ε]. Then, for almost every observation z under the observed-data law S.P_Z, the absolute difference between the sequential doubly-robust moment evaluated at η and at the true nuisance vector η₀ is bounded by the Lipschitz constant K_seqDR ε times the sum of the stage-0 regression error, the stage-1 regression error, and cross terms in which the stage-0 and stage-1 propensity-score errors are weighted by the corresponding outcome/regression residuals.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → ∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, ∀ᵐ (z : γ 0 × δ × γ 1 × δ × ℝ) ∂S.P_Z, |S.seqDRMomentFunctional η z S.θ₀ - S.seqDRMomentFunctional S.η₀ z S.θ₀| ≤ Causalean.Estimation.DTR.DTREstimationSystem.K_seqDR ε * (|η.μ₀_fn (Causalean.Estimation.DTR.projS₀ z) - S.μ₀_val (Causalean.Estimation.DTR.projS₀ z)| + |η.μ₁_fn (Causalean.Estimation.DTR.histH₁ z) - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)| + (|S.μ₁_val (Causalean.Estimation.DTR.histH₁ z) - S.μ₀_val (Causalean.Estimation.DTR.projS₀ z)| + |Causalean.Estimation.DTR.projY z - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)|) * |η.e₀_fn (Causalean.Estimation.DTR.projS₀ z) - S.e₀_val (Causalean.Estimation.DTR.projS₀ z)| + |Causalean.Estimation.DTR.projY z - S.μ₁_val (Causalean.Estimation.DTR.histH₁ z)| * |η.e₁_fn (Causalean.Estimation.DTR.histH₁ z) - S.e₁_val (Causalean.Estimation.DTR.histH₁ z)|)","l":607,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_stage0_ratio_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The stage-zero weighted regression contrast is Lipschitz in the stage-zero nuisance errors under overlap.","s":"∀ {ε e ê μ0 μ0h μ1 μ1h : ℝ}, 0 < ε → ε ≤ e → ε ≤ ê → |(μ1h - μ0h) / ê - (μ1 - μ0) / e| ≤ (|μ1h - μ1| + |μ0h - μ0|) / ε + |μ1 - μ0| * |ê - e| / ε ^ 2","l":288,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.seqDR_stage1_ratio_bound","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The stage-one weighted residual contrast is Lipschitz in the stage-one regression error and both propensity errors under overlap.","s":"∀ {ε e0 ê0 e1 ê1 y μ1 μ1h : ℝ}, 0 < ε → ε ≤ e0 → ε ≤ ê0 → ε ≤ e1 → ε ≤ ê1 → |(y - μ1h) / (ê0 * ê1) - (y - μ1) / (e0 * e1)| ≤ |μ1h - μ1| / ε ^ 2 + |y - μ1| * |ê0 - e0| / ε ^ 3 + |y - μ1| * |ê1 - e1| / ε ^ 3","l":335,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.split_stage_history_integral","k":"theorem","m":"Causalean.Estimation.DTR.RemainderIdentity.Helpers","a":"Estimation/DTR/RemainderIdentity","t":1,"d":"Additivity of stage-history integrals under the joint DTR data law. Consider a real-valued function f₀ of the stage-0 state that is measurable and integrable against the stage-0 history marginal law and a real-valued function f₁ of the stage-1 history (current state, previous treatment, previous state) that is measurable and integrable against the stage-1 history marginal law. Then the integral, against the joint law of the observed two-stage data tuple, of the sum of f₀ and f₁ each pulled back through its respective projection out of the data tuple equals the sum of the separate integrals of f₀ against the stage-0 marginal law and f₁ against the stage-1 marginal law.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) (f₀ : γ 0 → ℝ) (f₁ : γ 1 × δ × γ 0 → ℝ), Measurable f₀ → Measurable f₁ → MeasureTheory.Integrable f₀ S.P_H₀ → MeasureTheory.Integrable f₁ S.P_H₁ → ∫ (z : γ 0 × δ × γ 1 × δ × ℝ), f₀ (Causalean.Estimation.DTR.projS₀ z) + f₁ (Causalean.Estimation.DTR.histH₁ z) ∂S.P_Z = ∫ (s₀ : γ 0), f₀ s₀ ∂S.P_H₀ + ∫ (h : γ 1 × δ × γ 0), f₁ h ∂S.P_H₁","l":349,"f":"Causalean/Estimation/DTR/RemainderIdentity/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the observable stage-1 regression assigns to each sample realization the conditional mean of the outcome multiplied by the indicator of the system's target second-period treatment, divided by the conditional mean of that indicator given the second-period history.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → P.Ω → ℝ","l":211,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg_indD_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"On the regime-consistent path, the observable stage-1 regression agrees with the counterfactual stage-1 regression.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.stageOneReg ω * S.indD S.dbar 1 ω) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":236,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stageOneReg_memLp","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The observable stage-1 regression is square-integrable under strict overlap and a factual second moment.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.MemLp S.stageOneReg 2 P.μ","l":410,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stronglyMeasurable_e₀_val","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-0 propensity carried by an estimation system is strongly measurable on the stage-0 state space.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.StronglyMeasurable S.e₀_val","l":177,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stronglyMeasurable_e₁_val","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-1 propensity carried by an estimation system is strongly measurable on the stage-1 history space.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.StronglyMeasurable S.e₁_val","l":191,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stronglyMeasurable_μ₀_val","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-0 outcome regression carried by an estimation system is strongly measurable on the stage-0 state space.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.StronglyMeasurable S.μ₀_val","l":170,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.stronglyMeasurable_μ₁_val","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-1 outcome regression carried by an estimation system is strongly measurable on the stage-1 history space.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), MeasureTheory.StronglyMeasurable S.μ₁_val","l":184,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.theta_zero_factualS₀_integral","k":"theorem","m":"Causalean.Estimation.DTR.MeanZero","a":"Estimation/DTR","t":2,"d":"The DTR estimand θ₀ = E[Y(dbar)] lifts to an integral against P.μ: under DTR backdoor assumptions, θ₀ = ∫ ω, μ₀_val(factualS 0 ω) ∂P.μ, since μ₀_val ∘ factualS 0 is the σ(historyBundle 0)-conditional expectation of Y_of dbar and P.μ is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → S.θ₀ = ∫ (ω : P.Ω), S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) ∂P.μ","l":628,"f":"Causalean/Estimation/DTR/MeanZero.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.weighted_residual_integral_zero_stage0","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-0 weighted-residual integral vanishes. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε, satisfying the system's core identification assumptions (consistency, sequential exchangeability, positivity), and in which the observed outcome has finite second moment. For any weight function g on the stage-0 history that is measurable and for which the product g(S₀) · 1{D₀ = dbar 0} · (μ₁_val(S₁,D₀,S₀) − μ₀_val(S₀)) is integrable, then its expectation under P.μ is zero: the stage-0-weighted, treatment-indicator-gated gap between the stage-1 and stage-0 regression functions has zero mean.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∀ (g : γ 0 → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) P.μ → ∫ (ω : P.Ω), g (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) - S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))) ∂P.μ = 0","l":62,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.weighted_residual_integral_zero_stage1","k":"theorem","m":"Causalean.Estimation.DTR.ScorePullout","a":"Estimation/DTR","t":1,"d":"Stage-1 weighted-residual integral vanishes. Let S be a two-stage dynamic-treatment-regime estimation system with strict overlap at level ε, satisfying the system's core identification assumptions (consistency, sequential exchangeability, positivity), and in which the observed outcome has finite second moment. For any weight function g on the stage-1 history (S₁,D₀,S₀) that is measurable and for which the product g(S₁,D₀,S₀) · 1{D₀ = dbar 0} · 1{D₁ = dbar 1} · (Y − μ₁_val(S₁,D₀,S₀)) is integrable, then its expectation under P.μ is zero: the doubly indicator-gated stage-1 outcome residual, weighted by g, has zero mean.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → ∀ (g : γ 1 × δ × γ 0 → ℝ), Measurable g → MeasureTheory.Integrable (fun ω => g (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω))))) P.μ → ∫ (ω : P.Ω), g (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω * ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (S.dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω * (S.factualY ω - S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)))) ∂P.μ = 0","l":260,"f":"Causalean/Estimation/DTR/ScorePullout.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.η₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the true stagewise nuisance vector consists of that system's two outcome regressions and two treatment propensities.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ","l":372,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.θ₀","k":"def","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system, the target regime mean outcome is the potential-outcome dynamic-treatment-regime effect for the treatment regime selected by that system.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → ℝ","l":518,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.θ₀_eq_dtrEffect","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":1,"d":"For a two-stage dynamic treatment regime estimation system, its value-space DTR estimand θ₀ equals the potential-outcome DTR effect evaluated at the chosen regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.θ₀ = S.dtrEffect S.dbar","l":524,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₀_compat","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-0 value-space regression equals the counterfactual stage-0 regression under identification.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) =ᵐ[P.μ] fun ω => S.μ₀_val (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":198,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₀_val_memLp","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The true stage-zero regression representative is square-integrable under the stage-zero history marginal.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → MeasureTheory.MemLp S.μ₀_val 2 S.P_H₀","l":560,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_comp_eq_stageOneReg","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"The stage-1 value-space regression agrees almost everywhere with the observable stage-1 regression.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), (fun ω => S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)) =ᵐ[P.μ] S.stageOneReg","l":224,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_comp_mul_indD_eq","k":"theorem","m":"Causalean.Estimation.DTR.Setup","a":"Estimation/DTR","t":2,"d":"Composed observable μ₁_val, multiplied by the partial regime indicator indD dbar 1, agrees with the corresponding counterfactual conditional expectation. This is the consumer-facing form of stageOneReg_indD_eq.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ), S.Assumptions → (fun ω => S.μ₁_val (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω) * S.indD S.dbar 1 ω) =ᵐ[P.μ] fun ω => S.indD S.dbar 1 ω * (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven (S.Y_of S.dbar) P.μ ω","l":353,"f":"Causalean/Estimation/DTR/Setup.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.μ₁_val_memLp","k":"theorem","m":"Causalean.Estimation.DTR.ScoreL2.Helpers","a":"Estimation/DTR/ScoreL2","t":2,"d":"The true stage-one regression representative is square-integrable under the stage-one history marginal.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.StrictOverlap ε → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → MeasureTheory.MemLp S.μ₁_val 2 S.P_H₁","l":583,"f":"Causalean/Estimation/DTR/ScoreL2/Helpers.lean"},{"n":"Causalean.Estimation.DTR.DTREstimationSystem.ψ_seqDR","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a population outcome system, a treatment space, and the pair of first- and second-period state spaces, given a two-stage dynamic treatment-regime estimation system and an observed two-stage data tuple, the sequential doubly robust influence function is the system-specific sequential doubly robust moment evaluated at the system's true nuisance vector and target regime mean.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → {P : Causalean.PO.POSystem} → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.DTR.DTREstimationSystem P δ γ → γ 0 × δ × γ 1 × δ × ℝ → ℝ","l":391,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂","k":"structure","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"A stagewise nuisance vector for two-period dynamic-treatment-regime estimation: the baseline outcome regression and treatment propensity at the first stage, the second-stage outcome regression and treatment propensity given the second-period history, together with the measurability of all four nuisance functions.","s":"(δ : Type) → (γ : Fin 2 → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin 2) → MeasurableSpace (γ k)] → Type","l":111,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.ext","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"Two nuisance vectors are equal when all four stagewise components agree pointwise.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η η' : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ}, (∀ (s : γ 0), η.μ₀_fn s = η'.μ₀_fn s) → (∀ (s : γ 0), η.e₀_fn s = η'.e₀_fn s) → (∀ (h : γ 1 × δ × γ 0), η.μ₁_fn h = η'.μ₁_fn h) → (∀ (h : γ 1 × δ × γ 0), η.e₁_fn h = η'.e₁_fn h) → η = η'","l":194,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.ext_iff","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] {γ : Fin 2 → Type} [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] {η η' : Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ}, η = η' ↔ (∀ (s : γ 0), η.μ₀_fn s = η'.μ₀_fn s) ∧ (∀ (s : γ 0), η.e₀_fn s = η'.e₀_fn s) ∧ (∀ (h : γ 1 × δ × γ 0), η.μ₁_fn h = η'.μ₁_fn h) ∧ ∀ (h : γ 1 × δ × γ 0), η.e₁_fn h = η'.e₁_fn h","l":195,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instAdd","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the addition operation on two-stage dynamic-treatment-regime nuisance vectors is performed componentwise.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Add (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":153,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instAddCommGroup","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the additive commutative group structure on two-stage dynamic-treatment-regime nuisance vectors uses the zero vector, componentwise addition, componentwise negation, componentwise subtraction, natural-number scalar multiplication, and integer scalar multiplication, and satisfies the natural-zero rule, the natural-successor rule, the integer-zero rule, the positive-integer-successor rule, the negative-integer-successor rule, subtraction as addition of an inverse, associativity, the left-zero law, the right-zero law, inverse cancellation, and commutativity.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → AddCommGroup (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":210,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instModuleReal","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the real vector-space structure on two-stage dynamic-treatment-regime nuisance vectors uses componentwise scalar multiplication and satisfies multiplication by one, compatibility of successive scalar multiplications, multiplication of zero vectors, distribution over vector addition, distribution over scalar addition, and multiplication by the zero scalar.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Module ℝ (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":260,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instNeg","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the negation operation on two-stage dynamic-treatment-regime nuisance vectors is performed componentwise.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Neg (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":165,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instSMulReal","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the real scalar-multiplication operation on two-stage dynamic-treatment-regime nuisance vectors is performed componentwise.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → SMul ℝ (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":184,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instSub","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the subtraction operation on two-stage dynamic-treatment-regime nuisance vectors is performed componentwise.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Sub (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":172,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.DTRNuisanceVec₂.instZero","k":"instance","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"For a measurable treatment-history space with measurable singletons and two measurable stage-specific covariate spaces, the zero operation on two-stage dynamic-treatment-regime nuisance vectors sets every stage-specific regression and propensity component to zero.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → Zero (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":147,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.dml_DTR_estimator","k":"def","m":"Causalean.Estimation.DTR.DML","a":"Estimation/DTR","t":1,"d":"For a potential-outcome system whose sample space is standard Borel and whose measure is finite with a measurable treatment space in which every singleton is measurable and measurable stage-specific covariate spaces, a two-stage dynamic treatment-regime estimation system, an independent identically distributed sample whose observations comprise baseline covariates, first treatment, intermediate covariates, second treatment, and outcome, a one-shot sample split, a sequence of stagewise nuisance-function estimates indexed by sample size and the underlying random outcome, and a sample-size index, the one-shot double-machine-learning sequentially doubly robust estimator is the function of the underlying random outcome that averages, over the split's estimation fold, the sequential doubly robust moment at the target treatment regime, using the nuisance estimate at that sample size and target value zero.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ) → ℕ → P.Ω → ℝ","l":358,"f":"Causalean/Estimation/DTR/DML.lean"},{"n":"Causalean.Estimation.DTR.dml_DTR_isAsymLinear","k":"theorem","m":"Causalean.Estimation.DTR.DML","a":"Estimation/DTR","t":1,"d":"Asymptotic linearity of the one-shot DML DTR (n = 2) estimator — thm:est-dml-dtr-al. Assuming the DTR backdoor identification conditions hold, the theorem shows that the one-shot double/debiased-machine-learning estimator of the two-period dynamic-treatment-regime effect is asymptotically linear, with influence function ψ_seqDR, around the true effect θ₀. The population-truth propensities obey strict overlap — some ε ∈ (0, 1/2] sandwiches them a.s. at both stages, restated pointwise on the value-space propensity functions e₀_val, e₁_val; the factual outcome and every counterfactual outcome under a fixed regime are square-integrable; and the one-shot sample split's auxiliary-fold fraction |B(n)|/n converges to some c ∈ (0, 1). For every horizon n, the stage-0 nuisance learners μ̂₀, ê₀ and the stage-1 learners μ̂₁, ê₁ are each jointly measurable in the sample outcome and the covariate history at stage 0 and likewise at stage 1; the fitted propensities satisfy the same strict-overlap bound ε pointwise at both stages; each learner lies in L² of the covariate-history distribution at stage 0 and likewise at stage 1; and, viewed as a function of the sample outcome alone, each learner is measurable with respect to the auxiliary training fold's σ-algebra at stage 0 and likewise at stage 1, and jointly with the covariate in uncurried form at stage 0 and at stage 1. Finally, each stagewise estimation error converges to zero in L² at rate o_p(1), and every cross-stage product of an outcome-regression error with a propensity error vanishes at the doubly-robust rate o_p(n^{-1/2}).","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ}, S.Assumptions → S.StrictOverlap ε → ((∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε) → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (μ₀_hat e₀_hat : ℕ → P.Ω → γ 0 → ℝ) (μ₁_hat e₁_hat : ℕ → P.Ω → γ 1 × δ × γ 0 → ℝ) (h_mu0_meas : ∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) (h_e0_meas : ∀ (n : ℕ), Measurable fun p => e₀_hat n p.1 p.2) (h_mu1_meas : ∀ (n : ℕ), Measurable fun p => μ₁_hat n p.1 p.2) (h_e1_meas : ∀ (n : ℕ), Measurable fun p => e₁_hat n p.1 p.2), (∀ (n : ℕ) (ω : P.Ω), (∀ (s₀ : γ 0), ε ≤ e₀_hat n ω s₀ ∧ e₀_hat n ω s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ e₁_hat n ω h ∧ e₁_hat n ω h ≤ 1 - ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => μ₀_hat n ω s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => e₀_hat n ω s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => μ₁_hat n ω h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => e₁_hat n ω h) 2 S.P_H₁) → (∀ (n : ℕ), Measurable (μ₀_hat n)) → (∀ (n : ℕ), Measurable (e₀_hat n)) → (∀ (n : ℕ), Measurable (μ₁_hat n)) → (∀ (n : ℕ), Measurable (e₁_hat n)) → (∀ (n : ℕ), Measurable fun p => μ₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e₀_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => μ₁_hat n p.1 p.2) → (∀ (n : ℕ), Measurable fun p => e₁_hat n p.1 p.2) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun s₀ => μ₀_hat n ω s₀ - S.μ₀_val s₀) 2 S.P_H₀).toReal * (MeasureTheory.eLpNorm (fun h => e₁_hat n ω h - S.e₁_val h) 2 S.P_H₁).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun h => μ₁_hat n ω h - S.μ₁_val h) 2 S.P_H₁).toReal * (MeasureTheory.eLpNorm (fun s₀ => e₀_hat n ω s₀ - S.e₀_val s₀) 2 S.P_H₀).toReal) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.DTR.dml_DTR_estimator S sample split fun n ω => { μ₀_fn := μ₀_hat n ω, e₀_fn := e₀_hat n ω, μ₁_fn := μ₁_hat n ω, e₁_fn := e₁_hat n ω, μ₀_meas := Measurable.comp (h_mu0_meas n) (Measurable.prodMk measurable_const measurable_id), e₀_meas := Measurable.comp (h_e0_meas n) (Measurable.prodMk measurable_const measurable_id), μ₁_meas := Measurable.comp (h_mu1_meas n) (Measurable.prodMk measurable_const measurable_id), e₁_meas := Measurable.comp (h_e1_meas n) (Measurable.prodMk measurable_const measurable_id) }) S.θ₀ S.ψ_seqDR sample split.foldB","l":404,"f":"Causalean/Estimation/DTR/DML.lean"},{"n":"Causalean.Estimation.DTR.exists_stage0_lift","k":"theorem","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"Doob–Dynkin factorization through the stage-0 state. Consider a two-stage dynamic-treatment-regime potential-outcome system and a real-valued function g on the sample space. If g is measurable with respect to the σ-algebra generated by the stage-0 history — here just the initial state S₀, then g factors through the stage-0 state: there is a measurable function f on the stage-0 state space such that g equals f composed with the stage-0 state map.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P 2 δ γ) (g : P.Ω → ℝ), Measurable g → ∃ f, Measurable f ∧ g = fun ω => f (S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":39,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.Estimation.DTR.exists_stage1_lift","k":"theorem","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"Doob–Dynkin factorization through the stage-1 history. Consider a two-stage dynamic-treatment-regime potential-outcome system and a real-valued function g on the sample space. If g is measurable with respect to the σ-algebra generated by the stage-1 history — the current state, the previous treatment, and the previous state, then g factors through that history tuple: there is a measurable function f on the stage-1 history space such that g equals f composed with the map recording the current state, the previous treatment, and the previous state.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P 2 δ γ) (g : P.Ω → ℝ), Measurable g → ∃ f, Measurable f ∧ g = fun ω => f (S.factualS ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualD ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω, S.factualS ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ω)","l":69,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.Estimation.DTR.histH₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, given an observed two-stage data tuple, the stage-1 history is its second-period state, first-period treatment, and first-period state, in that order.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 1 × δ × γ 0","l":70,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.indEq","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space, given two treatment values, the real-valued equality indicator is one when they are equal and zero otherwise.","s":"{δ : Type} → δ → δ → ℝ","l":77,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.measurable_indEq","k":"theorem","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":2,"d":"On a treatment space whose one-point sets are measurable, comparing a varying treatment against a fixed reference treatment is a measurable real-valued function of the varying treatment.","s":"∀ {δ : Type} [inst : MeasurableSpace δ] [MeasurableSingletonClass δ] (d : δ), Measurable fun x => Causalean.Estimation.DTR.indEq x d","l":83,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projD₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, the first-treatment projection maps every observed two-stage data tuple to its first-period treatment.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → δ","l":58,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projD₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, the second-treatment projection maps every observed two-stage data tuple to its second-period treatment.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → δ","l":64,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projS₀","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, the initial-state projection maps every observed two-stage data tuple to its first-period state.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 0","l":55,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projS₁","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, the second-state projection maps every observed two-stage data tuple to its second-period state.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → γ 1","l":61,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.projY","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, the outcome projection maps every observed two-stage data tuple to its observed outcome.","s":"{δ : Type} → {γ : Fin 2 → Type} → γ 0 × δ × γ 1 × δ × ℝ → ℝ","l":67,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.seqDRGeneralMoment","k":"def","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":1,"d":"For a potential-outcome system whose sample space is standard Borel and whose measure is finite with a measurable treatment space in which every singleton is measurable and measurable stage-specific covariate spaces, a two-stage dynamic treatment-regime estimation system, and a real number for which every stage-0 and stage-1 true propensity score lies between that number and one minus that number, inclusive, the abstract general moment associated with the two-stage sequential doubly robust score has the system's true nuisance functions and target value, its two stagewise aggregate error seminorms, and that score as its moment function.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) → {ε : ℝ} → ((∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε) → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ 0 × δ × γ 1 × δ × ℝ) S.P_Z (Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ)","l":62,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDRMoment","k":"def","m":"Causalean.Estimation.DTR.SeqDRMoment","a":"Estimation/DTR","t":1,"d":"For a treatment space and the pair of first- and second-period state spaces, given a two-period target treatment regime, an observed two-stage data tuple, a stagewise nuisance vector, and a candidate regime mean, the two-stage sequential doubly robust moment is the baseline regression plus its first-stage and second-stage inverse-propensity-weighted residual corrections, minus the candidate mean.","s":"{δ : Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → {γ : Fin 2 → Type} → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → (Fin 2 → δ) → γ 0 × δ × γ 1 × δ × ℝ → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ → ℝ → ℝ","l":333,"f":"Causalean/Estimation/DTR/SeqDRMoment.lean"},{"n":"Causalean.Estimation.DTR.seqDR_bilinearRem","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":2,"d":"Sequential DR (DTR) satisfies BilinearRemainder with constant seqDR_rem_const ε.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → (∀ η ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε, MeasureTheory.MemLp (fun s₀ => η.μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀ ∧ MeasureTheory.MemLp (fun h => η.μ₁_fn h - S.μ₁_val h) 2 S.P_H₁ ∧ MeasureTheory.MemLp (fun s₀ => η.e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀ ∧ MeasureTheory.MemLp (fun h => η.e₁_fn h - S.e₁_val h) 2 S.P_H₁) → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise) C","l":119,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDR_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":1,"d":"Headline sequential DR (DTR) DML asymptotic-linearity theorem, derived from the abstract dml_chernozhukov_asymptoticLinear in Estimation/OrthogonalMoments/DMLChernozhukov.lean. Fix a dynamic-treatment-regime estimation system with strict two-stage propensity overlap and satisfying the DTR identification assumptions, and suppose the factual outcome and every counterfactual outcome under a fixed treatment history have finite second moment. Given an i.i.d. sample together with a one-shot cross-fitting split whose estimation-fold share converges to some constant strictly between $0$ and $1$, and a sequence of nuisance estimators η_hat that remain in the $ε$-overlap ball, with stagewise outcome-regression and propensity errors that are square-integrable, such that the resulting moment function is measurable against the sample and each cross-fitting fold, and is both integrable and square-integrable, and such that the individual L² nuisance-error rates vanish while their product is $o_P(n^{-1/2})$, then the resulting Chernozhukov one-step DML estimator is asymptotically linear at the true sequential-DR parameter, with influence function the sequential doubly-robust score evaluated at the truth.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (dbar : Fin 2 → δ), MeasureTheory.Integrable (fun ω => S.Y_of dbar ω ^ 2) P.μ) → ∀ (sample : Causalean.Stat.IIDSample P.Ω (γ 0 × δ × γ 1 × δ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → c < 1 → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.DTR.DTRNuisanceVec₂ δ γ), (∀ (n : ℕ) (ω : P.Ω), η_hat n ω ∈ Causalean.Estimation.DTR.DTREstimationSystem.H_ε ε) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).μ₀_fn s₀ - S.μ₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).μ₁_fn h - S.μ₁_val h) 2 S.P_H₁) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun s₀ => (η_hat n ω).e₀_fn s₀ - S.e₀_val s₀) 2 S.P_H₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun h => (η_hat n ω).e₁_fn h - S.e₁_val h) 2 S.P_H₁) → (∀ (n : ℕ), Measurable fun p => S.seqDRMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀) → (∀ (n : ℕ), Measurable fun p => S.seqDRMomentFunctional (η_hat n p.1) p.2 S.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.seqDRMomentFunctional (η_hat n ω) z S.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₁ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₂ (η_hat n ω) S.η₀)) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₁ (η_hat n ω) S.η₀) * ↑((Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).ρ₂ (η_hat n ω) S.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise) sample split η_hat) S.θ₀ (fun z => -(Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise).J₀_inv * S.seqDRMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":155,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.DTR.seqDR_meanZero","k":"theorem","m":"Causalean.Estimation.DTR.DTRInstance","a":"Estimation/DTR","t":2,"d":"Sequential DR (DTR) satisfies MeanZero.","s":"∀ {P : Causalean.PO.POSystem} {δ : Type} {γ : Fin 2 → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.DTR.DTREstimationSystem P δ γ) {ε : ℝ} (h_e_pointwise : (∀ (s₀ : γ 0), ε ≤ S.e₀_val s₀ ∧ S.e₀_val s₀ ≤ 1 - ε) ∧ ∀ (h : γ 1 × δ × γ 0), ε ≤ S.e₁_val h ∧ S.e₁_val h ≤ 1 - ε), S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.DTR.seqDRGeneralMoment S h_e_pointwise)","l":106,"f":"Causalean/Estimation/DTR/DTRInstance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.CanonicalGradientPairing","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A canonical-gradient pairing for a finite-dimensional LAN experiment. The map scoreMap embeds parameter directions into an ambient L²-type Hilbert space K; gradient belongs to the score range, the information is the score inner product, and target derivatives are inner products with the gradient. Consequently its efficient variance is ‖gradient‖².","s":"{H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → {K : Type u_3} → [inst_2 : NormedAddCommGroup K] → [inst_3 : InnerProductSpace ℝ K] → LinearMap.BilinForm ℝ H → (H →ₗ[ℝ] ℝ) → (H →ₗ[ℝ] K) → K → Prop","l":123,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A dominated finite-dimensional i.i.d. model is a probability law law θ for every parameter θ, together with one common dominating measure. The base parameter is 0.","s":"(X : Type u_3) → Type u_4 → [MeasurableSpace X] → Type (max u_3 u_4)","l":30,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel.iidLaw","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The n-observation i.i.d. law is the finite product of the one-observation law.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H → H → (n : ℕ) → MeasureTheory.Measure (Fin n → X)","l":51,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel.sqrtDensity","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The square root of the density of law θ with respect to the model's dominating measure.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H → H → X → ℝ","l":47,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"Quadratic-mean differentiability at the base parameter 0 with score score. The squared L² error in the first-order square-root-density expansion is little-o of ‖h‖², and the score is measurable and square-integrable under the base law.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → [MeasurableSpace H] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H → Type (max u_1 u_2)","l":58,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.centralSequence","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The normalized score sum on the canonical n-fold product sample.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [inst_2 : InnerProductSpace ℝ H] → [inst_3 : MeasurableSpace H] → {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M → (n : ℕ) → (Fin n → X) → H","l":99,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.centralSequence_weaklyConverges","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQMLAN","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Under the base product laws, the normalized i.i.d. score sums converge weakly to the centered Gaussian law whose covariance is the score second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M), Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges (fun n => M.iidLaw 0 n) qmd.centralSequence (Causalean.Stat.gaussianLimit qmd.score_measurable qmd.score_squareIntegrable)","l":543,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQMLAN.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.iidDQM_implies_LAN","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQMLAN","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A quadratic-mean differentiable dominated finite-dimensional i.i.d. model is locally asymptotically normal at its base parameter, with normalized score-sum central sequence and score-second-moment information form.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M), Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.localExperiment M) qmd.centralSequence (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd) (Causalean.Stat.gaussianLimit qmd.score_measurable qmd.score_squareIntegrable)","l":617,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQMLAN.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.iidLaw_probability","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Every finite product law of a dominated i.i.d. probability model is a probability measure.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] (M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H) (θ : H) (n : ℕ), MeasureTheory.IsProbabilityMeasure (M.iidLaw θ n)","l":283,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.iidLaw_rnDeriv_eq_prod","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The Radon--Nikodym derivative of a finite i.i.d. product law is the product of the one-observation Radon--Nikodym derivatives (of their absolutely continuous parts). No one-observation absolute-continuity assumption is needed.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] (M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H) (θ : H) (n : ℕ), (M.iidLaw θ n).rnDeriv (M.iidLaw 0 n) =ᵐ[M.iidLaw 0 n] fun x => ∏ i, (M.law θ).rnDeriv (M.law 0) (x i)","l":381,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.iid_logLikelihoodRatio_expansion","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQMLAN","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The generalized Radon--Nikodym log likelihood ratio of the finite product experiment admits the DQM quadratic expansion: normalized score sum minus half the Fisher-information quadratic form, with a remainder vanishing in base-law probability.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M) (h : H), Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability (fun n => M.iidLaw 0 n) (fun n x => (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.localExperiment M).logLikelihoodRatio n h x - inner ℝ h (qmd.centralSequence n x) + 1 / 2 * ((Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd) h) h) 0","l":606,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQMLAN.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.iid_logLikelihoodRatio_taylor","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQMLAN","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The triangular-array likelihood Taylor lemma: product-density factorization, the DQM L² increment approximation, and the vanishing singular mass imply the quadratic LAN expansion of the guarded log likelihood ratio.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M) (h : H), Filter.Tendsto (fun n => ↑n * ∫ (x : X), (qmd.sqrtLikelihoodIncrement n h x - (√↑n)⁻¹ * inner ℝ h (qmd.score x)) ^ 2 ∂M.law 0) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑n * ((M.law ((√↑n)⁻¹ • h)) {x | M.sqrtDensity 0 x = 0}).toReal) Filter.atTop (nhds 0) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability (fun n => M.iidLaw 0 n) (fun n x => (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.localExperiment M).logLikelihoodRatio n h x - inner ℝ h (qmd.centralSequence n x) + 1 / 2 * ((Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd) h) h) 0","l":490,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQMLAN.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"Fisher information is the bilinear second moment of the DQM score.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [inst_2 : InnerProductSpace ℝ H] → [FiniteDimensional ℝ H] → [inst_4 : MeasurableSpace H] → [BorelSpace H] → (M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M → LinearMap.BilinForm ℝ H","l":79,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information_apply","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Evaluating the Fisher-information form gives the second moment of the two scalar score projections.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M) (h g : H), ((Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd) h) g = ∫ (x : X), inner ℝ h (qmd.score x) * inner ℝ g (qmd.score x) ∂M.law 0","l":90,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information_isSymm_nonnegative","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The score second-moment form is symmetric and nonnegative.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M), (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd).IsSymm ∧ ∀ (h : H), 0 ≤ ((Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.information M qmd) h) h","l":270,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.localExperiment","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The canonical local experiment of an i.i.d. model uses the base product law at parameter zero and the local product law at parameter (sqrt n)⁻¹ • h.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [inst_2 : InnerProductSpace ℝ H] → [FiniteDimensional ℝ H] → [inst_4 : MeasurableSpace H] → [BorelSpace H] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment (fun n => Fin n → X) H","l":393,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.local_zeroBaseDensity_mass","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"For a dominated i.i.d. model that is quadratic-mean differentiable and a fixed local direction, the local probability mass outside the base-law support is little-o of one over the sample size.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M) (h : H), Filter.Tendsto (fun n => ↑n * ((M.law ((√↑n)⁻¹ • h)) {x | M.sqrtDensity 0 x = 0}).toReal) Filter.atTop (nhds 0)","l":576,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.score_integral_eq_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"DQM forces the score to have mean zero under the base law. This is the normalization identity obtained by differentiating the total mass of the model.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M), ∫ (x : X), qmd.score x ∂M.law 0 = 0","l":104,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.sqrtLikelihoodIncrement","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"Twice the relative square-root likelihood increment for one observation at local parameter h / sqrt n. It is set to zero on the zero set of the base square-root density; the local mass of that set is controlled separately.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [inst_2 : InnerProductSpace ℝ H] → [inst_3 : MeasurableSpace H] → {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M → ℕ → H → X → ℝ","l":403,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable.sqrtLikelihoodIncrement_L2","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDDQM","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"DQM makes the one-observation square-root likelihood increment equal to the projected score divided by sqrt n, with squared error o(1/n) under the base law.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] {M : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.DominatedIIDModel X H} (qmd : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IIDQuadraticMeanDifferentiable M) (h : H), Filter.Tendsto (fun n => ↑n * ∫ (x : X), (qmd.sqrtLikelihoodIncrement n h x - (√↑n)⁻¹ * inner ℝ h (qmd.score x)) ^ 2 ∂M.law 0) Filter.atTop (nhds 0)","l":504,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/IIDDQM.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A local experiment is locally asymptotically normal with central sequence centralSequence and information form information when the central sequence has the centered Gaussian characteristic function determined by information, and every fixed-direction log likelihood ratio equals the linear score term minus half the information quadratic form up to a remainder converging to zero in base-law probability.","s":"{Ω : ℕ → Type u_1} → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [FiniteDimensional ℝ H] → [inst_3 : MeasurableSpace H] → [BorelSpace H] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H → ((n : ℕ) → Ω n → H) → LinearMap.BilinForm ℝ H → MeasureTheory.Measure H → Prop","l":80,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsRegularEstimator","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A scalar estimator has regular limit limitLaw with derivative targetDerivative when, under every fixed local direction h, its base-centered statistic minus the deterministic local target shift converges weakly to the same law.","s":"{Ω : ℕ → Type u_1} → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [TopologicalSpace H] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H → ((n : ℕ) → Ω n → ℝ) → (H →ₗ[ℝ] ℝ) → MeasureTheory.Measure ℝ → Prop","l":108,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.JointWeakSubsequence","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A joint weak subsequential limit couples two row-indexed statistics whose marginal laws converge. The selected indices are strictly increasing, the joint laws converge against every bounded continuous test function, and the two marginals of the limit are the prescribed laws.","s":"{Ω : ℕ → Type u_1} → [inst : (n : ℕ) → MeasurableSpace (Ω n)] → {H : Type u_2} → [NormedAddCommGroup H] → [inst_2 : MeasurableSpace H] → (ℕ → ℕ) → ((n : ℕ) → MeasureTheory.Measure (Ω n)) → ((n : ℕ) → Ω n → ℝ) → ((n : ℕ) → Ω n → H) → MeasureTheory.Measure ℝ → MeasureTheory.Measure H → MeasureTheory.Measure (ℝ × H) → Prop","l":120,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A sequence of local statistical experiments with finite-dimensional direction space H. For row n, baseLaw n is the law at the base point and localLaw n h is the law at the fixed local direction h (normally the original parameter is displaced by h / sqrt n).","s":"(Ω : ℕ → Type u_3) → [(n : ℕ) → MeasurableSpace (Ω n)] → (H : Type u_4) → [Zero H] → Type (max u_3 u_4)","l":51,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment.logLikelihoodRatio","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The generalized log likelihood ratio of local direction h against the base law is the logarithm of the real Radon--Nikodym derivative of the local law's absolutely continuous part. Where that derivative is zero, the extended-real value -∞ is represented by the diverging finite truncation -n; DQM makes this exceptional event asymptotically negligible.","s":"{Ω : ℕ → Type u_1} → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → {H : Type u_2} → [inst : NormedAddCommGroup H] → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H → (n : ℕ) → H → Ω n → ℝ","l":69,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.RegularLANJointLimit","k":"structure","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A joint LAN limit couples the scalar estimator limit with the Gaussian central-sequence limit and satisfies the exponential-tilt identity forced by regularity under every fixed local direction. It is an intermediate conclusion of the LAN change-of-measure argument, not an assumption in the public convolution theorem.","s":"{H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : MeasurableSpace H] → LinearMap.BilinForm ℝ H → (H →ₗ[ℝ] ℝ) → MeasureTheory.Measure ℝ → MeasureTheory.Measure H → MeasureTheory.Measure (ℝ × H) → Prop","l":267,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A real triangular array converges in probability to c when, for every positive tolerance, the probability of an absolute error at least that tolerance tends to zero.","s":"{Ω : ℕ → Type u_1} → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → ((n : ℕ) → MeasureTheory.Measure (Ω n)) → ((n : ℕ) → Ω n → ℝ) → ℝ → Prop","l":44,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A row-indexed random element converges weakly to Q when its expectation against every bounded continuous real test function converges to the corresponding expectation under Q.","s":"{Ω : ℕ → Type u_1} → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → {H : Type u_2} → [TopologicalSpace H] → [inst : MeasurableSpace H] → ((n : ℕ) → MeasureTheory.Measure (Ω n)) → ((n : ℕ) → Ω n → H) → MeasureTheory.Measure H → Prop","l":32,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.abs_two_mul_log_one_add_half_sub_quadratic_le_cube","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LogTaylor","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"On [-1,1], the quadratic Taylor error of 2 log (1+w/2) is bounded by |w|³.","s":"∀ (w : ℝ), |w| ≤ 1 → |2 * Real.log (1 + w / 2) - w + 1 / 4 * w ^ 2| ≤ |w| ^ 3","l":21,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/LogTaylor.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.asymptoticVariance_ge_gradientNormSq_of_factorization","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Variance bound from convolution. If a probability limit law with finite second moment factors as the convolution of the efficient N(0, ‖gradient‖²) law and a residual probability law, then its variance is at least ‖gradient‖².","s":"∀ {K : Type u_1} [inst : NormedAddCommGroup K] (gradient : K) (limitLaw residualLaw : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure limitLaw] [MeasureTheory.IsProbabilityMeasure residualLaw], limitLaw = (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient).conv residualLaw → MeasureTheory.MemLp id 2 limitLaw → ‖gradient‖ ^ 2 ≤ ProbabilityTheory.variance id limitLaw","l":84,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.canonicalGradientPairing_of_mem_range","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Basic","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"Given an information form, a target derivative, a score embedding, and a candidate gradient, proof that the gradient is in the score range, that information is the score inner product, and that the derivative pairs with the gradient yield a canonical-gradient pairing.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {K : Type u_3} [inst_2 : NormedAddCommGroup K] [inst_3 : InnerProductSpace ℝ K] (information : LinearMap.BilinForm ℝ H) (targetDerivative : H →ₗ[ℝ] ℝ) (scoreMap : H →ₗ[ℝ] K), ∀ gradient ∈ scoreMap.range, (∀ (h g : H), (information h) g = inner ℝ (scoreMap h) (scoreMap g)) → (∀ (h : H), targetDerivative h = inner ℝ gradient (scoreMap h)) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.CanonicalGradientPairing information targetDerivative scoreMap gradient","l":138,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Basic.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The efficient scalar Gaussian law associated with a canonical gradient is the centered normal distribution with variance equal to the squared norm of that gradient.","s":"{K : Type u_3} → [NormedAddCommGroup K] → K → MeasureTheory.Measure ℝ","l":30,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw_memLp_two","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The efficient centered Gaussian law has a finite second moment.","s":"∀ {K : Type u_1} [inst : NormedAddCommGroup K] (gradient : K), MeasureTheory.MemLp id 2 (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient)","l":67,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw_probability","k":"instance","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The efficient Gaussian law is a probability measure.","s":"∀ {K : Type u_3} [inst : NormedAddCommGroup K] (gradient : K), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient)","l":36,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw_variance","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The variance of the efficient centered Gaussian law is exactly the squared norm of the canonical gradient.","s":"∀ {K : Type u_1} [inst : NormedAddCommGroup K] (gradient : K), ProbabilityTheory.variance id (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient) = ‖gradient‖ ^ 2","l":75,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.exists_jointWeakSubsequence","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Two measurable row-indexed statistics with weakly convergent probability marginals admit a joint weakly convergent subsequence, and every such joint limit has the prescribed marginals.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] {P : (n : ℕ) → MeasureTheory.Measure (Ω n)} {statistic : (n : ℕ) → Ω n → ℝ} {centralSequence : (n : ℕ) → Ω n → H} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H}, (∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (P n)) → MeasureTheory.IsProbabilityMeasure limitLaw → MeasureTheory.IsProbabilityMeasure scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges P statistic limitLaw → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges P centralSequence scoreLimit → ∃ jointLaw subsequence, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.JointWeakSubsequence subsequence P statistic centralSequence limitLaw scoreLimit jointLaw","l":145,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.exists_regularLANJointLimit","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Convolution","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"LAN, convergence of the estimator at the base law, and regularity under every fixed local alternative produce a joint subsequential limit satisfying the Gaussian-shift tilt identity. The tightness needed to choose a joint limit follows from the two convergent marginals in finite dimension.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {statistic : (n : ℕ) → Ω n → ℝ} {targetDerivative : H →ₗ[ℝ] ℝ} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsRegularEstimator E statistic targetDerivative limitLaw → ∃ jointLaw, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.RegularLANJointLimit information targetDerivative limitLaw scoreLimit jointLaw","l":337,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Convolution.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.exp_integral_tendsto_one_of_weaklyConverges","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LikelihoodNormalization","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"If real random variables converge weakly, their exponential limit has expectation one, and the row exponential expectations have asymptotic upper bound one, then those expectations tend to one. This is the nonnegative lower-semicontinuous Portmanteau step.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {P : (n : ℕ) → MeasureTheory.Measure (Ω n)} {X : (n : ℕ) → Ω n → ℝ} {Q : MeasureTheory.Measure ℝ}, (∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (P n)) → MeasureTheory.IsProbabilityMeasure Q → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges P X Q → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => Real.exp (X n ω)) (P n)) → ∫ (x : ℝ), Real.exp x ∂Q = 1 → (∀ (n : ℕ), ∫ (ω : Ω n), Real.exp (X n ω) ∂P n ≤ 1 + Real.exp (-↑n)) → Filter.Tendsto (fun n => ∫ (ω : Ω n), Real.exp (X n ω) ∂P n) Filter.atTop (nhds 1)","l":22,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/LikelihoodNormalization.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.iid_large_coordinate_measureReal_le","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"For one square-integrable row, the probability that at least one i.i.d. coordinate exceeds δ is bounded by n / δ² times the second moment on the tail |W| ≥ δ.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : X → ℝ), Measurable W → MeasureTheory.MemLp W 2 P → ∀ (n : ℕ) {δ : ℝ}, 0 < δ → (MeasureTheory.Measure.pi fun x => P).real {x | ∃ i, δ ≤ |W (x i)|} ≤ ↑n * (δ ^ 2)⁻¹ * ∫ (x : X) in {x | δ ≤ |W x|}, W x ^ 2 ∂P","l":291,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.iid_max_tendsto_zero_of_lindeberg","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"For an i.i.d. triangular row with measurable coordinates, finite second moments, and a Lindeberg tail condition, the probability that its largest absolute coordinate exceeds any positive threshold vanishes.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → X → ℝ), (∀ (n : ℕ), Measurable (W n)) → (∀ (n : ℕ), MeasureTheory.MemLp (W n) 2 P) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => ↑n * ∫ (x : X) in {x | ε ≤ |W n x|}, W n x ^ 2 ∂P) Filter.atTop (nhds 0)) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (MeasureTheory.Measure.pi fun x => P) {x | ∃ i, ε ≤ |W n (x i)|}) Filter.atTop (nhds 0)","l":564,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.iid_sum_approx_tendstoInProbability","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"For an i.i.d. product row, a centered sum of errors converges in probability to zero when the row error has total second moment tending to zero. A convergent scaled row mean supplies the stated deterministic limit.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → X → ℝ) (S : X → ℝ) (m : ℝ), (∀ (n : ℕ), Measurable (W n)) → (∀ (n : ℕ), MeasureTheory.MemLp (W n) 2 P) → MeasureTheory.MemLp S 2 P → ∫ (x : X), S x ∂P = 0 → Filter.Tendsto (fun n => ↑n * ∫ (x : X), (W n x - (√↑n)⁻¹ * S x) ^ 2 ∂P) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑n * ∫ (x : X), W n x ∂P) Filter.atTop (nhds m) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability (fun n => MeasureTheory.Measure.pi fun x => P) (fun n x => ∑ i, W n (x i) - (√↑n)⁻¹ * ∑ i, S (x i)) m","l":201,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.iid_sum_sq_tendstoInProbability_of_lindeberg","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The sum of squares in an infinitesimal i.i.d. triangular row obeys a weak law when its scaled second moment converges and its Lindeberg tail vanishes.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → X → ℝ) (q : ℝ), (∀ (n : ℕ), Measurable (W n)) → (∀ (n : ℕ), MeasureTheory.MemLp (W n) 2 P) → Filter.Tendsto (fun n => ↑n * ∫ (x : X), W n x ^ 2 ∂P) Filter.atTop (nhds q) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => ↑n * ∫ (x : X) in {x | ε ≤ |W n x|}, W n x ^ 2 ∂P) Filter.atTop (nhds 0)) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability (fun n => MeasureTheory.Measure.pi fun x => P) (fun n x => ∑ i, W n (x i) ^ 2) q","l":399,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.iid_truncated_sq_deviation_le","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The centered sum of the truncated squares W² 1{|W|<δ} under an i.i.d. product law obeys a Chebyshev bound controlled by n δ² E[W²].","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : X → ℝ), Measurable W → MeasureTheory.MemLp W 2 P → ∀ (n : ℕ) {δ ε : ℝ}, 0 < δ → 0 < ε → (MeasureTheory.Measure.pi fun x => P) {x | ε ≤ |(∑ i, if |W (x i)| < δ then W (x i) ^ 2 else 0) - ↑n * ∫ (y : X), if |W y| < δ then W y ^ 2 else 0 ∂P|} ≤ ENNReal.ofReal ((↑n * δ ^ 2 * ∫ (y : X), W y ^ 2 ∂P) / ε ^ 2)","l":347,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.integral_boundedContinuous_mul_exp_tendsto_of_normalized","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ExponentialTilt","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"Given probability laws in every row and in the limit, joint weak convergence, a negligible additive remainder, its measurability, row integrability of the exponential weights, integrability of the limiting weight, and normalization of their expectations, bounded complex observables converge under exponential tilting.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {S : Type u_2} [inst_1 : NormedAddCommGroup S] [inst_2 : NormedSpace ℝ S] [FiniteDimensional ℝ S] [inst_4 : MeasurableSpace S] [BorelSpace S] {P : (n : ℕ) → MeasureTheory.Measure (Ω n)} {Y : (n : ℕ) → Ω n → S} {Q : MeasureTheory.Measure S} {R : (n : ℕ) → Ω n → ℝ} (a : C(S, ℝ)) (f : BoundedContinuousFunction S ℂ), (∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (P n)) → MeasureTheory.IsProbabilityMeasure Q → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges P Y Q → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability P R 0 → (∀ (n : ℕ), AEMeasurable (R n) (P n)) → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => Real.exp (a (Y n ω) + R n ω)) (P n)) → MeasureTheory.Integrable (fun y => Real.exp (a y)) Q → Filter.Tendsto (fun n => ∫ (ω : Ω n), Real.exp (a (Y n ω) + R n ω) ∂P n) Filter.atTop (nhds (∫ (y : S), Real.exp (a y) ∂Q)) → Filter.Tendsto (fun n => ∫ (ω : Ω n), f (Y n ω) * ↑(Real.exp (a (Y n ω) + R n ω)) ∂P n) Filter.atTop (nhds (∫ (y : S), f y * ↑(Real.exp (a y)) ∂Q))","l":175,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ExponentialTilt.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.integral_exp_lanLogLikelihoodLimit","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The exponential moment of the affine Gaussian LAN log-likelihood limit equals one.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_3 : MeasurableSpace H] [BorelSpace H] {information : LinearMap.BilinForm ℝ H} {scoreLimit : MeasureTheory.Measure H}, MeasureTheory.IsProbabilityMeasure scoreLimit → (∀ (t : H), MeasureTheory.charFun scoreLimit t = Complex.exp (-(↑((information t) t) / 2))) → ∀ (h : H), ∫ (y : ℝ), Real.exp y ∂Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lanLogLikelihoodLimit information scoreLimit h = 1","l":538,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.integral_exp_logLikelihoodRatio_le","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The expectation of the exponential of the guarded log likelihood ratio is bounded by the local probability mass plus the artificial exp (-n) contribution on its zero-density set.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [BorelSpace H] (E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H) (n : ℕ) (h : H), ∫ (ω : Ω n), Real.exp (E.logLikelihoodRatio n h ω) ∂E.baseLaw n ≤ 1 + Real.exp (-↑n)","l":486,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.jointLimit_convolution_factorization","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Convolution","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"The Gaussian-shift tilt identity and a canonical-gradient representer imply that the scalar estimator marginal is the convolution of the efficient centered Gaussian law with a residual probability law. This is the limit-experiment form of the convolution argument.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [FiniteDimensional ℝ H] [inst_3 : MeasurableSpace H] [BorelSpace H] {K : Type u_3} [inst_5 : NormedAddCommGroup K] [inst_6 : InnerProductSpace ℝ K] {information : LinearMap.BilinForm ℝ H} {targetDerivative : H →ₗ[ℝ] ℝ} {scoreMap : H →ₗ[ℝ] K} {gradient : K} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H} {jointLaw : MeasureTheory.Measure (ℝ × H)}, MeasureTheory.IsProbabilityMeasure scoreLimit → (∀ (t : H), MeasureTheory.charFun scoreLimit t = Complex.exp (-(↑((information t) t) / 2))) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.RegularLANJointLimit information targetDerivative limitLaw scoreLimit jointLaw → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.CanonicalGradientPairing information targetDerivative scoreMap gradient → ∃ residualLaw, MeasureTheory.IsProbabilityMeasure residualLaw ∧ limitLaw = (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient).conv residualLaw","l":362,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Convolution.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lanLogLikelihoodLimit","k":"def","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"The limiting log likelihood ratio in direction h is the affine projection inner h Z - information(h,h)/2 of the LAN Gaussian central-sequence limit.","s":"{H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : MeasurableSpace H] → LinearMap.BilinForm ℝ H → MeasureTheory.Measure H → H → MeasureTheory.Measure ℝ","l":287,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lan_likelihoodRatio_integral_tendsto_one","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"In a LAN experiment, the exponential of the guarded log likelihood ratio has expectation tending to one under the base law. Equivalently, the asymptotically negligible singular part of the local law cannot lose mass in the Gaussian likelihood-ratio limit.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {scoreLimit : MeasureTheory.Measure H}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → ∀ (h : H), Filter.Tendsto (fun n => ∫ (ω : Ω n), Real.exp (E.logLikelihoodRatio n h ω) ∂E.baseLaw n) Filter.atTop (nhds 1)","l":598,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lan_logLikelihoodRatio_weaklyConverges","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.ConvolutionCore","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"A locally asymptotically normal experiment and a fixed local direction make its guarded log likelihood ratio converge weakly to the affine Gaussian limit.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {scoreLimit : MeasureTheory.Measure H}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → ∀ (h : H), Causalean.Estimation.Efficiency.AsymptoticLanConvolution.WeaklyConverges E.baseLaw (fun n ω => E.logLikelihoodRatio n h ω) (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lanLogLikelihoodLimit information scoreLimit h)","l":443,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/ConvolutionCore.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lan_tilt_identity_of_jointWeakSubsequence","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Convolution","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Along any joint weak subsequence of the base-law estimator and central sequence, LAN and regularity identify the exponentially tilted characteristic function. This is the scalar Le Cam third-lemma step; in particular, contiguity and control of the unbounded likelihood weight are conclusions of LAN rather than extra assumptions.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {statistic : (n : ℕ) → Ω n → ℝ} {targetDerivative : H →ₗ[ℝ] ℝ} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H} {jointLaw : MeasureTheory.Measure (ℝ × H)} {subsequence : ℕ → ℕ}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsRegularEstimator E statistic targetDerivative limitLaw → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.JointWeakSubsequence subsequence E.baseLaw statistic centralSequence limitLaw scoreLimit jointLaw → ∀ (t : ℝ) (h : H), ∫ (z : ℝ × H), Complex.exp (↑t * ↑z.1 * Complex.I) * Complex.exp ↑(inner ℝ h z.2 - 1 / 2 * (information h) h) ∂jointLaw = Complex.exp (↑t * ↑(targetDerivative h) * Complex.I) * MeasureTheory.charFun limitLaw t","l":270,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Convolution.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.lan_tilted_charFun_tendsto_of_jointWeakSubsequence","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Convolution","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Given a LAN experiment, a selected joint weak limit, measurability under its local laws, and normalized likelihood-ratio means, the local-law characteristic functions converge to the corresponding exponentially tilted joint-limit integrals.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {statistic : (n : ℕ) → Ω n → ℝ} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H} {jointLaw : MeasureTheory.Measure (ℝ × H)} {subsequence : ℕ → ℕ}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.JointWeakSubsequence subsequence E.baseLaw statistic centralSequence limitLaw scoreLimit jointLaw → (∀ (n : ℕ) (h : H), AEMeasurable (statistic n) (E.localLaw n h)) → (∀ (h : H), Filter.Tendsto (fun n => ∫ (ω : Ω n), Real.exp (E.logLikelihoodRatio n h ω) ∂E.baseLaw n) Filter.atTop (nhds 1)) → ∀ (t : ℝ) (h : H), Filter.Tendsto (fun n => ∫ (ω : Ω (subsequence n)), Complex.exp (↑t * ↑(statistic (subsequence n) ω) * Complex.I) ∂E.localLaw (subsequence n) h) Filter.atTop (nhds (∫ (z : ℝ × H), Complex.exp (↑t * ↑z.1 * Complex.I) * Complex.exp ↑(inner ℝ h z.2 - 1 / 2 * (information h) h) ∂jointLaw))","l":100,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Convolution.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.local_integral_sub_exp_logLikelihoodRatio_integral_tendsto_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LikelihoodNormalization","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"For a fixed local direction, a complex observable measurable under both the local and base laws, a uniform bound for that observable, and asymptotic likelihood-ratio normalization, its local-law integral and guarded likelihood-ratio base-law integral have vanishing difference.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} (h : H) {φ : (n : ℕ) → Ω n → ℂ}, (∀ (n : ℕ), AEMeasurable (φ n) (E.localLaw n h)) → (∀ (n : ℕ), AEMeasurable (φ n) (E.baseLaw n)) → (∃ C, 0 ≤ C ∧ ∀ (n : ℕ) (ω : Ω n), ‖φ n ω‖ ≤ C) → Filter.Tendsto (fun n => ∫ (ω : Ω n), Real.exp (E.logLikelihoodRatio n h ω) ∂E.baseLaw n) Filter.atTop (nhds 1) → Filter.Tendsto (fun n => ∫ (ω : Ω n), φ n ω ∂E.localLaw n h - ∫ (ω : Ω n), φ n ω * ↑(Real.exp (E.logLikelihoodRatio n h ω)) ∂E.baseLaw n) Filter.atTop (nhds 0)","l":101,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/LikelihoodNormalization.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.memLp_two_right_of_conv","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"If the convolution of two probability laws has a finite second moment and the left law has a finite second moment, then the right law also has a finite second moment.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp id 2 μ → MeasureTheory.MemLp id 2 (μ.conv ν) → MeasureTheory.MemLp id 2 ν","l":21,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.regular_asymptoticVariance_ge_gradientNormSq","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"In a finite-dimensional experiment with local asymptotic normality, a regular scalar estimator, a canonical-gradient representation, and a finite second moment for its limit law, the limit variance is at least the squared canonical-gradient norm.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {K : Type u_3} [inst_6 : NormedAddCommGroup K] [inst_7 : InnerProductSpace ℝ K] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {statistic : (n : ℕ) → Ω n → ℝ} {targetDerivative : H →ₗ[ℝ] ℝ} {scoreMap : H →ₗ[ℝ] K} {gradient : K} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsRegularEstimator E statistic targetDerivative limitLaw → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.CanonicalGradientPairing information targetDerivative scoreMap gradient → MeasureTheory.MemLp id 2 limitLaw → ‖gradient‖ ^ 2 ≤ ProbabilityTheory.variance id limitLaw","l":120,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.regular_convolution_limit","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Convolution","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"In a finite-dimensional experiment with local asymptotic normality, a scalar estimator regular under each fixed local alternative, and a canonical-gradient representation of its target derivative, the common estimator limit is the efficient centered Gaussian law convolved with a residual probability law.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] {H : Type u_2} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [inst_3 : FiniteDimensional ℝ H] [inst_4 : MeasurableSpace H] [inst_5 : BorelSpace H] {K : Type u_3} [inst_6 : NormedAddCommGroup K] [inst_7 : InnerProductSpace ℝ K] {E : Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LocalExperiment Ω H} {centralSequence : (n : ℕ) → Ω n → H} {information : LinearMap.BilinForm ℝ H} {statistic : (n : ℕ) → Ω n → ℝ} {targetDerivative : H →ₗ[ℝ] ℝ} {scoreMap : H →ₗ[ℝ] K} {gradient : K} {limitLaw : MeasureTheory.Measure ℝ} {scoreLimit : MeasureTheory.Measure H}, Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsLAN E centralSequence information scoreLimit → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.IsRegularEstimator E statistic targetDerivative limitLaw → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.CanonicalGradientPairing information targetDerivative scoreMap gradient → ∃ residualLaw, MeasureTheory.IsProbabilityMeasure residualLaw ∧ limitLaw = (Causalean.Estimation.Efficiency.AsymptoticLanConvolution.efficientGaussianLaw gradient).conv residualLaw","l":537,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Convolution.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.scaledL2Approx_lindeberg","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TriangularArray","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"An L² approximation of a triangular row by S / sqrt n transfers the fixed L² tail condition of S to the Lindeberg tail condition for the row.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → X → ℝ) (S : X → ℝ), (∀ (n : ℕ), Measurable (W n)) → (∀ (n : ℕ), MeasureTheory.MemLp (W n) 2 P) → MeasureTheory.MemLp S 2 P → Filter.Tendsto (fun n => ↑n * ∫ (x : X), (W n x - (√↑n)⁻¹ * S x) ^ 2 ∂P) Filter.atTop (nhds 0) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => ↑n * ∫ (x : X) in {x | ε ≤ |W n x|}, W n x ^ 2 ∂P) Filter.atTop (nhds 0)","l":24,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/TriangularArray.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.sum_log_taylor_remainder_tendstoInProbability","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.LogTaylor","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":1,"d":"If each row is a probability law, its log likelihood has the stated coordinate representation on the small-increment event, its quadratic sum converges in probability, and its largest coordinate vanishes in probability, the summed quadratic Taylor remainder converges in probability to zero.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (P : (n : ℕ) → MeasureTheory.Measure (Ω n)) (L : (n : ℕ) → Ω n → ℝ) (W : (n : ℕ) → Ω n → Fin n → ℝ) (q : ℝ), (∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (P n)) → (∀ (n : ℕ), ∀ᵐ (x : Ω n) ∂P n, (∀ (i : Fin n), |W n x i| ≤ 1) → L n x = ∑ i, 2 * Real.log (1 + W n x i / 2)) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability P (fun n x => ∑ i, W n x i ^ 2) q → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (P n) {x | ∃ i, ε ≤ |W n x i|}) Filter.atTop (nhds 0)) → Causalean.Estimation.Efficiency.AsymptoticLanConvolution.TendstoInProbability P (fun n x => L n x - ∑ i, W n x i + 1 / 4 * ∑ i, W n x i ^ 2) 0","l":56,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/LogTaylor.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.variance_ge_gradientNormSq_of_dense_scores","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"Compatibility with a paper's tangent-density argument: if every finite score collection gives a variance lower bound for an approximating gradient and those gradients converge in norm to the canonical gradient, then the limiting variance is bounded below by the full gradient norm.","s":"∀ {K : Type u_1} [inst : NormedAddCommGroup K] [NormedSpace ℝ K] (gradient : K) (finiteGradient : ℕ → K) (v : ℝ), (∀ (j : ℕ), ‖finiteGradient j‖ ^ 2 ≤ v) → Filter.Tendsto finiteGradient Filter.atTop (nhds gradient) → ‖gradient‖ ^ 2 ≤ v","l":106,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.variance_id_conv","k":"theorem","m":"Causalean.Estimation.Efficiency.AsymptoticLanConvolution.Variance","a":"Estimation/Efficiency/AsymptoticLanConvolution","t":2,"d":"For probability laws with finite second moments, the variance of their convolution is the sum of their variances.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp id 2 μ → MeasureTheory.MemLp id 2 ν → ProbabilityTheory.variance id (μ.conv ν) = ProbabilityTheory.variance id μ + ProbabilityTheory.variance id ν","l":49,"f":"Causalean/Estimation/Efficiency/AsymptoticLanConvolution/Variance.lean"},{"n":"Causalean.Estimation.Efficiency.BackdoorEstimationSystem.aipw_orthogonal_propensity_score","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"The role of the propensity score (Hahn).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, S.StrictOverlap ε → ∀ (α : γ → ℝ), Measurable α → MeasureTheory.Integrable (fun ω => (S.μ_val true (S.factualX ω) - S.μ_val false (S.factualX ω) - S.θ₀) * α (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => (fun x => α x * (1 - S.e_val x) / S.e_val x) (S.factualX ω) * (S.dVar.indicator true ω * (S.factualY ω - S.μ_val true (S.factualX ω)))) P.μ → MeasureTheory.Integrable (fun ω => (fun x => α x * S.e_val x / (1 - S.e_val x)) (S.factualX ω) * (S.dVar.indicator false ω * (S.factualY ω - S.μ_val false (S.factualX ω)))) P.μ → ∫ (z : γ × Bool × ℝ), S.ψ_AIPW z * (α (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z) * (Causalean.Estimation.ATE.BackdoorEstimationSystem.indA z - S.e_val (Causalean.Estimation.ATE.BackdoorEstimationSystem.projX z))) ∂S.P_Z = 0","l":196,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.Efficiency.BackdoorEstimationSystem.propensity_score_residual_integral_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.ATETangent","a":"Estimation/Efficiency","t":2,"d":"Any covariate-measurable multiplier has zero integral against the propensity-score residual.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) (h : γ → ℝ), Measurable h → MeasureTheory.Integrable (fun ω => h (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω))) P.μ → ∫ (ω : P.Ω), h (S.factualX ω) * (S.dVar.indicator true ω - S.e_val (S.factualX ω)) ∂P.μ = 0","l":82,"f":"Causalean/Estimation/Efficiency/ATETangent.lean"},{"n":"Causalean.Estimation.Efficiency.IsGradient","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space, a linear subspace designated as the tangent space, and two vectors, a reference influence function and a candidate gradient, the candidate is a gradient of the reference relative to the tangent space exactly when, for every vector in that subspace, its inner product with the candidate equals its inner product with the reference.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → Submodule ℝ H → H → H → Prop","l":44,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.IsPathwiseGradient","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space with a distinguished vector, a base law in an abstract collection of laws, a real-valued functional of laws, and a candidate influence-function vector, the candidate is a pathwise gradient of the functional at the base law exactly when, along every regular one-parameter submodel through that law, the functional is differentiable at zero and its derivative equals the inner product of the candidate vector with that submodel's score.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → (𝓛 → ℝ) → H → Prop","l":101,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.IsTangentSpace","k":"structure","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"A candidate closed subspace T of the Hilbert space is a genuine tangent space for the regular submodels through the base law P₀ when every submodel's score lies in T and T is contained in the closed linear span of all submodel scores — together these force T to equal that closed span, the tangent space generated by the available parametric perturbations.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → H → 𝓛 → Submodule ℝ H → Prop","l":87,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.RegularSubmodel","k":"structure","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"A regular submodel through the base law P₀ bundles a one-parameter path of laws, indexed by a real perturbation parameter, that passes through P₀ at parameter zero, together with its tangent direction, an L² score vector required to be mean-zero, i.e. orthogonal to the distinguished vector one.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → Type (max u_1 u_2)","l":54,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.effBound","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space, a tangent subspace admitting an orthogonal projection, and a reference influence function, the semiparametric efficiency bound is the squared norm of its efficient influence function.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → (T : Submodule ℝ H) → [T.HasOrthogonalProjection] → H → ℝ","l":68,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.effBound_eq_of_mem_sub","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"Tangent-shrinking corollary. If the reference influence function lies in the smaller tangent space T' ≤ T, then both the smaller and larger tangent spaces leave it fixed, so the efficiency bound is unchanged. Interpretation: knowing the propensity score shrinks the tangent space, but if ψ_AIPW already lives in the smaller space the efficiency bound does not move.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (T T' : Submodule ℝ H) [inst_2 : T.HasOrthogonalProjection] [inst_3 : T'.HasOrthogonalProjection], T' ≤ T → ∀ {g : H}, g ∈ T' → Causalean.Estimation.Efficiency.efficientIF T' g = g ∧ Causalean.Estimation.Efficiency.efficientIF T g = g ∧ Causalean.Estimation.Efficiency.effBound T' g = Causalean.Estimation.Efficiency.effBound T g","l":181,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.effBound_le_normSq","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Efficiency lower bound. For any function ψ that is a gradient at g, the efficiency bound is at most the squared norm of ψ.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → Causalean.Estimation.Efficiency.effBound T g ≤ ‖ψ‖ ^ 2","l":140,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF","k":"def","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space, a tangent subspace admitting an orthogonal projection, and a reference influence function, the efficient influence function is the orthogonal projection of the reference influence function onto that tangent subspace.","s":"{H : Type u_1} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → (T : Submodule ℝ H) → [T.HasOrthogonalProjection] → H → H","l":57,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"All pathwise gradients share the same projection onto a tangent space. Hence the efficient influence function efficientIF T g does not depend on the chosen pathwise gradient g: it is the canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → Causalean.Estimation.Efficiency.efficientIF T g = Causalean.Estimation.Efficiency.efficientIF T g'","l":202,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_self_of_isPathwiseGradient_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"A pathwise gradient lying in T equals its own projection. This recovers efficientIF_eq_self_of_mem, now as a corollary of \"g is the efficient influence function\": a gradient already in the tangent space is the canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → g ∈ T → Causalean.Estimation.Efficiency.efficientIF T g = g","l":227,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_self_of_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"If the reference gradient already lies in the tangent space, the efficient influence function equals it.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (T : Submodule ℝ H) [inst_2 : T.HasOrthogonalProjection] {g : H}, g ∈ T → Causalean.Estimation.Efficiency.efficientIF T g = g","l":174,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_eq_starProjection","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"efficientIF is the coerced star-projection (the projection seen as an endomorphism of H).","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.efficientIF T g = T.starProjection g","l":80,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_isGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function is itself a gradient of g.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.IsGradient T g (Causalean.Estimation.Efficiency.efficientIF T g)","l":105,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function lies in the tangent space.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] (g : H), Causalean.Estimation.Efficiency.efficientIF T g ∈ T","l":85,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.efficientIF_unique","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A gradient that lies in the tangent space is the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ψ ∈ T → ψ = Causalean.Estimation.Efficiency.efficientIF T g","l":163,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.hasDerivAt_tiltExp","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"The tilt-derivative lemma. Along the exponential tilt with a measurable score s that is bounded in absolute value by a constant M and has mean zero under P, the tilted expectation of a fixed function φ that is almost-everywhere strongly measurable and integrable against P is differentiable at t = 0, with derivative equal to ∫ φ·s dP. (Mean-zero s makes the normalizing-constant contribution vanish, so the derivative is the raw covariance ∫ φ·s dP.)","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s φ : Z → ℝ} {M : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → ∫ (z : Z), s z ∂P = 0 → MeasureTheory.AEStronglyMeasurable φ P → MeasureTheory.Integrable φ P → HasDerivAt (Causalean.Estimation.Efficiency.tiltExp P s φ) (∫ (z : Z), φ z * s z ∂P) 0","l":167,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.hasDerivAt_tilt_numerator","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"Numerator derivative. d/dt ∫ φ·e^{t·s} dP |₀ = ∫ φ·s dP, for bounded measurable score s and integrable φ.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s φ : Z → ℝ} {M : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.AEStronglyMeasurable φ P → MeasureTheory.Integrable φ P → HasDerivAt (fun t => ∫ (z : Z), φ z * Real.exp (t * s z) ∂P) (∫ (z : Z), φ z * s z ∂P) 0","l":103,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.inner_score_eq_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"Two pathwise gradients of the same functional pair identically with every submodel score: ⟪g, s⟫ = ⟪g', s⟫. (Both equal the pathwise derivative d/dt ψ(path t)|₀, which is unique.)","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → ∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel one P₀), inner ℝ g m.score = inner ℝ g' m.score","l":117,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.inner_sub_score_eq_zero_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The difference of two pathwise gradients is orthogonal to every submodel score.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → ∀ (m : Causalean.Estimation.Efficiency.RegularSubmodel one P₀), inner ℝ (g - g') m.score = 0","l":126,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.integral_tiltMeasure","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The expectation of h_fn under the tilted law equals tiltExp P s h_fn t. (No integrability hypothesis is needed: both sides reduce to the same P-integral of h_fn · e^{t·s} scaled by 1/c(t), and a non-integrable h_fn makes both the integral and tiltExp's numerator the same junk default.)","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → ∀ {h_fn : Z → ℝ}, ∫ (z : Z), h_fn z ∂Causalean.Estimation.Efficiency.tiltMeasure P s t = Causalean.Estimation.Efficiency.tiltExp P s h_fn t","l":260,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.isGradient_iff_sub_mem_orthogonal","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A vector is a gradient of g iff it differs from g by an element of the orthogonal complement of the tangent space.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} (g ψ : H), Causalean.Estimation.Efficiency.IsGradient T g ψ ↔ ψ - g ∈ Tᗮ","l":90,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.isGradient_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"Two pathwise gradients differ by an orthogonal-complement element, i.e. either is a TangentProjection-gradient of the other relative to any tangent space T. This is the bridge from the pathwise (HasDerivAt) layer to the inner product IsGradient algebra of TangentProjection.lean.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H} {T : Submodule ℝ H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → Causalean.Estimation.Efficiency.IsGradient T g g'","l":167,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.isPathwiseGradient_eq_efficientIF_of_mem","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The efficient influence function is the unique pathwise gradient in T. If g is any pathwise gradient and g' is a pathwise gradient lying in the tangent space T, then g' = efficientIF T g. Thus a pathwise gradient inside T is uniquely determined and equals the projection of every gradient — the efficient influence function / canonical gradient.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → g' ∈ T → g' = Causalean.Estimation.Efficiency.efficientIF T g","l":215,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.isProbabilityMeasure_tiltMeasure","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The tilted law tiltMeasure P s t is a probability measure.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.Efficiency.tiltMeasure P s t)","l":241,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.normSq_gradient_decomp","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Pythagoras for gradients. For any function ψ that is a gradient at g, the squared norm of ψ splits into the efficiency bound plus the squared norm of the remainder orthogonal to the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ‖ψ‖ ^ 2 = ‖Causalean.Estimation.Efficiency.efficientIF T g‖ ^ 2 + ‖ψ - Causalean.Estimation.Efficiency.efficientIF T g‖ ^ 2","l":122,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.norm_eq_iff_eq_efficientIF","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":1,"d":"Sharpness. For any function ψ that is a gradient at g, ψ attains the efficiency bound if and only if it equals the efficient influence function.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → (‖ψ‖ ^ 2 = Causalean.Estimation.Efficiency.effBound T g ↔ ψ = Causalean.Estimation.Efficiency.efficientIF T g)","l":148,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.orthogonalProjection_eq_of_isGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.TangentProjection","a":"Estimation/Efficiency","t":2,"d":"A gradient ψ projects onto the same efficient influence function as g.","s":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {g ψ : H}, Causalean.Estimation.Efficiency.IsGradient T g ψ → ↑(T.orthogonalProjection ψ) = Causalean.Estimation.Efficiency.efficientIF T g","l":110,"f":"Causalean/Estimation/Efficiency/TangentProjection.lean"},{"n":"Causalean.Estimation.Efficiency.orthogonalProjection_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"The projection of a pathwise gradient is a pathwise gradient. Suppose T is a tangent space for the regular submodels through the base law P₀, and suppose g reproduces the pathwise derivative of the functional ψ along every such submodel — it is a pathwise gradient of ψ at P₀. Then the orthogonal projection of g onto T, written efficientIF T g, is again a pathwise gradient of ψ.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {T : Submodule ℝ H} [inst_2 : T.HasOrthogonalProjection] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g : H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ (Causalean.Estimation.Efficiency.efficientIF T g)","l":183,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.scoreSet","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space with a distinguished vector and a base law in an abstract collection of laws, the score set is the set of score vectors of all regular one-parameter submodels through that base law.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → 𝓛 → Set H","l":71,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.sub_mem_orthogonal_of_isPathwiseGradient","k":"theorem","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":2,"d":"The difference of two pathwise gradients lies in the orthogonal complement of any tangent space T.","s":"∀ {𝓛 : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {one : H} {P₀ : 𝓛} {ψ : 𝓛 → ℝ} {g g' : H} {T : Submodule ℝ H}, Causalean.Estimation.Efficiency.IsTangentSpace one P₀ T → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g → Causalean.Estimation.Efficiency.IsPathwiseGradient one P₀ ψ g' → g - g' ∈ Tᗮ","l":134,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.tangentSpace","k":"def","m":"Causalean.Estimation.Efficiency.PathwiseGradient","a":"Estimation/Efficiency","t":1,"d":"Given a real normed inner-product space with a distinguished vector and a base law in an abstract collection of laws, the tangent space is the topological closure of the linear span of the score set of regular submodels through that base law.","s":"{𝓛 : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → H → 𝓛 → Submodule ℝ H","l":77,"f":"Causalean/Estimation/Efficiency/PathwiseGradient.lean"},{"n":"Causalean.Estimation.Efficiency.tiltExp","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued score function, a real-valued integrand, and a real perturbation parameter, the exponentially tilted expectation is $\\int \\phi(z)\\exp(ts(z))\\,dP(z)$ divided by $\\int \\exp(ts(z))\\,dP(z)$.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → (Z → ℝ) → ℝ → ℝ","l":60,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltMeasure","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued score function, and a real perturbation parameter, the exponentially tilted measure assigns density $\\exp(ts(z))/\\int\\exp(ts(u))\\,dP(u)$ relative to the given measure.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → MeasureTheory.Measure Z","l":70,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm","k":"def","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued score function, and a real perturbation parameter, the normalizing constant of the exponential tilt is $\\int \\exp(ts(z))\\,dP(z)$.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → ℝ","l":52,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm_pos","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"The normalizing constant c(t) = ∫ e^{t·s} dP is strictly positive.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → 0 < Causalean.Estimation.Efficiency.tiltNorm P s t","l":223,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tiltNorm_zero","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"At t = 0 the normalizing constant is the total mass = 1.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ}, Causalean.Estimation.Efficiency.tiltNorm P s 0 = 1","l":80,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.Efficiency.tilt_exp_integrable","k":"theorem","m":"Causalean.Estimation.Efficiency.Submodel","a":"Estimation/Efficiency","t":2,"d":"For a bounded measurable score, e^{t·s} is P-integrable (it is bounded by the constant e^{|t|·M} on the finite measure P).","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {P : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure P] {s : Z → ℝ} {M t : ℝ}, Measurable s → (∀ (z : Z), |s z| ≤ M) → MeasureTheory.Integrable (fun z => Real.exp (t * s z)) P","l":210,"f":"Causalean/Estimation/Efficiency/Submodel.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.AEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the measurable identification of the treatment node’s value space with the binary treatment space is the identity map.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.AIdx m g) ≃ᵐ Bool","l":405,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.AIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the treatment index is the observed treatment node of the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":384,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode","k":"inductive","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The finite backdoor witness-node type consists of the observed covariate node, the treatment node, the outcome node, the latent covariate-draw node, the treatment-noise node, and the outcome-noise node.","s":"Type","l":55,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.instFintype","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The finite enumeration of backdoor witness nodes consists of the six witness nodes, with every witness node included and a proof that this enumeration is exhaustive.","s":"Fintype Causalean.Estimation.MinimaxATE.Causal.WNode","l":77,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.instRepr","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The printable representation of a finite backdoor witness node is the node's corresponding label.","s":"Repr Causalean.Estimation.MinimaxATE.Causal.WNode","l":74,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WNode.repr","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The printable representation of a witness-graph node is its corresponding node label, with the observed covariate, treatment, outcome, the latent covariate draw, treatment noise, and outcome noise given their six respective labels.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → ℕ → Std.Format","l":62,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a covariate space, the value space assigned to each witness-graph node is the covariate space for the observed covariate, the binary space for treatment, the real line for the outcome, the covariate space for the latent covariate draw, the real line for treatment noise, and the real line for outcome noise.","s":"Type → Causalean.Estimation.MinimaxATE.Causal.WNode → Type","l":162,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_borel","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a covariate space equipped with a measurable structure and a standard-Borel structure and each witness node, the standard-Borel structure for that witness-node value space is the given structure for the observed covariate, the binary structure for treatment, the real-line structure for outcome, the given structure for the latent covariate draw, the real-line structure for treatment noise, and the real-line structure for outcome noise.","s":"∀ (C : Type) [inst : MeasurableSpace C] [StandardBorelSpace C] (n : Causalean.Estimation.MinimaxATE.Causal.WNode), StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":183,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_meas","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a covariate space equipped with a measurable structure and each witness node, the measurable-space structure for that witness-node value space is the given covariate measurable structure for the observed covariate, the binary measurable structure for treatment, the real-line measurable structure for outcome, the given covariate measurable structure for the latent covariate draw, the real-line measurable structure for treatment noise, and the real-line measurable structure for outcome noise.","s":"(C : Type) → [MeasurableSpace C] → (n : Causalean.Estimation.MinimaxATE.Causal.WNode) → MeasurableSpace (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":174,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.WΩ_nonempty","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a nonempty covariate space and each witness node, the nonemptiness certificate for that witness-node value space is the given certificate for the observed covariate, a binary value for treatment, a real value for outcome, the given certificate for the latent covariate draw, a real value for treatment noise, and a real value for outcome noise.","s":"∀ (C : Type) [Nonempty C] (n : Causalean.Estimation.MinimaxATE.Causal.WNode), Nonempty (Causalean.Estimation.MinimaxATE.Causal.WΩ C n)","l":192,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.XEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the measurable identification of the covariate node’s value space with the covariate space is the identity map.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.XIdx m g) ≃ᵐ C","l":421,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.XIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the covariate index is the observed covariate node of the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":398,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.YEquiv","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the measurable identification of the outcome node’s value space with the real line is the identity map.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).X (Causalean.Estimation.MinimaxATE.Causal.YIdx m g) ≃ᵐ ℝ","l":413,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.YIdx","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the outcome index is the observed outcome node of the induced potential-outcome system.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).V","l":391,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.causalATE","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the causal average treatment effect is the average treatment effect of the constructed finite backdoor potential-outcome system.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → ℝ","l":78,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.causalATE_eq_ate","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Causal identification bridge. Suppose the data-generating process (m, g) is valid and satisfies strict overlap: the propensity m lies strictly between 0 and 1 at every covariate value. Then the causal average treatment effect E[Y(1) − Y(0)] of the backdoor potential-outcome system built from (m, g) equals the finite observed-data contrast ate g = (1/|C|)·Σₓ(g(1,x) − g(0,x)).","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → Causalean.Estimation.MinimaxATE.Causal.causalATE m g = Causalean.Estimation.MinimaxATE.ate g","l":1571,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.covLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure, the covariate probability law assigns equal probability to every covariate value.","s":"(C : Type) → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → MeasureTheory.Measure C","l":242,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpBES","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable and with a standard Borel structure, a propensity function, an outcome-regression function, evidence that they form a valid data-generating process, and strict overlap of the propensity at every covariate value, the backdoor estimation system is the constructed potential-outcome system equipped with those supplied propensity and outcome-regression functions.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → [inst_4 : StandardBorelSpace C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → Causalean.Estimation.ATE.BackdoorEstimationSystem (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g) C","l":1549,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the backdoor potential-outcome system uses the constructed treatment, outcome, and covariate nodes as its treatment, outcome, and adjustment variables.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → Causalean.PO.POBackdoorSystem (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g) C","l":429,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpFixed","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the background assignment for the witness causal model assigns no fixed values.","s":"{C : Type} → [inst : Fintype C] → [inst_1 : Nonempty C] → [inst_2 : MeasurableSpace C] → [inst_3 : MeasurableSingletonClass C] → (m : C → ℝ) → (g : Bool → C → ℝ) → (Causalean.Estimation.MinimaxATE.Causal.dgpSCM m g).FixedValues","l":369,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function, the induced potential-outcome system is the one generated by the witness structural causal model with its empty background assignment.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.PO.POSystem","l":377,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_borel","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed potential-outcome system has a standard Borel sample space.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω","l":65,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function for both treatment arms, the population law of the constructed potential-outcome system is a probability measure.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":60,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpPO_standardBorel","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable and a standard-Borel structure, a propensity function, and an outcome-regression function for both treatment arms, the sample space of the constructed potential-outcome system has a standard-Borel structure.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, StandardBorelSpace (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω","l":72,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgpSCM","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite, nonempty covariate space equipped with a measurable structure whose singletons are measurable, a propensity function, and an outcome-regression function for the two treatment arms, the structural causal model has a uniformly distributed latent covariate, independent unit-interval treatment and outcome noises, treatment generated by thresholding its noise at the propensity, and outcome generated by thresholding its noise at the arm-specific outcome regression.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.SCM Causalean.Estimation.MinimaxATE.Causal.WNode (Causalean.Estimation.MinimaxATE.Causal.WΩ C)","l":252,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_P_X_eq_covLaw","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The factual covariate marginal of the constructed system is uniform on the finite covariate space.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Measure.map (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ = Causalean.Estimation.MinimaxATE.Causal.covLaw C","l":1513,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_adjustedCE_eq_g","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed adjusted conditional mean equals the supplied outcome regression in each treatment arm.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → ∀ (d : Bool), (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).adjustedCE d =ᵐ[(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ] fun ω => g d ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX ω)","l":1317,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_assumptions","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed finite backdoor system satisfies the standard backdoor assumptions.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [inst_4 : StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).Assumptions","l":1502,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_consistency","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed potential-outcome system satisfies consistency.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Consistency","l":91,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_integrable_Y0","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The control potential outcome is integrable because it is bounded Bernoulli-valued.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Integrable ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).YofD false) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1496,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_integrable_Y1","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The treated potential outcome is integrable because it is bounded Bernoulli-valued.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, MeasureTheory.Integrable ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).YofD true) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1490,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_overlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed propensity satisfies overlap whenever the supplied propensity is strictly between zero and one.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (∀ (x : C), m x ∈ Set.Ioo 0 1) → ∀ᵐ (ω : (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).Ω) ∂(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ, 0 < (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true ω ∧ (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true ω < 1","l":1466,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_propScore_eq_m","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The constructed treatment propensity equals the supplied propensity function given the covariate.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).propScore true =ᵐ[(Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ] fun ω => m ((Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).factualX ω)","l":1305,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.dgp_unconfoundedness","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Bridge","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Unconfoundedness A ⟂ (Y(1), Y(0)) | X. The treatment noise Ea and outcome noise Ey are independent latent roots given the covariate, so the realized treatment is conditionally independent of the potential-outcome bundle given X. Discharged via POSystem.ofSCM_condIndepCF_of_dSep (d-separation in the split graph + value correspondences).","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] [inst_4 : StandardBorelSpace C] {m : C → ℝ} {g : Bool → C → ℝ}, (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).CondIndepCF (Causalean.PO.RegimedVar.ofFactual (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).dVar) (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).cfBundle (Causalean.PO.RegimedVar.ofFactual (Causalean.Estimation.MinimaxATE.Causal.dgpBackdoor m g).xVar) (Causalean.Estimation.MinimaxATE.Causal.dgpPO m g).μ","l":1091,"f":"Causalean/Estimation/MinimaxATE/Causal/Bridge.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instDecidableEqWNode","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"","s":"DecidableEq Causalean.Estimation.MinimaxATE.Causal.WNode","l":58,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instDecidableRelWNodeWEdge","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The procedure deciding whether an ordered pair of witness nodes is an edge examines the finite witness-graph edge relation.","s":"DecidableRel Causalean.Estimation.MinimaxATE.Causal.wEdge","l":100,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instIsProbabilityMeasureCovLaw","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"For a finite, nonempty covariate space equipped with a measurable structure, the uniform covariate law is a probability measure.","s":"∀ (C : Type) [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Causal.covLaw C)","l":246,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.instIsProbabilityMeasureUnifLaw","k":"instance","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The uniform law on the closed unit interval is a probability measure.","s":"MeasureTheory.IsProbabilityMeasure Causalean.Estimation.MinimaxATE.Causal.unifLaw","l":236,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.outFun","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Given a covariate space, an outcome-regression function indexed by binary treatment and covariate value, a binary treatment value, a covariate value, and a real-valued outcome-noise draw, the real-valued outcome is one exactly when the noise draw is no greater than the corresponding outcome-regression value, and is zero otherwise.","s":"(C : Type) → (Bool → C → ℝ) → Bool → C → ℝ → ℝ","l":208,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.parentVal","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Given a covariate space, a child witness node, the supplied values of all parents of that child, a parent witness node, and an arrow from that parent to the child, the extracted parent value is that parent’s supplied value.","s":"(C : Type) → {c : Causalean.Estimation.MinimaxATE.Causal.WNode} → ((w : ↥((Causalean.initialSWIG Causalean.Estimation.MinimaxATE.Causal.wDAG).parents (Causalean.SWIGNode.random c))) → Causalean.swigΩ (Causalean.Estimation.MinimaxATE.Causal.WΩ C) ↑w) → {p : Causalean.Estimation.MinimaxATE.Causal.WNode} → Causalean.Estimation.MinimaxATE.Causal.wEdge p c → Causalean.Estimation.MinimaxATE.Causal.WΩ C p","l":222,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.treatFun","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Given a real-valued propensity score and a real-valued treatment-noise draw, the binary treatment assignment is true exactly when the noise draw is no greater than the propensity score.","s":"ℝ → ℝ → Bool","l":203,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.unifLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The latent-noise probability law is the uniform probability distribution on the closed unit interval $[0,1]$ of the real line.","s":"MeasureTheory.Measure ℝ","l":231,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wDAG","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The directed acyclic graph for the finite backdoor witness is the graph whose nodes are the witness nodes and whose arrows are the specified witness-graph edge relation.","s":"Causalean.DAG Causalean.Estimation.MinimaxATE.Causal.WNode","l":121,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wEdge","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The edge relation of the finite backdoor witness graph contains the arrow from the latent covariate draw to the observed covariate, the arrow from the observed covariate to treatment, the arrow from the observed covariate to outcome, the arrow from treatment to outcome, the arrow from treatment noise to treatment, and the arrow from outcome noise to outcome; every other ordered pair has no arrow.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → Causalean.Estimation.MinimaxATE.Causal.WNode → Prop","l":86,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wParent_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For any edge from node p to node c in the finite backdoor witness graph, the SWIG node for p is a parent of the SWIG node for c in the initial SWIG built from that graph.","s":"∀ {p c : Causalean.Estimation.MinimaxATE.Causal.WNode}, Causalean.Estimation.MinimaxATE.Causal.wEdge p c → Causalean.SWIGNode.random p ∈ (Causalean.initialSWIG Causalean.Estimation.MinimaxATE.Causal.wDAG).parents (Causalean.SWIGNode.random c)","l":215,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wSWIGGraph","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The single-world intervention graph for the finite backdoor witness has the witness directed acyclic graph, no fixed nodes, observed nodes for the covariate, treatment, and outcome, and unobserved nodes for the three latent noise variables.","s":"Causalean.SWIGGraph Causalean.Estimation.MinimaxATE.Causal.WNode","l":128,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wTopo","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"The topological ordering of a witness-graph node assigns rank zero to the latent covariate draw, rank one to treatment noise, rank two to outcome noise, rank three to the observed covariate, rank four to treatment, and rank five to the outcome.","s":"Causalean.Estimation.MinimaxATE.Causal.WNode → ℕ","l":104,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Causal.wTopo_lt","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Construction","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Every edge of the finite backdoor witness graph points strictly forward in the chosen topological order on its nodes.","s":"∀ (u v : Causalean.Estimation.MinimaxATE.Causal.WNode), Causalean.Estimation.MinimaxATE.Causal.wEdge u v → Causalean.Estimation.MinimaxATE.Causal.wTopo u < Causalean.Estimation.MinimaxATE.Causal.wTopo v","l":116,"f":"Causalean/Estimation/MinimaxATE/Causal/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr","k":"structure","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"General-constant-center construction data. This record packages two nonnegative Rademacher-bump magnitudes together with a constant nuisance center (m₀, g₀, g₁) with each coordinate strictly between zero and one, and four further inequalities — the propensity bump smaller than the treated center, the outcome bump at most the treated center, and two worst-case upper bounds keeping the perturbed propensity and treated-arm regression at most one — that together certify the perturbed propensity and outcome-regression functions built from this data stay in the unit interval.","s":"Type","l":43,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, a positive number of covariate pairs, and a nonnegative sample size, the null sample probability law is the joint law of that many independent observations from the null data-generating process.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":81,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null sample law is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QfalseG K n)","l":103,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QfalseG_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null n-sample law charges every point.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QfalseG K n) {ω} ≠ 0","l":131,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QpertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, a positive number of covariate pairs, a nonnegative sample size, and a binary sign vector over the pairs, the perturbed sample probability law is the joint law of that many independent observations from the sign-indexed perturbed data-generating process.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":88,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QpertG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.QpertG K n lam)","l":107,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, a positive number of covariate pairs, and a nonnegative sample size, the alternative sample probability law is the uniform mixture over all binary sign vectors of their perturbed sample laws.","s":"Causalean.Estimation.MinimaxATE.GenConstr → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":96,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QtrueG K n)","l":111,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.QtrueG_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QtrueG K n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPertG lam) (P.gPertG lam) (ω i)","l":150,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_gPertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals (g₁ − g₀) + g₁β(α+β)/(g₁² − β²), independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertG lam) = P.g₁ - P.g₀ + P.g₁ * P.β * (P.α + P.β) / (P.g₁ ^ 2 - P.β ^ 2)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_gapG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals g₁β(α+β)/(g₁² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertG lam) - Causalean.Estimation.MinimaxATE.ate P.ghatG = P.g₁ * P.β * (P.α + P.β) / (P.g₁ ^ 2 - P.β ^ 2)","l":124,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ate_ghatG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null estimate ĝ has ATE g₁ − g₀: every cell contributes g₁ − g₀.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K], Causalean.Estimation.MinimaxATE.ate P.ghatG = P.g₁ - P.g₀","l":64,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqDiv_QtrueG_QfalseG_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The χ² indistinguishability bound (general center): Γ ≤ 1, 2n²(Γ/2)² ≤ K·log 2 imply χ²(QtrueG‖QfalseG) ≤ 1.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (P.QtrueG K n) (P.QfalseG K n) ≤ 1","l":203,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqOverlapG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For a positive number of covariate pairs, general constant-center construction data, a first binary sign vector over the pairs, and a second binary sign vector over the pairs, the single-observation chi-squared second-moment overlap is the sum over all observed records of the product of their two perturbed masses divided by their null mass.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":58,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.chiSqOverlap_eqG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the per-cell coefficient Γ/K times the sum of pairwise sign agreements between lam and lam'.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapG lam lam' = 1 + P.Γ / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)","l":90,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.denomG_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The denominator 1 − (β/g₁)·Δ is positive: Δ ≤ 1 gives 1 − (β/g₁)·Δ ≥ 1 − β/g₁ > 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - P.β / P.g₁ * Causalean.Estimation.MinimaxATE.Δ lam x","l":112,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.g1sq_sub_betasq_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"g₁² − β² > 0 since 0 ≤ β < g₁.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 < P.g₁ ^ 2 - P.β ^ 2","l":46,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For a number of covariate pairs, general constant-center construction data, and a binary sign vector over those pairs, the perturbed outcome-regression function leaves the control arm at its constant center and sets the treated arm to its sign-perturbed numerator divided by one minus the scaled balanced sign perturbation.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":105,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG_true_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Gap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Cleared treated arm. Multiplying numerator and denominator by g₁ turns the inner fraction β/g₁ into the polynomial denominator g₁ − β·Δ: gλ(1,x) = (g₁² + α·g₁·Δ)/(g₁ − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertG lam true x = (P.g₁ ^ 2 + P.α * P.g₁ * Causalean.Estimation.MinimaxATE.Δ lam x) / (P.g₁ - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":50,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.gPertG_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The treated outcome arm: its pointwise deviation from ĝ(1) = g₁ is g₁(α+β)·Δ/(g₁ − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertG lam true x - P.g₁ = P.g₁ * (P.α + P.β) * Causalean.Estimation.MinimaxATE.Δ lam x / (P.g₁ - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":71,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ghatG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, the null outcome-regression function assigns its treated-arm center to every treated observation and its control-arm center to every control observation, at every paired-cell covariate value.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → Bool → Fin K × Bool → ℝ","l":94,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.inClassG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Class membership. Given the squared propensity-perturbation size (m₀·(β/g₁))² within the budget εm and the squared treated-arm outcome-regression perturbation bound g₁²(α+β)²/(g₁ − β)² within the budget εg, the Rademacher-perturbed data-generating process (mλ, gλ) lies in the structure-agnostic nuisance class ℱ(εg, εm) around the constant center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm (P.mPertG lam) (P.gPertG lam)","l":130,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.inClass_nullG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K : ℕ} {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm P.mhatG P.ghatG","l":74,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"For every positive number $K$ of paired cells, the paired-cell covariate space contains the first cell paired with the true binary position.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":55,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_gPertG_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The control outcome arm is unchanged (gλ(0,·) = g₀ = ĝ(0,·)), so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertG lam false) (P.ghatG false) = 0","l":62,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_gPertG_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Uniform bound on the treated arm's squared L² error: every pointwise term is ≤ g₁²(α+β)²/(g₁ − β)², so the average is too.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertG lam true) (P.ghatG true) ≤ P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2","l":81,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.l2sq_mPertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Membership","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The propensity perturbation has exact squared L² size (m₀·(β/g₁))²: it is a Rademacher bump of magnitude m₀·(β/g₁) on Δ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPertG lam) P.mhatG = (P.m₀ * (P.β / P.g₁)) ^ 2","l":47,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.mPertG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For a number of covariate pairs, general constant-center construction data, and a binary sign vector over those pairs, the perturbed propensity function multiplies the constant propensity center by one minus the scaled balanced sign perturbation.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → (Fin K → Bool) → Fin K × Bool → ℝ","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.mhatG","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, the null propensity function assigns the construction’s constant propensity center to every paired-cell covariate value.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.GenConstr → Fin K × Bool → ℝ","l":90,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.minimax_lower_bound_gen","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Structure-agnostic minimax lower bound (general constant center). Fix a constant nuisance center (m₀, g₀, g₁) ∈ (0,1)³ and Rademacher bump magnitudes (α, β). Suppose the squared propensity-perturbation size (m₀·β/g₁)² is within the budget εm and the squared outcome-regression perturbation size g₁²(α+β)²/(g₁−β)² is within the budget εg, with both budgets nonnegative. If further the per-cell overlap coefficient Γ is at most 1 and the sample size obeys the regime 2n²(Γ/2)² ≤ K·log 2, then for every measurable estimator, there is a data-generating process in the structure-agnostic class around this center on which the estimator misses the true ATE by s = g₁β(α+β)/(2(g₁²−β²)) with probability at least 1/4.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatG P.ghatG εg εm n est (P.g₁ * P.β * (P.α + P.β) / (2 * (P.g₁ ^ 2 - P.β ^ 2)))","l":240,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.minimax_lower_bound_mse_gen","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.MSE","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"Structure-agnostic minimax lower bound, expected-risk (MSE) form (general center). Under the same budget and regularity hypotheses as minimax_lower_bound_gen — the squared propensity-perturbation size within the budget εm, the squared outcome-regression perturbation size within the budget εg, both budgets nonnegative, the per-cell overlap coefficient Γ at most 1, and the sample size in the regime 2n²(Γ/2)² ≤ K·log 2 — every measurable estimator has the weaker expected-risk consequence: there is a data-generating process in the class on which the estimator's mean-squared error is at least s²/4, where s = g₁β(α+β)/(2(g₁²−β²)) ≍ √(εg·εm), obtained from the quantile bound by a Chebyshev (1−γ)-factor conversion at γ = 3/4.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (P.m₀ * (P.β / P.g₁)) ^ 2 ≤ εm → P.g₁ ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ - P.β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → ∃ p, (P.g₁ * P.β * (P.α + P.β) / (2 * (P.g₁ ^ 2 - P.β ^ 2))) ^ 2 / 4 ≤ Causalean.Estimation.MinimaxATE.nMSE p.property.valid n est","l":41,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/MSE.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.obsReal_pertG_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Denominator-free value of the perturbed observed mass obsReal mλ gλ at the four (d, y) combinations. The treated arm's denominator 1 − (β/g₁)Δ cancels against mλ = m₀·(1 − (β/g₁)Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPertG lam) (P.gPertG lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ * (P.g₁ + P.α * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ * (1 - P.g₁ - (P.α + P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) | false, true => (1 - P.m₀ + P.m₀ * (P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ | false, false => (1 - P.m₀ + P.m₀ * (P.β / P.g₁) * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀)","l":69,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.one_add_chiSqDiv_QtrueG_QfalseG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Mixture second-moment identity (general center).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.QtrueG K n) (P.QfalseG K n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapG lam lam' ^ n","l":167,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ratio_lt_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"β/g₁ < 1 since β < g₁.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), P.β / P.g₁ < 1","l":87,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"0 < β/g₁ is false in general (β may be 0); but 0 ≤ β/g₁ always.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 ≤ P.β / P.g₁","l":84,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.realG_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"An in-class DGP's n-sample miss probability at its own ATE is dominated by the minimax miss probability.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatG P.ghatG εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatG P.ghatG εg εm n est s","l":119,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.tvDist_QfalseG_QtrueG_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"Total-variation indistinguishability (general center).","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr) {K n : ℕ} [inst : NeZero K], P.Γ ≤ 1 → 2 * ↑n ^ 2 * (P.Γ / 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (P.QfalseG K n) (P.QtrueG K n) ≤ 1 / 2","l":223,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.validDGP_hatG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"The null constant-center data-generating process, with propensity m₀ and outcome regressions (g₀, g₁), is a valid finite observed-data model, i.e. all its component probabilities lie in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr), Causalean.Estimation.MinimaxATE.ValidDGP P.mhatG P.ghatG","l":122,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.validDGP_pertG","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.Construction","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.GenConstr) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPertG lam) (P.gPertG lam)","l":132,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.Γ","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":1,"d":"For general constant-center construction data, the per-cell chi-squared overlap coefficient is the sum of the three stated contributions from the treated-success, treated-failure, and control components of a paired covariate cell.","s":"Causalean.Estimation.MinimaxATE.GenConstr → ℝ","l":50,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.GenConstr.Γ_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterGeneral.LowerBound","a":"Estimation/MinimaxATE/ConstCenterGeneral","t":2,"d":"The per-cell overlap coefficient is nonnegative.","s":"∀ (P : Causalean.Estimation.MinimaxATE.GenConstr), 0 ≤ P.Γ","l":61,"f":"Causalean/Estimation/MinimaxATE/ConstCenterGeneral/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.InClass","k":"structure","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"Structure-agnostic nuisance class. A candidate data-generating process (m, g) belongs to the class ℱ(εg, εm) around fixed nuisance estimates (mhat, ghat) when it is a valid DGP, with propensity and both outcome-regression arms taking values in [0,1], each outcome-regression arm lies within squared L²(P_X) distance εg of the corresponding estimated arm, and the propensity lies within squared L²(P_X) distance εm of the estimated propensity.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (C → ℝ) → (Bool → C → ℝ) → Prop","l":147,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.InClassDGP","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite covariate space, a reference propensity function, a reference outcome-regression function, an outcome-regression error budget, and a propensity error budget, an in-class data-generating process is a pair of propensity and outcome-regression functions together with evidence that the pair belongs to the corresponding nuisance class.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type u_1","l":159,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.MinimaxRateThreshold","k":"structure","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"Minimax rate threshold (reusable vocabulary). An estimation problem over the finite model exhibits a rate threshold when a *single* estimator's worst-case miss is bounded above at one separation (sepUpper), while *every* estimator's worst-case miss is bounded below at a (smaller) separation (sepLower). When sepLower and sepUpper are of the same order, this certifies the minimax rate and that estimator attains it.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → ℕ → Type u_1","l":155,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.Obs","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a covariate space, one observed data record consists of a covariate value, a binary treatment indicator, and a binary outcome indicator.","s":"Type u_1 → Type (max u_1 0)","l":62,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.absolutelyContinuous_of_singleton_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Absolute continuity from full support. If ν charges every singleton, every measure is absolutely continuous w.r.t. ν.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω), (∀ (x : Ω), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":158,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.ate_gNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The ATE of the null DGP is g₁ − g₀.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ : ℝ}, Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) = g₁ - g₀","l":107,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.ate_gPert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The ATE of the perturbed DGP is (g₁ + δ) − g₀.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ δ : ℝ}, Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ) = g₁ + δ - g₀","l":115,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.chiSqDiv_obsPert_obsNull_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The single-observation χ² is nonnegative and equals m₀ δ²/(g₁(1−g₁)).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.obsLaw hvP) (Causalean.Estimation.MinimaxATE.obsLaw hvN) = m₀ * δ ^ 2 / (g₁ * (1 - g₁))","l":233,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.chiSqDiv_productLaw_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"χ² indistinguishability of the n-sample laws. In the regime n · m₀ δ²/(g₁(1−g₁)) ≤ log 2, the χ²-divergence of the n-sample perturbed law from the null is at most 1. Proved by tensorization (one_add_chiSqDiv_pi_iid) and (1 + x)^n ≤ exp(n x) ≤ 2.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} {n : ℕ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.productLaw hvP n) (Causalean.Estimation.MinimaxATE.productLaw hvN n) ≤ 1","l":243,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.gNull","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every covariate set, control-arm outcome mean $g_0$, and treated-arm outcome mean $g_1$, the null outcome-regression function assigns $g_1$ under treatment and $g_0$ under control at every covariate value.","s":"{C : Type u_1} → ℝ → ℝ → Bool → C → ℝ","l":77,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.gPert","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every covariate set, control-arm outcome mean $g_0$, treated-arm baseline outcome mean $g_1$, and real perturbation $\\delta$, the perturbed outcome-regression function assigns $g_1+\\delta$ under treatment and $g_0$ under control at every covariate value.","s":"{C : Type u_1} → ℝ → ℝ → ℝ → Bool → C → ℝ","l":82,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.inClass_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null DGP lies in the class around (mC, gNull) (zero nuisance error).","s":"∀ {C : Type u_1} [inst : Fintype C] {m₀ g₀ g₁ εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) → 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) εg εm (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)","l":137,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.inClass_pert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The perturbed DGP lies in the class around (mC, gNull) when δ² ≤ εg.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m₀ g₀ g₁ δ εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ) → δ ^ 2 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁) εg εm (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)","l":146,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.l2sq_gPert_gNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The squared L² distance between the perturbed and null treated arms is δ²; control arms agree. Hence each arm's error from gNull is ≤ εg once δ² ≤ εg.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {g₀ g₁ δ : ℝ} (d : Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ d) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁ d) = if d = true then δ ^ 2 else 0","l":123,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.mC","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every covariate set and real number $m_0$, the constant propensity function assigns $m_0$ to every covariate value.","s":"{C : Type u_1} → ℝ → C → ℝ","l":73,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.obsLaw_null_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null single-observation law charges every point (its mass is positive).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ : ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), (Causalean.Estimation.MinimaxATE.obsLaw hv) {z} ≠ 0","l":169,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.one_add_chiSqDiv_obsPert_obsNull","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Single-observation χ² (closed form). With μ = obsLaw pert, ν = obsLaw null, the (shifted) χ²-divergence is 1 + m₀ δ²/(g₁(1−g₁)).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → 1 + Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.obsLaw hvP) (Causalean.Estimation.MinimaxATE.obsLaw hvN) = 1 + m₀ * δ ^ 2 / (g₁ * (1 - g₁))","l":188,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.tvDist_productLaw_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Total-variation indistinguishability of the n-sample laws.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m₀ g₀ g₁ δ : ℝ} {n : ℕ} (hvN : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)) (hvP : Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)), 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.productLaw hvN n) (Causalean.Estimation.MinimaxATE.productLaw hvP n) ≤ 1 / 2","l":283,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.validDGP_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Validity of the null DGP.","s":"∀ {C : Type u_1} {m₀ g₀ g₁ : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gNull g₀ g₁)","l":91,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Parametric.validDGP_pert","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Validity of the perturbed DGP (using 0 ≤ δ and g₁ + δ ≤ 1).","s":"∀ {C : Type u_1} {m₀ g₀ g₁ δ : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → 0 ≤ δ → g₁ + δ ≤ 1 → Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.Parametric.mC m₀) (Causalean.Estimation.MinimaxATE.Parametric.gPert g₀ g₁ δ)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every positive number $K$ of paired cells and sample size $n$, the null $n$-sample law is the independent-product distribution of $n$ observations from the centered null data-generating process.","s":"(K n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":65,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.QfalseCont","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every measurable within-cell position space, positive number $K$ of paired cells, sample size $n$, and measure $\\rho$ on the position space, the continuous-covariate null law is the product of the finite-cell null $n$-sample law and the ancillary $n$-sample law generated by $\\rho$.","s":"{W : Type u_1} → [inst : MeasurableSpace W] → (K n : ℕ) → [NeZero K] → MeasureTheory.Measure W → MeasureTheory.Measure ((Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W))","l":68,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.QfalseCont_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate null law is a probability measure.","s":"∀ {K n : ℕ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ)","l":89,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The centered null n-sample law is a probability measure.","s":"∀ (K n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qfalse K n)","l":102,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qfalse_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null n-sample law charges every point (its mass is (8K)⁻ⁿ > 0).","s":"∀ {K n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.Qfalse K n) {ω} ≠ 0","l":74,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qpert","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every positive number $K$ of paired cells, nonnegative treated-outcome bump $\\alpha$, nonnegative propensity bump $\\beta$ satisfying the feasibility condition $\\alpha+2\\beta\\leq 1/2$, sample size $n$, and binary sign vector over the $K$ pairs, the sign-indexed perturbed $n$-sample law is the independent-product distribution of observations from the corresponding perturbed data-generating process.","s":"{K : ℕ} → {α β : ℝ} → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":76,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qpert_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Each perturbed n-sample law is a probability measure in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (n : ℕ) (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qpert hα hβ hαβ n lam)","l":106,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every positive number $K$ of paired cells, nonnegative treated-outcome bump $\\alpha$, nonnegative propensity bump $\\beta$ satisfying the feasibility condition $\\alpha+2\\beta\\leq 1/2$, and sample size $n$, the alternative $n$-sample law is the uniform mixture of the sign-indexed perturbed $n$-sample laws.","s":"{K : ℕ} → {α β : ℝ} → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":86,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.QtrueCont","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every measurable within-cell position space, positive number $K$ of paired cells, nonnegative treated-outcome bump $\\alpha$, nonnegative propensity bump $\\beta$ satisfying the feasibility condition $\\alpha+2\\beta\\leq1/2$, sample size $n$, and measure $\\rho$ on the position space, the continuous-covariate alternative law is the product of the finite-cell mixed alternative $n$-sample law and the ancillary $n$-sample law generated by $\\rho$.","s":"{K : ℕ} → {α β : ℝ} → {W : Type u_1} → [inst : MeasurableSpace W] → [NeZero K] → 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → (n : ℕ) → MeasureTheory.Measure W → MeasureTheory.Measure ((Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W))","l":76,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.QtrueCont_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate alternative law is a probability measure in the valid parameter regime.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ], MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ)","l":95,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The uniformly mixed alternative n-sample law is a probability measure in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n)","l":111,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.Qtrue_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2) (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) (ω i)","l":99,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.TwoPointWitness","k":"structure","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Two-point (Le Cam) witness for the structure-agnostic ATE lower bound: it bundles two n-sample data laws, indexed by a hypothesis label j : Bool, together with the data needed to run Le Cam's two-point method. For each hypothesis, Q j is a probability measure with true average treatment effect θ j; the two hypotheses' ATE values are separated by at least twice the half-scale s, while the total-variation distance between the two laws is bounded by c, and for every estimator the probability under Q j of missing θ j by s is dominated by the in-class minimax miss probability — the realizability condition that a mixture-of-in-class-DGPs construction discharges.","s":"(C : Type u_2) → [Fintype C] → [Nonempty C] → [MeasurableSpace C] → ℕ → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type u_2","l":60,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal","k":"structure","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Two-point Le Cam witness whose realizability target is minimaxMissCausal.","s":"(C : Type) → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → ℕ → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → Type","l":76,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.ValidDGP","k":"structure","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"A data-generating process (m, g) is valid when the propensity m takes values in the unit interval [0,1] and each outcome-regression arm g d also takes values in [0,1].","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → Prop","l":68,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr","k":"structure","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Cell-varying-center construction data for a Rademacher perturbation whose nuisance center may vary by paired cell (Jin–Syrgkanis 2024, Assumption 2, restricted to centers constant within a pair). It packages two bump-magnitude scalars, one on the treated outcome arm and one on the propensity, together with a nuisance center given by the pair-indexed functions m₀, g₀ and g₁ for the propensity and the two potential-outcome regressions, plus the inequalities certifying that both bump magnitudes are nonnegative, the center is pointwise strictly inside (0,1) for m₀, g₀ and g₁, the propensity bump is smaller than the treated-arm center, keeping the perturbation denominator positive, the treated-arm bump does not exceed the treated center, keeping the perturbed outcome regression nonnegative, and worst-case bounds forcing the perturbed outcome regression to stay at most one and the perturbed propensity to stay at most one.","s":"ℕ → Type","l":54,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every positive number $K$ of paired cells, cell-varying construction data, sample size $n$, the null $n$-sample law is the independent-product distribution of $n$ observations from the null cell-varying-center data-generating process.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":67,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QfalseV n)","l":91,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QfalseV_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null n-sample law charges every point.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QfalseV n) {ω} ≠ 0","l":118,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QpertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every positive number $K$ of paired cells, cell-varying construction data, sample size $n$, and binary sign vector over the pairs, the sign-indexed perturbed $n$-sample law is the independent-product distribution of $n$ observations from the corresponding perturbed cell-varying-center data-generating process.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":75,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QpertV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.QpertV n lam)","l":95,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every positive number $K$ of paired cells, cell-varying construction data, sample size $n$, the alternative $n$-sample law is the uniform mixture of the sign-indexed perturbed $n$-sample laws.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":84,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.QtrueV n)","l":99,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.QtrueV_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.QtrueV n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPertV lam) (P.gPertV lam) (ω i)","l":141,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gPertV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals the average over pairs j of 2g₁ⱼ(g₁ⱼ²+αβ)/(g₁ⱼ²−β²) − 2g₀ⱼ, independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, (2 * P.g₁ j * (P.g₁ j ^ 2 + P.α * P.β) / (P.g₁ j ^ 2 - P.β ^ 2) - 2 * P.g₀ j)","l":99,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gapV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals 2β(α+β)/(2K) times the sum over pairs j of g₁ⱼ/(g₁ⱼ² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) - Causalean.Estimation.MinimaxATE.ate P.ghatV = (↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)","l":114,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_gap_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The ATE gap is nonnegative.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam : Fin K → Bool), 0 ≤ Causalean.Estimation.MinimaxATE.ate (P.gPertV lam) - Causalean.Estimation.MinimaxATE.ate P.ghatV","l":130,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ate_ghatV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null estimate ĝ has ATE (1/2K) Σⱼ 2(g₁ⱼ − g₀ⱼ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K), Causalean.Estimation.MinimaxATE.ate P.ghatV = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, 2 * (P.g₁ j - P.g₀ j)","l":63,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqDiv_QtrueV_QfalseV_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The χ² indistinguishability bound (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.chiSqDiv (P.QtrueV n) (P.QfalseV n) ≤ 1","l":194,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqOverlapV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells, cell-varying construction data, and two binary sign vectors over the pairs, the single-observation chi-squared second-moment overlap is the sum, over all observed covariate, treatment, and outcome values, of the product of the two perturbed observed-data probabilities divided by the null observed-data probability.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":70,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.chiSqOverlap_eqV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the sum over pairs j of the per-pair coefficient ΓV j / K times the sign agreement between lam and lam' at pair j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapV lam lam' = 1 + ∑ j, P.ΓV j / ↑K * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))","l":101,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.denomV_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The denominator 1 − (β/g₁ j)·Δ is positive.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - P.β / P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x","l":131,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.g1sq_sub_betasq_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"g₁ⱼ² − β² > 0 since 0 ≤ β < g₁ⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 < P.g₁ j ^ 2 - P.β ^ 2","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells, cell-varying construction data, and binary sign vector over the pairs, the perturbed outcome-regression function assigns the control-arm center of the cell's pair under control and, under treatment, the sign-perturbed treated-arm center divided by one minus the signed propensity-to-treated-mean ratio.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":123,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV_true_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Cleared treated arm. gλ(1,x) = (g₁ x.1² + α·g₁ x.1·Δ)/(g₁ x.1 − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertV lam true x = (P.g₁ x.1 ^ 2 + P.α * P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) / (P.g₁ x.1 - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":50,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.gPertV_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The treated arm's pointwise deviation from ĝ(1,x) = g₁ x.1 is g₁ x.1·(α+β)·Δ/(g₁ x.1 − β·Δ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPertV lam true x - P.g₁ x.1 = P.g₁ x.1 * (P.α + P.β) * Causalean.Estimation.MinimaxATE.Δ lam x / (P.g₁ x.1 - P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":75,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ghatV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells and cell-varying construction data, the cell-varying outcome-regression center assigns the treated-arm center for a cell's pair under treatment and the control-arm center for that pair under control.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Bool → Fin K × Bool → ℝ","l":110,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.inClassV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Class membership. Suppose every pair's propensity-bump budget (m₀ⱼ·(β/g₁ⱼ))² ≤ εm holds and every pair's treated-arm budget g₁ⱼ²(α+β)²/(g₁ⱼ−β)² ≤ εg holds for a nonnegative outcome-error tolerance εg. Then the perturbed data-generating process (mλ, gλ) indexed by a Rademacher sign vector lam lies in the structure-agnostic nuisance class ℱ(εg, εm) around the cell-varying center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm (P.mPertV lam) (P.gPertV lam)","l":126,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.inClass_nullV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm P.mhatV P.ghatV","l":60,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"For every positive number $K$ of paired cells, the paired-cell covariate space contains the first cell paired with the true binary position.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":52,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_gPertV_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The control outcome arm is unchanged, so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertV lam false) (P.ghatV false) = 0","l":68,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_gPertV_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The treated arm's squared L² error is ≤ εg whenever every pair's bound g₁ⱼ²(α+β)²/(g₁ⱼ − β)² is ≤ εg.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εg : ℝ}, (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPertV lam true) (P.ghatV true) ≤ εg","l":86,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.l2sq_mPertV_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"The propensity perturbation's squared L² error is ≤ εm whenever every pair's bump magnitude (m₀ⱼ·(β/g₁ⱼ))² is ≤ εm.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] {εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPertV lam) P.mhatV ≤ εm","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mPertV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells, cell-varying construction data, and binary sign vector over the pairs, the perturbed propensity function assigns to each cell its pair's propensity center multiplied by one minus the propensity-to-treated-mean ratio times that cell's signed perturbation.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → (Fin K → Bool) → Fin K × Bool → ℝ","l":116,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mPertV_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The perturbed VaryingCenterCase1 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K], (∀ (j : Fin K), P.m₀ j * (1 + P.β / P.g₁ j) < 1) → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), P.mPertV lam x ∈ Set.Ioo 0 1","l":128,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mhatV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells and cell-varying construction data, the cell-varying propensity center assigns to each covariate cell the propensity-center value of that cell's pair.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Fin K × Bool → ℝ","l":105,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.mhatV_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The null VaryingCenterCase1 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) [NeZero K] (x : Fin K × Bool), P.mhatV x ∈ Set.Ioo 0 1","l":122,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.minimax_lower_bound_var","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"Structure-agnostic minimax lower bound (cell-varying center). Around a nuisance center P that varies across pairs (constant within each pair), suppose every pair's propensity-bump budget (m₀ⱼ·(β/g₁ⱼ))² ≤ εm holds and every pair's treated-arm budget g₁ⱼ²(α+β)²/(g₁ⱼ−β)² ≤ εg holds for nonnegative error tolerances εg, εm, and the per-pair overlap coefficients satisfy the total-mass bound Σⱼ Γⱼ/K ≤ 1 and the sample-size regularity budget (n²/2)·Σⱼ (Γⱼ/K)² ≤ log 2. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by half the cell-varying gap ate gλ − ate ĝ is at least 1/4.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatV P.ghatV εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)) / 2)","l":233,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.minimax_lower_bound_var_causal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"Causal-centered structure-agnostic minimax lower bound (Case 1). For the outcome-dominant cell-varying construction P, suppose the squared propensity perturbation stays within the budget εm, the squared outcome-regression perturbation stays within the budget εg, the perturbed propensity remains strictly below 1 in every cell, and both budgets are nonnegative. If in addition the aggregate separation budget across cells is at most 1 and the sample size satisfies the stated regularity regime relative to that budget, then for every measurable estimator, the causal-centered miss probability — of missing the true backdoor-identified ATE E[Y(1) − Y(0)] by at least half the displayed Case-1 separation gap — is at least 1/4 for some data-generating process in the class; the strict perturbed-overlap hypothesis is exactly what lets the observed-data Case-1 bound be re-centered onto the genuine causal estimand.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → (∀ (j : Fin K), P.m₀ j * (1 + P.β / P.g₁ j) < 1) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhatV P.ghatV εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β * (P.α + P.β)) * ∑ j, P.g₁ j / (P.g₁ j ^ 2 - P.β ^ 2)) / 2)","l":170,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.obsReal_pertV_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Denominator-free value of the perturbed observed mass at the four (d, y).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPertV lam) (P.gPertV lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ x.1 * (P.g₁ x.1 + P.α * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ x.1 * (1 - P.g₁ x.1 - (P.α + P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) | false, true => (1 - P.m₀ x.1 + P.m₀ x.1 * (P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ x.1 | false, false => (1 - P.m₀ x.1 + P.m₀ x.1 * (P.β / P.g₁ x.1) * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀ x.1)","l":81,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.one_add_chiSqDiv_QtrueV_QfalseV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Mixture second-moment identity (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.QtrueV n) (P.QfalseV n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapV lam lam' ^ n","l":158,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ratio_lt_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"β/g₁ j < 1 since β < g₁ j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), P.β / P.g₁ j < 1","l":102,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"0 ≤ β/g₁ j always.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 ≤ P.β / P.g₁ j","l":99,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.realV_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"An in-class DGP's miss probability at its own ATE is dominated by the minimax miss.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhatV P.ghatV εg εm n est s","l":107,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.realV_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"An in-class DGP's causal-centered miss probability is dominated by minimaxMissCausal.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhatV P.ghatV εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE m g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhatV P.ghatV εg εm n est s","l":155,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.tvDist_QfalseV_QtrueV_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Total-variation indistinguishability (cell-varying center).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.tvDist (P.QfalseV n) (P.QtrueV n) ≤ 1 / 2","l":215,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.validDGP_hatV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"The null cell-varying-center data-generating process, with propensity mhatV and outcome regressions ghatV, is a valid finite observed-data model, i.e. all its component probabilities lie in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K), Causalean.Estimation.MinimaxATE.ValidDGP P.mhatV P.ghatV","l":140,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.validDGP_pertV","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPertV lam) (P.gPertV lam)","l":150,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ΓV","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For every number $K$ of paired cells, cell-varying construction data, and pair $j$, the per-pair chi-squared-overlap coefficient is the sum of the three nonnegative terms determined by that pair's propensity center, treated-arm outcome center, and the two bump magnitudes.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr K → Fin K → ℝ","l":48,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr.ΓV_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":2,"d":"Γⱼ ≥ 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr K) (j : Fin K), 0 ≤ P.ΓV j","l":57,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2","k":"structure","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Second (propensity-dominant) cell-varying construction data for the same style of Rademacher perturbation as VarConstr but with the roles of the two nuisances swapped, so the propensity carries the larger deviation. It packages two bump-magnitude scalars, the larger on the propensity and the smaller on the treated outcome arm, together with a nuisance center given by the pair-indexed functions m₀, g₀ and g₁ for the propensity and the two potential-outcome regressions, plus the inequalities certifying that both bump magnitudes are nonnegative, the center is pointwise strictly inside (0,1) for m₀, g₀ and g₁, a worst-case bound forcing the perturbed treated outcome regression to stay at most one, which also keeps the perturbation denominator positive, the propensity bump coefficient not to exceed one, keeping the perturbed propensity nonnegative, and the perturbed propensity to stay at most one.","s":"ℕ → Type","l":57,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.D2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells, a binary sign vector indexing a perturbation, and a covariate cell, the treated-arm denominator is $1+(\\beta/g_{1j})\\Delta-\\alpha\\beta$, with $j$ the pair containing that cell and $\\Delta$ its signed perturbation.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Fin K × Bool → ℝ","l":108,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.D2_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The denominator D = 1 + (β/g₁)·Δ − αβ is positive (at least g₁ > 0).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), 0 < P.D2 lam x","l":134,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and a sample size, provided that there is at least one pair, the null sample law is the joint distribution of that many independent observed records generated from the construction's unperturbed propensity and outcome regressions.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":61,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.Qfalse2 n)","l":86,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qfalse2_singleton_ne_zero","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null n-sample law charges every point.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.Qfalse2 n) {ω} ≠ 0","l":113,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qpert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells, a sample size, and a binary sign vector indexing a perturbation, provided that there is at least one pair, the perturbed sample law is the joint distribution of that many independent observed records generated from the corresponding perturbed propensity and outcome regressions.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → (Fin K → Bool) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":69,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qpert2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Each perturbed sample law is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K] (lam : Fin K → Bool), MeasureTheory.IsProbabilityMeasure (P.Qpert2 n lam)","l":90,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and a sample size, provided that there is at least one pair, the alternative sample law is the uniform mixture, over all binary sign vectors, of the corresponding perturbed independent-sample laws.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (n : ℕ) → [NeZero K] → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool))","l":78,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2_isProb","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The Rademacher mixture of perturbed sample laws is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (n : ℕ) [inst : NeZero K], MeasureTheory.IsProbabilityMeasure (P.Qtrue2 n)","l":94,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.Qtrue2_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The alternative law's .real point mass: a uniform mixture over sign vectors.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (P.Qtrue2 n).real {ω} = ∑ lam, (2 ^ K)⁻¹ * ∏ i, Causalean.Estimation.MinimaxATE.obsReal (P.mPert2 lam) (P.gPert2 lam) (ω i)","l":136,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.alphabeta_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"αβ ≤ 1 (a consequence of hgU and g₁ⱼ > 0).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), P.α * P.β ≤ 1","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gPert2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Exact ATE of the perturbed construction. For any Rademacher sign vector lam, the average treatment effect of the perturbed outcome regression equals the average over pairs j of 2g₁ⱼ³(1−αβ)/(g₁ⱼ²(1−αβ)²−β²) − 2g₀ⱼ, independent of lam.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, (2 * P.g₁ j ^ 3 * (1 - P.α * P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2) - 2 * P.g₀ j)","l":146,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gap2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any Rademacher sign vector lam, the gap between the perturbed construction's ATE and the null estimate's ATE equals 2β/(2K) times the sum over pairs j of g₁ⱼ(αg₁ⱼ²(1−αβ) + β)/(g₁ⱼ²(1−αβ)² − β²).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) - Causalean.Estimation.MinimaxATE.ate P.ghat2 = (↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)","l":161,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_gap2_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The ATE gap is nonnegative.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam : Fin K → Bool), 0 ≤ Causalean.Estimation.MinimaxATE.ate (P.gPert2 lam) - Causalean.Estimation.MinimaxATE.ate P.ghat2","l":179,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ate_ghat2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null estimate ĝ has ATE (1/2K) Σⱼ 2(g₁ⱼ − g₀ⱼ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K), Causalean.Estimation.MinimaxATE.ate P.ghat2 = (↑(Fintype.card (Fin K × Bool)))⁻¹ * ∑ j, 2 * (P.g₁ j - P.g₀ j)","l":91,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqDiv_Qtrue2_Qfalse2_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The χ² indistinguishability bound (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.chiSqDiv (P.Qtrue2 n) (P.Qfalse2 n) ≤ 1","l":189,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqOverlapV2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and two binary sign vectors indexing perturbations, the single-observation chi-squared second-moment overlap is the sum, over every possible observed record $z$, of $q_{\\lambda}(z)q_{\\lambda'}(z)/p_0(z)$, where $q_{\\lambda}$ and $q_{\\lambda'}$ are the two perturbed record probabilities and $p_0$ is the unperturbed record probability.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":69,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.chiSqOverlap_eq2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any two Rademacher sign vectors lam and lam' indexing perturbed data-generating processes, the single-observation χ² overlap between them equals one plus the sum over pairs j of the propensity-dominant coefficient ΓV2 j / K times the sign agreement between lam and lam' at pair j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (lam lam' : Fin K → Bool), P.chiSqOverlapV2 lam lam' = 1 + ∑ j, P.ΓV2 j / ↑K * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))","l":110,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.clearedDenom_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The cleared denominator g₁ⱼ(1 − αβ) + β·σ is positive.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K) (σ : ℝ), σ = 1 ∨ σ = -1 → 0 < P.g₁ j * (1 - P.α * P.β) + P.β * σ","l":81,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.denomE_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The pair denominator Eⱼ = g₁ⱼ²(1 − αβ)² − β² > 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 < P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2","l":59,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.g1_one_sub_ab_sub_beta","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"g₁ⱼ(1 − αβ) − β ≥ g₁ⱼ² > 0: clearing hgU by g₁ⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), P.g₁ j ^ 2 ≤ P.g₁ j * (1 - P.α * P.β) - P.β","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and a binary sign vector indexing a perturbation, the perturbed outcome-regression function equals the baseline control-arm regression for control observations and the baseline treated-arm regression divided by the treated-arm denominator for treated observations.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":164,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2_true_eq2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Gap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Cleared treated arm. gλ(1,x) = g₁ x.1² / (g₁ x.1·(1 − αβ) + β·Δ) — no nested fractions, so field_simp can clear it given the (positive) denominator.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPert2 lam true x = P.g₁ x.1 ^ 2 / (P.g₁ x.1 * (1 - P.α * P.β) + P.β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":69,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.gPert2_true_sub","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The treated arm's pointwise deviation from ĝ(1,x) = g₁ x.1 is β·(α g₁ x.1 − Δ)/D.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.gPert2 lam true x - P.g₁ x.1 = P.β * (P.α * P.g₁ x.1 - Causalean.Estimation.MinimaxATE.Δ lam x) / P.D2 lam x","l":71,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ghat2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells, the cell-varying outcome-regression center assigns the treated-arm baseline outcome regression to treated observations and the control-arm baseline outcome regression to control observations, using the baseline associated with the cell's pair.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Bool → Fin K × Bool → ℝ","l":149,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.inClass2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Class membership. Fix a sign pattern lam and nuisance-error budgets εg, εm. If the squared quantity (m₀ⱼ·κⱼ)² is at most εm at every cell j, the worst-case squared deviation of the perturbed treated-arm outcome regression is at most εg at every cell j, and εg is nonnegative, then the perturbed propensity and outcome-regression pair (mλ, gλ) lies in the structure-agnostic nuisance class ℱ(εg, εm) around the cell-varying center (m̂, ĝ).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εg εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → 0 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm (P.mPert2 lam) (P.gPert2 lam)","l":137,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.inClass_null2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null estimate is itself in the class (zero nuisance error).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm P.mhat2 P.ghat2","l":54,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.instNonemptyFinBoolProd","k":"instance","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"For every positive number $K$ of paired cells, the paired-cell covariate space contains the first cell paired with the true binary position.","s":"∀ {K : ℕ} [NeZero K], Nonempty (Fin K × Bool)","l":46,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_gPert2_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The control outcome arm is unchanged, so its L² error is 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPert2 lam false) (P.ghat2 false) = 0","l":64,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_gPert2_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The treated arm's squared L² error is ≤ εg whenever every pair satisfies β²·(α g₁ⱼ + 1)²/(1 − β/g₁ⱼ − αβ)² ≤ εg.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εg : ℝ}, (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.gPert2 lam true) (P.ghat2 true) ≤ εg","l":84,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.l2sq_mPert2_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Membership","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The propensity perturbation's squared L² error is exactly the average of (m₀ⱼ·κⱼ)², hence ≤ εm whenever every pair satisfies (m₀ⱼ·κⱼ)² ≤ εm.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] {εm : ℝ}, (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (P.mPert2 lam) P.mhat2 ≤ εm","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and a binary sign vector indexing a perturbation, the perturbed propensity function assigns each covariate cell the baseline propensity times $(1+\\alpha g_{1j}\\Delta)$ times the treated-arm denominator, where $j$ is the cell's pair and $\\Delta$ is its signed perturbation.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → (Fin K → Bool) → Fin K × Bool → ℝ","l":156,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Affine collapse of the propensity. For any Rademacher sign vector lam and cell x, the perturbed propensity at x equals m₀(x.1)·(1 + κ(x.1)·Δ(lam,x)), an exactly affine function of the perturbation.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool), P.mPert2 lam x = P.m₀ x.1 * (1 + P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x)","l":172,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mPert2_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The perturbed VaryingCenterCase2 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K], (∀ (j : Fin K), P.κ j < 1) → (∀ (j : Fin K), P.m₀ j * (1 + P.κ j) < 1) → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), P.mPert2 lam x ∈ Set.Ioo 0 1","l":267,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mhat2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells, the cell-varying propensity center assigns to each covariate cell the construction's baseline propensity for that cell's pair.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K × Bool → ℝ","l":144,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.mhat2_strictOverlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"The null VaryingCenterCase2 witness has strict propensity overlap.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) [NeZero K] (x : Fin K × Bool), P.mhat2 x ∈ Set.Ioo 0 1","l":261,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.minimax_lower_bound_var2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"Structure-agnostic minimax lower bound (second / propensity-dominant construction). Fix a cell-varying nuisance center P, sample size n, and nuisance-error budgets εg, εm, and suppose the treated-arm outcome bump magnitude is strictly positive. If the perturbed propensity's squared deviation from its center is at most εm in every cell, the perturbed treated-arm outcome regression's squared deviation from its center is at most εg in every cell, εm is nonnegative, εg is nonnegative, the normalized per-pair χ² coefficients sum to at most 1, and n²/2 times the sum of their squares is at most log 2, then for every measurable estimator, the worst-case-over-class probability that the estimator's error exceeds half of the displayed strictly positive separation gap is at least 1/4.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, 0 < P.β → (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhat2 P.ghat2 εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)) / 2)","l":228,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.minimax_lower_bound_var2_causal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Causal-centered structure-agnostic minimax lower bound (Case 2). For the propensity-dominant cell-varying construction, a strictly positive treated-arm bump and strict perturbed-propensity overlap imply that every measurable estimator has causal-centered miss probability at least 1/4 at half of the displayed strictly positive ATE separation.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ}, 0 < P.β → (∀ (j : Fin K), (P.m₀ j * P.κ j) ^ 2 ≤ εm) → (∀ (j : Fin K), P.β ^ 2 * (P.α * P.g₁ j + 1) ^ 2 / (1 - P.β / P.g₁ j - P.α * P.β) ^ 2 ≤ εg) → (∀ (j : Fin K), P.κ j < 1) → (∀ (j : Fin K), P.m₀ j * (1 + P.κ j) < 1) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhat2 P.ghat2 εg εm n est (((↑(Fintype.card (Fin K × Bool)))⁻¹ * (2 * P.β) * ∑ j, P.g₁ j * (P.α * P.g₁ j ^ 2 * (1 - P.α * P.β) + P.β) / (P.g₁ j ^ 2 * (1 - P.α * P.β) ^ 2 - P.β ^ 2)) / 2)","l":312,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.obsReal_pert2_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Denominator-free value of the perturbed observed mass at the four (d, y).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (P.mPert2 lam) (P.gPert2 lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => P.m₀ x.1 * (P.g₁ x.1 + P.α * P.g₁ x.1 ^ 2 * Causalean.Estimation.MinimaxATE.Δ lam x) | true, false => P.m₀ x.1 * (1 + P.α * P.g₁ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * (P.D2 lam x - P.g₁ x.1) | false, true => (1 - P.m₀ x.1 - P.m₀ x.1 * P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * P.g₀ x.1 | false, false => (1 - P.m₀ x.1 - P.m₀ x.1 * P.κ x.1 * Causalean.Estimation.MinimaxATE.Δ lam x) * (1 - P.g₀ x.1)","l":81,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.one_add_chiSqDiv_Qtrue2_Qfalse2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Mixture second-moment identity (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], 1 + Causalean.Stat.chiSqDiv (P.Qtrue2 n) (P.Qfalse2 n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * P.chiSqOverlapV2 lam lam' ^ n","l":153,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ratio_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"0 ≤ β/g₁ j.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.β / P.g₁ j","l":115,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.real2_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"An in-class DGP's miss probability at its own ATE is dominated by the minimax miss.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss P.mhat2 P.ghat2 εg εm n est s","l":102,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.real2_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"An in-class Case-2 DGP's causal-centered miss probability is dominated by minimaxMissCausal.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K] {εg εm : ℝ} {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass P.mhat2 P.ghat2 εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE m g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal P.mhat2 P.ghat2 εg εm n est s","l":297,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.tvDist_Qfalse2_Qtrue2_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.LowerBound","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Total-variation indistinguishability (second construction).","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) {n : ℕ} [inst : NeZero K], ∑ j, P.ΓV2 j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV2 j / ↑K) ^ 2 ≤ Real.log 2 → Causalean.Stat.tvDist (P.Qfalse2 n) (P.Qtrue2 n) ≤ 1 / 2","l":210,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/LowerBound.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.validDGP_hat2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"The null DGP (m̂, ĝ) is valid.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K), Causalean.Estimation.MinimaxATE.ValidDGP P.mhat2 P.ghat2","l":186,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.validDGP_pert2","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For any Rademacher sign vector lam indexing the perturbation, the perturbed propensity and outcome-regression functions define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (P.mPert2 lam) (P.gPert2 lam)","l":194,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ΓV2","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and one pair of cells, the pair's chi-squared-overlap coefficient is $m_{0j}\\alpha^2g_{1j}^3 + m_{0j}(\\beta/g_{1j}+\\alpha g_{1j}(1-g_{1j})-\\alpha^2\\beta g_{1j})^2/(1-g_{1j}) + m_{0j}^2\\kappa_j^2/(1-m_{0j})$.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K → ℝ","l":47,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.ΓV2_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.ChiSqOverlap","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"Γⱼ ≥ 0.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.ΓV2 j","l":57,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.κ","k":"def","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":1,"d":"For a propensity-dominant construction with a specified number of paired covariate cells and a pair of cells, the propensity bump coefficient is $\\beta/g_{1j}+\\alpha g_{1j}-\\alpha^2\\beta g_{1j}$, where the construction supplies the bump magnitudes and the treated-arm outcome-regression center.","s":"{K : ℕ} → Causalean.Estimation.MinimaxATE.VarConstr2 K → Fin K → ℝ","l":101,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.VarConstr2.κ_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase2.Construction","a":"Estimation/MinimaxATE/VaryingCenterCase2","t":2,"d":"0 ≤ κⱼ.","s":"∀ {K : ℕ} (P : Causalean.Estimation.MinimaxATE.VarConstr2 K) (j : Fin K), 0 ≤ P.κ j","l":118,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase2/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.absolutelyContinuous_of_singleton_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Absolute continuity from full support. If ν charges every singleton, every measure is absolutely continuous w.r.t. ν (a ν-null set must be empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω), (∀ (x : Ω), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipwScoreFin","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"For a finite covariate set, a fitted propensity function, fitted outcome-regression functions for the two treatment arms, and an observed covariate, treatment, and binary outcome record, the fixed-center augmented inverse-probability-weighted score is the fitted treated-minus-control contrast plus the treated residual divided by the fitted propensity when treated, or minus the control residual divided by one minus that propensity when untreated.","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → Causalean.Estimation.MinimaxATE.Obs C → ℝ","l":62,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipwScore_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Score bound. The AIPW score is bounded by B = 1 + 2/ε whenever the center nuisances are valid and mhat is ε-bounded off {0,1}.","s":"∀ {C : Type u_1} [Nonempty C] (mhat : C → ℝ) (ghat : Bool → C → ℝ), Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), |Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z| ≤ 1 + 2 / ε","l":265,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_attains_minimax_rate","k":"def","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"For a cell-varying construction over a nonzero finite number of paired covariate cells, a positive sample size, and nonnegative outcome-regression and propensity-error budgets, suppose for every pair $j$, the squared propensity perturbation $(m_{0j}\\beta/g_{1j})^2$ is at most $\\varepsilon_m$, for every pair $j$, the squared treated-arm outcome-regression perturbation $g_{1j}^2(\\alpha+\\beta)^2/(g_{1j}-\\beta)^2$ is at most $\\varepsilon_g$, the normalized overlap coefficients satisfy $\\sum_j \\Gamma_j/K\\leq 1$, and their normalized squared sum satisfies $(n^2/2)\\sum_j(\\Gamma_j/K)^2\\leq\\log 2$. For a positive overlap constant such that the fitted propensity is between $\\varepsilon$ and $1-\\varepsilon$ in every covariate cell, and a separation $s$ strictly larger than $\\varepsilon^{-1}2\\sqrt{\\varepsilon_g}\\sqrt{\\varepsilon_m}$, the minimax-rate threshold certificate specifies the fixed-center augmented inverse-probability-weighted estimator, its upper miss-probability bound at $s$, and the construction-induced lower miss-probability bound for every measurable estimator.","s":"{K : ℕ} → (P : Causalean.Estimation.MinimaxATE.VarConstr K) → [inst : NeZero K] → {n : ℕ} → 0 < n → {εg εm : ℝ} → (∀ (j : Fin K), (P.m₀ j * (P.β / P.g₁ j)) ^ 2 ≤ εm) → (∀ (j : Fin K), P.g₁ j ^ 2 * (P.α + P.β) ^ 2 / (P.g₁ j - P.β) ^ 2 ≤ εg) → 0 ≤ εg → 0 ≤ εm → ∑ j, P.ΓV j / ↑K ≤ 1 → ↑n ^ 2 / 2 * ∑ j, (P.ΓV j / ↑K) ^ 2 ≤ Real.log 2 → {ε : ℝ} → 0 < ε → (∀ (x : Fin K × Bool), ε ≤ P.mhatV x ∧ ε ≤ 1 - P.mhatV x) → {s : ℝ} → ε⁻¹ * (2 * √εg * √εm) < s → Causalean.Estimation.MinimaxATE.MinimaxRateThreshold P.mhatV P.ghatV εg εm n","l":177,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_bias_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Doubly-robust bias bound. Given a positive slack ε such that the fitted propensity mhat stays at least ε away from both 0 and 1 at every covariate value — the center has uniform overlap, the absolute bias of the population AIPW mean relative to the true ATE is at most ε⁻¹ times the product of the combined treated/control L²(P_X) outcome-regression error and the L²(P_X) propensity error.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (mhat : C → ℝ) (ghat : Bool → C → ℝ) {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ ε⁻¹ * ((√(Causalean.Estimation.MinimaxATE.l2sq (g true) (ghat true)) + √(Causalean.Estimation.MinimaxATE.l2sq (g false) (ghat false))) * √(Causalean.Estimation.MinimaxATE.l2sq m mhat))","l":160,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_bias_identity","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Doubly-robust bias identity. Assume the fitted propensity mhat takes values strictly between 0 and 1 at every covariate value. Then the population mean of the AIPW score under the true data-generating process (m, g), minus the true average treatment effect, equals a finite doubly-robust remainder built from cell-by-cell products of the propensity error m − mhat and the outcome-regression errors g − ghat on each treatment arm.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (mhat : C → ℝ) (ghat : Bool → C → ℝ), (∀ (x : C), 0 < mhat x) → (∀ (x : C), mhat x < 1) → ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g = (↑(Fintype.card C))⁻¹ * ∑ x, ((1 - m x / mhat x) * (ghat true x - g true x) - (1 - (1 - m x) / (1 - mhat x)) * (ghat false x - g false x))","l":106,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_inclass_bias_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"In-class bias bound. For a positive overlap threshold ε such that the fixed propensity estimate mhat stays at least ε away from both 0 and 1 at every covariate value, and for any data-generating process (m, g) lying in the structure-agnostic nuisance class around the fixed estimates (mhat, ghat) with error budgets εg, εm, the plug-in bias of the fixed-center AIPW estimator's population score is at most ε⁻¹·2·√εg·√εm.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm m g → |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ ε⁻¹ * (2 * √εg * √εm)","l":108,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_mean_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":2,"d":"The mean of the AIPW estimator equals the single-observation population mean of the score (mean of an i.i.d. average), for n > 0.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ) {n : ℕ}, 0 < n → ∫ (sample : Fin n → Causalean.Estimation.MinimaxATE.Obs C), Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n sample ∂Causalean.Estimation.MinimaxATE.productLaw hv n = ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z","l":51,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_minimaxMiss_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":1,"d":"Worst-case (minimax) miss bound for AIPW. Suppose the fixed nuisance estimates (mhat, ghat) are themselves valid, for a positive overlap threshold ε such that mhat stays at least ε away from 0 and 1 everywhere, nonnegative error budgets εg, εm, a positive sample size n, and a separation s exceeding the uniform bias bound b = ε⁻¹·2·√εg·√εm. Then the worst-case-over-class miss probability of the fixed-center AIPW estimator at separation s is at most ((1+2/ε)²/n)/(s−b)².","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → 0 ≤ εg → 0 ≤ εm → ∀ {n : ℕ}, 0 < n → ∀ {s : ℝ}, ε⁻¹ * (2 * √εg * √εm) < s → Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) s ≤ (1 + 2 / ε) ^ 2 / ↑n / (s - ε⁻¹ * (2 * √εg * √εm)) ^ 2","l":133,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_nMiss_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Optimality","a":"Estimation/MinimaxATE","t":2,"d":"Per-DGP Chebyshev miss bound. If the plug-in bias is ≤ b and the variance is ≤ V, then for any separation s > b the miss probability is ≤ V / (s − b)².","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ) {n : ℕ}, 0 < n → ∀ {b V s : ℝ}, |∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z - Causalean.Estimation.MinimaxATE.ate g| ≤ b → ProbabilityTheory.variance (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) (Causalean.Estimation.MinimaxATE.productLaw hv n) ≤ V → b < s → Causalean.Estimation.MinimaxATE.nMiss hv n (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) s ≤ V / (s - b) ^ 2","l":73,"f":"Causalean/Estimation/MinimaxATE/Optimality.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_pop_mean","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Population mean of the AIPW score under the DGP (m, g) is the finite sum Σ_z obsReal m g z · ψ_AIPW(z; m̂, ĝ).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ), ∫ (z : Causalean.Estimation.MinimaxATE.Obs C), Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z ∂Causalean.Estimation.MinimaxATE.obsLaw hv = ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z * Causalean.Estimation.MinimaxATE.aipwScoreFin mhat ghat z","l":91,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.aipw_var_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"Variance bound. Suppose the true data-generating process (m, g) is valid, the fixed nuisance center (mhat, ghat) is itself a valid data-generating process, and that center has uniform overlap: the fitted propensity mhat stays at least a positive constant ε away from both 0 and 1 at every covariate value. Then on an n-observation i.i.d. sample from the true single-observation law, the variance of the fixed-center AIPW sample-average estimator is at most (1 + 2/ε)²/n.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [MeasurableSingletonClass C] [inst_3 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (mhat : C → ℝ) (ghat : Bool → C → ℝ), Causalean.Estimation.MinimaxATE.ValidDGP mhat ghat → ∀ {ε : ℝ}, 0 < ε → (∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x) → ∀ (n : ℕ), ProbabilityTheory.variance (Causalean.Estimation.MinimaxATE.estAIPW mhat ghat n) (Causalean.Estimation.MinimaxATE.productLaw hv n) ≤ (1 + 2 / ε) ^ 2 / ↑n","l":300,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.ancillaryLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every measurable within-cell position space, measure $\\rho$ on that space, and sample size $n$, the ancillary $n$-sample law is the independent-product measure with one-draw factor $\\rho$.","s":"{W : Type u_1} → [inst : MeasurableSpace W] → MeasureTheory.Measure W → (n : ℕ) → MeasureTheory.Measure (Fin n → W)","l":54,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.ancillaryLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"For every measurable within-cell position space, probability measure $\\rho$ on that space, and sample size $n$, the ancillary $n$-sample law is a probability measure.","s":"∀ {W : Type u_1} [inst : MeasurableSpace W] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.ancillaryLaw ρ n)","l":60,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite covariate space and an outcome-regression function for binary treatment, the average treatment effect is the uniform average over covariate values of the treated-arm regression minus the control-arm regression.","s":"{C : Type u_1} → [Fintype C] → (Bool → C → ℝ) → ℝ","l":135,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_gPerturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Exact ATE of the perturbed construction. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 and any Rademacher sign vector lam, the average treatment effect of the perturbed construction equals 2β(α+β)/(1−4β²), independent of lam.","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) = 2 * β * (α + β) / (1 - 4 * β ^ 2)","l":81,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_gap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 and any Rademacher sign vector lam, the gap between the perturbed construction's average treatment effect and the null estimate's average treatment effect (which is zero) equals 2β(α+β)/(1−4β²).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ate (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) - Causalean.Estimation.MinimaxATE.ate Causalean.Estimation.MinimaxATE.ghat = 2 * β * (α + β) / (1 - 4 * β ^ 2)","l":110,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.ate_ghat","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Gap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The null estimate ĝ ≡ 1/2 has zero ATE: every cell contributes 1/2 − 1/2 = 0.","s":"∀ {K : ℕ}, Causalean.Estimation.MinimaxATE.ate Causalean.Estimation.MinimaxATE.ghat = 0","l":44,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Gap.lean"},{"n":"Causalean.Estimation.MinimaxATE.bddAbove_nMissCausal_range","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Each in-class causal-centered miss probability is bounded above by 1.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] (mhat : C → ℝ) (ghat : Bool → C → ℝ) (εg εm : ℝ) (n : ℕ) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) (s : ℝ), BddAbove (Set.range fun p => (Causalean.Estimation.MinimaxATE.productLaw p.property.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE (↑p).1 (↑p).2|})","l":50,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.bddAbove_nMiss_range","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"Each in-class miss probability is bounded above by 1 (it is a probability), so the range of nMiss over the class is bounded above.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] (mhat : C → ℝ) (ghat : Bool → C → ℝ) (εg εm : ℝ) (n : ℕ) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) (s : ℝ), BddAbove (Set.range fun p => Causalean.Estimation.MinimaxATE.nMiss p.property.valid n est s)","l":243,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqDiv_QtrueCont_QfalseCont_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate alternative is chi-squared close to the null in the finite-cell regime.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ) (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) ≤ 1","l":104,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqDiv_Qtrue_Qfalse_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The χ² indistinguishability bound. In the regime 2n²γ² ≤ K·log 2 (with γ = α²+2αβ+3β², 2γ ≤ 1), the χ²-divergence of the alternative mixture from the null is at most 1.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) (Causalean.Estimation.MinimaxATE.Qfalse K n) ≤ 1","l":157,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqOverlap","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For a number of paired covariate cells, two real-valued bump magnitudes, and two binary sign vectors indexing perturbations, the one-observation chi-squared second-moment overlap is the sum, over every possible observed record $z$, of $q_{\\lambda}(z)q_{\\lambda'}(z)/p_0(z)$, where $q_{\\lambda}$ and $q_{\\lambda'}$ are the two perturbed record probabilities and $p_0$ is the unperturbed record probability.","s":"{K : ℕ} → ℝ → ℝ → (Fin K → Bool) → (Fin K → Bool) → ℝ","l":49,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.chiSqOverlap_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Closed form of the one-observation χ² overlap. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, the regime keeping every perturbed nuisance in [0,1], and any two Rademacher sign vectors lam, lam' indexing perturbed laws, the one-observation χ² second-moment overlap of the two perturbed laws relative to the null equals 1 + (2(α²+2αβ+3β²)/K)·Σⱼ signOf(lam j)·signOf(lam' j).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam lam' : Fin K → Bool), Causalean.Estimation.MinimaxATE.chiSqOverlap α β lam lam' = 1 + 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)","l":83,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.denom_pos","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The treated-arm denominator in the perturbed outcome regression is positive in the valid parameter regime.","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ β → α + 2 * β ≤ 1 / 2 → 0 ≤ α → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), 0 < 1 - 2 * β * Causalean.Estimation.MinimaxATE.Δ lam x","l":119,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.estAIPW","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"For a finite covariate set, a fitted propensity function, fitted outcome-regression functions for the two treatment arms, a sample size, and a sample of observed covariate, treatment, and binary-outcome records, the fixed-center augmented inverse-probability-weighted estimator is the arithmetic mean of the corresponding fixed-center scores.","s":"{C : Type u_1} → (C → ℝ) → (Bool → C → ℝ) → (n : ℕ) → (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ","l":82,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.exists_center_overlap","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":2,"d":"Uniform lower bound on the center weights. With mhat strictly inside (0,1) on the finite type C, there is ε > 0 with ε ≤ mhat x and ε ≤ 1 − mhat x for all x.","s":"∀ {C : Type u_1} [Fintype C] [Nonempty C] (mhat : C → ℝ), (∀ (x : C), 0 < mhat x) → (∀ (x : C), mhat x < 1) → ∃ ε > 0, ∀ (x : C), ε ≤ mhat x ∧ ε ≤ 1 - mhat x","l":144,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.explicitWitness","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every positive number $K$ of paired cells, sample size $n$, nonnegative treated-outcome bump $\\alpha$, nonnegative propensity bump $\\beta$ satisfying the feasibility condition $\\alpha+2\\beta\\leq 1/2$, outcome-error budget $\\varepsilon_g$ at least $(\\alpha+\\beta)^2/(1-2\\beta)^2$, propensity-error budget $\\varepsilon_m$ at least $\\beta^2$, nonnegative error budgets, and total variation distance at most $1/2$ between the null and mixed alternative sample laws, the explicit two-point witness has the centered null and uniformly mixed perturbation as its two experiments and is valid for the stated nuisance budgets.","s":"{K n : ℕ} → {α β εg εm : ℝ} → [inst : NeZero K] → (hα : 0 ≤ α) → (hβ : 0 ≤ β) → (hαβ : α + 2 * β ≤ 1 / 2) → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2 → Causalean.Estimation.MinimaxATE.TwoPointWitness (Fin K × Bool) n Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm","l":132,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.explicit_minimax_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative error tolerances εg, εm, and suppose the centered null law and the uniform mixture of perturbed laws are statistically close, at total-variation distance at most 1/2. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by s = β(α+β)/(1−4β²) is at least 1/4.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est (β * (α + β) / (1 - 4 * β ^ 2))","l":192,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.gPerturbed","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For any natural number of pairs, real parameters governing the outcome and propensity perturbations, and a Boolean orientation chosen for each pair, the perturbed outcome-regression function assigns $1/2$ in the control arm and, in the treated arm at each cell, the ratio of $1/2$ plus the outcome magnitude times that cell's bump to one minus twice the propensity magnitude times that bump.","s":"{K : ℕ} → ℝ → ℝ → (Fin K → Bool) → Bool → Fin K × Bool → ℝ","l":108,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.gPerturbed_true_sub_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The treated outcome arm: its pointwise deviation from ĝ(1) = 1/2 is (α+β)·Δ/(1 − 2β·Δ).","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.gPerturbed α β lam true x - 1 / 2 = (α + β) * Causalean.Estimation.MinimaxATE.Δ lam x / (1 - 2 * β * Causalean.Estimation.MinimaxATE.Δ lam x)","l":67,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.ghat","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For any natural number of pairs, the centered outcome-regression function assigns the value $1/2$ to every paired covariate cell in each of the two treatment arms.","s":"{K : ℕ} → Bool → Fin K × Bool → ℝ","l":101,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.inClass_null","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The centered null nuisance functions are in the nuisance class whenever the budgets are nonnegative.","s":"∀ {K : ℕ} {εg εm : ℝ}, 0 ≤ εg → 0 ≤ εm → Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat","l":57,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.inClass_perturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Class membership. For nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg, the perturbed data-generating process (mλ, gλ) indexed by a Rademacher sign vector lam lies in the structure-agnostic nuisance class ℱ(εg, εm) around the centered estimates (m̂, ĝ) = (1/2, 1/2).","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K] {εg εm : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam)","l":128,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.ingster_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Ingster","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For a nonnegative coefficient γ with 2γ ≤ 1 satisfying the regularity budget 2n²γ² ≤ K·log 2, the uniform double average, over pairs of Rademacher sign vectors lam, lam' : Fin K → Bool, of (1 + (2γ/K)·Σⱼ signOf(lam j)·signOf(lam' j))^n is at most 2.","s":"∀ (K n : ℕ) [NeZero K] {γ : ℝ}, 0 ≤ γ → 2 * γ ≤ 1 → 2 * ↑n ^ 2 * γ ^ 2 ≤ ↑K * Real.log 2 → ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * (1 + 2 * γ / ↑K * ∑ j, Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j)) ^ n ≤ 2","l":37,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Ingster.lean"},{"n":"Causalean.Estimation.MinimaxATE.ingster_bound_general","k":"theorem","m":"Causalean.Estimation.MinimaxATE.VaryingCenterCase1.Ingster","a":"Estimation/MinimaxATE/VaryingCenterCase1","t":1,"d":"For nonnegative per-coordinate weights d j summing to at most 1 satisfying the regularity budget (n²/2)·Σⱼ (d j)² ≤ log 2, the uniform double average, over pairs of Rademacher sign vectors lam, lam' : Fin K → Bool, of (1 + Σⱼ d j·signOf(lam j)·signOf(lam' j))^n is at most 2.","s":"∀ (K n : ℕ) [NeZero K] {d : Fin K → ℝ}, (∀ (j : Fin K), 0 ≤ d j) → ∑ j, d j ≤ 1 → ↑n ^ 2 / 2 * ∑ j, d j ^ 2 ≤ Real.log 2 → ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * (1 + ∑ j, d j * (Causalean.Estimation.MinimaxATE.signOf (lam j) * Causalean.Estimation.MinimaxATE.signOf (lam' j))) ^ n ≤ 2","l":42,"f":"Causalean/Estimation/MinimaxATE/VaryingCenterCase1/Ingster.lean"},{"n":"Causalean.Estimation.MinimaxATE.integrable_sq_rnDeriv_QtrueCont_QfalseCont","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The squared density deviation for the continuous experiment is integrable under the null.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), MeasureTheory.Integrable (fun z => (((Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ).rnDeriv (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) z).toReal - 1) ^ 2) (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ)","l":122,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite covariate space, a first real-valued covariate function, and a second real-valued covariate function, the squared uniform $L^2$ distance is the uniform average of their squared pointwise difference.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → ℝ","l":141,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_bump","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Constant-magnitude (Rademacher) bump. If a sign function σ satisfies (σ x)² = 1 at every covariate value, then on a nonempty finite covariate space, the squared L²(P_X) distance between b shifted by δ·σ and b itself equals δ². This is exactly what makes a Rademacher-bump perturbation land on the boundary of the nuisance class ℱ(ε,·) when δ = √ε.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] (b : C → ℝ) (δ : ℝ) (σ : C → ℝ), (∀ (x : C), σ x ^ 2 = 1) → Causalean.Estimation.MinimaxATE.l2sq (fun x => b x + δ * σ x) b = δ ^ 2","l":51,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_comm","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq is symmetric.","s":"∀ {C : Type u_1} [inst : Fintype C] (a b : C → ℝ), Causalean.Estimation.MinimaxATE.l2sq a b = Causalean.Estimation.MinimaxATE.l2sq b a","l":36,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_gPerturbed_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The control outcome arm is unchanged (gλ(0,·) = 1/2 = ĝ(0,·)), so its L² error is 0.","s":"∀ {K : ℕ} {α β : ℝ} (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.gPerturbed α β lam false) (Causalean.Estimation.MinimaxATE.ghat false) = 0","l":58,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_gPerturbed_true_le","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Uniform bound on the treated arm's squared L² error: every pointwise term is ≤ (α+β)²/(1 − 2β)², so the average is too.","s":"∀ {K : ℕ} {α β : ℝ} [NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.gPerturbed α β lam true) (Causalean.Estimation.MinimaxATE.ghat true) ≤ (α + β) ^ 2 / (1 - 2 * β) ^ 2","l":79,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_mPerturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Membership","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The propensity perturbation has exact squared L² size β²: it is a Rademacher bump of magnitude β on Δ.","s":"∀ {K : ℕ} {β : ℝ} [NeZero K] (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.l2sq (Causalean.Estimation.MinimaxATE.mPerturbed β lam) Causalean.Estimation.MinimaxATE.mhat = β ^ 2","l":44,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Membership.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq is nonnegative.","s":"∀ {C : Type u_1} [inst : Fintype C] (a b : C → ℝ), 0 ≤ Causalean.Estimation.MinimaxATE.l2sq a b","l":44,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.l2sq_self","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Bump","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"l2sq of a function with itself is zero.","s":"∀ {C : Type u_1} [inst : Fintype C] (a : C → ℝ), Causalean.Estimation.MinimaxATE.l2sq a a = 0","l":32,"f":"Causalean/Estimation/MinimaxATE/Reduction/Bump.lean"},{"n":"Causalean.Estimation.MinimaxATE.mPerturbed","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For any natural number of pairs, a real perturbation magnitude, and a Boolean orientation chosen for each pair, the perturbed propensity function assigns to each cell $1/2$ minus the magnitude times that cell's paired-cell Rademacher bump.","s":"{K : ℕ} → ℝ → (Fin K → Bool) → Fin K × Bool → ℝ","l":104,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.mhat","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For any natural number of pairs, the centered propensity function assigns probability $1/2$ to every paired covariate cell.","s":"{K : ℕ} → Fin K × Bool → ℝ","l":98,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimaxMiss","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space with a measurable structure, a reference propensity function, a reference outcome-regression function, an outcome-regression error budget, a propensity error budget, a nonnegative sample size, an estimator based on that sample, and a real-valued threshold, the minimax miss probability is the supremum, over all data-generating processes in the specified nuisance class, of the probability that the estimator differs from that process’s true average treatment effect by at least the threshold.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":229,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimaxMissCausal","k":"def","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":1,"d":"For a finite nonempty covariate space, with a measurable structure in which every singleton is measurable, centered propensity and outcome-regression functions, two real radii with no sign restrictions, a sample size, an estimator based on that many observed treatment--outcome--covariate records, and a real error threshold, the causal-centered minimax miss probability is the supremum, over all valid observed-data distributions in the nuisance-function class determined by those centers and radii, of the probability that the estimator's absolute error from that distribution's average potential-outcome treatment effect is at least the threshold.","s":"{C : Type} → [Fintype C] → [Nonempty C] → [inst : MeasurableSpace C] → [MeasurableSingletonClass C] → (C → ℝ) → (Bool → C → ℝ) → ℝ → ℝ → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":44,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimax_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Structure-agnostic minimax lower bound (unconditional). Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative outcome- and propensity-error tolerances εg, εm, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. Then for any measurable estimator of the average treatment effect from n i.i.d. paired-cell observations, the worst-case-over-class probability that it misses the true ATE by s = β(α+β)/(1−4β²), over the structure-agnostic nuisance class centered at the constant estimates (m̂, ĝ) = (1/2, 1/2), is at least 1/4 — the doubly-robust product rate is unbeatable.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est (β * (α + β) / (1 - 4 * β ^ 2))","l":204,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.minimax_lower_bound_mse","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.MSE","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Structure-agnostic minimax lower bound, expected-risk (MSE) form. Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2 meeting the Rademacher perturbation budgets β² ≤ εm and (α+β)²/(1−2β)² ≤ εg for nonnegative error tolerances εg, εm, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. Then for any measurable estimator of the average treatment effect, there is a data-generating process in the structure-agnostic nuisance class on which its mean-squared error is at least s²/4, with s = β(α+β)/(1−4β²) ≍ √(εg·εm) — the doubly-robust product rate is an unbeatable expected-squared-error floor. This is the weaker, (1−γ)-factored consequence (γ = 3/4) of the quantile bound, in the form used by Balakrishnan et al.","s":"∀ {K n : ℕ} {α β εg εm : ℝ} [inst : NeZero K], 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → β ^ 2 ≤ εm → (α + β) ^ 2 / (1 - 2 * β) ^ 2 ≤ εg → 0 ≤ εg → 0 ≤ εm → 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ}, Measurable est → ∃ p, (β * (α + β) / (1 - 4 * β ^ 2)) ^ 2 / 4 ≤ Causalean.Estimation.MinimaxATE.nMSE p.property.valid n est","l":46,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/MSE.lean"},{"n":"Causalean.Estimation.MinimaxATE.mixture_two_point_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.WitnessMixture","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Mixture two-point lower bound (one-sided separation). Suppose a null data-generating process (m₀, g₀) lies in the structure-agnostic nuisance class and a finite family of alternative data-generating processes (mlam i, glam i) also lie in the class, with nonnegative mixture weights w summing to 1, where every alternative's average treatment effect is at least 2s away from the null's, and the null's n-sample law and the weighted mixture of the alternatives' n-sample laws are at total-variation distance at most c. Then for any measurable estimator of the average treatment effect, the worst-case-over-class probability that it misses the true ATE by s is at least (1 − c)/2. Unlike TwoPointWitness, the alternatives need not share a common ATE — only 2 s ≤ |ate g₀ − ate (glam i)| per component.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {ι : Type u_2} [inst_3 : Fintype ι] {m₀ : C → ℝ} {g₀ : Bool → C → ℝ} (hnull : Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm m₀ g₀) {mlam : ι → C → ℝ} {glam : ι → Bool → C → ℝ} (hin : ∀ (i : ι), Causalean.Estimation.MinimaxATE.InClass mhat ghat εg εm (mlam i) (glam i)) (w : ι → ENNReal), ∑ i, w i = 1 → ∀ {s c : ℝ}, (∀ (i : ι), 2 * s ≤ |Causalean.Estimation.MinimaxATE.ate g₀ - Causalean.Estimation.MinimaxATE.ate (glam i)|) → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.productLaw hnull.valid n) (Causalean.Stat.mixture w fun i => Causalean.Estimation.MinimaxATE.productLaw (hin i).valid n) ≤ c → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est s","l":58,"f":"Causalean/Estimation/MinimaxATE/Reduction/WitnessMixture.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMSE","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space with a measurable structure, a propensity function, an outcome-regression function, evidence that the functions define valid probabilities, a nonnegative sample size, and an estimator based on that sample, the mean-squared error is the expected squared deviation of the estimator from the true average treatment effect.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ","l":193,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space with a measurable structure, a propensity function, an outcome-regression function, evidence that the functions define valid probabilities, a nonnegative sample size, an estimator based on that sample, and a real-valued threshold, the miss probability is the probability that the estimator differs from the true average treatment effect by at least the threshold.","s":"{C : Type u_1} → [Fintype C] → [MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → ((Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ) → ℝ → ℝ","l":183,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMissCausal_le_minimaxMissCausal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"A specific in-class DGP's causal-centered miss probability is dominated by the causal-centered minimax miss.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {n : ℕ} {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ} (p : Causalean.Estimation.MinimaxATE.InClassDGP mhat ghat εg εm), (Causalean.Estimation.MinimaxATE.productLaw p.property.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.Causal.causalATE (↑p).1 (↑p).2|} ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est s","l":64,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"A specific in-class DGP's miss probability is ≤ minimaxMiss.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} {n : ℕ} {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ} (p : Causalean.Estimation.MinimaxATE.InClassDGP mhat ghat εg εm), Causalean.Estimation.MinimaxATE.nMiss p.property.valid n est s ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est s","l":256,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.nMiss_sq_le_nMSE","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"Chebyshev/Markov bridge. For a valid data-generating process (m, g) and a nonnegative separation threshold s, the squared threshold times the probability that an estimator misses the true average treatment effect by at least s, on n i.i.d. draws, is at most the estimator's mean-squared error. This is the quantitative form of the paper's observation that the quantile risk lower bound implies the expected-risk one (𝔐ₙ,γ ≥ ρ ⟹ minimax 𝔼-risk ≥ (1−γ)ρ`).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] [MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (n : ℕ) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ} {s : ℝ}, 0 ≤ s → s ^ 2 * Causalean.Estimation.MinimaxATE.nMiss hv n est s ≤ Causalean.Estimation.MinimaxATE.nMSE hv n est","l":207,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.neg_one_le_Δ","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump is always at least minus one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), -1 ≤ Causalean.Estimation.MinimaxATE.Δ lam x","l":92,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space with a measurable structure, a propensity function, an outcome-regression function, and evidence that these functions define valid probabilities, the single-observation probability measure is the measure associated with the corresponding finite probability mass function.","s":"{C : Type u_1} → [Fintype C] → [inst : MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → MeasureTheory.Measure (Causalean.Estimation.MinimaxATE.Obs C)","l":120,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"For every finite, nonempty covariate space equipped with a $\\sigma$-algebra, propensity function $m$, binary-treatment outcome-regression function $g$, and evidence that these functions constitute a valid data-generating process, the corresponding single-observation law is a probability measure.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.obsLaw hv)","l":128,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsLaw_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The one-observation law assigns each observed point exactly its finite observed-data mass.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (z : Causalean.Estimation.MinimaxATE.Obs C), (Causalean.Estimation.MinimaxATE.obsLaw hv).real {z} = Causalean.Estimation.MinimaxATE.obsReal m g z","l":40,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsPMF","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space, a propensity function, an outcome-regression function, and evidence that these functions define valid probabilities, the single-observation probability mass function assigns each record its nonnegative real-valued mass.","s":"{C : Type u_1} → [Fintype C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → PMF (Causalean.Estimation.MinimaxATE.Obs C)","l":108,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite covariate space, a propensity function, an outcome-regression function for binary treatment, and an observed record, the record’s real-valued probability mass is the uniform covariate mass times the conditional treatment probability times the conditional outcome probability.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (Bool → C → ℝ) → Causalean.Estimation.MinimaxATE.Obs C → ℝ","l":75,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_nonneg","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"Nonnegativity of obsReal for a valid DGP.","s":"∀ {C : Type u_1} [inst : Fintype C] {m : C → ℝ} {g : Bool → C → ℝ}, Causalean.Estimation.MinimaxATE.ValidDGP m g → ∀ (z : Causalean.Estimation.MinimaxATE.Obs C), 0 ≤ Causalean.Estimation.MinimaxATE.obsReal m g z","l":98,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_perturbed_eq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSqOverlap","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The perturbed observed-data mass has an explicit denominator-free value at every treatment-outcome cell.","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool) (x : Fin K × Bool) (d y : Bool), Causalean.Estimation.MinimaxATE.obsReal (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam) (x, d, y) = (↑(Fintype.card (Fin K × Bool)))⁻¹ * match d, y with | true, true => (1 / 2 + α * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | true, false => (1 / 2 - (α + 2 * β) * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | false, true => (1 / 2 + β * Causalean.Estimation.MinimaxATE.Δ lam x) / 2 | false, false => (1 / 2 + β * Causalean.Estimation.MinimaxATE.Δ lam x) / 2","l":61,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSqOverlap.lean"},{"n":"Causalean.Estimation.MinimaxATE.obsReal_sum","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"The total mass of obsReal is 1 (for nonempty C): summing over Y gives 1, then over D gives 1, then over X gives card C · (1/card C) = 1.","s":"∀ {C : Type u_1} [inst : Fintype C] [Nonempty C] (m : C → ℝ) (g : Bool → C → ℝ), ∑ z, Causalean.Estimation.MinimaxATE.obsReal m g z = 1","l":83,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.one_add_chiSqDiv_Qtrue_Qfalse","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Mixture second-moment identity. On the finite product space, 1 + χ²(Qtrue‖Qfalse) is the uniform average over Rademacher sign pairs of the single-observation overlap raised to the sample size n.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 1 + Causalean.Stat.chiSqDiv (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) (Causalean.Estimation.MinimaxATE.Qfalse K n) = ∑ lam, ∑ lam', (2 ^ K)⁻¹ * (2 ^ K)⁻¹ * Causalean.Estimation.MinimaxATE.chiSqOverlap α β lam lam' ^ n","l":116,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.parametric_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Parametric","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Parametric Ω(1/n) minimax lower bound for structure-agnostic ATE estimation. Around a constant nuisance center (m₀, g₀, g₁) with all three coordinates strictly between 0 and 1, with a nonnegative treated-arm shift δ satisfying g₁ + δ ≤ 1 that lies inside the outcome-error budget δ² ≤ εg for a nonnegative propensity-error tolerance εm, and in the sample-size regime n·m₀δ²/(g₁(1−g₁)) ≤ log 2, for any measurable estimator of the average treatment effect the worst-case-over-class probability that it misses the true ATE by s = δ/2 is at least 1/4.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [MeasurableSingletonClass C] {n : ℕ} {m₀ g₀ g₁ δ εg εm : ℝ}, 0 < m₀ → m₀ < 1 → 0 < g₀ → g₀ < 1 → 0 < g₁ → g₁ < 1 → 0 ≤ δ → g₁ + δ ≤ 1 → δ ^ 2 ≤ εg → 0 ≤ εm → ↑n * (m₀ * δ ^ 2 / (g₁ * (1 - g₁))) ≤ Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss (fun x => m₀) (fun d x => if d = true then g₁ else g₀) εg εm n est (δ / 2)","l":310,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Parametric.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw","k":"def","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":1,"d":"For a finite nonempty covariate space with a measurable structure, a propensity function, an outcome-regression function, evidence that the functions define valid probabilities, and a nonnegative sample size, the sample probability law is the joint law of that many independent observations from the single-observation law.","s":"{C : Type u_1} → [Fintype C] → [inst : MeasurableSpace C] → [Nonempty C] → {m : C → ℝ} → {g : Bool → C → ℝ} → Causalean.Estimation.MinimaxATE.ValidDGP m g → (n : ℕ) → MeasureTheory.Measure (Fin n → Causalean.Estimation.MinimaxATE.Obs C)","l":167,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw_isProb","k":"instance","m":"Causalean.Estimation.MinimaxATE.Model","a":"Estimation/MinimaxATE","t":2,"d":"For every finite, nonempty covariate space equipped with a $\\sigma$-algebra, propensity function $m$, binary-treatment outcome-regression function $g$, evidence that these functions constitute a valid data-generating process, and sample size $n$, the corresponding independent-sample law is a probability measure.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : Nonempty C] {m : C → ℝ} {g : Bool → C → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (n : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Estimation.MinimaxATE.productLaw hv n)","l":176,"f":"Causalean/Estimation/MinimaxATE/Model.lean"},{"n":"Causalean.Estimation.MinimaxATE.productLaw_real_singleton","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The .real product point mass of an n-sample DGP law factorizes over draws.","s":"∀ {K n : ℕ} [inst : NeZero K] {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hv : Causalean.Estimation.MinimaxATE.ValidDGP m g) (ω : Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)), (Causalean.Estimation.MinimaxATE.productLaw hv n).real {ω} = ∏ i, Causalean.Estimation.MinimaxATE.obsReal m g (ω i)","l":67,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.real_le_minimaxMiss","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The miss probability of any in-class data-generating process is bounded by the minimax miss probability.","s":"∀ {K n : ℕ} {εg εm : ℝ} [inst : NeZero K] {m : Fin K × Bool → ℝ} {g : Bool → Fin K × Bool → ℝ} (hin : Causalean.Estimation.MinimaxATE.InClass Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm m g) (est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) → ℝ) (s : ℝ), (Causalean.Estimation.MinimaxATE.productLaw hin.valid n).real {x | s ≤ |est x - Causalean.Estimation.MinimaxATE.ate g|} ≤ Causalean.Estimation.MinimaxATE.minimaxMiss Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat εg εm n est s","l":121,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For a Boolean-valued position, the sign function equals $1$ when the position is true and $-1$ when it is false.","s":"Bool → ℝ","l":51,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_false","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The sign of the false Boolean value is minus one.","s":"Causalean.Estimation.MinimaxATE.signOf false = -1","l":56,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Every Boolean sign is either one or minus one.","s":"∀ (b : Bool), Causalean.Estimation.MinimaxATE.signOf b = 1 ∨ Causalean.Estimation.MinimaxATE.signOf b = -1","l":63,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_sq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The square of every Boolean sign is one.","s":"∀ (b : Bool), Causalean.Estimation.MinimaxATE.signOf b ^ 2 = 1","l":59,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signOf_true","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The sign of the true Boolean value is one.","s":"Causalean.Estimation.MinimaxATE.signOf true = 1","l":54,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For every number $K$ of paired cells, the sign-mixture weight function assigns equal probability to each of the $2^K$ binary sign vectors.","s":"(K : ℕ) → (Fin K → Bool) → ENNReal","l":71,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight_sum","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ExplicitWitness","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The uniform weights over sign vectors have total mass one.","s":"∀ (K : ℕ), ∑ lam, Causalean.Estimation.MinimaxATE.signWeight K lam = 1","l":95,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ExplicitWitness.lean"},{"n":"Causalean.Estimation.MinimaxATE.signWeight_toReal","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The real-valued uniform sign weight (2^K)⁻¹.","s":"∀ {K : ℕ} (lam : Fin K → Bool), (Causalean.Estimation.MinimaxATE.signWeight K lam).toReal = (2 ^ K)⁻¹","l":91,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.tvDist_QfalseCont_QtrueCont_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The continuous-covariate null and alternative are indistinguishable up to total variation one half.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.QfalseCont K n ρ) (Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ) ≤ 1 / 2","l":178,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.tvDist_Qfalse_Qtrue_le_half","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ChiSquaredCore","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"Total-variation indistinguishability. In the same regime, the null and alternative n-sample laws are statistically 1/2-close in total variation — discharging the hypothesis carried abstractly in ExplicitWitness.lean.","s":"∀ {K n : ℕ} {α β : ℝ} [inst : NeZero K] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → Causalean.Stat.tvDist (Causalean.Estimation.MinimaxATE.Qfalse K n) (Causalean.Estimation.MinimaxATE.Qtrue hα hβ hαβ n) ≤ 1 / 2","l":182,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ChiSquaredCore.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitnessCausal_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"Le Cam lower bound for a causal-centered two-point witness.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal C n mhat ghat εg εm) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - W.c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est W.s","l":92,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitnessCausal_quarter","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Causal.Minimax","a":"Estimation/MinimaxATE/Causal","t":2,"d":"A causal-centered witness with c ≤ 1/2 yields a 1/4 minimax miss lower bound.","s":"∀ {C : Type} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] [inst_3 : MeasurableSingletonClass C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitnessCausal C n mhat ghat εg εm), W.c ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMissCausal mhat ghat εg εm n est W.s","l":108,"f":"Causalean/Estimation/MinimaxATE/Causal/Minimax.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitness_lower_bound","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":1,"d":"Structure-agnostic two-point lower bound. Given a two-point (Le Cam) witness W packaging two statistically close n-sample laws with separated true average-treatment-effect values, for any measurable estimator, the worst-case-over-class probability that it misses the true ATE by W.s is at least (1 − W.c)/2. The proof is two_point_lower_bound_of_tvDist_le applied to the two witness laws, followed by the realizability domination.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitness C n mhat ghat εg εm) {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → (1 - W.c) / 2 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est W.s","l":94,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.twoPointWitness_quarter","k":"theorem","m":"Causalean.Estimation.MinimaxATE.Reduction.Witness","a":"Estimation/MinimaxATE/Reduction","t":2,"d":"A witness with total-variation budget c ≤ 1/2 forces every estimator to miss the true ATE by W.s with probability at least 1/4 somewhere in the class.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : Nonempty C] [inst_2 : MeasurableSpace C] {n : ℕ} {mhat : C → ℝ} {ghat : Bool → C → ℝ} {εg εm : ℝ} (W : Causalean.Estimation.MinimaxATE.TwoPointWitness C n mhat ghat εg εm), W.c ≤ 1 / 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs C) → ℝ}, Measurable est → 1 / 4 ≤ Causalean.Estimation.MinimaxATE.minimaxMiss mhat ghat εg εm n est W.s","l":114,"f":"Causalean/Estimation/MinimaxATE/Reduction/Witness.lean"},{"n":"Causalean.Estimation.MinimaxATE.two_point_lower_bound_continuous","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.ContinuousX","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Fix nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, in the sample-size regime 2n²γ² ≤ K·log 2 with γ = α²+2αβ+3β² and 2γ ≤ 1. For any measurable estimator est of the average treatment effect that observes both the finite paired-cell data and n independent ancillary continuous within-cell positions drawn from an arbitrary probability space, the larger of the two miss probabilities — under the mixed alternative law and under the null law — that est misses the corresponding true ATE (2β(α+β)/(1−4β²) or 0) by β(α+β)/(1−4β²) is at least 1/4.","s":"∀ {K n : ℕ} {α β : ℝ} {W : Type u_1} [inst : MeasurableSpace W] [inst_1 : NeZero K] (ρ : MeasureTheory.Measure W) [MeasureTheory.IsProbabilityMeasure ρ] (hα : 0 ≤ α) (hβ : 0 ≤ β) (hαβ : α + 2 * β ≤ 1 / 2), 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ≤ 1 → 2 * ↑n ^ 2 * (α ^ 2 + 2 * α * β + 3 * β ^ 2) ^ 2 ≤ ↑K * Real.log 2 → ∀ {est : (Fin n → Causalean.Estimation.MinimaxATE.Obs (Fin K × Bool)) × (Fin n → W) → ℝ}, Measurable est → 1 / 4 ≤ max ((Causalean.Estimation.MinimaxATE.QtrueCont hα hβ hαβ n ρ).real {x | β * (α + β) / (1 - 4 * β ^ 2) ≤ |est x - 2 * β * (α + β) / (1 - 4 * β ^ 2)|}) ((Causalean.Estimation.MinimaxATE.QfalseCont K n ρ).real {x | β * (α + β) / (1 - 4 * β ^ 2) ≤ |est x - 0|})","l":205,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/ContinuousX.lean"},{"n":"Causalean.Estimation.MinimaxATE.validDGP_hat","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"The null construction's centered propensity and outcome-regression functions, both fixed at one half, define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ}, Causalean.Estimation.MinimaxATE.ValidDGP Causalean.Estimation.MinimaxATE.mhat Causalean.Estimation.MinimaxATE.ghat","l":128,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.validDGP_perturbed","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"Given nonnegative bump magnitudes α and β with α + 2β ≤ 1/2, the perturbed propensity and outcome-regression functions indexed by a Rademacher sign vector lam define a valid finite observed-data model, i.e. take values in [0,1].","s":"∀ {K : ℕ} {α β : ℝ}, 0 ≤ α → 0 ≤ β → α + 2 * β ≤ 1 / 2 → ∀ (lam : Fin K → Bool), Causalean.Estimation.MinimaxATE.ValidDGP (Causalean.Estimation.MinimaxATE.mPerturbed β lam) (Causalean.Estimation.MinimaxATE.gPerturbed α β lam)","l":134,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.yReal","k":"def","m":"Causalean.Estimation.MinimaxATE.Achievability.AIPWEstimator","a":"Estimation/MinimaxATE/Achievability","t":1,"d":"For a binary outcome, its real-valued representation is one for a success and zero otherwise.","s":"Bool → ℝ","l":58,"f":"Causalean/Estimation/MinimaxATE/Achievability/AIPWEstimator.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ","k":"def","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":1,"d":"For any natural number of pairs, a choice of Boolean orientation for each pair, and a cell consisting of a pair and a Boolean position, the paired-cell Rademacher bump is the product of the sign of the position and the sign chosen for that pair.","s":"{K : ℕ} → (Fin K → Bool) → Fin K × Bool → ℝ","l":69,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_le_one","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump is always at most one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x ≤ 1","l":86,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_mem","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump only takes the values one and minus one.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x = 1 ∨ Causalean.Estimation.MinimaxATE.Δ lam x = -1","l":80,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.MinimaxATE.Δ_sq","k":"theorem","m":"Causalean.Estimation.MinimaxATE.ConstCenterHalf.Construction","a":"Estimation/MinimaxATE/ConstCenterHalf","t":2,"d":"The Rademacher bump has square one at every paired cell.","s":"∀ {K : ℕ} (lam : Fin K → Bool) (x : Fin K × Bool), Causalean.Estimation.MinimaxATE.Δ lam x ^ 2 = 1","l":75,"f":"Causalean/Estimation/MinimaxATE/ConstCenterHalf/Construction.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure and a continuous real-linear operator on real-valued square-integrable functions, the complex lift is the continuous complex-linear operator on complex-valued square-integrable functions that applies the original operator separately to real and imaginary parts. It first forms the continuous real-linear operator that combines the separately transformed components, then bundles that operator as a complex-linear map.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ)","l":357,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLiftFun","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, a continuous real-linear operator on real-valued square-integrable functions, and a complex-valued square-integrable function, the function-level complex lift is the complex-valued function obtained by applying the operator separately to the real and imaginary parts and then combining the resulting real-valued functions as real part plus $i$ times imaginary part.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → ↥(MeasureTheory.Lp ℂ 2 μ) → ↥(MeasureTheory.Lp ℂ 2 μ)","l":231,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_adjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The complex lift commutes with taking adjoints.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), ContinuousLinearMap.adjoint (Causalean.Estimation.NPIV.Complexification.complexLift A) = Causalean.Estimation.NPIV.Complexification.complexLift (ContinuousLinearMap.adjoint A)","l":486,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The action equation for complexLift: it agrees with complexLiftFun A on every input.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (f : ↥(MeasureTheory.Lp ℂ 2 μ)), (Causalean.Estimation.NPIV.Complexification.complexLift A) f = Causalean.Estimation.NPIV.Complexification.ιLp (A (Causalean.Estimation.NPIV.Complexification.reLp f)) + Complex.I • Causalean.Estimation.NPIV.Complexification.ιLp (A (Causalean.Estimation.NPIV.Complexification.imLp f))","l":390,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Self-adjointness is preserved by the complex lift. Direct corollary of complexLift_adjoint (rewrite A.adjoint = A inside the RHS).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)}, IsSelfAdjoint A → IsSelfAdjoint (Causalean.Estimation.NPIV.Complexification.complexLift A)","l":503,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.complexLift_real","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Restriction of the complex lift to the real subspace recovers A. reLp_comp_ιLp gives reLp (ιLp g) = g and imLp_comp_ιLp gives imLp (ιLp g) = 0, so complexLift_apply collapses to ιLp (A g) + I • ιLp (A 0) = ιLp (A g).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.complexLift A) (Causalean.Estimation.NPIV.Complexification.ιLp g) = Causalean.Estimation.NPIV.Complexification.ιLp (A g)","l":401,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.imLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, the imaginary-part operator maps each complex-valued square-integrable function to its pointwise imaginary part, viewed as a real-valued square-integrable function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":114,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.imLp_comp_ιLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"The imaginary part vanishes on the image of ιLp (pointwise RCLike.im_ofReal : RCLike.im (r : ℂ) = 0).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.imLp (Causalean.Estimation.NPIV.Complexification.ιLp f) = 0","l":146,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.instAlgebraRealLpCLM","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, the real-scalar algebra structure on the algebra of continuous complex-linear operators on complex-valued square-integrable functions is obtained by restricting the usual complex scalar algebra to real scalars.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Algebra ℝ (↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ))","l":83,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, the real-part operator maps each complex-valued square-integrable function to its pointwise real part, viewed as a real-valued square-integrable function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℂ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":105,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_add_smul_imLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Reconstruction identity: every complex L²-class is the complex combination of its real and imaginary parts re-embedded via ιLp. Pointwise this is RCLike.re_add_im : (r.re : ℂ) + I * r.im = r.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℂ 2 μ)), Causalean.Estimation.NPIV.Complexification.ιLp (Causalean.Estimation.NPIV.Complexification.reLp f) + Complex.I • Causalean.Estimation.NPIV.Complexification.ιLp (Causalean.Estimation.NPIV.Complexification.imLp f) = f","l":191,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_comp_ιLp","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"reLp is a left-inverse of ιLp.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.reLp (Causalean.Estimation.NPIV.Complexification.ιLp f) = f","l":129,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.reLp_complexLift_real","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Corollary of complexLift_real: projecting back via reLp gives exactly A g. Direct from reLp_comp_ιLp.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) (g : ↥(MeasureTheory.Lp ℝ 2 μ)), Causalean.Estimation.NPIV.Complexification.reLp ((Causalean.Estimation.NPIV.Complexification.complexLift A) (Causalean.Estimation.NPIV.Complexification.ιLp g)) = A g","l":411,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, a continuous real-linear operator on real-valued square-integrable functions, and a real-valued function of a real argument, the real continuous-functional-calculus operator applies the complex continuous functional calculus to the complex lift of the operator with the symbol extended from the real part, then restricts the result back to real-valued square-integrable functions. It first constructs the complex functional-calculus operator, then bundles its real-valued restriction as a continuous real-linear operator.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)) → (ℝ → ℝ) → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":547,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Action equation for realCFC — the basic rewrite rule used by spectral proofs.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.realCFC A f) g = Causalean.Estimation.NPIV.Complexification.reLp ((cfc (fun z => ↑(f z.re)) (Causalean.Estimation.NPIV.Complexification.complexLift A)) (Causalean.Estimation.NPIV.Complexification.ιLp g))","l":598,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_id","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"realCFC of the identity symbol recovers A.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), (Causalean.Estimation.NPIV.Complexification.realCFC A id) g = A g","l":613,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"realCFC preserves self-adjointness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → IsSelfAdjoint (Causalean.Estimation.NPIV.Complexification.realCFC A f)","l":645,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_mul","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"Symbol-multiplication law for realCFC.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f g : ℝ → ℝ), Continuous f → Continuous g → (Causalean.Estimation.NPIV.Complexification.realCFC A fun x => f x * g x) = Causalean.Estimation.NPIV.Complexification.realCFC A f ∘SL Causalean.Estimation.NPIV.Complexification.realCFC A g","l":837,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_norm_le","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Norm-via-spectrum bound for realCFC. For a self-adjoint bounded operator A and a continuous real-valued symbol f, if f is bounded in absolute value by a nonnegative constant c on the real spectrum of A, then the operator realCFC A f is a contraction up to c: for every vector g, ‖realCFC A f g‖ ≤ c · ‖g‖.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → ∀ (f : ℝ → ℝ), Continuous f → ∀ (c : ℝ), 0 ≤ c → (∀ x ∈ spectrum ℝ A, |f x| ≤ c) → ∀ (g : ↥(MeasureTheory.Lp ℝ 2 μ)), ‖(Causalean.Estimation.NPIV.Complexification.realCFC A f) g‖ ≤ c * ‖g‖","l":1037,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.realCFC_resolvent_mul_self","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"Resolvent symbol for realCFC. For a self-adjoint bounded operator A whose real spectrum lies in the nonnegative reals, and for a strictly positive regularization parameter λ, composing the real-functional-calculus operator for the affine symbol x ↦ λ + x with that for the resolvent symbol x ↦ (λ + x)⁻¹ yields the identity operator on the ambient L² space.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (A : ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)), IsSelfAdjoint A → (∀ x ∈ spectrum ℝ A, 0 ≤ x) → ∀ (lambda : ℝ), 0 < lambda → ((Causalean.Estimation.NPIV.Complexification.realCFC A fun x => lambda + x) ∘SL Causalean.Estimation.NPIV.Complexification.realCFC A fun x => (lambda + x)⁻¹) = ContinuousLinearMap.id ℝ ↥(MeasureTheory.Lp ℝ 2 μ)","l":886,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.ιLp","k":"def","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space equipped with a measure, the real-to-complex embedding maps each real-valued square-integrable function to the complex-valued square-integrable function having that real part and zero imaginary part.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℂ 2 μ)","l":122,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.Complexification.ιLp_isometry","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Complexification","a":"Estimation/NPIV/Operator","t":2,"d":"ιLp is an isometric embedding. Follows from RCLike.ofRealLI being a LinearIsometry and the fact that compLpL of a norm-one CLM preserves norms on L²; concretely, ‖ιLp f‖ = ‖f‖ reduces pointwise to ‖(r : ℂ)‖ = ‖r‖ (RCLike.norm_ofReal).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f : ↥(MeasureTheory.Lp ℝ 2 μ)), ‖Causalean.Estimation.NPIV.Complexification.ιLp f‖ = ‖f‖","l":169,"f":"Causalean/Estimation/NPIV/Operator/Complexification.lean"},{"n":"Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps","k":"structure","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"The TRAE-DR remainder hypotheses bundle the sufficient conditions under which the doubly-robust TRAE estimator, built from primal nuisance estimators ĥ_n and dual nuisance estimators q̂_n over a cross-fitting split, is asymptotically linear: the fitted primal nuisance is L²-consistent for the truth in probability, likewise for the fitted dual nuisance, both fitted nuisances stay in their respective candidate classes at every sample size and outcome, the √-scaled mixed-bias integral between the two nuisance errors is asymptotically negligible, the centered empirical-process remainder from plugging the fitted nuisances into the oracle score vanishes at the √-rate, and the oracle score has finite variance under the observation law.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → [MeasureTheory.IsProbabilityMeasure μ] → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → {q₀ : S.𝒵 → ℝ} → S.DualSolution q₀ → {P_W : MeasureTheory.Measure S.𝒲} → [MeasureTheory.IsProbabilityMeasure P_W] → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → S.𝒳 → ℝ) → (ℕ → Ω → S.𝒵 → ℝ) → Prop","l":119,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒲","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"For a measurable sample space with a measure and an inverse-problem system, the measurable structure on the observation space is the measurable structure specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒲","l":47,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒳","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"For a measurable sample space with a measure and an inverse-problem system, the measurable structure on the covariate space is the measurable structure specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒳","l":52,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.instMeasurableSpace_𝒵","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"For a measurable sample space with a measure and an inverse-problem system, the measurable structure on the instrument space is the measurable structure specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒵","l":57,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.mean_zero_of_DualSolution","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Mean-zero of the oracle score under the observation law P_W.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {P_W : MeasureTheory.Measure S.𝒲}, MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ∂P_W = 0","l":99,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.mean_zero_of_DualSolution_mu","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Mean-zero of the oracle score under the structural law μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∫ (ω : Ω), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ (S.W ω) ∂μ = 0","l":75,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.mixed_bias_sufficient","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":2,"d":"Sufficient condition for the mixed-bias hypothesis using either operator side (the displayed min{·,·} form in thm:est-trae-dr-al-criterion):","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} {P_W : MeasureTheory.Measure S.𝒲} [MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ) (bnd : ℕ → Ω → ℝ), (∀ (n : ℕ) (ω : Ω), |∫ (ω' : Ω), (q₀ (S.zOf (S.W ω')) - q_hat n ω (S.zOf (S.W ω'))) * (h_hat n ω (S.xOf (S.W ω')) - S.h₀ (S.xOf (S.W ω'))) ∂μ| ≤ bnd n ω) → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * bnd n ω) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * ∫ (ω' : Ω), (q₀ (S.zOf (S.W ω')) - q_hat n ω (S.zOf (S.W ω'))) * (h_hat n ω (S.xOf (S.W ω')) - S.h₀ (S.xOf (S.W ω'))) ∂μ) (fun x => 1) μ","l":397,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_asymp_normal","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"TRAE-DR asymptotic normality — thm:est-trae-dr-asymp-normal. Under the dual-solution hypothesis on q₀, primal nuisance estimators ĥ_n, indexed by sample size and outcome, satisfying — together with the paired dual estimators — the bundled TRAE-DR remainder conditions, and the law bridge identifying the pushforward of μ along the observation map W with the observation law P_W, suppose further that the √n-rescaled estimator sequence is almost-everywhere measurable at every sample size and the normalized influence-function partial sum is almost-everywhere measurable at every sample size. Then the rescaled TRAE-DR estimator converges in distribution, along the estimation-fold sizes, to the centered Gaussian law with variance σ₀² := ∫ ρ₀(w)² dP_W.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → ∀ (h_meas_θ : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W)) μ h_meas_θ","l":51,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_estimator","k":"def","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, a probability law on its observation space, an independent and identically distributed sample under the two measures, a one-shot split of that sample, a sequence of data-dependent primal nuisance estimators, a sequence of data-dependent dual nuisance estimators, a sample size, and a realization of the underlying sample space, the one-shot TRAE doubly robust estimator is the average over the split's evaluation fold of the system's doubly robust pseudo-outcome evaluated using the two nuisance estimators at that sample size and realization.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → S.𝒳 → ℝ) → (ℕ → Ω → S.𝒵 → ℝ) → ℕ → Ω → ℝ","l":62,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_estimator_eq_avg_phi","k":"theorem","m":"Causalean.Estimation.NPIV.DR.Estimator","a":"Estimation/NPIV/DR","t":1,"d":"Estimator unfolding. For primal nuisance estimators ĥ_n, indexed by sample size and by outcome, paired with dual nuisance estimators q̂_n, an inverse-problem system, an i.i.d. sample, and a one-shot cross-fitting split, the one-shot TRAE doubly robust estimator evaluated at sample size n and outcome ω equals the average, over the evaluation fold B(n), of the pointwise doubly-robust pseudo-outcome φ_{ĥ_n,q̂_n} computed at each fold member's observation.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {P_W : MeasureTheory.Measure S.𝒲} (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ) (n : ℕ) (ω : Ω), Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω = (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, S.phiVal (h_hat n ω) (q_hat n ω) (sample.Z i ω)","l":84,"f":"Causalean/Estimation/NPIV/DR/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_isAsymLinear","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"TRAE-DR asymptotic-linearity criterion — thm:est-trae-dr-al-criterion. Fix a linear inverse-problem functional system S, an i.i.d. sample, and a one-shot cross-fitting split, and suppose q₀ solves the associated dual moment equation. Given primal nuisance estimators ĥ_n, indexed by sample size and outcome, paired with dual nuisance estimators satisfying the bundled L²-consistency, candidate-membership, mixed-bias, and empirical-process remainder conditions, together with the law bridge identifying the pushforward of μ along the observation map W with the observation law P_W and measurability of the oracle score ρ₀, the TRAE-DR estimator is asymptotically linear at the structural target θ₀, with mean-zero, finite-variance influence function ρ₀ := φ_{h₀,q₀} − θ₀ and vanishing √n-rescaled remainder, indexed along the estimation folds.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) sample split.foldB","l":192,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_studentized","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"Studentized TRAE-DR convergence. Fix a dual solution q₀ of the inverse-problem system S together with a first-stage nuisance-estimator sequence ĥ_n, and suppose the sample's W-marginal is identified as P_W. Assume the efficient influence function ρ₀ is measurable and the rescaled estimator sequence, its normalized-sum representation, and the studentized statistic itself are all almost-everywhere measurable at every sample size. If the asymptotic standard deviation σ₀ is strictly positive with σ₀² = ∫ ρ₀(w)² dP_W and a variance-estimator sequence σ̂_n converges to σ₀ in probability, then the studentized statistic √|B(n)| · (θ̂_n − θ₀) / σ̂_n converges in distribution to the standard normal law N(0, 1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → ∀ (σ_hat_n : ℕ → Ω → ℝ) (σ₀ : ℝ), 0 < σ₀ → σ₀ ^ 2 = ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W → Causalean.Stat.Tendsto_inProb σ_hat_n (fun x => σ₀) μ → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → ∀ (h_studentized_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) μ), Causalean.Stat.Tendsto_dist (fun n ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) (Causalean.Stat.gaussianMeasure 0 1) μ h_studentized_meas","l":106,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.trae_dr_wald_coverage","k":"theorem","m":"Causalean.Estimation.NPIV.DR.AsymptoticNormal","a":"Estimation/NPIV/DR","t":1,"d":"Wald asymptotic coverage — thm:est-trae-dr-asymp-normal. Under the same setup as trae_dr_studentized — a dual solution q₀ of the inverse-problem system S, a first-stage nuisance-estimator sequence ĥ_n, the sample's W-marginal identified as P_W, measurability of ρ₀, of the rescaled estimator, of its normalized-sum representation, and of the studentized statistic, and a variance-estimator sequence σ̂_n that is consistent in probability for a strictly positive σ₀ satisfying σ₀² = ∫ ρ₀(w)² dP_W — fix any strictly positive real number z. Then provided the coverage probability of the Wald interval θ̂_n ± z · σ̂_n / √|B(n)| and the probability that the studentized statistic lands in [-z, z] become asymptotically equal, the Wald-interval coverage probability converges to the standard-normal mass 2 Φ(z) − 1 on [-z, z].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ} (hq₀ : S.DualSolution q₀) {P_W : MeasureTheory.Measure S.𝒲} [inst_2 : MeasureTheory.IsProbabilityMeasure P_W] (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) (split : Causalean.Stat.OneShotSplit sample) (h_hat : ℕ → Ω → S.𝒳 → ℝ) (q_hat : ℕ → Ω → S.𝒵 → ℝ), Causalean.Estimation.NPIV.DR.TRAEDRRemainderHyps S hq₀ sample split h_hat q_hat → MeasureTheory.Measure.map S.W μ = P_W → ∀ (σ_hat_n : ℕ → Ω → ℝ) (σ₀ : ℝ), 0 < σ₀ → σ₀ ^ 2 = ∫ (w : S.𝒲), Causalean.Estimation.NPIV.DR.ρ₀ S q₀ w ^ 2 ∂P_W → Causalean.Stat.Tendsto_inProb σ_hat_n (fun x => σ₀) μ → Measurable (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat) S.θ₀ split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (Causalean.Estimation.NPIV.DR.ρ₀ S q₀) split.foldB n) μ) → (∀ (n : ℕ), AEMeasurable (fun ω => √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω) μ) → ∀ (z : ℝ), 0 < z → Filter.Tendsto (fun n => (μ {ω | |Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀| ≤ z * σ_hat_n n ω / √↑(split.foldB n).card}).toReal - (μ {ω | √↑(split.foldB n).card * (Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀) / σ_hat_n n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (μ {ω | |Causalean.Estimation.NPIV.DR.trae_dr_estimator S sample split h_hat q_hat n ω - S.θ₀| ≤ z * σ_hat_n n ω / √↑(split.foldB n).card}).toReal) Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":181,"f":"Causalean/Estimation/NPIV/DR/AsymptoticNormal.lean"},{"n":"Causalean.Estimation.NPIV.DR.ρ₀","k":"def","m":"Causalean.Estimation.NPIV.DR.AsymptoticLinear","a":"Estimation/NPIV/DR","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, a dual nuisance function on the instrument space, and an observation, the oracle score is the doubly robust pseudo-outcome formed from the system's primal nuisance function and the supplied dual nuisance function at that observation, minus the system's scalar target.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":65,"f":"Causalean/Estimation/NPIV/DR/AsymptoticLinear.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem","k":"structure","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"A linear inverse-problem functional system packages an observation space together with measurable treatment-side and instrument-side covariate projections of the observed data, measurable closed candidate sets for the primal and dual nuisances, each closed under subtraction, observation-level moment maps that are linear in their nuisance argument, jointly measurable and integrable against every candidate, whose cross product q(Z)h(X) is integrable as well, and a primal nuisance in the candidate set satisfying the inverse-problem moment equation against every dual candidate (def:est-trae-system).","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max (max (max u_1 (u_2 + 1)) (u_3 + 1)) (u_4 + 1))","l":43,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.DualSolution","k":"structure","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"DualSolution S q₀ says q₀ ∈ Qbar and q₀ solves the dual moment identity E[m_e(W; h)] = E[q₀(Z) h(X)] for every h ∈ Hbar (def:est-trae-dual-solution).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒵 → ℝ) → Prop","l":170,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.X","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, and an outcome in the sample space, the treatment-side covariate random variable is the system's covariate projection evaluated at its observation generated by that outcome.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → Ω → S.𝒳","l":141,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.Z","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, and an outcome in the sample space, the instrument-side covariate random variable is the system's instrument projection evaluated at its observation generated by that outcome.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → Ω → S.𝒵","l":144,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.phi","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, a primal nuisance function on the covariate space, a dual nuisance function on the instrument space, and an outcome in the sample space, the pseudo-outcome along the random observation is the pointwise doubly robust pseudo-outcome evaluated at the observation generated by that outcome.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → Ω → ℝ","l":162,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.phiVal","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, a primal nuisance function on the covariate space, a dual nuisance function on the instrument space, and an observation, the pointwise doubly robust pseudo-outcome is the treatment-side moment plus the outcome-side moment minus the product of the dual nuisance at the observation's instrument projection and the primal nuisance at its covariate projection.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":151,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.Θ","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure, an inverse-problem system, a primal nuisance function on the covariate space, and a dual nuisance function on the instrument space, the doubly robust functional is the expectation, under the measure, of the doubly robust pseudo-outcome along the system's random observation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → ℝ","l":183,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.InverseProblemSystem.θ₀","k":"def","m":"Causalean.Estimation.NPIV.Setup","a":"Estimation/NPIV","t":1,"d":"For a measurable sample space with a measure and an inverse-problem system, the scalar target is the expectation, under the measure, of the system's treatment-side moment evaluated at the system's observation and primal nuisance function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.InverseProblemSystem Ω μ → ℝ","l":147,"f":"Causalean/Estimation/NPIV/Setup.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem","k":"structure","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"An operator system: an InverseProblemSystem enriched with the L² data needed to define the concrete population projection operator T = Π_{Q̄} ∘ E[· | σ(Z)]. On top of the base system it packages the σ-algebra generated by the instrument Z, with the facts that it is coarser than the ambient σ-algebra on the sample space and coincides with the σ-algebra pulled back along Z, that the ambient probability measure is finite, and closed submodules of the ambient L²(μ) space realizing the primal and dual candidate function classes, the dual one further required to admit an orthogonal projection; finally, every candidate primal or dual function is square-integrable once composed with the covariate or instrument map, with its L² class coherently landing inside the matching submodule.","s":"(Ω : Type u_1) → [m₀ : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max (max (max u_1 (u_2 + 1)) (u_3 + 1)) (u_4 + 1))","l":65,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.QbarAEMeasurableMZ","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, this predicate asserts that every dual candidate, evaluated at the instrument of a sampled observation, is almost-everywhere strongly measurable with respect to the instrument-generated σ-algebra.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → Prop","l":241,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and an ambient $L^2$ random variable, the NPIV projection operator is the orthogonal projection onto the closed dual candidate space of that variable's conditional expectation given the instrument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ↥(MeasureTheory.Lp ℝ 2 μ)","l":151,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_add","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"T distributes over addition (companion to T_sub).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.T (f + g) = S.T f + S.T g","l":200,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_inner_eq_integral","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"Population identity for T. Consider an operator system S — an inverse-problem system enriched with an L² realization of the candidate sets — and fix two primal candidate functions h, h' in the primal candidate set Hbar together with a dual candidate function f in the dual candidate set Qbar. Writing T for the operator that conditions on the σ-algebra generated by the instrument Z and then orthogonally projects onto the L² realization of Qbar, the L²(μ) inner product of T applied to the lifted residual h − h' against the lifted f equals the expectation of the pointwise product (h(X) − h'(X))·f(Z).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) {h h' : S.𝒳 → ℝ} (hh : h ∈ S.Hbar) (hh' : h' ∈ S.Hbar) {f : S.𝒵 → ℝ} (hf : f ∈ S.Qbar), inner ℝ (S.T (S.hL2 hh - S.hL2 hh')) (S.qL2 hf) = ∫ (ω : Ω), (h (S.xOf (S.W ω)) - h' (S.xOf (S.W ω))) * f (S.zOf (S.W ω)) ∂μ","l":270,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.T_sub","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"T distributes over subtraction.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.T (f - g) = S.T f - S.T g","l":184,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tadjoint","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the adjoint operator is the Hilbert-space adjoint of its bundled NPIV projection operator.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":87,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tlin","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the bundled NPIV operator is the continuous linear map that conditions an ambient $L^2$ random variable on the instrument and then orthogonally projects it onto the closed dual candidate space.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":63,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tlin_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":2,"d":"Coherence: applying the bundled CLM Tlin to f agrees with the function-level operator T from Causalean/Estimation/NPIV/Operator.lean. The proof unfolds both sides and uses Submodule.starProjection_apply = orthogonalProjectionFn.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f : ↥(MeasureTheory.Lp ℝ 2 μ)), S.Tlin f = S.T f","l":77,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tstar_T","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the self-adjoint positive-semidefinite composite is its bundled NPIV operator followed by its Hilbert-space adjoint.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":98,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.Tstar_T_isSelfAdjoint","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For any NPIV operator system, the composite operator T† T, obtained by composing the second-stage conditional-expectation operator with its adjoint, is self-adjoint.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), IsSelfAdjoint S.Tstar_T","l":108,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStep","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the conditional-expectation operator maps every ambient $L^2$ random variable to its conditional expectation given the σ-algebra generated by the instrument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ↥(MeasureTheory.Lp ℝ 2 μ)","l":136,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStepL","k":"def","m":"Causalean.Estimation.NPIV.Operator.Adjoint","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the bundled conditional-expectation operator is the continuous linear map sending each ambient $L^2$ random variable to its conditional expectation given the σ-algebra generated by the instrument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":50,"f":"Causalean/Estimation/NPIV/Operator/Adjoint.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.condExpStep_sub","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"condExpStep distributes over subtraction (linearity of condExpL2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), S.condExpStep (f - g) = S.condExpStep f - S.condExpStep g","l":176,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.hL2","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and a covariate function belonging to its primal candidate set, the lifted primal candidate is the $L^2$ equivalence class of that function evaluated at the covariates.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {h : S.𝒳 → ℝ} → h ∈ S.Hbar → ↥(MeasureTheory.Lp ℝ 2 μ)","l":114,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.qL2","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and an instrument function belonging to its dual candidate set, the lifted dual candidate is the $L^2$ equivalence class of that function evaluated at the instruments.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {q : S.𝒵 → ℝ} → q ∈ S.Qbar → ↥(MeasureTheory.Lp ℝ 2 μ)","l":122,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.qbar_aestrongly_measurable_mZ","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"q ∘ zOf ∘ W is m_Z-strongly measurable for every q ∈ Qbar. Direct consequence of m_Z_eq : m_Z = comap (zOf ∘ W) inst𝒵 and meas_of_Qbar: the canonical map (zOf ∘ W) : Ω → 𝒵 is measurable from comap (zOf ∘ W) inst𝒵 to inst𝒵 (comap_measurable), so its composition with the inst𝒵-measurable q : 𝒵 → ℝ is m_Z-measurable, hence m_Z-strongly measurable into ℝ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), S.QbarAEMeasurableMZ","l":251,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.strongNorm","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and an ambient $L^2$ random variable, the strong norm is that variable's $L^2$ norm.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ℝ","l":369,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.strongNorm_sq_hL2_eq_integral","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"Squared strong norm of a lifted Hbar member, unfolded as the integral of the squared representative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) {h : S.𝒳 → ℝ} (hh : h ∈ S.Hbar), S.strongNorm (S.hL2 hh) ^ 2 = ∫ (ω : Ω), h (S.xOf (S.W ω)) ^ 2 ∂μ","l":395,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and a real regularization level, the ambient Tikhonov bilinear form sends two $L^2$ random variables $u,v$ to $\\langle Tu,Tv\\rangle+\\lambda\\langle u,v\\rangle$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ℝ","l":74,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and a real regularization level, the restricted Tikhonov bilinear form is the ambient Tikhonov bilinear form evaluated on two elements of the closed primal candidate subspace.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ℝ → ↥S.Hbar_L2 →L[ℝ] ↥S.Hbar_L2 →L[ℝ] ℝ","l":105,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for the restricted bilinear form.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥S.Hbar_L2), ((S.tikhonovBilinSub lambda) u) v = ((S.tikhonovBilin lambda) ↑u) ↑v","l":115,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilinSub_isCoercive","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Coercivity of tikhonovBilinSub for 0 < λ. Witness constant: λ. Proof: bilin u u = ‖T u‖² + λ ‖u‖² ≥ λ ‖u‖² since ‖T u‖² ≥ 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → IsCoercive (S.tikhonovBilinSub lambda)","l":122,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for tikhonovBilin.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥(MeasureTheory.Lp ℝ 2 μ)), ((S.tikhonovBilin lambda) u) v = inner ℝ (S.Tlin u) (S.Tlin v) + lambda * inner ℝ u v","l":89,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovBilin_symm","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Symmetry of tikhonovBilin.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (lambda : ℝ) (u v : ↥(MeasureTheory.Lp ℝ 2 μ)), ((S.tikhonovBilin lambda) u) v = ((S.tikhonovBilin lambda) v) u","l":99,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system whose closed primal candidate subspace admits orthogonal projection and a real regularization level, the population Tikhonov minimiser is the unique variational solution in that subspace when $\\lambda>0$, and is the zero $L^2$ element when $\\lambda\\le0$. It first equips the candidate subspace with its complete Hilbert-space structure; under $\\lambda>0$, it obtains the coercive restricted Tikhonov form, constructs the associated variational equivalence, and maps the target functional to its representing candidate element, whose inverse image is returned as the minimiser; when $\\lambda\\le0$, it returns zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → [S.Hbar_L2.HasOrthogonalProjection] → ℝ → ↥(MeasureTheory.Lp ℝ 2 μ)","l":165,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_mem","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"The minimiser lies in Hbar_L2.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] (lambda : ℝ), S.tikhonovMinimiserL2 lambda ∈ S.Hbar_L2","l":191,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_optimality","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"Variational identity for the Tikhonov minimiser. For a strictly positive Tikhonov regularization level λ and any function v in the closed primal candidate subspace Hbar_L2, the population Tikhonov minimiser h*_λ at level λ satisfies the identity ⟪T h*_λ, T v⟫ + λ · ⟪h*_λ, v⟫ = ⟪T h₀, T v⟫, where T is the projection-composed conditional-expectation operator and h₀ is the L² class of the structural function.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → ∀ {v : ↥(MeasureTheory.Lp ℝ 2 μ)}, v ∈ S.Hbar_L2 → inner ℝ (S.Tlin (S.tikhonovMinimiserL2 lambda)) (S.Tlin v) + lambda * inner ℝ (S.tikhonovMinimiserL2 lambda) v = inner ℝ (S.Tlin (S.hL2 S.h₀_mem)) (S.Tlin v)","l":200,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovMinimiserL2_strong_convexity","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"Population strong convexity at the Tikhonov minimiser (L² level). For a strictly positive Tikhonov regularization level λ and any function h in the closed primal candidate subspace Hbar_L2, the non-negative excess λ‖h − h*_λ‖² + ‖T(h − h*_λ)‖² — the amount by which the quadratic Tikhonov objective at h exceeds its value at the population minimiser h*_λ — is bounded above by ‖T(h − h₀)‖² − ‖T(h*_λ − h₀)‖² + λ(‖h‖² − ‖h*_λ‖²), where T is the projection-composed conditional-expectation operator and h₀ is the structural function's L² class.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) [inst_1 : S.Hbar_L2.HasOrthogonalProjection] {lambda : ℝ}, 0 < lambda → ∀ {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ S.Hbar_L2 → lambda * ‖h - S.tikhonovMinimiserL2 lambda‖ ^ 2 + ‖S.Tlin (h - S.tikhonovMinimiserL2 lambda)‖ ^ 2 ≤ ‖S.Tlin (h - S.hL2 S.h₀_mem)‖ ^ 2 - ‖S.Tlin (S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem)‖ ^ 2 + lambda * (‖h‖ ^ 2 - ‖S.tikhonovMinimiserL2 lambda‖ ^ 2)","l":244,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovTargetSub","k":"def","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, the Tikhonov target functional sends each element of the closed primal candidate subspace to the inner product of its transformed image with the transformed structural function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ↥S.Hbar_L2 →L[ℝ] ℝ","l":137,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.tikhonovTargetSub_apply","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.Tikhonov","a":"Estimation/NPIV/Operator","t":2,"d":"Pointwise formula for the target functional.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (v : ↥S.Hbar_L2), S.tikhonovTargetSub v = inner ℝ (S.Tlin (S.hL2 S.h₀_mem)) (S.Tlin ↑v)","l":151,"f":"Causalean/Estimation/NPIV/Operator/Tikhonov.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.weakNorm","k":"def","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system and an ambient $L^2$ random variable, the weak norm is the $L^2$ norm of its projected conditional expectation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ↥(MeasureTheory.Lp ℝ 2 μ) → ℝ","l":373,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.OperatorSystem.weakNorm_le_strongNorm","k":"theorem","m":"Causalean.Estimation.NPIV.Operator","a":"Estimation/NPIV/Operator","t":2,"d":"The projected conditional-expectation norm is bounded by the ambient L² norm.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) (f : ↥(MeasureTheory.Lp ℝ 2 μ)), S.weakNorm f ≤ S.strongNorm f","l":379,"f":"Causalean/Estimation/NPIV/Operator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"An estimator, indexed by sample size and randomness, is a TRAE primal estimator relative to a nuisance fold when, at every sample size and realization, it belongs to the statistical candidate class, attains an empirical sup-min objective on that fold no worse than at any other candidate in the class, and is jointly measurable in the randomness and the covariate argument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (ℕ → Ω → S.𝒳 → ℝ) → Prop","l":158,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.LocalizedRegimeBundle","k":"structure","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regime","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"A localized regime bundle packages a countable index set and a measurable family of loss functions on a nonempty covariate space, together with a localization norm, nonnegative on every class member and a measurable sample embedding into that covariate space. It further records a localized empirical-process regime whose critical radius is at most the target localization scale δ_n, is strictly positive, and is a fixed point of the process's expected-supremum function at that scale. Finally, uniformly over every radius at least δ_n, it requires that the empirical Rademacher process on the star-hull is almost-surely bounded and that the corresponding empirical Rademacher complexity is integrable.","s":"(Ω : Type u_2) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → ℕ → ℝ → Type (max 1 u_2)","l":32,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regime.lean"},{"n":"Causalean.Estimation.NPIV.Primal.LocalizedRegimes","k":"structure","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regime","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"This structure bundles four localized regime witnesses, one for each function class entering the empirical-process and centred-regulariser arguments — the product, moment, critic, and candidate classes, with the deterministic hypotheses that the observation variable has the stated law, that the population Tikhonov solution is realizable in the statistical candidate class, that the moment, candidate, and critic maps are almost-surely uniformly bounded, and that every candidate admits a critic whose L² lift realizes the projected residual. It also records that the empirical sup-min objective is bounded above by, and attained at, the population sup objective; diameter caps on the candidate class and the critic class; and nonnegative proportionality constants together with a matching radius lower bound for the moment-class pair differences and for the product-class cross terms. For the candidate class it further supplies an interpretation triple: an index into the H-bundle whose norm controls the pairwise loss gap, and whose composed evaluation equals the difference of squared candidate losses. For the product class it supplies both a single-candidate interpretation triple bounded by the localization scale and a pair-difference interpretation triple scaling with the candidate gap. For the moment class it likewise supplies a single-critic interpretation triple and a pair-difference interpretation triple.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → Causalean.Stat.IIDSample Ω S.𝒲 μ P_W → {β lambda : ℝ} → (sc : Causalean.Estimation.NPIV.SourceCondition S β) → Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc → ℕ → ℝ → Type (max (max (max 1 u_2) u_3) u_4)","l":105,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regime.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAEClasses","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"This structure packages a nonempty statistical candidate class for the primal nuisance that sits inside the ambient closed candidate set, together with a nonempty statistical critic class that sits inside the ambient closed critic set, for the primal Tikhonov-regularized adversarial optimization (def:est-trae-population-criterion, line 139).","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → Type (max u_2 u_3)","l":88,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"This structure augments the deterministic TRAE primal rate hypotheses with a discharged high-probability empirical-process / centred-regulariser event, bounding the right-hand side of the population strong-convexity inequality by an explicit deterministic-looking expression, abstracting away the localized empirical-process and centred-regulariser arguments that the underlying localized wrapper supplies.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → (split : Causalean.Stat.OneShotSplit sample) → (lambda : ℝ) → ℝ → ℝ → (ℕ → ℝ) → (h_hat : ℕ → Ω → S.𝒳 → ℝ) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Type u_3","l":158,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.TRAERatePrimalHyps","k":"structure","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"This structure bundles the deterministic hypotheses of the primal TRAE rate theorem: a β-source condition for the target nuisance, a Tikhonov bias certificate at the chosen regularization level, realizability of the resulting population Tikhonov solution in the statistical candidate class, closedness of the critic class for primal residuals, mean-square continuity of the moment map on critics, and almost-sure uniform boundedness of the moment map and the candidate and critic functions.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → (split : Causalean.Stat.OneShotSplit sample) → (lambda : ℝ) → ℝ → ℝ → (ℕ → ℝ) → (h_hat : ℕ → Ω → S.𝒳 → ℝ) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Type u_3","l":91,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.centred_regulariser_bound_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regulariser","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"The centred empirical regularizer gap is bounded uniformly over all sample sizes by the localized rate from the squared candidate-difference event. For every sample size n, suppose a localized-regimes witness at the corresponding fold size and localization rate delta n is available, the regularization weight satisfies lambda ≥ 0, and the confidence level satisfies 0 < ζ < 1. Then there is an event of probability at least 1 - ζ on which, for every split size with at least one fold-A observation, the empirical-vs- population regularizer discrepancy between the estimator and the population Tikhonov candidate is controlled by an explicit bound built from the fold's diameter, critical radius, and localization rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → ∀ {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_reg, MeasurableSet Aζ_reg ∧ μ Aζ_reg ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_reg, ∀ (n : ℕ), 1 ≤ split.n₁ n → |lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2 - (S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2 - S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2))| ≤ lambda * (4 * ((regimes n).H_diameter + delta n) * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_H.regime.ψ (split.n₁ n)) + (regimes n).bundle_H.regime.b * √(2 * Real.log (2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":55,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regulariser.lean"},{"n":"Causalean.Estimation.NPIV.Primal.combined_metric_rate","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Combined-metric rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → ∃ Aζ K, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ K ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)","l":290,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_critic_argmax_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Deterministic argmax-localization bridge. If f_emp empirically beats the closedness witness for the same candidate h, and the population-vs-empirical loss difference is controlled by R, then the empirical critic is within squared L² distance R of the closedness witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} [MeasureTheory.IsProbabilityMeasure μ] {lambda : ℝ} {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f_emp : S.𝒵 → ℝ} (hf_emp : f_emp ∈ TC.F) {n : ℕ} {ω : Ω} {R : ℝ}, S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω → 2 * ∫ (ω' : Ω), S.m (S.W ω') f_closed ∂μ - 2 * ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_closed (S.zOf (S.W ω')) ∂μ - ∫ (ω' : Ω), f_closed (S.zOf (S.W ω')) ^ 2 ∂μ - (2 * ∫ (ω' : Ω), S.m (S.W ω') f_emp ∂μ - 2 * ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_emp (S.zOf (S.W ω')) ∂μ - ∫ (ω' : Ω), f_emp (S.zOf (S.W ω')) ^ 2 ∂μ) ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω - Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω + R → ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf_emp)‖ ^ 2 ≤ R","l":379,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_critic_argmax_localized_from_components","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Componentwise version of empirical_critic_argmax_localized.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} [MeasureTheory.IsProbabilityMeasure μ] {lambda : ℝ} {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f_emp : S.𝒵 → ℝ} (hf_emp : f_emp ∈ TC.F) {n : ℕ} {ω : Ω} {Rm_closed RHF_closed RF_closed Rm_emp RHF_emp RF_emp : ℝ}, S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_closed n ω ≤ Causalean.Estimation.NPIV.Primal.innerObjective S sample split lambda h f_emp n ω → |(↑(split.n₁ n))⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f_closed - ∫ (ω' : Ω), S.m (S.W ω') f_closed ∂μ| ≤ Rm_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f_closed (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_closed (S.zOf (S.W ω')) ∂μ| ≤ RHF_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, f_closed (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f_closed (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ RF_closed → |(↑(split.n₁ n))⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f_emp - ∫ (ω' : Ω), S.m (S.W ω') f_emp ∂μ| ≤ Rm_emp → |(↑(split.n₁ n))⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f_emp (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f_emp (S.zOf (S.W ω')) ∂μ| ≤ RHF_emp → |(↑(split.n₁ n))⁻¹ * ∑ k, f_emp (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f_emp (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ RF_emp → ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf_emp)‖ ^ 2 ≤ 2 * Rm_closed + 2 * RHF_closed + RF_closed + (2 * Rm_emp + 2 * RHF_emp + RF_emp)","l":639,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_process_event_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Discharge of empirical_process_event from localized_uniform_deviation — explicit-rate form. Given a localized-regime bundle for the weak-norm, regularizer, and cross function classes at each fold-A sample size and a nonnegative Tikhonov regularization weight lambda, for every confidence level ζ strictly between 0 and 1 there is an event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the weak-norm estimation excess plus lambda times the strong-norm estimation excess is bounded by the sum of the explicit per-n empirical-process rate and the explicit per-n centred-regularizer rate — each an additive combination of critical radii and a √(log/n) deviation term.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → ∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (8 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)) + lambda * (4 * ((regimes n).H_diameter + delta n) * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_H.regime.ψ (split.n₁ n)) + (regimes n).bundle_H.regime.b * √(2 * Real.log (2 ^ (n + 2) / ζ) / ↑(split.n₁ n)))","l":83,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.empirical_process_event_of_absorption","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Empirical-process event in the Rate.lean shape, conditional on absorption. Given a localized-regime bundle for the weak-norm, regularizer, and cross classes at each fold-A sample size, a nonnegative regularization weight lambda, an absorption hypothesis providing, for every confidence level ζ in (0, 1), a nonnegative constant K_ep bounding the explicit per-n empirical-process rate by K_ep times the population shape R² + δ_n·y + δ_n² + λ·δ_n·x + λ·δ_n², for every ω and every n with 1 ≤ split.n₁ n, and a small-n slack hypothesis extending the same K_ep-domination, for every set Aζ and every nonnegative K_ep, to every n with split.n₁ n = 0 and every ω ∈ Aζ, then for every confidence level ζ in (0, 1) there is an event of probability at least 1 − ζ and a nonnegative constant K_ep such that, for every ω in the event and every n, the weak-norm estimation excess plus lambda times the strong-norm estimation excess is bounded by K_ep times the population shape.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)), 0 ≤ lambda → (∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ K_ep, 0 ≤ K_ep ∧ ∀ (ω : Ω) (n : ℕ), 1 ≤ split.n₁ n → Causalean.Estimation.NPIV.Primal.explicitRate regimes is_estimator ω n ζ ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → (∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∀ (Aζ : Set Ω) (K_ep : ℝ), 0 ≤ K_ep → ∀ ω ∈ Aζ, ∀ (n : ℕ), ¬1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → ∀ (ζ : ℝ), 0 < ζ → ζ < 1 → ∃ Aζ K_ep, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ K_ep ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 tb.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n","l":257,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_inequality_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Localized empirical-process inequality for the primal NPIV estimator. Given, for every n, a localized-regime bundle for the weak-norm, regularizer, and cross function classes at that fold-A sample size, and a confidence level ζ strictly between 0 and 1, there is an event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the weak-norm estimation excess ‖T(ĥ_n − h_0)‖² − ‖T(h*_λ − h_0)‖² is bounded by the empirical regularizer gap λ · (‖h*‖²_{A(n)} − ‖ĥ‖²_{A(n)}) plus a localized envelope built from the per-n critical radii scaled by δ_n and a √(log(1/ζ)/n) deviation term, where ‖h‖²_{A(n)} := (split.n₁ n)⁻¹ ∑_{k < split.n₁ n} h(X_k)² is the fold-A empirical second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_ep, MeasurableSet Aζ_ep ∧ μ Aζ_ep ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_ep, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 ≤ lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":227,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_master_event_from_localized","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Master localized empirical-process event for the primal NPIV analysis. Given, for every sample size n, a localized-regime bundle for the weak-norm, regularizer, and cross function classes at that fold-A sample size, and a confidence level ζ strictly between 0 and 1, there is a single event of probability at least 1 − ζ on which, simultaneously for every n with 1 ≤ split.n₁ n, the population weak-objective excess plus the empirical regularizer excess is bounded by the empirical sup-objective excess plus a localized envelope built from the regimes' critical radii and a √(log(1/ζ)/n) term.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} (is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat) (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∃ Aζ_master, MeasurableSet Aζ_master ∧ μ Aζ_master ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ Aζ_master, ∀ (n : ℕ), 1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) - (S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2)) ≤ Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda (h_hat n ω) n ω - Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda tb.h_lambda_star_fun n ω + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":733,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_per_n_inequality_from_deviations","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"EP per-n inequality from an objective-level localized modulus (Helper B). Pure analytic per-n step, with no probability statement. Given a sample size n with 1 ≤ split.n₁ n, a localized-regime bundle at that fold-A sample size, a confidence level ζ strictly between 0 and 1, and the pointwise objective-level inequality, at sample point ω, bounding the population weak-objective-plus-regularizer excess by the empirical sup-objective excess plus the localized envelope, the weak-norm estimation excess is bounded by the empirical regularizer gap plus the same localized envelope, the empirical sup-objective excess having been eliminated using the optimality of ĥ_n (is_estimator.opt) against tb.h_lambda_star_fun. This statement is intentionally not vacuous: the hypothesis still contains the empirical sup-objective excess, which the proof cancels rather than assumes away.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} (is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat) (sc : Causalean.Estimation.NPIV.SourceCondition S β) (tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) {ζ : ℝ}, 0 < ζ → ζ < 1 → ∀ (n : ℕ), 1 ≤ split.n₁ n → ∀ (ω : Ω), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) - (S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2)) ≤ Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda (h_hat n ω) n ω - Causalean.Estimation.NPIV.Primal.supObjective S TC sample split lambda tb.h_lambda_star_fun n ω + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n))) → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 tb.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 ≤ lambda * ((↑(split.n₁ n))⁻¹ * ∑ k, tb.h_lambda_star_fun (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑(split.n₁ n))⁻¹ * ∑ k, h_hat n ω (S.xOf (sample.Z (↑k) ω)) ^ 2) + (16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_HF.regime.ψ (split.n₁ n)) + 16 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_mF.regime.ψ (split.n₁ n)) + 8 * delta n * Causalean.Stat.Concentration.criticalRadius ((regimes n).bundle_F.regime.ψ (split.n₁ n)) + (4 * (regimes n).bundle_HF.regime.b + 4 * (regimes n).bundle_mF.regime.b + 2 * (regimes n).bundle_F.regime.b) * √(2 * Real.log (4 * 2 ^ (n + 1) / ζ) / ↑(split.n₁ n)))","l":146,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.ep_pop_inner_at_closedness_witness","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPPerN","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"EP closedness-witness population identity (Helper B1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} [MeasureTheory.IsProbabilityMeasure μ] {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f : S.𝒵 → ℝ} (hf : f ∈ TC.F), S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf) → 2 * ∫ (ω : Ω), S.m (S.W ω) f ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f (S.zOf (S.W ω)) ^ 2 ∂μ = S.weakNorm (S.hL2 (TC.H_subset hh) - S.hL2 S.h₀_mem) ^ 2","l":24,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPPerN.lean"},{"n":"Causalean.Estimation.NPIV.Primal.explicitRate","k":"def","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"For an NPIV operator system, candidate and critic classes, an observation-law measure, an independent sample, a one-shot split of that sample, a real regularization level and source exponent, a sequence of localization radii, a sequence of covariate estimators, a source condition, its Tikhonov bias-bound bundle, localized-regime bundles at every nuisance-fold sample size, a TRAE primal estimator, a sample realization, a sample-size index, and a real confidence level, the explicit rate is the sum of the localized empirical-process critical-radius and envelope terms, plus the regularization contribution, at that index and confidence level.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} → {P_W : MeasureTheory.Measure S.𝒲} → {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} → {split : Causalean.Stat.OneShotSplit sample} → {lambda β : ℝ} → {delta : ℕ → ℝ} → {h_hat : ℕ → Ω → S.𝒳 → ℝ} → {sc : Causalean.Estimation.NPIV.SourceCondition S β} → {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} → ((n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb (split.n₁ n) (delta n)) → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Ω → ℕ → ℝ → ℝ","l":188,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.innerIntegrand","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an NPIV operator system, a real regularization level, a covariate candidate function, an instrument critic function, and an observation, the empirical integrand is $2\\{m(w;f)-h(x)f(z)\\}-f(z)^2+\\lambda h(x)^2$, where $x$ and $z$ are that observation's covariate and instrument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → ℝ → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → S.𝒲 → ℝ","l":109,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.innerObjective","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an NPIV operator system, an independent sample, a one-shot sample split, a real regularization level, a covariate candidate function, an instrument critic function, a sample-size index, and a sample realization, the fold-A inner objective is the average empirical integrand over the nuisance fold at that index.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (S.𝒳 → ℝ) → (S.𝒵 → ℝ) → ℕ → Ω → ℝ","l":123,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒲","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an inverse-problem system, the observation space is equipped with the measurable space specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒲","l":60,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒳","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an inverse-problem system, the covariate space is equipped with the measurable space specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒳","l":68,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.instMeasurableSpace_𝒵","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an inverse-problem system, the instrument space is equipped with the measurable space specified by that system.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) → MeasurableSpace S.𝒵","l":76,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.integral_comp_law_W","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventsBase","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Integrating a measurable function under the law of the observed variable is the same as integrating its pullback over the ambient sample space.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {P_W : MeasureTheory.Measure S.𝒲}, MeasureTheory.Measure.map S.W μ = P_W → ∀ {g : S.𝒲 → ℝ}, Measurable g → ∫ (w : S.𝒲), g w ∂P_W = ∫ (ω : Ω), g (S.W ω) ∂μ","l":51,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventsBase.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the critic class star(F) — controls |(1/n) Σ f(Z_i)² − E[f(Z)²]| uniformly over f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, f (S.zOf (sample.Z (↑k) ω)) ^ 2 - ∫ (ω' : Ω), f (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_F.regime.ψ n) + regime.bundle_F.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the squared critic class star(F). This is the non-peeled building block for the Foster pair-gap bridge.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f₁ ∈ TC.F, ∀ f₂ ∈ TC.F, |(↑n)⁻¹ * ∑ k, (f₁ (S.zOf (sample.Z (↑k) ω)) ^ 2 - f₂ (S.zOf (sample.Z (↑k) ω)) ^ 2) - ∫ (ω' : Ω), f₁ (S.zOf (S.W ω')) ^ 2 - f₂ (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 4 * (regime.F_L2_const * regime.F_diameter) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_F.regime.ψ n) + regime.bundle_F.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":88,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_F_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the squared critic class star(F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair f₁, f₂ in the critic class TC.F, on which the gap between the empirical and population second-moment differences of f₁ and f₂ is bounded by 8 · F_L2_const · ‖f₁ − f₂‖_strong · δ_n + 5 · δ_n², where the strong norm is taken in the critic's L² embedding and F_L2_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (f₁ : S.𝒵 → ℝ) (hf₁ : f₁ ∈ TC.F) (f₂ : S.𝒵 → ℝ) (hf₂ : f₂ ∈ TC.F), |(↑n)⁻¹ * ∑ k, (f₁ (S.zOf (sample.Z (↑k) ω)) ^ 2 - f₂ (S.zOf (sample.Z (↑k) ω)) ^ 2) - ∫ (ω' : Ω), f₁ (S.zOf (S.W ω')) ^ 2 - f₂ (S.zOf (S.W ω')) ^ 2 ∂μ| ≤ 8 * regime.F_L2_const * S.strongNorm (S.qL2 (TC.F_subset hf₁) - S.qL2 (TC.F_subset hf₂)) * δ_n + 5 * δ_n ^ 2","l":171,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_H","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventH","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Fixed-diameter localized deviation event for the candidate class star(H). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair h₁, h₂ in the primal hypothesis class TC.H, on which the gap between the fold's empirical second-moment difference (1/n) Σ_k h₁(X_k)² − (1/n) Σ_k h₂(X_k)² and its population counterpart E[h₁(X)²] − E[h₂(X)²] is bounded by 4 · (H_diameter + δ_n) · criticalRadius(ψ n) plus a √(2·log(1/δ)/n) deviation term, where H_diameter and the critical-radius regime come from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h₁ ∈ TC.H, ∀ h₂ ∈ TC.H, |(↑n)⁻¹ * ∑ k, h₁ (S.xOf (sample.Z (↑k) ω)) ^ 2 - (↑n)⁻¹ * ∑ k, h₂ (S.xOf (sample.Z (↑k) ω)) ^ 2 - (∫ (ω' : Ω), h₁ (S.xOf (S.W ω')) ^ 2 ∂μ - ∫ (ω' : Ω), h₂ (S.xOf (S.W ω')) ^ 2 ∂μ)| ≤ 4 * (regime.H_diameter + δ_n) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_H.regime.ψ n) + regime.bundle_H.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventH.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the product class star(H · F) — controls |(1/n) Σ h(X_i) f(Z_i) − E[h(X) f(Z)]| uniformly over h ∈ TC.H, f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h ∈ TC.H, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, h (S.xOf (sample.Z (↑k) ω)) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), h (S.xOf (S.W ω')) * f (S.zOf (S.W ω')) ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_HF.regime.ψ n) + regime.bundle_HF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the cross class star(H · F) — produces a *single* μ-event simultaneously valid for every triple (h₁, h₂, f) ∈ TC.H × TC.H × TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ h₁ ∈ TC.H, ∀ h₂ ∈ TC.H, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, (h₁ (S.xOf (sample.Z (↑k) ω)) - h₂ (S.xOf (sample.Z (↑k) ω))) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), (h₁ (S.xOf (S.W ω')) - h₂ (S.xOf (S.W ω'))) * f (S.zOf (S.W ω')) ∂μ| ≤ 4 * (regime.HF_pair_const * regime.H_diameter * δ_n + δ_n) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_HF.regime.ψ n) + regime.bundle_HF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":91,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_HF_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventHF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the cross class star(H · F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair h₁, h₂ in the primal class TC.H and every critic f in TC.F, on which the gap between the empirical and population means of (h₁ − h₂) · f is bounded by 8 · HF_pair_const · δ_n² · ‖h₁ − h₂‖_strong + 5 · δ_n², where the strong norm is taken in the primal L² embedding and HF_pair_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (h₁ : S.𝒳 → ℝ) (hh₁ : h₁ ∈ TC.H) (h₂ : S.𝒳 → ℝ) (hh₂ : h₂ ∈ TC.H), ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, (h₁ (S.xOf (sample.Z (↑k) ω)) - h₂ (S.xOf (sample.Z (↑k) ω))) * f (S.zOf (sample.Z (↑k) ω)) - ∫ (ω' : Ω), (h₁ (S.xOf (S.W ω')) - h₂ (S.xOf (S.W ω'))) * f (S.zOf (S.W ω')) ∂μ| ≤ 8 * regime.HF_pair_const * δ_n ^ 2 * S.strongNorm (S.hL2 (TC.H_subset hh₁) - S.hL2 (TC.H_subset hh₂)) + 5 * δ_n ^ 2","l":218,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventHF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side localized deviation event for the moment class star(m ∘ F) — controls |(1/n) Σ m(W_i; f) − E[m(W; f)]| uniformly over f ∈ TC.F.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f ∈ TC.F, |(↑n)⁻¹ * ∑ k, S.m (sample.Z (↑k) ω) f - ∫ (ω' : Ω), S.m (S.W ω') f ∂μ| ≤ 4 * δ_n * Causalean.Stat.Concentration.criticalRadius (regime.bundle_mF.regime.ψ n) + regime.bundle_mF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":26,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF_pair","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Ω-side fixed-diameter pair-form localized deviation event for the moment class star(m ∘ F). This is the non-peeled building block for the Foster pair-gap bridge; the peeled variant can reuse the same interpretation at dyadic radii below mF_L2_const * F_diameter.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ f₁ ∈ TC.F, ∀ f₂ ∈ TC.F, |(↑n)⁻¹ * ∑ k, (S.m (sample.Z (↑k) ω) f₁ - S.m (sample.Z (↑k) ω) f₂) - ∫ (ω' : Ω), S.m (S.W ω') f₁ - S.m (S.W ω') f₂ ∂μ| ≤ 4 * (regime.mF_L2_const * regime.F_diameter) * Causalean.Stat.Concentration.criticalRadius (regime.bundle_mF.regime.ψ n) + regime.bundle_mF.regime.b * √(2 * Real.log (1 / δ) / ↑n)","l":89,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_for_mF_pair_peeled","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventMF","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Peeled pair-form localized deviation event for the moment class star(m ∘ F). Given a positive sample size n and a confidence level δ in (0, 1], there is a single event of probability at least 1 − δ, valid simultaneously for every pair f₁, f₂ in the critic class TC.F, on which the gap between the empirical and population means of m(·; f₁) − m(·; f₂) is bounded by 8 · mF_L2_const · δ_n · ‖f₁ − f₂‖_strong + 5 · δ_n², where the strong norm is taken in the critic's L² embedding and mF_L2_const comes from the supplied localized-regime witness.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {β lambda : ℝ} {sc : Causalean.Estimation.NPIV.SourceCondition S β} {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} {n : ℕ} {δ_n : ℝ} (regime : Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample sc tb n δ_n), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (f₁ : S.𝒵 → ℝ) (_hf₁ : f₁ ∈ TC.F) (f₂ : S.𝒵 → ℝ) (_hf₂ : f₂ ∈ TC.F), |(↑n)⁻¹ * ∑ k, (S.m (sample.Z (↑k) ω) f₁ - S.m (sample.Z (↑k) ω) f₂) - ∫ (ω' : Ω), S.m (S.W ω') f₁ - S.m (S.W ω') f₂ ∂μ| ≤ 8 * regime.mF_L2_const * δ_n * S.strongNorm (S.qL2 (TC.F_subset _hf₁) - S.qL2 (TC.F_subset _hf₂)) + 5 * δ_n ^ 2","l":166,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventMF.lean"},{"n":"Causalean.Estimation.NPIV.Primal.localized_omega_event_sharp_for_bundle","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.LocalizedEventsBase","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Ω-side Foster–Syrgkanis sharp localized event for a single NPIV LocalizedRegimeBundle. Consider an IID sample drawn from a probability space, where the pushforward of the ambient measure under the observed-variable map equals the stated observation law $P_W$ and the sample size n is positive. Fix a confidence parameter ζ with $0 < ζ ≤ 1$, a localized rate $δ_n$ with $δ_n > 0$, and a radius Rmax with δ_n ≤ Rmax. If for every dyadic level K with Rmax ≤ δ_n · 2 ^ K, the corresponding complexity term stays below δ_n ^ 2, then there is an event of probability at least 1 - ζ on which, for every index in the regime bundle whose function norm is at most Rmax, the empirical average of that function over the first n sample draws deviates from its population mean by at most 8 * δ_n * norm + 5 * δ_n ^ 2.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} {δ_n Rmax : ℝ} (B : Causalean.Estimation.NPIV.Primal.LocalizedRegimeBundle S.𝒲 P_W n δ_n), MeasureTheory.Measure.map S.W μ = P_W → 0 < n → ∀ {ζ : ℝ}, 0 < ζ → ζ ≤ 1 → 0 < δ_n → δ_n ≤ Rmax → (∀ (K : ℕ), Rmax ≤ δ_n * 2 ^ K → B.regime.b * √(2 * Real.log (2 * (↑K + 1) / ζ) / ↑n) ≤ δ_n ^ 2) → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal ζ ∧ ∀ ω ∈ E, ∀ (i : B.ι), B.norm (B.F i) ≤ Rmax → |(↑n)⁻¹ * ∑ k, B.F i (B.X (sample.Z (↑k) ω)) - ∫ (ω' : Ω), B.F i (B.X (S.W ω')) ∂μ| ≤ 8 * δ_n * B.norm (B.F i) + 5 * δ_n ^ 2","l":61,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/LocalizedEventsBase.lean"},{"n":"Causalean.Estimation.NPIV.Primal.measure_iInter_nat_ge_one_sub_tsum_of_ge","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Regulariser","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"If each event in a countable family has probability at least one minus its assigned error, then their intersection has probability at least one minus the sum of those errors.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {E : ℕ → Set Ω} {a : ℕ → ENNReal}, (∀ (n : ℕ), MeasurableSet (E n)) → (∀ (n : ℕ), μ (E n) ≥ 1 - a n) → μ (⋂ n, E n) ≥ 1 - ∑' (n : ℕ), a n","l":27,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Regulariser.lean"},{"n":"Causalean.Estimation.NPIV.Primal.measure_inter_ge_one_sub_add_of_ge","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"The intersection of two high-probability events has probability at least one minus the sum of their two failure probabilities.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A B : Set Ω} {a b : ENNReal}, MeasurableSet A → MeasurableSet B → μ A ≥ 1 - a → μ B ≥ 1 - b → μ (A ∩ B) ≥ 1 - (a + b)","l":58,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.populationShape","k":"def","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EventAssembly","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"For an NPIV operator system, candidate and critic classes, an observation-law measure, an independent sample, a one-shot split of that sample, a real regularization level and source exponent, a sequence of localization radii, a sequence of covariate estimators, a source condition, its Tikhonov bias-bound bundle, a TRAE primal estimator, a sample realization, and a sample-size index, the population shape is $R^2+\\delta_n y+\\delta_n^2+\\lambda\\delta_n x+\\lambda\\delta_n^2$, where $R$ is the weak-norm discrepancy between the population Tikhonov solution and the structural function, $y$ is the weak-norm estimation discrepancy, and $x$ is its strong-norm counterpart.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} → {P_W : MeasureTheory.Measure S.𝒲} → {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} → {split : Causalean.Stat.OneShotSplit sample} → {lambda β : ℝ} → {delta : ℕ → ℝ} → {h_hat : ℕ → Ω → S.𝒳 → ℝ} → {sc : Causalean.Estimation.NPIV.SourceCondition S β} → {tb : Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc} → Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat → Ω → ℕ → ℝ","l":224,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EventAssembly.lean"},{"n":"Causalean.Estimation.NPIV.Primal.population_closedness_critic_gap_eq","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.EPMasterEvent","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":2,"d":"Population curvature at the closedness critic. For a fixed candidate h, the closedness witness f_closed is the population maximizer of the quadratic adversarial criterion, and the drop at any critic f is exactly the squared L² critic gap.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} [MeasureTheory.IsProbabilityMeasure μ] {h : S.𝒳 → ℝ} (hh : h ∈ TC.H) {f_closed : S.𝒵 → ℝ} (hf_closed : f_closed ∈ TC.F) {f : S.𝒵 → ℝ} (hf : f ∈ TC.F), S.T (S.hL2 S.h₀_mem - S.hL2 (TC.H_subset hh)) = S.qL2 (TC.F_subset hf_closed) → 2 * ∫ (ω : Ω), S.m (S.W ω) f_closed ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f_closed (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f_closed (S.zOf (S.W ω)) ^ 2 ∂μ - (2 * ∫ (ω : Ω), S.m (S.W ω) f ∂μ - 2 * ∫ (ω : Ω), h (S.xOf (S.W ω)) * f (S.zOf (S.W ω)) ∂μ - ∫ (ω : Ω), f (S.zOf (S.W ω)) ^ 2 ∂μ) = ‖S.qL2 (TC.F_subset hf_closed) - S.qL2 (TC.F_subset hf)‖ ^ 2","l":334,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/EPMasterEvent.lean"},{"n":"Causalean.Estimation.NPIV.Primal.strong_rate_from_combined","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Strong rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → ∀ {Aζ : Set Ω} {K : ℝ}, 0 ≤ K → (∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)) → ∃ C_strong, 0 ≤ C_strong ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min β 1)","l":378,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.supObjective","k":"def","m":"Causalean.Estimation.NPIV.Primal.Estimator","a":"Estimation/NPIV/Primal","t":1,"d":"For an NPIV operator system, candidate and critic classes, an independent sample, a one-shot sample split, a real regularization level, a covariate candidate function, a sample-size index, and a sample realization, the TRAE primal sup-min objective is the supremum of the fold-A inner objective over the critic class.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → Causalean.Estimation.NPIV.Primal.TRAEClasses S → {P_W : MeasureTheory.Measure S.𝒲} → (sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W) → Causalean.Stat.OneShotSplit sample → ℝ → (S.𝒳 → ℝ) → ℕ → Ω → ℝ","l":138,"f":"Causalean/Estimation/NPIV/Primal/Estimator.lean"},{"n":"Causalean.Estimation.NPIV.Primal.trae_primal_rate","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"TRAE primal rate from localized regimes. Assume the base deterministic TRAE primal rate hypotheses hold (source condition and Tikhonov bias bound), and that a localized-regimes witness is available at every fold-A sample size, with regularization parameter 0 < lambda < 2 and confidence level 0 < ζ < 1. Suppose further that the localized discharge rate is absorbed by a constant multiple of the population shape at every confidence level and every eligible fold size, and that the same absorption inequality also holds, with an explicit weak/strong-norm correction term, at every fold size too small for the localized regime. Then there exist nonnegative constants C_strong, C_weak and an event of probability at least 1 - ζ on which, for every fold-A sample size n, both the strong-norm rate ‖ĥ_n − h₀‖²_{L²(P_X)} ≤ C_strong · (δ_n²/λ + ‖w₀‖_{L²(P_X)} · λ^min(β,1)) and the weak-norm rate ‖T(ĥ_n − h₀)‖²_{L²(P_Z)} ≤ C_weak · (δ_n² + ‖w₀‖_{L²(P_X)} · λ^min(β+1,2)) hold simultaneously.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (base : Causalean.Estimation.NPIV.Primal.TRAERatePrimalHyps S TC sample split lambda β ζ delta h_hat is_estimator) [MeasureTheory.IsProbabilityMeasure μ] (regimes : (n : ℕ) → Causalean.Estimation.NPIV.Primal.LocalizedRegimes S TC sample base.source_condition base.tikhonov_bias (split.n₁ n) (delta n)), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → (∀ (ζ' : ℝ), 0 < ζ' → ζ' < 1 → ∃ K_ep, 0 ≤ K_ep ∧ ∀ (ω : Ω) (n : ℕ), 1 ≤ split.n₁ n → Causalean.Estimation.NPIV.Primal.explicitRate regimes is_estimator ω n ζ' ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → (∀ (ζ' : ℝ), 0 < ζ' → ζ' < 1 → ∀ (Aζ : Set Ω) (K_ep : ℝ), 0 ≤ K_ep → ∀ ω ∈ Aζ, ∀ (n : ℕ), ¬1 ≤ split.n₁ n → S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 - S.weakNorm (S.hL2 base.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + lambda * (S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω))) ^ 2 - S.strongNorm (S.hL2 base.tikhonov_bias.h_lambda_star_mem) ^ 2) ≤ K_ep * Causalean.Estimation.NPIV.Primal.populationShape is_estimator ω n) → ∃ C_strong C_weak Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ C_strong ∧ 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 base.source_condition.w₀_mem) * lambda ^ min β 1) ∧ S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 base.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":827,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.trae_primal_rate_from_empirical_process","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":1,"d":"Non-iterated TRAE primal rate theorem (thm:est-trae-rate-theorem, lines 174–248 of trae_inverse_problems.tex). Assume the abstract deterministic TRAE primal rate hypotheses — source condition, Tikhonov bias bound, and localized empirical-process event — hold, the regularization parameter satisfies 0 < lambda < 2, and the confidence level satisfies 0 < ζ < 1. Then there exist nonnegative constants C_strong, C_weak and an event of probability at least 1 - ζ on which, for every fold-A sample size n, both the strong-norm rate ‖ĥ_n − h₀‖²_{L²(P_X)} ≤ C_strong · (δ_n²/λ + ‖w₀‖_{L²(P_X)} · λ^min(β,1)) and the weak-norm rate ‖T(ĥ_n − h₀)‖²_{L²(P_Z)} ≤ C_weak · (δ_n² + ‖w₀‖_{L²(P_X)} · λ^min(β+1,2)) hold simultaneously.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → lambda < 2 → 0 < ζ → ζ < 1 → ∃ C_strong C_weak Aζ, MeasurableSet Aζ ∧ μ Aζ ≥ 1 - ENNReal.ofReal ζ ∧ 0 ≤ C_strong ∧ 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_strong * (delta n ^ 2 / lambda + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min β 1) ∧ S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":767,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.weak_rate_from_combined","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.Rate","a":"Estimation/NPIV/Primal","t":2,"d":"Weak rate (helper, deterministic).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {TC : Causalean.Estimation.NPIV.Primal.TRAEClasses S} {P_W : MeasureTheory.Measure S.𝒲} {sample : Causalean.Stat.IIDSample Ω S.𝒲 μ P_W} {split : Causalean.Stat.OneShotSplit sample} {lambda β ζ : ℝ} {delta : ℕ → ℝ} {h_hat : ℕ → Ω → S.𝒳 → ℝ} {is_estimator : Causalean.Estimation.NPIV.Primal.IsTRAEPrimalEstimator S TC sample split lambda h_hat} (hyps : Causalean.Estimation.NPIV.Primal.TRAERatePrimalAbstractHyps S TC sample split lambda β ζ delta h_hat is_estimator), 0 < lambda → ∀ {Aζ : Set Ω} {K : ℝ}, 0 ≤ K → (∀ ω ∈ Aζ, ∀ (n : ℕ), lambda * S.strongNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 + S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 hyps.tikhonov_bias.h_lambda_star_mem) ^ 2 ≤ K * (S.weakNorm (S.hL2 hyps.tikhonov_bias.h_lambda_star_mem - S.hL2 S.h₀_mem) ^ 2 + delta n ^ 2)) → ∃ C_weak, 0 ≤ C_weak ∧ ∀ ω ∈ Aζ, ∀ (n : ℕ), S.weakNorm (S.hL2 (TC.H_subset (is_estimator.mem_H n ω)) - S.hL2 S.h₀_mem) ^ 2 ≤ C_weak * (delta n ^ 2 + S.strongNorm (S.hL2 hyps.source_condition.w₀_mem) * lambda ^ min (β + 1) 2)","l":602,"f":"Causalean/Estimation/NPIV/Primal/Rate.lean"},{"n":"Causalean.Estimation.NPIV.Primal.young_cross_envelope","k":"theorem","m":"Causalean.Estimation.NPIV.Primal.EmpiricalProcessEvent.Algebra","a":"Estimation/NPIV/Primal/EmpiricalProcessEvent","t":1,"d":"Young / AM-GM cross-term envelope. For nonnegative reals R, δ, w, and κ, the mixed sum δ · (R + w + δ) + R · κ is bounded above by the quadratic envelope R² + δ · w + (3/2) · δ² + (1/2) · κ².","s":"∀ (R δ w κ : ℝ), 0 ≤ R → 0 ≤ δ → 0 ≤ w → 0 ≤ κ → δ * (R + w + δ) + R * κ ≤ R ^ 2 + δ * w + 3 / 2 * δ ^ 2 + 1 / 2 * κ ^ 2","l":58,"f":"Causalean/Estimation/NPIV/Primal/EmpiricalProcessEvent/Algebra.lean"},{"n":"Causalean.Estimation.NPIV.SourceCondition","k":"structure","m":"Causalean.Estimation.NPIV.SourceCondition","a":"Estimation/NPIV","t":1,"d":"The β-source condition represents the true primal nuisance as the image, under a nonnegative spectral power of the normal NPIV operator, of an admissible witness function in the primal candidate class, via the spectral identity expressing the nuisance as that power of the operator applied to the witness, inside the L² space.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → Type u_3","l":51,"f":"Causalean/Estimation/NPIV/SourceCondition.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition","k":"structure","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Spectral β-source condition at the primal nuisance h₀: strengthens SourceCondition by additionally requiring that the primal candidate subspace coincides with the full ambient L² space, which is what lets the Lax–Milgram Tikhonov minimiser on the candidate class agree with the resolvent expression needed to run the spectral argument.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Estimation.NPIV.OperatorSystem Ω μ → ℝ → Type u_3","l":100,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Hbar_L2_hasProj","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Closedness witness for the full primal candidate subspace.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), S.Hbar_L2.HasOrthogonalProjection","l":125,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Tstar_T_isPositive","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"T†T is a positive operator.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ), S.Tstar_T.IsPositive","l":131,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.Tstar_T_spectrum_nonneg","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Positivity of T†T: its real spectrum lies in [0, ∞).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (_sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), ∀ x ∈ spectrum ℝ S.Tstar_T, 0 ≤ x","l":138,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.biasConst","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"For a spectral source condition, the bias constant is $(\\max\\{1,\\lVert T^*T\\rVert+1\\})^\\beta$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {β : ℝ} → Causalean.Estimation.NPIV.SpectralSourceCondition S β → ℝ","l":178,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.biasConst_nonneg","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"The uniform Tikhonov bias constant is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), 0 ≤ sc.biasConst","l":190,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.continuous_sourceSymbol","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"The source symbol is continuous whenever the source exponent is nonnegative.","s":"∀ {β : ℝ}, 0 ≤ β → Continuous (Causalean.Estimation.NPIV.SpectralSourceCondition.sourceSymbol β)","l":156,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.sourceSymbol","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"For a real source exponent, the source symbol is the function sending each real number $x$ to $(\\max\\{x,0\\})^{\\beta/2}$.","s":"ℝ → ℝ → ℝ","l":148,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.spectralPower","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"For a spectral source condition, the spectral-power operator is the real functional-calculus transform of the adjoint-product NPIV operator by the symbol $x\\mapsto(\\max\\{x,0\\})^{\\beta/2}$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} → {β : ℝ} → Causalean.Estimation.NPIV.SpectralSourceCondition S β → ↥(MeasureTheory.Lp ℝ 2 μ) →L[ℝ] ↥(MeasureTheory.Lp ℝ 2 μ)","l":163,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.spectral_identity_h₀","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":2,"d":"Restated spectral identity using spectralPower.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β), S.hL2 S.h₀_mem = sc.spectralPower (S.hL2 sc.w₀_mem)","l":171,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.strong_bias","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Strong-metric Tikhonov bias bound. Given a spectral β-source condition sc linking the structural function h₀ to a coefficient w₀ through the spectral power operator (T†T)^{β/2}, for any strictly positive Tikhonov regularization level λ, the squared strong-metric ($L^2(P_X)$) distance between the Tikhonov minimiser h*_λ at level λ and h₀ is bounded by biasConst · ‖w₀‖² · λ^{min(β,2)}, where biasConst is a constant determined by T†T and β.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → ‖S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem‖ ^ 2 ≤ sc.biasConst * ‖S.hL2 sc.w₀_mem‖ ^ 2 * lambda ^ min β 2","l":816,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.tikhonovMinimiserL2_eq_resolvent","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Resolvent identification of the Lax–Milgram minimiser. Given a spectral β-source condition sc whose primal candidate set coincides with the whole ambient L² space, for any strictly positive Tikhonov regularization level λ, the population Tikhonov minimiser at level λ equals the resolvent expression obtained by applying the real functional calculus of T†T to the symbol x ↦ x/(λ+x), evaluated at the L² class of the structural function h₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → S.tikhonovMinimiserL2 lambda = (Causalean.Estimation.NPIV.Complexification.realCFC S.Tstar_T fun x => x / (lambda + x)) (S.hL2 S.h₀_mem)","l":468,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.SpectralSourceCondition.weak_bias","k":"theorem","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Weak-metric Tikhonov bias bound. Given the same spectral β-source condition sc linking h₀ to w₀ through (T†T)^{β/2}, for any strictly positive Tikhonov regularization level λ, the squared weak-metric ($L^2(P_Z)$) norm of the operator T applied to the Tikhonov-minimiser bias h*_λ − h₀ is bounded by biasConst · ‖w₀‖² · λ^{min(β+1,2)}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {S : Causalean.Estimation.NPIV.OperatorSystem Ω μ} {β : ℝ} (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) {lambda : ℝ}, 0 < lambda → ‖S.Tlin (S.tikhonovMinimiserL2 lambda - S.hL2 S.h₀_mem)‖ ^ 2 ≤ sc.biasConst * ‖S.hL2 sc.w₀_mem‖ ^ 2 * lambda ^ min (β + 1) 2","l":993,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.TikhonovBiasBound","k":"structure","m":"Causalean.Estimation.NPIV.SourceCondition","a":"Estimation/NPIV","t":1,"d":"The Tikhonov bias-bound bundle records, at a positive regularization level, the properties of the corresponding population Tikhonov solution: a nonnegative constant for which its squared distance to the true nuisance in the strong candidate-space norm and its squared distance to the true nuisance in the weak operator-image norm are each bounded by that constant times a power of the regularization level, together with the population strong-convexity inequality it satisfies relative to every other candidate.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β : ℝ) → ℝ → Causalean.Estimation.NPIV.SourceCondition S β → Type u_3","l":84,"f":"Causalean/Estimation/NPIV/SourceCondition.lean"},{"n":"Causalean.Estimation.NPIV.TikhonovPullback","k":"structure","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"Function-level pullback datum for the discharge. The user provides a Hbar-element h_lambda_star_fun whose L² class equals the Lax–Milgram minimiser tikhonovMinimiserL2 S λ constructed in Operator/Tikhonov.lean. This single pullback is the only function-level commitment needed: the bias and convexity inequalities all live at the L² level and transport along this equation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β : ℝ) → ℝ → Causalean.Estimation.NPIV.SpectralSourceCondition S β → Type u_3","l":1207,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.instAlgebraRealLpCLM","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"For a measurable sample space and its measure, the real-scalar algebra is the algebra of bounded complex-linear operators on the corresponding complex-valued $L^2$ space.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Algebra ℝ (↥(MeasureTheory.Lp ℂ 2 μ) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 μ))","l":76,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.mixed_bias_identity","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":1,"d":"Mixed-bias / DR identity — prop:est-trae-mixed-bias. Fix a linear inverse-problem functional system, and suppose q₀ solves the associated dual moment equation. Then for any primal candidate function h in the closed candidate set Hbar and any dual candidate function q in the closed candidate set Qbar, the bias of the doubly-robust functional Θ(h, q) relative to the true target θ₀ equals the expectation of the product (q₀(Z) − q(Z))·(h(X) − h₀(X)) of the dual and primal nuisance errors.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {h : S.𝒳 → ℝ}, h ∈ S.Hbar → ∀ {q : S.𝒵 → ℝ}, q ∈ S.Qbar → S.Θ h q - S.θ₀ = ∫ (ω : Ω), (q₀ (S.Z ω) - q (S.Z ω)) * (h (S.X ω) - S.h₀ (S.X ω)) ∂μ","l":42,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.NPIV.tikhonov_bias_from_spectral","k":"def","m":"Causalean.Estimation.NPIV.Operator.SpectralCalculus","a":"Estimation/NPIV/Operator","t":1,"d":"For an NPIV operator system, a real source exponent, a real regularization level, a strictly positive regularization level, a spectral source condition, and a pullback of the $L^2$ Tikhonov minimiser to a primal candidate function, the construction returns a Tikhonov bias-bound bundle for that system, exponent, level, and the source condition underlying the spectral condition.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.NPIV.OperatorSystem Ω μ) → (β lambda : ℝ) → 0 < lambda → (sc : Causalean.Estimation.NPIV.SpectralSourceCondition S β) → Causalean.Estimation.NPIV.TikhonovPullback S β lambda sc → Causalean.Estimation.NPIV.TikhonovBiasBound S β lambda sc.toSourceCondition","l":1226,"f":"Causalean/Estimation/NPIV/Operator/SpectralCalculus.lean"},{"n":"Causalean.Estimation.NPIV.Θ_h₀_eq_θ₀","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":2,"d":"DR cancellation at the truth on the primal side: Θ(h₀, q) = θ₀ for every q ∈ Qbar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {q : S.𝒵 → ℝ}, q ∈ S.Qbar → S.Θ S.h₀ q = S.θ₀","l":171,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.NPIV.Θ_q₀_eq_θ₀","k":"theorem","m":"Causalean.Estimation.NPIV.MixedBias","a":"Estimation/NPIV","t":2,"d":"DR cancellation at the truth on the dual side: Θ(h, q₀) = θ₀ for every h ∈ Hbar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (S : Causalean.Estimation.NPIV.InverseProblemSystem Ω μ) {q₀ : S.𝒵 → ℝ}, S.DualSolution q₀ → ∀ {h : S.𝒳 → ℝ}, h ∈ S.Hbar → S.Θ h q₀ = S.θ₀","l":183,"f":"Causalean/Estimation/NPIV/MixedBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Bias_n","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.SecondOrderBias","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system, a target-direction derivative bundle at its true nuisance function, a target-direction derivative bundle at a nuisance function, and a target estimate, the loss-gradient nuisance bias is the population integral of the first bundle evaluated at the target estimate minus the corresponding population integral of the second bundle.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀ → {g : G} → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S g → Θ → ℝ","l":57,"f":"Causalean/Estimation/OrthogonalLearning/Population/SecondOrderBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"On a measurable covariate space, given a nuisance-function vector at the truth, the bounded-direction nuisance slice is the set of all nuisance-function vectors for which the outcome-regression difference from the truth is uniformly bounded over both treatment arms and covariate values, and the propensity-score difference from the truth is uniformly bounded over covariate values.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Set (Causalean.Estimation.ATE.NuisanceVec γ)","l":59,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For an orthogonal statistical-learning system and a real clamp radius, the centered clamped target-minimization condition states that every candidate target has auxiliary population risk at least that of the designated target, where its excess loss relative to the designated target is clamped to $[-b,b]$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → ℝ → Prop","l":409,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"For a CATE estimation system with a standard Borel unit space and finite population measure, a candidate target set, a candidate target, an evaluation map from candidate targets to functions of the covariates, and a real clamp radius, the clamped DR-Learner target-minimization condition states that, for every target in the candidate set, the population risk obtained by clamping the true-nuisance squared pseudo-outcome error to the interval $[-b,b]$ is no greater for the designated target than for that target.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → {Θ : Type u_2} → Set Θ → Θ → (Θ → γ → ℝ) → ℝ → Prop","l":120,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRCriticalRadius","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For a CATE estimation system with a standard Borel unit space and finite population measure, a candidate target set, an evaluation map from candidate targets to functions of the covariates, a sequence indexing targets in the candidate set, a nuisance-function vector, a norm on real-valued observed-data functions, and a family of real complexity envelopes, the DR-Learner critical-radius condition requires that each envelope is sub-root and that at every sample size it upper-bounds the population Rademacher complexity of the indexed centered quadratic augmented inverse-probability-weighted loss class.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → {Θ : Type u_2} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → (Θ_set : Set Θ) → (Θ → γ → ℝ) → (ℕ → ↑Θ_set) → Causalean.Estimation.ATE.NuisanceVec γ → ((γ × Bool × ℝ → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → Prop","l":47,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DREvalBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For a candidate target set, an evaluation map from candidate targets to functions of the covariates, and a real bound, the DR-Learner evaluation-boundedness condition requires that the bound is nonnegative and that the absolute evaluation of every candidate target at every covariate value is at most the bound.","s":"{γ : Type u_1} → {Θ : Type u_2} → Set Θ → (Θ → γ → ℝ) → ℝ → Prop","l":56,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"On a measurable covariate space, for a nuisance-function vector and a real bound, the DR-Learner outcome-regression boundedness condition requires that the bound is nonnegative and that for either treatment arm and every covariate value, the absolute conditional-mean outcome regression is at most the bound.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → ℝ → Prop","l":87,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For a CATE estimation system with a standard Borel unit space and finite population measure, a candidate target set, a nuisance-function vector, and a real overlap floor, the DR-Learner nuisance-overlap condition requires: the floor is strictly positive; it is at most one half; and at every covariate value the nuisance propensity score lies between the floor and one minus the floor.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → {Θ : Type u_2} → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → Set Θ → Causalean.Estimation.ATE.NuisanceVec γ → ℝ → Prop","l":76,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DROutcomeBounded","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For a CATE estimation system with a standard Borel unit space and finite population measure and a real bound, the DR-Learner outcome-boundedness condition requires that the bound is nonnegative and that the observed outcome has absolute value at most the bound for almost every observation under the system's observed-data law.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → ℝ → Prop","l":64,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"For a CATE estimation system with a standard Borel unit space and finite population measure, a candidate target set, a candidate target, and an evaluation map from candidate targets to functions of the covariates, the DR-Learner target-minimization condition states that, for every target in the candidate set, the population squared error between the true-nuisance augmented inverse-probability-weighted pseudo-outcome and the evaluation of the designated target is no greater than the corresponding squared error for that target.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.CATE.CATEEstimationSystem P γ → {Θ : Type u_2} → Set Θ → Θ → (Θ → γ → ℝ) → Prop","l":100,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DiffQuotientEnvelopeG","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system, the nuisance-direction difference-quotient envelope condition states that every admissible nuisance function has a positive neighborhood radius and an integrable envelope such that for almost every observation and every nonzero perturbation smaller than that radius, the absolute nuisance-direction loss difference quotient is bounded by the envelope.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":79,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.DiffQuotientEnvelopeTheta","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system and a nuisance function, the target-direction difference-quotient envelope condition states that every candidate target has a positive neighborhood radius and an integrable envelope such that for almost every observation and every nonzero perturbation smaller than that radius, the absolute target-direction loss difference quotient is bounded by the envelope.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Prop","l":67,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.EvalDirDeriv","k":"structure","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"A pointwise directional-derivative bundle for a candidate evaluation map along the segment from a base point to a candidate parameter: it packages the derivative's value at each covariate together with the fact that the map's difference quotient along the segment converges to that value as the perturbation parameter vanishes, the derivative's measurability in the covariate, and a uniform bound on the derivative over the parameter set.","s":"{Θ : Type u_1} → [inst : NormedAddCommGroup Θ] → [InnerProductSpace ℝ Θ] → {γ : Type u_2} → [MeasurableSpace γ] → Set Θ → Θ → (Θ → γ → ℝ) → Type (max u_1 u_2)","l":59,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.FirstOrderInequality","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system and target-direction derivative data for its loss at the distinguished nuisance function, the population first-order inequality holds exactly when, for every target in the system's target class, the integral under the population observation law of the corresponding target directional derivative is nonnegative. This is the population KKT condition characterising the distinguished target as a minimizer of population risk at the distinguished nuisance over the target class.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀ → Prop","l":122,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.G_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"For an orthogonal statistical-learning system, the nuisance perturbation-closure condition holds exactly when every nuisance function in its nuisance class and every scalar between zero and one produce a point on the line segment from the distinguished nuisance function to that nuisance function which also belongs to the nuisance class.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":109,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasDirDerivG","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Pointwise directional derivative of g ↦ ℓ z θ₀ g along the segment from g₀ to g, packaged with the pointwise tendsto witness and measurability.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Type (max u_2 u_4)","l":72,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Directional derivative of the learning-system loss in the target coordinate, at a fixed nuisance g. Bundles a candidate directional-derivative function dℓ_θ, giving a real number for each target θ and observation z, the witness that for every target θ in the system's target class and every observation z, the loss's difference quotient along the segment from θ₀ to θ at nuisance g converges to dℓ_θ θ z as the step size shrinks to zero, and measurability of dℓ_θ θ in the observation for every target θ.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Type (max u_2 u_3)","l":52,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Population.DirectionalDeriv","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Mixed target-nuisance directional derivative of the learning-system loss, at the truth (θ₀, g₀). Bundles a target-direction directional-derivative bundle Dθ_at g anchored at every accessible nuisance value g, a real-valued mixed directional-derivative function dℓ_θg of the target, the nuisance, and the observation, the witness that for every target θ and nuisance g in the system's classes and every observation z, the target-direction derivative anchored at the perturbed nuisance g₀ + t(g - g₀) converges, as t → 0, to dℓ_θg θ g z, and measurability of dℓ_θg θ g in the observation for every target θ and nuisance g.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Type (max (max u_2 u_3) u_4)","l":87,"f":"Causalean/Estimation/OrthogonalLearning/Population/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LearningSystem","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Orthogonal statistical-learning system. Bundles a convex target class Θ_set containing a distinguished target θ₀, a nuisance class G_set containing a distinguished nuisance g₀, a jointly measurable loss ℓ, and the condition that θ₀ minimizes the population risk over the target class at the true nuisance g₀.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_2) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Θ : Type u_3) → [inst : NormedAddCommGroup Θ] → [InnerProductSpace ℝ Θ] → (G : Type u_4) → [inst : AddCommGroup G] → [Module ℝ G] → Type (max (max u_2 u_3) u_4)","l":43,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LearningSystem.L","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"Given an orthogonal statistical-learning system, a target value, and a nuisance value, the population risk is the integral, under the population observation law, of the system's loss at that target and nuisance.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Θ → G → ℝ","l":88,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Local","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, an independent and identically distributed sample with the system's population law, a one-shot sample split, a rate sequence, a confidence tolerance, and a nuisance function, the local empirical-process modulus condition holds exactly when, for every sample size, there exists an event that is measurable, has measure at least $1-\\delta^+$, with subtraction truncated at zero, and on which, uniformly over the target class, the population excess risk minus the fold-B empirical excess risk is at most $\\rho_n$ times the distance from the distinguished target plus $\\rho_n^2$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → ℝ) → ℝ → G → Prop","l":45,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Local.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulusRandom","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, an independent and identically distributed sample with the system's population law, a one-shot sample split, a rate sequence, a confidence tolerance, and a sample-dependent nuisance estimator, the random-nuisance local empirical-process modulus condition holds exactly when, for every sample size, there exists an event that is measurable, has measure at least $1-\\delta^+$, with subtraction truncated at zero, and on which, uniformly over the target class, the population excess risk at the realized nuisance estimate minus the fold-B empirical excess risk is at most $\\rho_n$ times the distance from the distinguished target plus $\\rho_n^2$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → ℝ) → ℝ → (ℕ → Ω → G) → Prop","l":60,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For an orthogonal statistical-learning system, an independent and identically distributed sample from its observation law, a one-shot sample split, a nuisance function, a sequence indexing candidate targets, a norm on real-valued observation functions, a family of real complexity envelopes, and a real bound, the localized Rademacher regime requires: the bound is nonnegative; the centered loss is bounded in absolute value by that bound for every observation and candidate target; each complexity envelope is sub-root; and at every sample size, the envelope upper-bounds the population Rademacher complexity of the indexed centered-loss class on the split's validation fold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (_S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit _S_iid → G → (ℕ → ↑S.Θ_set) → ((Z → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → ℝ → Prop","l":86,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"For an orthogonal statistical-learning system, an independent and identically distributed sample from its observation law, a one-shot sample split, a nuisance function, a sequence indexing candidate targets, a norm on real-valued observation functions, a family of real complexity envelopes, and a real bound, the almost-everywhere localized Rademacher regime requires: the bound is nonnegative; for almost every observation under the observation law, the centered loss is bounded in absolute value by that bound for every candidate target; each complexity envelope is sub-root; and at every sample size, the envelope upper-bounds the population Rademacher complexity of the indexed centered-loss class on the split's validation fold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (_S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit _S_iid → G → (ℕ → ↑S.Θ_set) → ((Z → ℝ) → ℝ) → (ℕ → ℝ → ℝ) → ℝ → Prop","l":131,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system and a nuisance function, the target-continuity condition for the loss holds exactly when, for every observation, the loss as a function of the target is continuous on the system's target class at that fixed nuisance function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → Prop","l":174,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system and a bundle of mixed target--nuisance directional derivatives of its loss, the mixed-score dominated-convergence bridge states that, for every candidate target and admissible nuisance function, the integrated centered difference quotient of the target-direction derivative along the nuisance perturbation converges, as the nonzero perturbation tends to zero, to the observation-law integral of the corresponding mixed directional derivative.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S → Prop","l":91,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"For an orthogonal statistical-learning system and a bundle of mixed target--nuisance directional derivatives of its loss, the Neyman-orthogonality condition for the loss states that, for every target in the candidate target set and every nuisance function in the admissible nuisance set, the observation-law integral of the mixed directional derivative at those target and nuisance directions equals zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S → Prop","l":58,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv","k":"structure","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"A nuisance derivative bundle records, for the doubly robust pseudo-outcome, its directional derivative in each nuisance direction as the limit of the corresponding difference quotient, together with that derivative's measurability.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.ATE.NuisanceVec γ → Type u_1","l":90,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.RademacherBound","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, an independent and identically distributed sample with the system's population law, a one-shot sample split, a nuisance function, a sequence of targets in the target class, and a real sequence, the Rademacher-complexity bound holds exactly when, for every sample size, the proposed bound is nonnegative and the Rademacher complexity of the centred loss class indexed by that target sequence on fold B, computed under the ambient sample measure using the zeroth observation coordinate, is at most the proposed bound.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → G → (ℕ → ↑S.Θ_set) → (ℕ → ℝ) → Prop","l":110,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.PluginERM","a":"Estimation/OrthogonalLearning","t":1,"d":"This predicate says that a sample-size-indexed target estimator, evaluated against a plug-in nuisance estimate on the estimation fold, is an approximate empirical-risk minimizer: the optimization slack is nonnegative at every sample size, the estimator's value always lies in the target parameter set, and its fold-B empirical risk at the plug-in nuisance is within that slack of the risk at every other point in the target set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → (ℕ → Ω → Θ) → (ℕ → Ω → G) → (ℕ → ℝ) → Prop","l":60,"f":"Causalean/Estimation/OrthogonalLearning/PluginERM.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.LinftyDevTailBound","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Given a measure on a sample space, a nonempty finite coordinate set, a sample-indexed vector-valued deviation field, a deviation threshold, and a confidence tolerance, the high-probability sup-norm tail-bound condition holds exactly when there exists an event that is measurable, has measure at least $1-\\delta^+$, with subtraction truncated at zero, and on which the maximum absolute coordinate of the deviation field is at most the threshold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → {p : ℕ} → Finset.univ.Nonempty → (Ω → EuclideanSpace ℝ (Fin p)) → ℝ → ℝ → Prop","l":264,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"For a finite coordinate dimension and a designated support set of coordinates, the restricted cone consists of exactly those coefficient vectors whose coordinate one-norm outside the support is no greater than three times their coordinate one-norm on the support.","s":"{p : ℕ} → Finset (Fin p) → Set (EuclideanSpace ℝ (Fin p))","l":79,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.RSC","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"For a finite coordinate dimension, an empirical-risk function, a specified gradient-like vector-valued function, a reference coefficient vector, a support set, and a real curvature constant, restricted strong convexity means that, for every vector in the restricted cone of the support, the empirical-risk increase minus the inner product of that perturbation with the specified vector-valued function at the reference vector is at least one half the curvature constant times the perturbation's squared Euclidean norm.","s":"{p : ℕ} → (EuclideanSpace ℝ (Fin p) → ℝ) → (EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) → EuclideanSpace ℝ (Fin p) → Finset (Fin p) → ℝ → Prop","l":43,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/RSC.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM","k":"structure","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Sparse plug-in regularised ERM (predicate form). Given an empirical risk with the plug-in nuisance already absorbed and a candidate parameter estimate, this predicate records that the ℓ₁-penalty level is nonnegative and that the candidate minimises the ℓ₁-penalised empirical risk over the entire ambient parameter space, with no restriction to a support set.","s":"{p : ℕ} → (EuclideanSpace ℝ (Fin p) → ℝ) → EuclideanSpace ℝ (Fin p) → ℝ → Prop","l":148,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Full","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"For a finite coordinate dimension and a coefficient vector, the full coordinate one-norm is the sum of the absolute values of all its coordinates.","s":"{p : ℕ} → EuclideanSpace ℝ (Fin p) → ℝ","l":60,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Full_eq","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":2,"d":"l1Full decomposes along any subset and its complement in Finset.univ.","s":"∀ {p : ℕ} (ν : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)), Causalean.Estimation.OrthogonalLearning.Sparse.l1Full ν = Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀ + Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν (Finset.univ \\ S₀)","l":99,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"For a finite coordinate dimension, a coefficient vector, and a set of coordinates, the restricted coordinate one-norm is the sum of the absolute values of that vector's coordinates in the specified set.","s":"{p : ℕ} → EuclideanSpace ℝ (Fin p) → Finset (Fin p) → ℝ","l":53,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm_supp_le_card_sqrt_mul_l2norm","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Cauchy–Schwarz on the support. The ℓ¹ norm of a vector ν restricted to a finite index set S₀ is bounded by √|S₀| times its full ℓ² norm.","s":"∀ {p : ℕ} (ν : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)), Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀ ≤ √↑S₀.card * ‖ν‖","l":107,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.linftyDev","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"For a finite coordinate dimension, a proof that its coordinate set is nonempty, and a coefficient vector, the coordinate infinity-norm deviation is the largest absolute coordinate value of that vector.","s":"{p : ℕ} → Finset.univ.Nonempty → EuclideanSpace ℝ (Fin p) → ℝ","l":65,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.mem_RestrictedCone_iff","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.Setup","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Restricted cone membership unfolded. A vector ν lies in the restricted cone RestrictedCone S₀ exactly when its ℓ¹ norm off the support S₀ is at most three times its ℓ¹ norm on S₀.","s":"∀ {p : ℕ} (S₀ : Finset (Fin p)) (ν : EuclideanSpace ℝ (Fin p)), ν ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ↔ Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν (Finset.univ \\ S₀) ≤ 3 * Causalean.Estimation.OrthogonalLearning.Sparse.l1Norm ν S₀","l":90,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.sparse_plugin_guarantee","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"Deterministic sparse plug-in ERM guarantee. Let empRiskFn be a convex empirical risk function on EuclideanSpace ℝ (Fin p), with population gradient popGrad and empirical gradient map gradEmp. Assume the support S₀ of the truth θ₀ has cardinality s, that θ₀ vanishes off S₀, that the restricted-strong-convexity modulus σn is strictly positive and empRiskFn is σn-restricted-strongly-convex at θ₀ over S₀, and that the empirical gradient at θ₀ satisfies the subgradient basic inequality for empRiskFn. Given a penalty level with lambda > 0, suppose the index set Fin p is nonempty so that lambda is at least twice the sup-norm deviation of the empirical gradient from the population gradient at θ₀, that the population gradient vanishes at the truth (the first-order condition), and that θhat is a sparse plug-in regularised empirical-risk minimizer at penalty level lambda. Then the estimation error θhat − θ₀ lies in the restricted cone around S₀, and its Euclidean norm is at most 12 · lambda · √s / σn.","s":"∀ {p : ℕ} (empRiskFn : EuclideanSpace ℝ (Fin p) → ℝ) (popGrad : EuclideanSpace ℝ (Fin p)) (gradEmp : EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) (θ₀ θhat : EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)) (s : ℕ), S₀.card = s → (∀ i ∉ S₀, θ₀.ofLp i = 0) → ConvexOn ℝ Set.univ empRiskFn → ∀ (σn : ℝ), 0 < σn → Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity empRiskFn gradEmp θ₀ S₀ σn → (∀ (θ : EuclideanSpace ℝ (Fin p)), empRiskFn θ - empRiskFn θ₀ ≥ inner ℝ (gradEmp θ₀) (θ - θ₀)) → ∀ (lambda : ℝ), 0 < lambda → ∀ (hp : Finset.univ.Nonempty), lambda ≥ 2 * Causalean.Estimation.OrthogonalLearning.Sparse.linftyDev hp (gradEmp θ₀ - popGrad) → popGrad = 0 → Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM empRiskFn θhat lambda → θhat - θ₀ ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ∧ ‖θhat - θ₀‖ ≤ 12 * lambda * √↑s / σn","l":49,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Sparse.sparse_plugin_guarantee_highProb","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Sparse.PluginGuarantee","a":"Estimation/OrthogonalLearning/Sparse","t":1,"d":"High-probability sparse plug-in ERM guarantee. Let empRiskFn(ω) be a random empirical risk function on EuclideanSpace ℝ (Fin p) with random empirical gradient map gradEmp(ω), and popGrad the population gradient. Assume the support S₀ of the truth θ₀ has cardinality s, that θ₀ vanishes off S₀, and that empRiskFn(ω) is convex on the ambient space for every ω. Suppose the restricted-strong- convexity modulus σn is strictly positive and empRiskFn(ω) is σn-restricted-strongly-convex at θ₀ over S₀ for every ω, and that the empirical gradient at θ₀ satisfies the subgradient basic inequality for empRiskFn(ω), for every ω. Given a penalty level with lambda > 0, and assuming the index set Fin p is nonempty, that the population gradient vanishes at the truth, and that θhat(ω) is a sparse plug-in regularised empirical-risk minimizer of empRiskFn(ω) at penalty level lambda, for every ω. Suppose further that the deviation of the empirical gradient from popGrad at θ₀ obeys a sup-norm tail bound ρ at confidence level δ, with lambda at least twice that tail level, lambda ≥ 2ρ. Then there is an event of probability at least 1 - δ on which, for every ω in it, the estimation error θhat ω − θ₀ lies in the restricted cone around S₀ and its Euclidean norm is at most 12 · lambda · √s / σn.","s":"∀ {p : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (empRiskFn : Ω → EuclideanSpace ℝ (Fin p) → ℝ) (popGrad : EuclideanSpace ℝ (Fin p)) (gradEmp : Ω → EuclideanSpace ℝ (Fin p) → EuclideanSpace ℝ (Fin p)) (θ₀ : EuclideanSpace ℝ (Fin p)) (θhat : Ω → EuclideanSpace ℝ (Fin p)) (S₀ : Finset (Fin p)) (s : ℕ), S₀.card = s → (∀ i ∉ S₀, θ₀.ofLp i = 0) → (∀ (ω : Ω), ConvexOn ℝ Set.univ (empRiskFn ω)) → ∀ (σn : ℝ), 0 < σn → (∀ (ω : Ω), Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedStrongConvexity (empRiskFn ω) (gradEmp ω) θ₀ S₀ σn) → (∀ (ω : Ω) (θ : EuclideanSpace ℝ (Fin p)), empRiskFn ω θ - empRiskFn ω θ₀ ≥ inner ℝ (gradEmp ω θ₀) (θ - θ₀)) → ∀ (lambda : ℝ), 0 < lambda → ∀ (hp : Finset.univ.Nonempty), popGrad = 0 → (∀ (ω : Ω), Causalean.Estimation.OrthogonalLearning.Sparse.SparsePluginERM (empRiskFn ω) (θhat ω) lambda) → ∀ (ρ δ : ℝ), Causalean.Estimation.OrthogonalLearning.Sparse.LinftyDevTailBound μ hp (fun ω => gradEmp ω θ₀ - popGrad) ρ δ → lambda ≥ 2 * ρ → ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, θhat ω - θ₀ ∈ Causalean.Estimation.OrthogonalLearning.Sparse.RestrictedCone S₀ ∧ ‖θhat ω - θ₀‖ ≤ 12 * lambda * √↑s / σn","l":283,"f":"Causalean/Estimation/OrthogonalLearning/Sparse/PluginGuarantee.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, a nuisance function, and a real bound, the uniform bounded-loss condition holds exactly when, for every observation and every target in the target class, the absolute loss at that nuisance function is at most the bound.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → ℝ → Prop","l":155,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, a nuisance function, and a real bound, the almost-everywhere uniform bounded-loss condition holds exactly when, outside a set of population probability zero, every target in the target class has absolute loss at that nuisance function at most the bound.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → G → ℝ → Prop","l":163,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.anchor_mem_boundedNuisanceDirs","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"The anchor η₀ belongs to BoundedNuisanceDirs η₀ (with Cμ = Ce = 0).","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η₀ : Causalean.Estimation.ATE.NuisanceVec γ), η₀ ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs η₀","l":83,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.badDataSet","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an orthogonal statistical-learning system, an independent and identically distributed sample with the system's population law, a one-shot sample split, a sample size, a loss bound, a confidence tolerance, a candidate complexity-bound sequence, and a nuisance function, the product-sample bad event is the set of fold-B observation vectors for which twice the centred-loss Rademacher complexity plus $2b\\sqrt{2\\log(1/\\delta)/m}$, where $m$ is the fold-B size, does not exceed the centred-loss uniform deviation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → (split : Causalean.Stat.OneShotSplit S_iid) → (n : ℕ) → ℝ → ℝ → (ℕ → ℝ) → G → Set (Fin (split.foldB n).card → Z)","l":91,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.badDataSet_mass_le","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Mass bound. The product-space bad set has Measure.pi P_Z-mass ≤ δ. The proof uses the deterministic fixed-nuisance construction from Rademacher.lean: McDiarmid concentration, symmetrization, and separable lifting.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < (split.foldB n).card → (MeasureTheory.Measure.pi fun x => P_Z) (Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R g) ≤ ENNReal.ofReal δ","l":151,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drBias_le_product","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.SecondOrderBias","a":"Estimation/CATE/OrthogonalLearning","t":1,"d":"DR-Learner second-order product bias bound. For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions, with the true nuisance η₀ lying in the strict-overlap band at some margin ε > 0, fix a convex candidate target class in an inner-product space with a real-valued evaluation map, and suppose θ₀ belongs to this class, every candidate's evaluation is measurable, and θ₀'s evaluation agrees pointwise with the true value-space CATE. Fix a candidate nuisance h that also lies in the strict-overlap band at the same margin, a candidate target θ̂, and a nonnegative constant B bounding the evaluation-map directional derivative at θ̂. Assume the arm-wise outcome-regression fit of h, the AIPW pseudo-outcome discrepancy between h and the truth, and that discrepancy weighted by the directional derivative are all integrable, the outcome-regression error of h and the propensity error of h are square-integrable arm by arm, and the loss-gradient integrand at the true nuisance and the loss-gradient integrand at the candidate nuisance h are each integrable against the observation law. Then the loss-gradient nuisance bias Bias_n, evaluated between the true and candidate nuisance directional derivatives at θ̂, is bounded in absolute value by (2B/ε) times the sum over treatment arms of the outcome-regression L² error times the propensity L² error.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), ∀ h ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε, ∀ (θhat : Θ) {B : ℝ}, 0 ≤ B → (∀ (x : γ), |D.dEval θhat x| ≤ B) → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => h.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → MeasureTheory.Integrable (fun ω => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) * D.dEval θhat (S.factualX ω)) P.μ → (∀ (a : Bool), MeasureTheory.MemLp (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => h.e_fn x - S.e_val x) 2 S.P_X → MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ θhat z) S.P_Z → MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ θhat z) S.P_Z → |Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h) θhat| ≤ 2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => h.e_fn x - S.e_val x) 2 S.P_X).toReal","l":56,"f":"Causalean/Estimation/CATE/OrthogonalLearning/SecondOrderBias.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drLearningSystem","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"Given a CATE estimation system with a standard Borel unit space and finite population measure, a real inner-product target space, a convex candidate target set, a designated candidate target belonging to that set, an evaluation map whose evaluation is measurable for every candidate, the condition that the designated target evaluates pointwise to the system's conditional average treatment effect, and the condition that it minimizes the true-nuisance population squared pseudo-outcome risk over the candidate set, the DR-Learner orthogonal statistical-learning system has observed-data law given by the CATE system, nuisance functions given by outcome regressions and a propensity score, and squared augmented inverse-probability-weighted pseudo-outcome loss.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) → (Θ : Type u_2) → [inst_3 : NormedAddCommGroup Θ] → [inst_4 : InnerProductSpace ℝ Θ] → (Θ_set : Set Θ) → Convex ℝ Θ_set → (θ₀ : Θ) → θ₀ ∈ Θ_set → (eval : Θ → γ → ℝ) → (∀ (θ : Θ), Measurable (eval θ)) → (∀ (x : γ), eval θ₀ x = S.τ_val x) → Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval → Causalean.Estimation.OrthogonalLearning.LearningSystem P.Ω P.μ (γ × Bool × ℝ) S.P_Z Θ (Causalean.Estimation.ATE.NuisanceVec γ)","l":142,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv","k":"def","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"For a potential-outcome system with a standard-Borel sample space and a finite measure, a covariate space with its σ-algebra, a CATE estimation system, its causal identification assumptions, a strictly positive overlap bound, and membership of the system's true nuisance in the corresponding overlap class, together with a target inner-product space, a convex target-parameter set, a target parameter in that set, a measurable target-evaluation rule that equals the true conditional treatment effect at the target parameter, a statement that this parameter minimizes the doubly robust target criterion, an evaluation-rule directional-derivative bundle, and a nuisance directional-derivative bundle, the mixed directional-derivative bundle for the resulting doubly robust learning system is given by the displayed closed-form derivatives.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) → S.Assumptions → {ε : ℝ} → 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → (Θ : Type u_2) → [inst_3 : NormedAddCommGroup Θ] → [inst_4 : InnerProductSpace ℝ Θ] → (Θ_set : Set Θ) → (Θ_convex : Convex ℝ Θ_set) → (θ₀ : Θ) → (θ₀_mem : θ₀ ∈ Θ_set) → (eval : Θ → γ → ℝ) → (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) → (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) → (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) → Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval → Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀ → Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes)","l":122,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drNeymanOrthog","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"DR-Learner loss orthogonality under a DCT bridge (prop:est-osl-dr-loss-orthogonal). For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions, fix a margin ε > 0 such that the true nuisance η₀ lies in the strict-overlap slice at that margin, i.e. the propensity score is bounded away from 0 and 1. Fix a convex candidate target class inside an inner-product space together with a real-valued evaluation map on it, and suppose θ₀ belongs to this class, every candidate's evaluation is measurable, θ₀'s evaluation agrees pointwise with the true value-space CATE, and θ₀ minimizes the population AIPW pseudo-outcome squared-loss risk against the true nuisance over the candidate class. Given directional-derivative data for the evaluation map and for the doubly robust pseudo-outcome that together assemble a closed-form mixed directional derivative for the loss, if a dominated-convergence bridge licenses passing the limit defining the integrated mixed target/nuisance score through the integral, then this closed-form derivative witnesses that the DR-Learner squared loss is Neyman-orthogonal: its integrated mixed directional derivative vanishes at the truth (θ₀, η₀) for every admissible target and bounded nuisance direction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND) → Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND)","l":736,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.drNeymanOrthog_witness","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":1,"d":"DR-Learner loss orthogonality (prop:est-osl-dr-loss-orthogonal). For a CATE estimation system built on a potential-outcome model that satisfies the back-door identification assumptions with the propensity score bounded away from 0 and 1 by some margin (strict overlap), fix a convex candidate target class inside an inner-product space together with a real-valued evaluation map on it, and suppose the candidate θ₀ belongs to this class, every candidate's evaluation is measurable, θ₀'s evaluation agrees pointwise with the true value-space CATE, and θ₀ minimizes the population AIPW pseudo-outcome squared-loss risk against the true nuisance over the candidate class. If, for the resulting DR-Learner learning system, a dominated-convergence bridge licenses passing the limit defining the mixed target/nuisance directional derivative through the integral and the integrated target-directional score has vanishing derivative, at every point of the bounded nuisance slice and for every candidate target, along the segment toward that point from the true nuisance, then the DR-Learner squared loss is Neyman-orthogonal: its integrated mixed directional derivative between target and nuisance directions vanishes at the truth (θ₀, η₀) for every admissible target and nuisance direction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → (∃ ε > 0, S.StrictOverlap ε) → ∀ (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (M : Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes)), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) M → (∀ θ ∈ Θ_set, ∀ η ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs S.η₀, Filter.Tendsto (fun t => (∫ (z : γ × Bool × ℝ), (M.Dθ_at (S.η₀ + t • (η - S.η₀))).dℓ_θ θ z ∂S.P_Z - ∫ (z : γ × Bool × ℝ), (M.Dθ_at S.η₀).dℓ_θ θ z ∂S.P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)) → Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) M","l":193,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_loss_uniformly_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Bridge: DR-Learner squared loss is bounded under (M_Θ, M_Y, M_μ, ε).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2)","l":94,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_scoreFlat","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"The score-flatness hypothesis for drMixedDirDeriv.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND) → ∀ θ ∈ Θ_set, ∀ η ∈ Causalean.Estimation.OrthogonalLearning.BoundedNuisanceDirs S.η₀, Filter.Tendsto (fun t => (∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (S.η₀ + t • (η - S.η₀))).dℓ_θ θ z ∂S.P_Z - ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ θ z ∂S.P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)","l":646,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.dr_scoreZero_of_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.DRLearner.Analytic","a":"Estimation/CATE/OrthogonalLearning/DRLearner","t":2,"d":"Discharge of hScoreZero for bounded nuisance directions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ), S.Assumptions → ∀ {ε : ℝ}, 0 < ε → S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (θ₀ : Θ) (eval : Θ → γ → ℝ) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀), ∀ θ ∈ Θ_set, ∀ (η : Causalean.Estimation.ATE.NuisanceVec γ), (∃ Cμ, ∀ (b : Bool) (x : γ), |(η - S.η₀).μ_fn b x| ≤ Cμ) → (∃ Ce, ∀ (x : γ), |(η - S.η₀).e_fn x| ≤ Ce) → ∫ (z : γ × Bool × ℝ), ND.dPhi η z * D.dEval θ z.1 ∂S.P_Z = 0","l":483,"f":"Causalean/Estimation/CATE/OrthogonalLearning/DRLearner/Analytic.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.empRiskFoldB","k":"def","m":"Causalean.Estimation.OrthogonalLearning.PluginERM","a":"Estimation/OrthogonalLearning","t":1,"d":"For a sample space, its σ-algebra and measure; an observation space and its measure; a target inner-product space; and a nuisance vector space, a learning system, an independent and identically distributed sample, a one-shot sample split, a sample size, a realized sample point, a target parameter, and a nuisance value, the estimation-fold empirical risk is the average loss over the split's estimation fold at that parameter and nuisance value.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit S_iid → ℕ → Ω → Θ → G → ℝ","l":45,"f":"Causalean/Estimation/OrthogonalLearning/PluginERM.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an independent and identically distributed sample, a one-shot sample split, and a sample size, the fold-B coordinate map sends each sample realization to its fold-B observations, indexed in their canonical finite order.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) → (split : Causalean.Stat.OneShotSplit S_iid) → (n : ℕ) → Ω → Fin (split.foldB n).card → Z","l":82,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord_law","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Fold-B coordinate law. μ.map (foldBCoord n) = Measure.pi P_Z. This identifies the validation-fold coordinates as an i.i.d. product sample.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (split : Causalean.Stat.OneShotSplit S_iid) (n : ℕ), MeasureTheory.Measure.map (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n) μ = MeasureTheory.Measure.pi fun x => P_Z","l":127,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldBCoord_meas","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"The fold-B coordinate map is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} (split : Causalean.Stat.OneShotSplit S_iid) (n : ℕ), Measurable (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n)","l":556,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.foldB_pi_law","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Fold-B joint-law identification (public alias). For an IID sample S split via split, at any fold size n, the joint distribution of the fold-B subsample, mapped from μ via ω ↦ (i ↦ S.Z i ω) indexed by i ∈ split.foldB n, equals the product measure Measure.pi (fun _ ↦ P).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (n : ℕ), MeasureTheory.Measure.map (fun ω i => S.Z (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P","l":90,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_drLearner","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner bounded-loss Rademacher modulus. Consider the doubly robust orthogonal-learning system for conditional treatment effects, cross-fitted via a one-shot sample split, whose truth-identifying candidate belongs to the target set, has a measurable evaluation map, and recovers the population CATE function through that evaluation map. Suppose the candidate evaluation maps are uniformly bounded, the observed outcome is almost-surely bounded, the realised nuisance outcome-regression is uniformly bounded on both treatment arms, and the realised propensity stays away from 0 and 1 by a fixed margin; suppose also that the resulting doubly robust loss is continuous in the candidate over the target set at that realised nuisance, a sequence R n bounds the Rademacher complexity of the centred loss class on the estimation fold, the truth-identifying candidate still minimizes the clamped population loss at the realised nuisance, and the confidence level δ lies strictly above 0 and at most 1. Then there is a nonnegative constant b for which the system obeys the local empirical-process modulus condition at the explicit rate ρ n = √(2 R n + 2b · √(2 log(1/δ) / |B(n)|)) on nonempty estimation folds, with boundary value ρ n = √(2b) on empty folds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ h","l":258,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_localized_drLearner","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner localized modulus, sharp Foster–Syrgkanis form. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm that is invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius criticalRadius (ψ m), and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Then there is a nonnegative envelope b such that the DR-Learner system satisfies the localized empirical-process modulus predicate at rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) on nonempty validation folds, falling back to √(2b) when the fold is empty.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] [Countable ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → (∀ (F F' : γ × Bool × ℝ → ℝ), F =ᵐ[S.P_Z] F' → norm F = norm F') → ∀ {Rmax L : ℝ}, 0 ≤ L → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ L * ‖θ - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀‖) → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm S.P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → γ × Bool × ℝ) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => S.P_Z)) → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → 2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2 * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ h","l":84,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_localized_drLearner_bounded","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearner","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":2,"d":"DR-Learner bounded-loss localized modulus.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∃ b, 0 ≤ b ∧ Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split (fun _n => √(2 * (2 * b))) δ h","l":308,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearner.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_bounded_rademacher","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Bounded-loss Rademacher bridge theorem. Assume b is nonnegative, that the loss magnitude is uniformly bounded by b over the parameter set, and that the loss is continuous in the parameter on the parameter set. Given a sequence R n that is nonnegative and upper-bounds the population Rademacher complexity of the centred loss class on the fold-B sample at every sample size, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with rate ρ n := √(2 · b) when the fold-B sample is empty and ρ n := √(2 · R n + 2 · b · √(2 · log(1/δ) / |foldB n|)) otherwise.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 ≤ b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ g","l":397,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_bounded_rademacher_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere bounded-loss bridge. Assume b is nonnegative, that the loss magnitude is bounded by b at P_Z-almost-every observation, over the parameter set, and that the loss is continuous in the parameter on the parameter set. Given a sequence R n that is nonnegative and upper-bounds the population Rademacher complexity of the centred loss class on the fold-B sample at every sample size, and that the target parameter minimizes, over the parameter set, the population risk of the loss — evaluated at the model's baseline nuisance and clamped to [-b, b], then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with rate ρ n := √(2 · b) when the fold-B sample is empty and ρ n := √(2 · R n + 2 · b · √(2 · log(1/δ) / |foldB n|)) otherwise.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 ≤ b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLossAE S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → (∀ θ ∈ S.Θ_set, ∫ (z : Z), max (-b) (min b (S.ℓ z S.θ₀ S.g₀)) ∂P_Z ≤ ∫ (z : Z), max (-b) (min b (S.ℓ z θ S.g₀)) ∂P_Z) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card))) δ g","l":885,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_bounded","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Uniform-bound fallback bridge. Under a localized Rademacher regime — a uniform bound b on the centred pointwise loss difference together with a sub-root Rademacher- complexity envelope ψ, and given that the population excess risk at every admissible parameter is likewise bounded by b, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with the constant envelope ρ n := √(2 · b).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] (g : G) (idx : ℕ → ↑S.Θ_set) {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime S S_iid split g idx norm ψ b → (∀ θ ∈ S.Θ_set, |S.L θ g - S.L S.θ₀ g| ≤ b) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun _n => √(2 * b)) δ g","l":238,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_bounded_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere uniform-bound fallback bridge. Under an almost-everywhere localized Rademacher regime — a P_Z-a.e. bound b on the centred pointwise loss difference together with a sub-root Rademacher-complexity envelope ψ, and given that the population excess risk at every admissible parameter is likewise bounded by b, then for any confidence level 0 < δ ≤ 1 the local empirical-process modulus condition holds, with the constant envelope ρ n := √(2 · b).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] (g : G) (idx : ℕ → ↑S.Θ_set) {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE S S_iid split g idx norm ψ b → (∀ θ ∈ S.Θ_set, |S.L θ g - S.L S.θ₀ g| ≤ b) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun _n => √(2 * b)) δ g","l":319,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_sharp","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Localized critical-radius modulus, countable-class Foster–Syrgkanis form. Fix a countable, densely-indexed target class on which the loss is continuous in the parameter for every observation and a localized Rademacher regime holds — a uniform centred-loss bound and a sub-root Rademacher-complexity envelope ψ. Suppose the Lipschitz constant L is nonnegative, that the chosen function-space norm of the centred loss difference at each parameter is bounded by L · ‖θ − θ₀‖, that the loss at each parameter is measurable and integrable under P_Z, and that the same centred loss difference has diameter at most Rmax in that norm. Assume the critical radius of ψ at every fold-B sample size satisfies criticalRadius (ψ m) ≤ Rmax and criticalRadius (ψ m) > 0, together with the sub-root fixed-point inequality ψ m (criticalRadius (ψ m)) ≤ criticalRadius (ψ m) ^ 2; that ψ upper-bounds the population Rademacher complexity of the star-hulled centred loss class on a fold-B-sized sample; and the technical regularity conditions that the empirical Rademacher supremum is bounded above and the upper empirical Rademacher complexity process is integrable. For any confidence level 0 < δ ≤ 1, assume the Foster–Syrgkanis peeling-absorption condition that at every dyadic shell count covering Rmax, the McDiarmid concentration slack at the union-bound-adjusted confidence is dominated by the squared critical radius. Then the local empirical-process modulus condition holds, with envelope ρ n := (8 · L + 3) · criticalRadius (ψ |foldB n|) (falling back to √(2 · b) when the fold-B sample is empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] [Countable ↑S.Θ_set] (g : G), (∀ (z : Z), Continuous fun θ => S.ℓ z (↑θ) g) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ {norm : (Z → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ} {b L Rmax : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegime S S_iid split g idx norm ψ b → 0 ≤ L → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ L * ‖θ - S.θ₀‖) → (∀ θ ∈ S.Θ_set, Measurable fun z => S.ℓ z θ g) → (∀ θ ∈ S.Θ_set, MeasureTheory.Integrable (fun z => S.ℓ z θ g) P_Z) → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → Z) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => P_Z)) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ g","l":426,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_sharp_ae","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Almost-everywhere countable-class localized bridge. Fix a countable, densely-indexed target class on which the loss is continuous in the parameter for every observation and the chosen function-space norm is invariant under P_Z-a.e. equality of its argument. Assume an almost-everywhere localized Rademacher regime holds — a uniform centred-loss bound and a sub-root Rademacher-complexity envelope ψ, that the Lipschitz constant L is nonnegative, that the norm of the centred loss difference at each parameter is bounded by L · ‖θ − θ₀‖, that the loss at each parameter is measurable and integrable under P_Z, and that the same centred loss difference has diameter at most Rmax in that norm. Assume the critical radius of ψ at every fold-B sample size satisfies criticalRadius (ψ m) ≤ Rmax and criticalRadius (ψ m) > 0, together with the sub-root fixed-point inequality ψ m (criticalRadius (ψ m)) ≤ criticalRadius (ψ m) ^ 2; that ψ upper-bounds the population Rademacher complexity of the star-hulled centred loss class on a fold-B-sized sample; the technical regularity conditions that the empirical Rademacher supremum is bounded above and the upper empirical Rademacher complexity process is integrable; and that the population-optimal parameter minimizes the auxiliary population risk with the centred excess loss clamped to [-b, b]. For any confidence level 0 < δ ≤ 1, assume the Foster–Syrgkanis peeling-absorption condition that at every dyadic shell count covering Rmax, the McDiarmid concentration slack at the union-bound-adjusted confidence is dominated by the squared critical radius. Then the local empirical-process modulus condition holds, with envelope ρ n := (8 · L + 3) · criticalRadius (ψ |foldB n|) (falling back to √(2 · b) when the fold-B sample is empty).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [Nonempty ↑S.Θ_set] [Countable ↑S.Θ_set] (g : G), (∀ (z : Z), Continuous fun θ => S.ℓ z (↑θ) g) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ {norm : (Z → ℝ) → ℝ}, (∀ (F F' : Z → ℝ), F =ᵐ[P_Z] F' → norm F = norm F') → ∀ {ψ : ℕ → ℝ → ℝ} {b L Rmax : ℝ}, Causalean.Estimation.OrthogonalLearning.LocalizedRademacherRegimeAE S S_iid split g idx norm ψ b → 0 ≤ L → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ L * ‖θ - S.θ₀‖) → (∀ θ ∈ S.Θ_set, Measurable fun z => S.ℓ z θ g) → (∀ θ ∈ S.Θ_set, MeasureTheory.Integrable (fun z => S.ℓ z θ g) P_Z) → (∀ θ ∈ S.Θ_set, (norm fun z => S.ℓ z θ g - S.ℓ z S.θ₀ g) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → Z) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => S.ℓ z (↑θ) g - S.ℓ z S.θ₀ g) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => P_Z)) → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes S b → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun n => if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) δ g","l":970,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_of_localized_singleton","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Localized","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Trivial finite class. When Θ_set = {θ₀} the modulus inequality holds with ρ n := 0 (the centred excess risk is identically zero). Mirrors localEmpProcessModulus_singleton in the global-Rademacher bridge.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [MeasureTheory.IsProbabilityMeasure μ] (g : G), S.Θ_set = {S.θ₀} → ∀ {δ : ℝ}, 0 < δ → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun x => 0) δ g","l":1456,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Localized.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_random_of_bounded_rademacher","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Cross-fit lift of the empirical-process modulus to a random nuisance. Let ĥ be a fold-A-measurable random nuisance sequence. Assume b is strictly positive, that at every sample size and sample point the loss magnitude at the realised nuisance ĥ n ω is uniformly bounded by b, and that the loss at that realised nuisance is continuous in the parameter. Given a sequence R n that Rademacher-controls the centred loss class at every realisation ĥ n ω, and a confidence level 0 < δ ≤ 1. Suppose further that the σ-algebra m_A n generating the fold-A information is coarser than the ambient σ-algebra, that fold A is independent of the fold-B coordinate block under this σ-algebra, and that the nuisance-indexed bad event depends jointly measurably on the fold-A outcome and the fold-B sample. Then the random-nuisance local empirical-process modulus condition holds, at rate ρ n := √(2·b) on empty folds and ρ n := √(2·R n + 2·b·√(2·log(1/δ)/|foldB n|)) otherwise, evaluated at the random nuisance ĥ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P_Z] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (ĥ : ℕ → Ω → G), (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S (ĥ n ω) b) → (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S (ĥ n ω)) → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), (∀ (n : ℕ) (ω : Ω), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split (ĥ n ω) idx R) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (m_A : ℕ → MeasurableSpace Ω), (∀ (n : ℕ), m_A n ≤ inferInstance) → (∀ (n : ℕ), ProbabilityTheory.Indep (m_A n) (MeasurableSpace.comap (Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n) inferInstance) μ) → (∀ (n : ℕ), MeasurableSet {p | p.2 ∈ Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R (ĥ n p.1)}) → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulusRandom S S_iid split (Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R) δ ĥ","l":566,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.localEmpProcessModulus_singleton","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.Rademacher","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"Trivial finite class. When Θ_set = {θ₀} (the class collapses to the truth), the modulus inequality holds with ρ n := 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) [MeasureTheory.IsProbabilityMeasure μ] (g : G), S.Θ_set = {S.θ₀} → ∀ {δ : ℝ}, 0 < δ → Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split (fun x => 0) δ g","l":1146,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.modulusRadius","k":"def","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Given an independent and identically distributed sample, a one-shot sample split, a loss bound, a confidence tolerance, and a complexity-bound sequence, the deterministic modulus-radius sequence assigns to every sample size $n$ the value $\\sqrt{2b}$ when fold B is empty, and otherwise the value $\\sqrt{2R_n+2b\\sqrt{2\\log(1/\\delta)/m}}$, where $m$ is the fold-B size.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z} → Causalean.Stat.OneShotSplit S_iid → ℝ → ℝ → (ℕ → ℝ) → ℕ → ℝ","l":115,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.modulus_of_not_badData","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomNuisance","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":2,"d":"On-event modulus. Off the bad set, the modulus inequality holds uniformly over Θ_set for a fixed nuisance g, using the same bounded-loss Rademacher radius as the deterministic bridge in Rademacher.lean.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) [MeasureTheory.IsProbabilityMeasure μ] (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {b : ℝ}, 0 < b → ∀ (g : G), Causalean.Estimation.OrthogonalLearning.UniformlyBoundedLoss S g b → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset S g → ∀ (idx : ℕ → ↑S.Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound S S_iid split g idx R → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ) (ω : Ω), 0 < (split.foldB n).card → Causalean.Estimation.OrthogonalLearning.foldBCoord S_iid split n ω ∉ Causalean.Estimation.OrthogonalLearning.badDataSet S S_iid split n b δ R g → ∀ θ ∈ S.Θ_set, S.L θ g - S.L S.θ₀ g - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω θ g - Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω S.θ₀ g) ≤ Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R n * ‖θ - S.θ₀‖ + Causalean.Estimation.OrthogonalLearning.modulusRadius split b δ R n ^ 2","l":233,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.neymanOrthog_iff_score_deriv_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.Population.NeymanOrthogonal","a":"Estimation/OrthogonalLearning/Population","t":1,"d":"Score reformulation of Neyman orthogonality. Assume the DCT-bridge condition — the integrated centred target-direction difference quotient along a nuisance perturbation converges to the integrated mixed directional derivative as the perturbation size tends to zero. Then Neyman orthogonality of the loss is equivalent to the score map g' ↦ D_θ L(θ₀, g')[ν_θ] having zero first derivative at g₀, for every admissible target direction ν_θ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (M : Causalean.Estimation.OrthogonalLearning.HasMixedDirDeriv S), Causalean.Estimation.OrthogonalLearning.MixedScoreDCTBridge S M → (Causalean.Estimation.OrthogonalLearning.NeymanOrthogLoss S M ↔ ∀ θ ∈ S.Θ_set, ∀ g ∈ S.G_set, Filter.Tendsto (fun t => (∫ (z : Z), (M.Dθ_at (S.g₀ + t • (g - S.g₀))).dℓ_θ θ z ∂P_Z - ∫ (z : Z), (M.Dθ_at S.g₀).dℓ_θ θ z ∂P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0))","l":111,"f":"Causalean/Estimation/OrthogonalLearning/Population/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_highProb","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerEndToEnd","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner end-to-end high-probability oracle inequality. Consider the same doubly robust orthogonal-learning system, one-shot sample split, and boundedness / overlap / continuity / Rademacher package as in localEmpProcessModulus_drLearner — truth-identifying admissibility and evaluation correctness, uniform bounds on the candidate evaluations, the outcome, and the realised nuisance regression together with propensity overlap, and loss continuity, a Rademacher bound R n, the clamped-loss minimizer property, and a confidence level δ in (0, 1] . Assume in addition that a plug-in empirical-risk-minimisation estimator sequence τhat, evaluated against the same realised nuisance on every cross-fitting fold, attains the empirical risk up to an optimization slack r_opt n, that the population risk is strongly convex at the realised nuisance with modulus σ > 0, and that the truth-identifying candidate satisfies the first-order optimality inequality for the population risk's directional derivative at the truth. Then there is a nonnegative constant b such that, for every sample size n, with P.μ-probability at least 1 − δ the estimation error obeys ‖τhat n ω − θ₀‖² ≤ (4(1+σ)/σ²) · ρ_{n,δ}² + (4/σ) · Bias_n + (4/σ) · r_opt n, where ρ_{n,δ} is the modulus rate realised by localEmpProcessModulus_drLearner.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).g₀) (Dθ_at_h : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h) (τhat : ℕ → P.Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), Dθ_at_h.dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card)) ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth Dθ_at_h (τhat n ω) + 4 / σ * r_opt n","l":49,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerEndToEnd.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_highProb_honest","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerHonest","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Product-bias DR-Learner oracle inequality. Consider a CATE estimation system satisfying the underlying backdoor identification assumptions, with an evaluation functional that is measurable in its parameter and recovers the true CATE at a parameter θ₀ belonging to the constraint set, and suppose the evaluation functional, the outcome, and a fixed plug-in nuisance's conditional-mean function are uniformly bounded while that nuisance's propensity score satisfies ε-overlap. Assume further the DR-loss is continuous in the parameter, its localized empirical process admits a Rademacher complexity bound, and a clamped version of θ₀ still minimizes the loss, the confidence level lies in $(0,1]$, the overlap parameter is positive, and both the true and the plug-in nuisance lie in the ε-overlap slice, and that the estimator sequence stays within the constraint set as an approximate sample-split plug-in empirical-risk minimizer (slack r_opt) whose excess population risk satisfies a strong-convexity-type lower bound with constant σ>0 and a first-order orthogonality inequality at the true nuisance's directional derivative. Finally, a battery of boundedness and integrability conditions bounds the directional-derivative envelope and controls the two nuisance-error terms entering the second-order bias decomposition. Then for every sample fold n, with probability at least 1-δ the squared estimation error of the DR-Learner is bounded by an oracle/Rademacher term plus an explicit second-order product-bias term (2B/ε)·Σ_a ‖Δμ_a‖₂·‖Δe‖₂ in the two nuisance L²-errors, plus an optimization slack.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.LossContinuousOnΘset (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ (R : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.RademacherBound (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split h idx R → Causalean.Estimation.OrthogonalLearning.DRClampedThetaMinimizes S Θ_set θ₀ eval ((M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε), h ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε → ∀ (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀) (τhat : ℕ → P.Ω → Θ), (∀ (n : ℕ) (ω : P.Ω), τhat n ω ∈ Θ_set) → ∀ (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) → ∀ {B : ℝ}, 0 ≤ B → (∀ θ ∈ Θ_set, ∀ (x : γ), |D.dEval θ x| ≤ B) → (∀ (a : Bool), MeasureTheory.Integrable (fun ω => h.μ_fn a (S.factualX ω)) P.μ) → MeasureTheory.Integrable (fun ω => Causalean.Estimation.CATE.phi_eta (S.factualZ ω) h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω)) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω') h - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) * D.dEval (τhat n ω) (S.factualX ω')) P.μ) → (∀ (a : Bool), MeasureTheory.MemLp (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X) → MeasureTheory.MemLp (fun x => h.e_fn x - S.e_val x) 2 S.P_X → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ (τhat n ω) z) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at h).dℓ_θ (τhat n ω) z) S.P_Z) → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * √(if (split.foldB n).card = 0 then 2 * b else 2 * R n + 2 * b * √(2 * Real.log (1 / δ) / ↑(split.foldB n).card)) ^ 2 + 4 / σ * (2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => h.μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => h.e_fn x - S.e_val x) 2 S.P_X).toReal) + 4 / σ * r_opt n","l":53,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerHonest.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_drLearner_random_nuisance","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.DRLearnerRandomNuisance","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner oracle inequality at a random cross-fitted nuisance. Consider a CATE estimation system satisfying the backdoor identification assumptions, with an evaluation functional measurable in its parameter and recovering the true CATE at a parameter θ₀ in the constraint set, and suppose the overlap parameter is positive while the true nuisance and every realization of the random, fold-A-estimated nuisance ĥ n ω lie in the ε-overlap slice. Assume the estimator sequence stays in the constraint set as an approximate sample-split plug-in empirical-risk minimizer with respect to the random nuisance (slack r_opt), whose excess population risk at each realized ĥ n ω obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the true nuisance's directional derivative, and that on a high-probability event the centred excess empirical risk at the realized estimator and nuisance is controlled by a rate ρ n. Finally, a battery of boundedness and integrability conditions governs the directional-derivative envelope and bounds the two random nuisance-error terms. Then for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle/Rademacher term (4(1+σ)/σ²)·(ρ n)² plus a random second-order product-bias term (4/σ)·(2B/ε)·Σ_a ‖(ĥ n ω).μ_fn a − μ_val a‖₂·‖(ĥ n ω).e_fn − e_val‖₂ evaluated at the realized nuisance, plus an optimization slack (4/σ)·r_opt n.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (hA : S.Assumptions) (Θ : Type u_2) [inst_3 : NormedAddCommGroup Θ] [inst_4 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {ε : ℝ} (hε_pos : 0 < ε) (h_overlap_η₀ : S.η₀ ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) (D : Causalean.Estimation.OrthogonalLearning.EvalDirDeriv Θ_set θ₀ eval) (ND : Causalean.Estimation.OrthogonalLearning.NuisanceDirDeriv S.η₀) (ĥ : ℕ → P.Ω → Causalean.Estimation.ATE.NuisanceVec γ), (∀ (n : ℕ) (ω : P.Ω), ĥ n ω ∈ Causalean.Estimation.ATE.BackdoorEstimationSystem.H_ε ε) → ∀ (τhat : ℕ → P.Ω → Θ), (∀ (n : ℕ) (ω : P.Ω), τhat n ω ∈ Θ_set) → ∀ (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat ĥ r_opt → ∀ (σ : ℝ), 0 < σ → (∀ (n : ℕ) (ω : P.Ω), ∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ (ĥ n ω) ≥ ∫ (z : γ × Bool × ℝ), ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (ĥ n ω)).dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀) → ∀ (ρ : ℕ → ℝ) {δ : ℝ}, (∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L (τhat n ω) (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ (ĥ n ω) - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split n ω (τhat n ω) (ĥ n ω) - Causalean.Estimation.OrthogonalLearning.empRiskFoldB (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split n ω θ₀ (ĥ n ω)) ≤ ρ n * ‖τhat n ω - θ₀‖ + ρ n ^ 2) → ∀ {B : ℝ}, 0 ≤ B → (∀ θ ∈ Θ_set, ∀ (x : γ), |D.dEval θ x| ≤ B) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.Integrable (fun ω' => (ĥ n ω).μ_fn a (S.factualX ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => Causalean.Estimation.CATE.phi_eta (S.factualZ ω') (ĥ n ω) - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (Causalean.Estimation.CATE.phi_eta (S.factualZ ω') (ĥ n ω) - Causalean.Estimation.CATE.phi₀ S (S.factualZ ω')) * D.dEval (τhat n ω) (S.factualX ω')) P.μ) → (∀ (n : ℕ) (ω : P.Ω) (a : Bool), MeasureTheory.MemLp (fun x => (ĥ n ω).μ_fn a x - S.μ_val a x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (ĥ n ω).e_fn x - S.e_val x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at S.η₀).dℓ_θ (τhat n ω) z) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => ((Causalean.Estimation.OrthogonalLearning.drMixedDirDeriv S hA hε_pos h_overlap_η₀ Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes D ND).Dθ_at (ĥ n ω)).dℓ_θ (τhat n ω) z) S.P_Z) → ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * (2 * B / ε * ∑ a, (MeasureTheory.eLpNorm (fun x => (ĥ n ω).μ_fn a x - S.μ_val a x) 2 S.P_X).toReal * (MeasureTheory.eLpNorm (fun x => (ĥ n ω).e_fn x - S.e_val x) 2 S.P_X).toReal) + 4 / σ * r_opt n","l":57,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/DRLearnerRandomNuisance.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_localized_drLearner_highProb","k":"theorem","m":"Causalean.Estimation.CATE.OrthogonalLearning.LocalEmpProcess.LocalizedDRLearnerEndToEnd","a":"Estimation/CATE/OrthogonalLearning/LocalEmpProcess","t":1,"d":"DR-Learner localized high-probability oracle inequality. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius, and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Finally, suppose the estimator sequence is an approximate sample-split plug-in empirical-risk minimizer at the fixed nuisance h (slack r_opt), whose excess population risk obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the truth's directional derivative. Then there is a nonnegative envelope b such that, for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle term (4(1+σ)/σ²)·(ρ n)² at the sharp localized Foster–Syrgkanis rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) (falling back to √(2b) on an empty validation fold), plus the nuisance bias term (4/σ)·Bias_n and an optimization slack (4/σ)·r_opt n.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.CATE.CATEEstimationSystem P γ) (Θ : Type u_2) [inst_4 : NormedAddCommGroup Θ] [inst_5 : InnerProductSpace ℝ Θ] (Θ_set : Set Θ) (Θ_convex : Convex ℝ Θ_set) [Nonempty ↑Θ_set] [Countable ↑Θ_set] (θ₀ : Θ) (θ₀_mem : θ₀ ∈ Θ_set) (eval : Θ → γ → ℝ) (eval_meas : ∀ (θ : Θ), Measurable (eval θ)) (eval_θ₀ : ∀ (x : γ), eval θ₀ x = S.τ_val x) (θ₀_minimizes : Causalean.Estimation.OrthogonalLearning.DRThetaMinimizes S Θ_set θ₀ eval) (S_iid : Causalean.Stat.IIDSample P.Ω (γ × Bool × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit S_iid) {M_Θ M_Y M_μ ε : ℝ}, Causalean.Estimation.OrthogonalLearning.DREvalBounded Θ_set eval M_Θ → Causalean.Estimation.OrthogonalLearning.DROutcomeBounded S M_Y → ∀ (h : Causalean.Estimation.ATE.NuisanceVec γ), Causalean.Estimation.OrthogonalLearning.DRNuisanceMuBounded h M_μ → Causalean.Estimation.OrthogonalLearning.DRNuisanceOverlap S Θ_set h ε → Causalean.Estimation.OrthogonalLearning.CenteredClampedThetaMinimizes (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2) → (∀ (z : γ × Bool × ℝ), Continuous fun θ => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h) → ∀ (idx : ℕ → ↑(Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set), DenseRange idx → ∀ {norm : (γ × Bool × ℝ → ℝ) → ℝ} {ψ : ℕ → ℝ → ℝ}, Causalean.Estimation.OrthogonalLearning.DRCriticalRadius S Θ_set eval (fun k => ⟨↑(idx k), (idx k).property⟩) h norm ψ → (∀ (F F' : γ × Bool × ℝ → ℝ), F =ᵐ[S.P_Z] F' → norm F = norm F') → ∀ {Rmax L : ℝ}, 0 ≤ L → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ L * ‖θ - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀‖) → (∀ θ ∈ (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).Θ_set, (norm fun z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) ≤ Rmax) → (∀ (m : ℕ), Causalean.Stat.Concentration.criticalRadius (ψ m) ≤ Rmax) → (∀ (m : ℕ), 0 < Causalean.Stat.Concentration.criticalRadius (ψ m)) → (∀ (m : ℕ), ψ m (Causalean.Stat.Concentration.criticalRadius (ψ m)) ≤ Causalean.Stat.Concentration.criticalRadius (ψ m) ^ 2) → (∀ (m : ℕ), Causalean.Stat.Concentration.RademacherUpperBound (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm S.P_Z id m (ψ m)) → (∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → γ × Bool × ℝ) (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r p (S_fin k)|)) → (∀ (m : ℕ) (r : ℝ), MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity m (Causalean.Stat.Concentration.starHullZeroOut (fun θ z => (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (↑θ) h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).ℓ z (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).θ₀ h) norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => S.P_Z)) → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → (∀ (n K : ℕ), 0 < (split.foldB n).card → Rmax ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) * 2 ^ K → 2 * (M_Θ + 2 * M_μ + 2 * (M_Y + M_μ) / ε) ^ 2 * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑(split.foldB n).card) ≤ Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card) ^ 2) → ∀ (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).g₀) (Dθ_at_h : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) h) (τhat : ℕ → P.Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) S_iid split τhat (fun x x_1 => h) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ Θ_set, (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ h - (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes).L θ₀ h ≥ ∫ (z : γ × Bool × ℝ), Dθ_at_h.dℓ_θ θ z ∂S.P_Z + σ / 2 * ‖θ - θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth → ∃ b, 0 ≤ b ∧ ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ P.μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖τhat n ω - θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * (if (split.foldB n).card = 0 then √(2 * b) else (8 * L + 3) * Causalean.Stat.Concentration.criticalRadius (ψ (split.foldB n).card)) ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n (Causalean.Estimation.OrthogonalLearning.drLearningSystem S Θ Θ_set Θ_convex θ₀ θ₀_mem eval eval_meas eval_θ₀ θ₀_minimizes) Dθ_truth Dθ_at_h (τhat n ω) + 4 / σ * r_opt n","l":39,"f":"Causalean/Estimation/CATE/OrthogonalLearning/LocalEmpProcess/LocalizedDRLearnerEndToEnd.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_plugin_ERM","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.OracleInequality","a":"Estimation/OrthogonalLearning","t":1,"d":"Conservative oracle inequality for the orthogonal sample-split plug-in ERM. Assume the realised nuisance at every sample point admits the directional-derivative structure needed for the first-order expansion of the loss, that the estimator is a sample-split plug-in ERM with optimization slack r_opt, and that the strong-convexity modulus σ is strictly positive. Suppose the population risk at the realised nuisance is σ-strongly convex around θ₀ in the chosen norm, and that the first-order inequality holds at the truth (θ₀, g₀). If, at the specific sample point (n, ω), the local empirical-process modulus inequality is realised for the estimator's target value against the realised nuisance, then the squared target-space estimation error is bounded by 4(1+σ)/σ² · (ρ n)² + (4/σ) · Bias_n + (4/σ) · r_opt n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀) (ghat : ℕ → Ω → G) (Dθ_hat : (n : ℕ) → (ω : Ω) → Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S (ghat n ω)) (θhat : ℕ → Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM S S_iid split θhat ghat r_opt → ∀ (σ : ℝ), 0 < σ → (∀ (n : ℕ) (ω : Ω), ∀ θ ∈ S.Θ_set, S.L θ (ghat n ω) - S.L S.θ₀ (ghat n ω) ≥ ∫ (z : Z), (Dθ_hat n ω).dℓ_θ θ z ∂P_Z + σ / 2 * ‖θ - S.θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality S Dθ_truth → ∀ (ρ : ℕ → ℝ) (n : ℕ) (ω : Ω), S.L (θhat n ω) (ghat n ω) - S.L S.θ₀ (ghat n ω) - (Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω (θhat n ω) (ghat n ω) - Causalean.Estimation.OrthogonalLearning.empRiskFoldB S S_iid split n ω S.θ₀ (ghat n ω)) ≤ ρ n * ‖θhat n ω - S.θ₀‖ + ρ n ^ 2 → ‖θhat n ω - S.θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n S Dθ_truth (Dθ_hat n ω) (θhat n ω) + 4 / σ * r_opt n","l":69,"f":"Causalean/Estimation/OrthogonalLearning/OracleInequality.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.oracle_inequality_plugin_ERM_highProb","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.OracleInequality","a":"Estimation/OrthogonalLearning","t":1,"d":"High-probability conservative oracle inequality for the orthogonal sample-split plug-in ERM. Fix a single nuisance value g. Assume the estimator is a sample-split plug-in ERM at this fixed nuisance, with optimization slack r_opt, and that the strong-convexity modulus σ is strictly positive. Suppose the population risk at g is σ-strongly convex around θ₀ in the chosen norm, and that the first-order inequality holds at the truth (θ₀, g₀). If the local empirical-process modulus condition holds at rate ρ and confidence level δ for the fixed nuisance g, then for every sample size n there is an event of probability at least 1 - δ on which the squared target-space estimation error is bounded by 4(1+σ)/σ² · (ρ n)² + (4/σ) · Bias_n + (4/σ) · r_opt n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {Θ : Type u_3} [inst_2 : NormedAddCommGroup Θ] [inst_3 : InnerProductSpace ℝ Θ] {G : Type u_4} [inst_4 : AddCommGroup G] [inst_5 : Module ℝ G] (S : Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G) (S_iid : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit S_iid) (Dθ_truth : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S S.g₀) (g : G) (Dθ_at_g : Causalean.Estimation.OrthogonalLearning.HasDirDerivTheta S g) (θhat : ℕ → Ω → Θ) (r_opt : ℕ → ℝ), Causalean.Estimation.OrthogonalLearning.SampleSplitPluginERM S S_iid split θhat (fun x x_1 => g) r_opt → ∀ (σ : ℝ), 0 < σ → (∀ θ ∈ S.Θ_set, S.L θ g - S.L S.θ₀ g ≥ ∫ (z : Z), Dθ_at_g.dℓ_θ θ z ∂P_Z + σ / 2 * ‖θ - S.θ₀‖ ^ 2) → Causalean.Estimation.OrthogonalLearning.FirstOrderInequality S Dθ_truth → ∀ (ρ : ℕ → ℝ) (δ : ℝ), Causalean.Estimation.OrthogonalLearning.LocalEmpProcessModulus S S_iid split ρ δ g → ∀ (n : ℕ), ∃ E, MeasurableSet E ∧ μ E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ‖θhat n ω - S.θ₀‖ ^ 2 ≤ 4 * (1 + σ) / σ ^ 2 * ρ n ^ 2 + 4 / σ * Causalean.Estimation.OrthogonalLearning.Bias_n S Dθ_truth Dθ_at_g (θhat n ω) + 4 / σ * r_opt n","l":180,"f":"Causalean/Estimation/OrthogonalLearning/OracleInequality.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.randomParam_event_le","k":"theorem","m":"Causalean.Estimation.OrthogonalLearning.LocalEmpProcess.RandomParam","a":"Estimation/OrthogonalLearning/LocalEmpProcess","t":1,"d":"Cross-fit conditioning keystone. Let Y be measurable with pushforward law μ.map Y = ν, and let m_A be a sub-σ-algebra with m_A ≤ mΩ that is independent of the σ-algebra generated by Y. Suppose the family of \"bad\" subsets of β indexed by ω has an m_A-measurable joint dependence on (ω, ·) and each individual bad set has ν-mass at most δ. Then the random event where Y ω falls in the ω-dependent bad set also has μ-mass at most δ.","s":"∀ {Ω : Type u_1} {β : Type u_2} [mΩ : MeasurableSpace Ω] [mβ : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → β}, Measurable Y → ∀ {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.Measure.map Y μ = ν → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap Y mβ) μ → ∀ {δ : ℝ} (Bad : Ω → Set β), MeasurableSet {p | p.2 ∈ Bad p.1} → (∀ (ω : Ω), ν (Bad ω) ≤ ENNReal.ofReal δ) → μ {ω | Y ω ∈ Bad ω} ≤ ENNReal.ofReal δ","l":54,"f":"Causalean/Estimation/OrthogonalLearning/LocalEmpProcess/RandomParam.lean"},{"n":"Causalean.Estimation.OrthogonalLearning.Θ_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalLearning.Setup","a":"Estimation/OrthogonalLearning","t":1,"d":"For an orthogonal statistical-learning system, the target perturbation-closure condition holds exactly when every target in its target class and every scalar between zero and one produce a point on the line segment from the distinguished target to that target which also belongs to the target class. This condition is weaker than convexity and supports directional-derivative hypotheses without requiring the whole target class to be convex.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {Θ : Type u_3} → [inst_2 : NormedAddCommGroup Θ] → [inst_3 : InnerProductSpace ℝ Θ] → {G : Type u_4} → [inst_4 : AddCommGroup G] → [inst_5 : Module ℝ G] → Causalean.Estimation.OrthogonalLearning.LearningSystem Ω μ Z P_Z Θ G → Prop","l":101,"f":"Causalean/Estimation/OrthogonalLearning/Setup.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Automatic debiasing representer. For a regression-nuisance moment system, a function α₀ on the covariate space that is measurable and integrable against the covariate measure, and that represents the Gateaux derivative of the population moment at the truth, in every direction, as the L²-inner product of α₀ against the regression target evaluated in that direction.","s":"Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys → Type u_2","l":112,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a linear regression-functional system, the population linear functional maps each regression function in that system's admissible class to the integral of its linear moment function under the system's observation measure.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → ℝ","l":103,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m_add","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Additivity of L_of_m on integrable summands.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (γ₁ γ₂ : S.H_γ), MeasureTheory.Integrable (fun z => S.m_lin z γ₁) S.P_Z → MeasureTheory.Integrable (fun z => S.m_lin z γ₂) S.P_Z → Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S (γ₁ + γ₂) = Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ₁ + Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ₂","l":109,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m_smul","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"ℝ-homogeneity of L_of_m.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (c : ℝ) (γ : S.H_γ), Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S (c • γ) = c * Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S γ","l":119,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear regression-functional system. Bundles an observation space with an induced regression-argument space, a regression class (a real vector space of nuisance parameters) paired with an evaluation map into the argument space, an observation-level functional linear in the nuisance argument, and a regression truth; the evaluation map is required additive and homogeneous in the nuisance argument, the observation-level functional is likewise additive and homogeneous in that argument and measurable in the observation for every fixed nuisance value, the regression-argument law is the pushforward of the observation law under the projection, and the regression residual at the truth, weighted by any measurable integrable function of the regression argument, integrates to zero.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":39,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Regression-nuisance moment system. Bundles an observation space with an induced covariate space, an outcome variable, a regression target functional on a normed nuisance class, the true nuisance and true scalar parameter, a population moment functional of the parameter and nuisance, and an observation-level moment kernel that averages to it; the regression target is required additive and homogeneous in its nuisance argument, the population moment vanishes at the truth, the supplied Gateaux derivative of the moment at the truth is itself additive and homogeneous, the kernel is measurable and integrates against the observation measure to the population moment, the covariate measure is the pushforward of the observation measure under the projection, and the regression residual at the truth, weighted by any measurable integrable function of the covariates, integrates to zero.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":42,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a regression-nuisance moment system, a regression nuisance function, a candidate representer on the covariate space, and a scalar target value, the automatically debiased population moment is the expectation of the automatically debiased score under the system's observation distribution.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → (S.X → ℝ) → ℝ → ℝ","l":141,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_directional_g_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the regression direction. For a regression-nuisance moment system with Riesz representer and any perturbation ν_g of the regression nuisance, the Gateaux derivative of the population debiased moment in the g-direction at the truth vanishes — equivalently, this is the representer identity for the perturbation ν_g.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (rep : Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter S) (ν_g : S.H), S.D_g_M ν_g - ∫ (x : S.X), rep.α₀ x * S.γ_target ν_g x ∂S.P_X = 0","l":176,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_directional_α_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the representer direction. For any perturbation ν_α of the representer, assume ν_α is measurable and the ν_α-weighted regression-residual product at the truth is integrable. Then the directional derivative of the population debiased moment in the α-direction at the truth vanishes: the integral of the weighted regression residual is zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (ν_α : S.X → ℝ), Measurable ν_α → MeasureTheory.Integrable (fun z => ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z)) ∂S.P_Z = 0","l":187,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment_meanZero_at_truth","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Mean-zero of the debiased moment at the truth. Given a regression-nuisance moment system with representer rep, assume the α₀-weighted regression-residual product at the truth is integrable and the baseline moment integrand at the truth is integrable. Then the automatically debiased population moment, evaluated at the true nuisance and true parameter, equals zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) (rep : Causalean.Estimation.OrthogonalMoments.AutoDebias.AutoDebiasRepresenter S), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → MeasureTheory.Integrable (fun z => S.m S.g₀ z S.θ₀) S.P_Z → Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedMoment S S.g₀ rep.α₀ S.θ₀ = 0","l":155,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.autoDebiasedScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a regression-nuisance moment system, a regression nuisance function, a candidate representer on the covariate space, a scalar target value, and an observation, the automatically debiased score is the baseline moment contribution plus the candidate representer evaluated at the observation's covariates times the observed-outcome residual from the regression target.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → (S.X → ℝ) → ℝ → S.Z → ℝ","l":128,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.finDiffRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a regression-nuisance moment system, a sample-indexed auxiliary sample space, observation-valued sample data, sample-indexed target estimates, sample-indexed nuisance estimates, finite-difference scales, a candidate nuisance-direction function, sample-index sets, a sample size, and a realized auxiliary sample point, the finite-difference representer loss is the empirical second moment of the candidate's regression target on the specified index set minus twice the centered finite-difference approximation to the moment's nuisance derivative at the corresponding estimates.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → {Ω : Type u_1} → (ℕ → Ω → S.Z) → (ℕ → Ω → ℝ) → (ℕ → Ω → S.H) → (ℕ → ℝ) → S.H → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":217,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.genRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.General","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a regression-nuisance moment system and a candidate regression function in its nuisance class, the general Riesz loss is the covariate-distribution mean of the squared regression target of that function minus twice the directional derivative of the population moment in that function's direction.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.RegNuisanceMomentSys) → S.H → ℝ","l":201,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/General.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDMLEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Given a linear regression-function system, a Riesz representation of its target functional, a nonnegative neighborhood radius, measurability of the linear Riesz score for every joint nuisance and target value, an independent and identically distributed sample, a one-shot split of that sample, and a sequence of joint nuisance estimators indexed by sample size and randomness, the one-shot linear automatic-debiasing estimator maps each sample size and randomness realization to a real-valued estimate.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X → (ε : ℝ) → 0 ≤ ε → (∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) → (sample : Causalean.Stat.IIDSample Ω S.Z μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) → ℕ → Ω → ℝ","l":229,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDML_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Linear Auto-DML asymptotic-linearity wrapper. Assume ε is nonnegative and that the linear Riesz score is measurable in the observation for every nuisance and target value. Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂, suppose the Auto-DML moment has mean zero at the truth, the baseline score is square-integrable (finite variance), and the population moment at η̂ is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable. Finally suppose the L² score difference between the estimated and true nuisance is o_P(1), and the product of the two nuisance-error rates decays at the parametric rate o_P(n^{-1/2}). Then the one-shot linear Auto-DML estimator is asymptotically linear at the target value L_of_m S S.g₀, with influence function the baseline linear Riesz score scaled by the inverse Jacobian factor, indexed over the fold-B subsample.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) (sample : Causalean.Stat.IIDSample Ω S.Z μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → Ω → Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S), Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas) → MeasureTheory.Integrable (fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z ^ 2) S.P_Z → ∀ {Crem : ℝ}, (∀ (n : ℕ) (ω : Ω), |∫ (z : S.Z), (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ∂S.P_Z| ≤ Crem * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)) → (∀ (n : ℕ), Measurable fun p => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n p.1) p.2 (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ), Measurable fun ω z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ), Measurable fun p => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n p.1) p.2 (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) S.P_Z) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (η_hat n ω) z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ - (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀ z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀) 2 S.P_Z).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ (η_hat n ω) (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoDMLEstimator S rep ε hε_nn h_score_meas sample split η_hat) (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) (fun z => -(Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).J₀_inv * Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z) sample split.foldB","l":255,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Given a measure on the sample space, a linear regression-function system, a Riesz representation of that system's target functional, a nonnegative neighborhood radius, and measurability of the linear Riesz score for every joint nuisance and scalar target value, the linear automatic-debiasing general moment has that score, its true regression and Riesz representer as nuisance truth, and the target functional evaluated at the true regression as scalar truth.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X → (ε : ℝ) → 0 ≤ ε → (∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z) → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ S.Z S.P_Z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S)","l":64,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a linear regression-function system, the joint nuisance object for its linear automatic-debiasing moment is a pair comprising a regression function from the system's regression class and a candidate Riesz representer, which is a real-valued function of the covariates.","s":"Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys → Type (max u_2 u_3)","l":46,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance.instAddCommGroup","k":"instance","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → AddCommGroup (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S)","l":54,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance.instModule","k":"instance","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → Module ℝ (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S)","l":59,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Bilinear remainder bound for the linear Auto-DML moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z), ∃ C, ∀ η ∈ (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).H_ε, Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_int_pred✝ S rep η → MeasureTheory.MemLp (fun x => S.γ_target η.1 x - S.γ_target S.g₀ x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.2 x - rep.α₀ x) 2 S.P_X → |∫ (z : S.Z), (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).m η z (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).θ₀ ∂S.P_Z| ≤ C * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₁ η (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀) * ↑((Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).ρ₂ η (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas).η₀)","l":155,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linAuto_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.DML","a":"Estimation/OrthogonalMoments/AutoDebias","t":2,"d":"Mean-zero of the linear Auto-DML moment at the truth. Specialises linRieszScore_meanZero.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ε : ℝ) (hε_nn : 0 ≤ ε) (h_score_meas : ∀ (η : Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoNuisance S) (θ : ℝ), Measurable fun z => Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S η.1 η.2 θ z), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → Causalean.Estimation.OrthogonalMoments.MeanZero (Causalean.Estimation.OrthogonalMoments.AutoDebias.linAutoGeneralMoment μ S rep ε hε_nn h_score_meas)","l":111,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/DML.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a linear regression-functional system and a candidate regression function in its admissible class, the linear Riesz loss is the covariate-distribution mean of the squared regression target of that function minus twice the system's population linear functional evaluated at it.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → ℝ","l":182,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss_FOC_iff_representer","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"First-order condition for Riesz loss minimizers (Prop 4, first half). Assume the squared regression functional along every perturbed line α₀_idx + t • ν is integrable, that the linear moment integrand m_lin(·, ν) is integrable for every direction ν, and that the product γ_target α₀_idx · γ_target ν is integrable for every direction ν. Then α₀_idx is a directional minimizer of the Riesz loss along every line through it if and only if it indexes a Riesz representer, i.e. L_of_m S ν = ∫ γ_target α₀_idx · γ_target ν dP_X for every ν.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (α₀_idx : S.H_γ), (∀ (ν : S.H_γ) (t : ℝ), MeasureTheory.Integrable (fun x => S.γ_target (α₀_idx + t • ν) x ^ 2) S.P_X) → (∀ (ν : S.H_γ), MeasureTheory.Integrable (fun z => S.m_lin z ν) S.P_Z) → (∀ (ν : S.H_γ), MeasureTheory.Integrable (fun x => S.γ_target α₀_idx x * S.γ_target ν x) S.P_X) → ((∀ (ν : S.H_γ) (t : ℝ), Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α₀_idx ≤ Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S (α₀_idx + t • ν)) ↔ ∀ (ν : S.H_γ), Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S ν = ∫ (x : S.X), S.γ_target α₀_idx x * S.γ_target ν x ∂S.P_X)","l":259,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss_excess_eq_l2dist","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Excess Riesz loss equals the squared L²(P_X) distance to the representer. Let α₀_idx index the Riesz representer via rep.α₀ = γ_target α₀_idx pointwise. Assume (γ_target α) ^ 2 is integrable, (γ_target α₀_idx) ^ 2 is integrable, the product γ_target α · γ_target α₀_idx is integrable, and the squared difference (γ_target α − γ_target α₀_idx) ^ 2 is integrable. Then the excess linear Riesz loss of α over α₀_idx equals the squared L²(P_X) distance between γ_target α and γ_target α₀_idx.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (α α₀_idx : S.H_γ) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X), (∀ (x : S.X), rep.α₀ x = S.γ_target α₀_idx x) → MeasureTheory.Integrable (fun x => S.γ_target α x ^ 2) S.P_X → MeasureTheory.Integrable (fun x => S.γ_target α₀_idx x ^ 2) S.P_X → MeasureTheory.Integrable (fun x => S.γ_target α x * S.γ_target α₀_idx x) S.P_X → MeasureTheory.Integrable (fun x => (S.γ_target α x - S.γ_target α₀_idx x) ^ 2) S.P_X → Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α - Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszLoss S α₀_idx = ∫ (x : S.X), (S.γ_target α x - S.γ_target α₀_idx x) ^ 2 ∂S.P_X","l":196,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"For a linear regression-functional system, a regression function in its admissible class, a candidate Riesz representer on the covariate space, a scalar target value, and an observation, the linear Riesz score is the generic Riesz score specialized to that system's regression target, population functional, covariate projection, and observed outcome.","s":"(S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) → S.H_γ → (S.X → ℝ) → ℝ → S.Z → ℝ","l":127,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_directional_g_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the regression direction. For a linear regression-functional system with Riesz representer and any perturbation ν_g of the regression nuisance, the Gateaux derivative of the population debiased moment in the g-direction at the truth vanishes — equivalently, this is the representer identity for the perturbation ν_g.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X) (ν_g : S.H_γ), ∫ (z : S.Z), S.m_lin z ν_g ∂S.P_Z - ∫ (x : S.X), rep.α₀ x * S.γ_target ν_g x ∂S.P_X = 0","l":156,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_directional_α_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Directional zero in the representer direction. For any perturbation ν_α of the representer, assume ν_α is measurable and the ν_α-weighted regression-residual product at the truth is integrable. Then the population mean of the ν_α-weighted regression residual at the truth is zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (ν_α : S.X → ℝ), Measurable ν_α → MeasureTheory.Integrable (fun z => ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), ν_α (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z)) ∂S.P_Z = 0","l":169,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.AutoDebias.Linear","a":"Estimation/OrthogonalMoments/AutoDebias","t":1,"d":"Mean-zero of the debiased linear score at the truth. Given the linear regression-functional system with Riesz representer rep, assume the α₀-weighted regression-residual product at the truth is integrable. Then the population mean of the linear Riesz score, evaluated at the true regression function and the representer's α₀, equals zero.","s":"∀ (S : Causalean.Estimation.OrthogonalMoments.AutoDebias.LinRegFnSys) (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation S.H_γ S.γ_target (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S) S.P_X), MeasureTheory.Integrable (fun z => rep.α₀ (S.proj_X z) * (S.Y_obs z - S.γ_target S.g₀ (S.proj_X z))) S.P_Z → ∫ (z : S.Z), Causalean.Estimation.OrthogonalMoments.AutoDebias.linRieszScore S S.g₀ rep.α₀ (Causalean.Estimation.OrthogonalMoments.AutoDebias.L_of_m S S.g₀) z ∂S.P_Z = 0","l":139,"f":"Causalean/Estimation/OrthogonalMoments/AutoDebias/Linear.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.BilinearRemainder","k":"def","m":"Causalean.Estimation.OrthogonalMoments.RemainderBound","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment system and a real constant, the bilinear-remainder condition requires that, for every nuisance value in the system's admissible perturbation set, the absolute population moment at the true scalar target is at most $C$ times the product of its two seminorm distances from the true nuisance.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → ℝ → Prop","l":36,"f":"Causalean/Estimation/OrthogonalMoments/RemainderBound.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.DiffQuotientEnvelope","k":"def","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment system, the difference-quotient envelope condition requires that, for every admissible nuisance value, there is a positive radius and an integrable envelope function such that, almost everywhere under the data distribution, every nonzero displacement within that radius has its moment difference quotient along the line from the true nuisance to that value bounded in norm by the envelope.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":46,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"A general moment bundles a score function of a nuisance, an observation, and a scalar parameter, a truth nuisance η₀ and truth parameter θ₀, a set of admissible nuisance perturbations, and a pair of bilinear seminorms used to bound product-rate remainders, subject to: the score is jointly measurable in the observation for every nuisance and parameter value, the truth nuisance belongs to the perturbation set, and the population moment's parameter-derivative at the truth (its Jacobian) is nonzero, so that its inverse is well-defined.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_2) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (H : Type u_3) → [inst : AddCommGroup H] → [Module ℝ H] → Type (max u_2 u_3)","l":36,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment.J₀_inv","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment system, the inverse population Jacobian is the reciprocal of the system's nonzero population Jacobian.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → ℝ","l":83,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.GeneralMoment.J₀_mul_J₀_inv","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"Jacobian times its inverse is one. For a general orthogonal-moment system, the population Jacobian times its inverse equals one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), M.J₀ * M.J₀_inv = 1","l":90,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.H_ε_PerturbClosed","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment system, perturbation-set segment closure states that, for every nuisance value in its admissible perturbation set and every weight in the closed unit interval, the corresponding point on the line segment from the true nuisance to that value also belongs to the admissible set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":109,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.HasDirDeriv","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.DirectionalDeriv","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment, a candidate pointwise directional-derivative function dM of the score along the line segment from the truth to a perturbed nuisance, evaluated at each observation, together with the witnesses that for every perturbation in the admissible set and every observation, the score's difference quotient along that segment tends to dM's value there as the step size shrinks to zero, and that dM at each perturbation is measurable in the observation.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Type (max u_2 u_3)","l":35,"f":"Causalean/Estimation/OrthogonalMoments/DirectionalDeriv.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearMoment","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"A linear moment is a general moment whose score is affine in the scalar target parameter.","s":"(Ω : Type u_4) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Z : Type u_5) → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (H : Type u_6) → [inst : AddCommGroup H] → [Module ℝ H] → Type (max u_5 u_6)","l":116,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearSmootherOp","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Linear-smoother operator (Def def:est-cate-second-stage, smoother form). A second-stage regression operator extended with an abstract array of smoothing weights indexed by sample size, randomness scope, query point, and data tuple, from which the weighted-sum representation of the operator's output can be built; the linear-combination identity itself is not required here but recorded separately as a predicate below.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (γ : Type u_2) → [MeasurableSpace γ] → Type (max u_1 u_2)","l":43,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.LinearSmootherOp.IsLinearSmoother","k":"def","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a linear-smoother operator, a sample size, a randomness realization, a query point, a finite index set for the data fold, real weights on that index set, and the corresponding covariate-treatment-outcome data tuples, the linear-smoother condition states that, for every real-valued pseudo-outcome function, the operator's estimate equals the weighted sum of that function over the fold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → {ι : Type u_3} → Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ → ℕ → Ω → γ → Finset ι → (ι → ℝ) → (ι → γ × Bool × ℝ) → Prop","l":72,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.MeanZero","k":"def","m":"Causalean.Estimation.OrthogonalMoments.MomentFunctional","a":"Estimation/OrthogonalMoments","t":1,"d":"For a general moment system, the mean-zero condition states that the population expectation of its score at the true nuisance and true scalar target is zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → Prop","l":103,"f":"Causalean/Estimation/OrthogonalMoments/MomentFunctional.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","k":"def","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a general moment system equipped with a directional derivative of that moment, Neyman orthogonality means that, for every nuisance value in its admissible perturbation set, the population integral of the directional derivative is zero.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) → Causalean.Estimation.OrthogonalMoments.HasDirDeriv M → Prop","l":38,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.RieszRepresentation","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"A Riesz representation for a continuous linear functional on a regression class under a covariate measure: a function α₀ on the covariate space that is measurable, integrable against the covariate measure, and that represents the functional's value at every element of the regression class as the covariate-measure integral of α₀ against that element's evaluation.","s":"{X : Type u_3} → [inst : MeasurableSpace X] → (H_γ : Type u_4) → [inst_1 : AddCommGroup H_γ] → [Module ℝ H_γ] → (H_γ → X → ℝ) → (H_γ → ℝ) → MeasureTheory.Measure X → Type u_3","l":48,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","k":"structure","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Abstract bundle for a second-stage regression operator (Def def:est-cate-second-stage): an operator mapping a sample size, a randomness scope, a real-valued pseudo-outcome function of a data tuple, and a query point to a real-valued estimate, together with the minimal requirement that for every sample size and constant pseudo-outcome, the map from randomness scope and query point to the operator's value is jointly measurable; stronger measurability, and any linearity of the operator in its function input, are deferred to concrete instances or the separate IsLinearInInput predicate.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (γ : Type u_2) → [MeasurableSpace γ] → Type (max u_1 u_2)","l":47,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.IsLinearInInput","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"For a second-stage regression operator, linearity in the pseudo-outcome input means that, for every sample size, randomness realization, pair of real-valued pseudo-outcome functions, and query point, the estimate for their pointwise sum equals the sum of their separate estimates. Linear smoothers satisfy this predicate; kernel-or-tree mean estimators with random splits need not satisfy it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → Prop","l":81,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.oracleEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a second-stage regression operator and a fixed real-valued pseudo-outcome function, the oracle estimator maps each sample size, randomness realization, and query point to the operator's estimate using that fixed pseudo-outcome.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ × Bool × ℝ → ℝ) → ℕ → Ω → γ → ℝ","l":92,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator.oracleRiskScale","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a second-stage regression operator, a fixed real-valued pseudo-outcome function, a target function, a query point, and a sample size, the oracle pointwise risk scale is the square root of the expected squared difference, over the operator's randomness, between the oracle estimator and the target at that query point.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ × Bool × ℝ → ℝ) → (γ → ℝ) → γ → ℕ → ℝ","l":100,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.Stable","k":"def","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a second-stage regression operator, a target function, a sequence of distances between pseudo-outcomes, a query point, and a conditional-bias identification criterion, stability means that for every estimated pseudo-outcome sequence, true pseudo-outcome, and claimed conditional-bias sequence satisfying the criterion, convergence of the distance to zero in probability implies that the operator discrepancy after subtracting the smoothed bias is negligible in probability relative to the oracle pointwise risk scale.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {γ : Type u_2} → [inst_1 : MeasurableSpace γ] → Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ → (γ → ℝ) → (ℕ → Ω → ℝ) → γ → ((ℕ → Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → Ω → γ → ℝ) → Prop) → Prop","l":122,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.WeightedNorm","k":"def","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Given a finite index set, real weights on that set, a real-valued function on its indices, and a real exponent, the weighted empirical norm is the normalized absolute-weight power mean of the function's absolute values, with the indicated exponent.","s":"{ι : Type u_1} → Finset ι → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ","l":103,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.WeightedNorm_nonneg","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":2,"d":"The weighted norm is non-negative.","s":"∀ {ι : Type u_1} (B : Finset ι) (w g : ι → ℝ) (p : ℝ), 0 ≤ Causalean.Estimation.OrthogonalMoments.WeightedNorm B w g p","l":120,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.bilinear_remainder_of_smoothness","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.RemainderBound","a":"Estimation/OrthogonalMoments","t":1,"d":"Smoothness bridge to a bilinear remainder bound. For a general moment M with directional derivative D satisfying Neyman orthogonality, given an envelope g that is P_Z-integrable, suppose the linearization residual obeys a uniform second-order envelope with constant K: for every perturbation η in the nuisance neighborhood, |m(η,·,θ₀) − m(η₀,·,θ₀) − dM(η,·)| ≤ K·ρ₁(η,η₀)·ρ₂(η,η₀)·g almost everywhere, with the directional derivative integrable at every such η, the moment integrable at the baseline η₀, the moment integrable at every perturbed η, and the moment having population mean zero. Then some constant C makes the population moment obey the bilinear remainder bound C·ρ₁(η,η₀)·ρ₂(η,η₀) uniformly over the nuisance neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) (D : Causalean.Estimation.OrthogonalMoments.HasDirDeriv M), Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal M D → ∀ {K : ℝ} {g : Z → ℝ}, MeasureTheory.Integrable g P_Z → (∀ η ∈ M.H_ε, ∀ᵐ (z : Z) ∂P_Z, |M.m η z M.θ₀ - M.m M.η₀ z M.θ₀ - D.dM η z| ≤ K * ↑(M.ρ₁ η M.η₀) * ↑(M.ρ₂ η M.η₀) * g z) → (∀ η ∈ M.H_ε, MeasureTheory.Integrable (fun z => D.dM η z) P_Z) → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀) P_Z → (∀ η ∈ M.H_ε, MeasureTheory.Integrable (fun z => M.m η z M.θ₀) P_Z) → Causalean.Estimation.OrthogonalMoments.MeanZero M → ∃ C, Causalean.Estimation.OrthogonalMoments.BilinearRemainder M C","l":46,"f":"Causalean/Estimation/OrthogonalMoments/RemainderBound.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.DMLChernozhukov","a":"Estimation/OrthogonalMoments","t":1,"d":"For a measurable population space with a population measure, a measurable observed-data space with its observed-data law, and a real vector space of nuisance values, a general moment system, an independent and identically distributed sample, a one-shot evaluation-fold split of that sample, a sequence of nuisance estimators, and a sample-size index, the Chernozhukov one-step double-machine-learning estimator maps each population state to the true target minus the inverse Jacobian times the evaluation-fold empirical mean of the moment evaluated at that target and at the estimated nuisance.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → Ω → H) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Estimation/OrthogonalMoments/DMLChernozhukov.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dmlCrossFitEstimator","k":"def","m":"Causalean.Estimation.OrthogonalMoments.DMLCrossFit","a":"Estimation/OrthogonalMoments","t":1,"d":"For a measurable population space with a population measure, a measurable observed-data space with its observed-data law, and a real vector space of nuisance values, a general moment system, an independent and identically distributed sample, a number of folds, a K-fold split of that sample, a sequence of fold-specific nuisance estimators, and a sample-size index, the K-fold cross-fitted Chernozhukov double-machine-learning estimator maps each population state to the true target minus the inverse Jacobian times the average, across folds, of each fold's empirical mean moment evaluated at the true target and that fold's nuisance estimate.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → {H : Type u_3} → [inst_2 : AddCommGroup H] → [inst_3 : Module ℝ H] → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H → (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) → {K : ℕ} → Causalean.Stat.KFoldSplit sample K → (ℕ → Fin K → Ω → H) → ℕ → Ω → ℝ","l":440,"f":"Causalean/Estimation/OrthogonalMoments/DMLCrossFit.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dml_chernozhukov_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.DMLChernozhukov","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the Chernozhukov DML estimator. Drops the zero-centering hypothesis hθ_zero from dml_asymptoticLinear. Given a general moment M with mean zero at the truth, assume the truth-evaluated score is square-integrable (finite variance). Given an i.i.d. sample with a one-shot fold split whose fold-B fraction converges to a strictly positive limit c > 0 along card (foldB n) / n → c, and a sequence of cross-fitted nuisance estimators η̂, suppose the population moment at η̂ is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable, fold-A-measurable in ω, and, at every fold and sample point, integrable and square-integrable. Finally suppose the L² score difference between the estimated and true nuisance is o_P(1), and the product of the two nuisance-error rates decays at the parametric rate o_P(n^{-1/2}). Then the Chernozhukov one-step estimator is asymptotically linear at the truth M.θ₀, with influence function −J₀⁻¹ · m(η₀, ·, θ₀) and asymptotic variance J₀⁻¹ Σ J₀⁻ᵀ where Σ := ∫ m(η₀, z, θ₀)² dP_Z, indexed over split.foldB.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), Causalean.Estimation.OrthogonalMoments.MeanZero M → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀ ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → Ω → H) {Crem : ℝ}, (∀ (n : ℕ) (ω : Ω), |∫ (z : Z), M.m (η_hat n ω) z M.θ₀ ∂P_Z| ≤ Crem * ↑(M.ρ₁ (η_hat n ω) M.η₀) * ↑(M.ρ₂ (η_hat n ω) M.η₀)) → (∀ (n : ℕ), Measurable fun p => M.m (η_hat n p.1) p.2 M.θ₀) → (∀ (n : ℕ), Measurable fun ω z => M.m (η_hat n ω) z M.θ₀) → (∀ (n : ℕ), Measurable fun p => M.m (η_hat n p.1) p.2 M.θ₀) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n ω) z M.θ₀) P_Z) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n ω) z M.θ₀ ^ 2) P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => M.m (η_hat n ω) z M.θ₀ - M.m M.η₀ z M.θ₀) 2 P_Z).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n ω) M.η₀) * ↑(M.ρ₂ (η_hat n ω) M.η₀)) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator M sample split η_hat) M.θ₀ (fun z => -M.J₀_inv * M.m M.η₀ z M.θ₀) sample split.foldB","l":83,"f":"Causalean/Estimation/OrthogonalMoments/DMLChernozhukov.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.dml_crossFit_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.DMLCrossFit","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the K-fold cross-fitted Chernozhukov DML estimator. Same Chernozhukov form as dml_chernozhukov_asymptoticLinear, but with K folds. Given a general moment M with mean zero at the truth, assume the truth-evaluated score is square-integrable (finite variance), that there are at least two folds, K > 1, and a sequence of per-fold cross-fitted nuisance estimators η̂. Suppose the population moment at η̂, trained on each fold's complement, is bounded by a constant times the product of the two bilinear-remainder seminorms, at every fold count, fold index, and sample point. Assume the technical regularity package that the moment at η̂ is jointly measurable at every fold, measurable with respect to the fold's training complement, and, at every fold count, fold index, and sample point, integrable and square-integrable. Finally suppose that, at every fold, the L² score difference between the fold's estimated and true nuisance is o_P(1), each of the two nuisance-error rates is individually o_P(1), and their product decays at the parametric rate o_P(n^{-1/2}). Then the K-fold cross-fitted estimator is asymptotically linear at the truth M.θ₀ with influence function −J₀⁻¹ · m(η₀, ·, θ₀), indexed over the full sample (the fold-level sub-aggregations sum to a full-sample average asymptotically).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] [StandardBorelSpace Ω] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H), Causalean.Estimation.OrthogonalMoments.MeanZero M → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀ ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) {K : ℕ}, 1 < K → ∀ (split : Causalean.Stat.KFoldSplit sample K) (η_hat : ℕ → Fin K → Ω → H) {Crem : ℝ}, (∀ (n : ℕ) (k : Fin K) (ω : Ω), |∫ (z : Z), M.m (η_hat n k ω) z M.θ₀ ∂P_Z| ≤ Crem * ↑(M.ρ₁ (η_hat n k ω) M.η₀) * ↑(M.ρ₂ (η_hat n k ω) M.η₀)) → (∀ (n : ℕ) (k : Fin K), Measurable fun p => M.m (η_hat n k p.1) p.2 M.θ₀) → (∀ (n : ℕ) (k : Fin K), Measurable fun ω z => M.m (η_hat n k ω) z M.θ₀) → (∀ (n : ℕ) (k : Fin K), Measurable fun p => M.m (η_hat n k p.1) p.2 M.θ₀) → (∀ (n : ℕ) (k : Fin K) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n k ω) z M.θ₀) P_Z) → (∀ (n : ℕ) (k : Fin K) (ω : Ω), MeasureTheory.Integrable (fun z => M.m (η_hat n k ω) z M.θ₀ ^ 2) P_Z) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => M.m (η_hat n k ω) z M.θ₀ - M.m M.η₀ z M.θ₀) 2 P_Z).toReal) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n k ω) M.η₀)) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₂ (η_hat n k ω) M.η₀)) (fun x => 1) μ) → (∀ (k : Fin K), Causalean.Stat.IsLittleOp (fun n ω => ↑(M.ρ₁ (η_hat n k ω) M.η₀) * ↑(M.ρ₂ (η_hat n k ω) M.η₀)) (fun n => ↑n ^ (-(1 / 2))) μ) → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlCrossFitEstimator M sample split η_hat) M.θ₀ (fun z => -M.J₀_inv * M.m M.η₀ z M.θ₀) sample fun n => Finset.range n","l":467,"f":"Causalean/Estimation/OrthogonalMoments/DMLCrossFit.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.integratedMoment_diffQuotient_tendsto_zero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal","a":"Estimation/OrthogonalMoments","t":1,"d":"Abstract DCT bridge. Given a general moment M with directional-derivative structure D, assume Neyman orthogonality — the population directional derivative vanishes at every admissible nuisance perturbation, and an L¹(P_Z) envelope dominating the difference quotient of the moment near t = 0. Suppose η lies in the admissible nuisance neighborhood M.H_ε, that the moment along the segment from η₀ to η is integrable at every nonzero t, and that the moment at η₀ is integrable. Then the integrated difference quotient (∫ m(η₀ + t · (η − η₀)) dP_Z − ∫ m(η₀) dP_Z) / t tends to zero as t → 0 along the punctured neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {H : Type u_3} [inst_2 : AddCommGroup H] [inst_3 : Module ℝ H] (M : Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z H) (D : Causalean.Estimation.OrthogonalMoments.HasDirDeriv M), Causalean.Estimation.OrthogonalMoments.NeymanOrthogonal M D → Causalean.Estimation.OrthogonalMoments.DiffQuotientEnvelope M → ∀ {η : H}, η ∈ M.H_ε → (∀ (t : ℝ), t ≠ 0 → MeasureTheory.Integrable (fun z => M.m (M.η₀ + t • (η - M.η₀)) z M.θ₀) P_Z) → MeasureTheory.Integrable (fun z => M.m M.η₀ z M.θ₀) P_Z → Filter.Tendsto (fun t => (∫ (z : Z), M.m (M.η₀ + t • (η - M.η₀)) z M.θ₀ ∂P_Z - ∫ (z : Z), M.m M.η₀ z M.θ₀ ∂P_Z) / t) (nhdsWithin 0 {0}ᶜ) (nhds 0)","l":58,"f":"Causalean/Estimation/OrthogonalMoments/NeymanOrthogonal.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.oracle_expansion","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.SecondStageOperator","a":"Estimation/OrthogonalMoments","t":1,"d":"Oracle expansion for the DR-Learner (Thm thm:est-cate-dr-oracle, abstract operator-level form). Given a second-stage regression operator op that is stable at the query point x for target function target, with respect to a distance d_n between pseudo-outcomes and a caller-supplied conditional-bias identification predicate BiasIdent, suppose d_n converges to zero in probability under μ, i.e. the first-stage pseudo-outcome estimate is consistent, and suppose the estimated pseudo-outcome fHat_n, the true pseudo-outcome f, and the claimed conditional bias bHat_n satisfy BiasIdent. Then the discrepancy between the operator applied to fHat_n and to f, minus the operator applied to bHat_n, is o_p of the oracle risk scale under μ: the operator-level oracle expansion holds modulo o_p(R^*_n(x)).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] (op : Causalean.Estimation.OrthogonalMoments.SecondStageOperator Ω μ γ) (target : γ → ℝ) (x : γ) (d_n : ℕ → Ω → ℝ) (fHat_n : ℕ → Ω → γ × Bool × ℝ → ℝ) (f : γ × Bool × ℝ → ℝ) (bHat_n : ℕ → Ω → γ → ℝ) (BiasIdent : (ℕ → Ω → γ × Bool × ℝ → ℝ) → (γ × Bool × ℝ → ℝ) → (ℕ → Ω → γ → ℝ) → Prop), Causalean.Estimation.OrthogonalMoments.Stable op target d_n x BiasIdent → Causalean.Stat.Tendsto_inProb d_n (fun x => 0) μ → BiasIdent fHat_n f bHat_n → Causalean.Stat.IsLittleOp (fun n ω => op.evalAt n ω (fHat_n n ω) x - op.evalAt n ω f x - op.evalAt n ω (fun z => bHat_n n ω z.1) x) (fun n => op.oracleRiskScale f target x n) μ","l":158,"f":"Causalean/Estimation/OrthogonalMoments/SecondStageOperator.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.parametricMoment","k":"def","m":"Causalean.Estimation.OrthogonalMoments.Parametric","a":"Estimation/OrthogonalMoments","t":1,"d":"For a measurable population space with a population measure and a measurable observed-data space with its observed-data law, a real-valued parametric moment function, a target parameter, a nonzero scalar Jacobian, and the condition that the moment is measurable at every parameter value, the no-nuisance parametric moment system is the general moment system whose score is the supplied parametric moment and whose nuisance space contains only a single element.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {Z : Type u_2} → [inst_1 : MeasurableSpace Z] → {P_Z : MeasureTheory.Measure Z} → (m_par : ℝ → Z → ℝ) → ℝ → (J₀ : ℝ) → J₀ ≠ 0 → (∀ (θ : ℝ), Measurable (m_par θ)) → Causalean.Estimation.OrthogonalMoments.GeneralMoment Ω μ Z P_Z Unit","l":40,"f":"Causalean/Estimation/OrthogonalMoments/Parametric.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.parametric_asymptoticLinear","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Parametric","a":"Estimation/OrthogonalMoments","t":1,"d":"Asymptotic linearity of the parametric one-step estimator. Fix a score m_par with a nonzero scalar Jacobian J₀ at the true parameter θ₀, where m_par θ is measurable for every θ. If the score m_par θ₀ has population mean zero and it has finite second moment, then for an i.i.d. sample and a one-shot fold split of it, the parametric one-step estimator built from m_par and J₀ is asymptotically linear at θ₀ with influence function ψ(z) := −J₀⁻¹·m_par(θ₀, z).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Z : Type u_2} [inst_1 : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} (m_par : ℝ → Z → ℝ) (θ₀ J₀ : ℝ) (hJ : J₀ ≠ 0) (m_meas : ∀ (θ : ℝ), Measurable (m_par θ)), ∫ (z : Z), m_par θ₀ z ∂P_Z = 0 → MeasureTheory.Integrable (fun z => m_par θ₀ z ^ 2) P_Z → ∀ (sample : Causalean.Stat.IIDSample Ω Z μ P_Z) (split : Causalean.Stat.OneShotSplit sample), Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator (Causalean.Estimation.OrthogonalMoments.parametricMoment m_par θ₀ J₀ hJ m_meas) sample split fun x x_1 => ()) θ₀ (fun z => -J₀⁻¹ * m_par θ₀ z) sample split.foldB","l":64,"f":"Causalean/Estimation/OrthogonalMoments/Parametric.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore","k":"def","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"For an observed-data space and a covariate space, a regression-function vector space, a map assigning each regression function its value at every covariate, a real-valued linear functional of that regression function, an observed-data-to-covariate map, an observed outcome map, a regression function, a Riesz representer, a scalar target, and an observed-data realization, the generic Riesz orthogonal score is the functional evaluated at the regression function plus the representer times its observed residual, minus the target.","s":"{Z : Type u_2} → {X : Type u_3} → {H_γ : Type u_4} → [inst : AddCommGroup H_γ] → [Module ℝ H_γ] → (H_γ → X → ℝ) → (H_γ → ℝ) → (Z → X) → (Z → ℝ) → H_γ → (X → ℝ) → ℝ → Z → ℝ","l":66,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore_bilinearRem","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":2,"d":"Bilinear remainder identity for the Riesz score.","s":"∀ {Z : Type u_2} [inst : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {X : Type u_3} [inst_1 : MeasurableSpace X] {H_γ : Type u_4} [inst_2 : AddCommGroup H_γ] [inst_3 : Module ℝ H_γ] {γ_target : H_γ → X → ℝ} {L : H_γ → ℝ} {P_X : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P_Z] (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation H_γ γ_target L P_X) (γ₀ γ : H_γ) (α : X → ℝ) (proj_X : Z → X) (Y_obs : Z → ℝ), P_X = MeasureTheory.Measure.map proj_X P_Z → Measurable proj_X → ∫ (z : Z), rep.α₀ (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → ∫ (z : Z), α (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → MeasureTheory.Integrable (fun z => α (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z))) P_Z → MeasureTheory.Integrable (fun z => α (proj_X z) * γ_target γ (proj_X z)) P_Z → MeasureTheory.Integrable (fun z => α (proj_X z) * γ_target γ₀ (proj_X z)) P_Z → MeasureTheory.Integrable (fun x => rep.α₀ x * γ_target γ x) P_X → MeasureTheory.Integrable (fun x => rep.α₀ x * γ_target γ₀ x) P_X → MeasureTheory.Integrable α P_X → MeasureTheory.Integrable (γ_target γ) P_X → MeasureTheory.Integrable (γ_target γ₀) P_X → ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ α (L γ₀) z ∂P_Z - ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ₀ rep.α₀ (L γ₀) z ∂P_Z = -∫ (x : X), (α x - rep.α₀ x) * (γ_target γ x - γ_target γ₀ x) ∂P_X","l":188,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.rieszScore_meanZero","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.Riesz","a":"Estimation/OrthogonalMoments","t":1,"d":"Mean-zero of the Riesz score at the truth. Given a Riesz representation rep of a linear functional L on the regression class H_γ under P_X, with true regression function γ₀, and observed data given by the covariate projection proj_X and the outcome Y_obs, suppose the representer residual α₀(proj_X z)·(Y_obs z − γ_target γ₀ (proj_X z)) has population mean zero under P_Z — which holds when γ_target γ₀ is the conditional expectation of Y_obs given proj_X, since residuals are orthogonal to all square-integrable functions of proj_X. Then the orthogonal score rieszScore evaluated at the truth (γ₀, α₀, L γ₀) integrates to zero under P_Z:","s":"∀ {Z : Type u_2} [inst : MeasurableSpace Z] {P_Z : MeasureTheory.Measure Z} {X : Type u_3} [inst_1 : MeasurableSpace X] {H_γ : Type u_4} [inst_2 : AddCommGroup H_γ] [inst_3 : Module ℝ H_γ] {γ_target : H_γ → X → ℝ} {L : H_γ → ℝ} {P_X : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P_Z] (rep : Causalean.Estimation.OrthogonalMoments.RieszRepresentation H_γ γ_target L P_X) (γ₀ : H_γ) (proj_X : Z → X) (Y_obs : Z → ℝ), ∫ (z : Z), rep.α₀ (proj_X z) * (Y_obs z - γ_target γ₀ (proj_X z)) ∂P_Z = 0 → ∫ (z : Z), Causalean.Estimation.OrthogonalMoments.rieszScore γ_target L proj_X Y_obs γ₀ rep.α₀ (L γ₀) z ∂P_Z = 0","l":89,"f":"Causalean/Estimation/OrthogonalMoments/Riesz.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.smoother_bias_holder","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Single-function Hölder bound for a linear smoother. Assume op realises a linear smoother at sample size n, randomness ω, and query point x — its evaluation of any function on the fold B is the weighted sum Σ w_i · f(xs_i), and that the absolute weights sum to at most c_n. Then the absolute smoothed bias of g (evaluated on the γ-component of the data) at x is bounded by c_n times the weighted L¹ norm of g ∘ xs.1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] {ι : Type u_3} (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ) (n : ℕ) (ω : Ω) (x : γ) (g : γ → ℝ) (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n : ℝ), op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → |op.evalAt n ω (fun z => g z.1) x| ≤ c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g (xs i).1) 1","l":135,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.OrthogonalMoments.smoother_bias_product_holder","k":"theorem","m":"Causalean.Estimation.OrthogonalMoments.LinearSmoother","a":"Estimation/OrthogonalMoments","t":1,"d":"Product Hölder bound for a linear smoother (Prop prop:est-cate-linear-smoother-bound). Assume op realises a linear smoother at sample size n, randomness ω, and query point x, that the absolute weights sum to at most c_n, and that p and q are Hölder-conjugate exponents, 1/p + 1/q = 1. Then the absolute smoothed bias of the product g₁ · g₂ (evaluated on the γ-component of the data) at x is bounded by c_n times the weighted L^p norm of g₁ ∘ xs.1 times the weighted L^q norm of g₂ ∘ xs.1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {γ : Type u_2} [inst_1 : MeasurableSpace γ] {ι : Type u_3} (op : Causalean.Estimation.OrthogonalMoments.LinearSmootherOp Ω μ γ) (n : ℕ) (ω : Ω) (x : γ) (g₁ g₂ : γ → ℝ) (B : Finset ι) (w : ι → ℝ) (xs : ι → γ × Bool × ℝ) (c_n p q : ℝ), op.IsLinearSmoother n ω x B w xs → ∑ i ∈ B, |w i| ≤ c_n → p.HolderConjugate q → |op.evalAt n ω (fun z => g₁ z.1 * g₂ z.1) x| ≤ c_n * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g₁ (xs i).1) p * Causalean.Estimation.OrthogonalMoments.WeightedNorm B w (fun i => g₂ (xs i).1) q","l":214,"f":"Causalean/Estimation/OrthogonalMoments/LinearSmoother.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance","k":"structure","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":1,"d":"A partially linear nuisance bundles an outcome regression and a treatment regression on the covariates, requiring each to be measurable.","s":"(γ : Type u_1) → [MeasurableSpace γ] → Type u_1","l":35,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.ext","k":"theorem","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":1,"d":"Two nuisance pairs η and η' are equal whenever their outcome-regression components agree at every covariate value and their treatment-regression components agree at every covariate value, so the two pairs are equal.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.PLR.PLRNuisance γ}, (∀ (x : γ), η.lFn x = η'.lFn x) → (∀ (x : γ), η.mFn x = η'.mFn x) → η = η'","l":100,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.ext_iff","k":"theorem","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] {η η' : Causalean.Estimation.PLR.PLRNuisance γ}, η = η' ↔ (∀ (x : γ), η.lFn x = η'.lFn x) ∧ ∀ (x : γ), η.mFn x = η'.mFn x","l":104,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instAdd","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the addition operation on partially linear nuisance pairs assigns to each pair of nuisance pairs the pair obtained by adding their outcome regressions and their treatment regressions pointwise.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Add (Causalean.Estimation.PLR.PLRNuisance γ)","l":60,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instAddCommGroup","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the additive commutative group structure on partially linear nuisance pairs supplies the zero pair, componentwise addition, componentwise negation, componentwise subtraction, repeated addition by natural numbers, repeated addition and negation by integers, the zero-multiple law, the successor-multiple law, the zero integer-multiple law, the positive successor integer-multiple law, the negative successor integer-multiple law, subtraction as addition of a negative, associativity of addition, the left zero law, the right zero law, cancellation of a pair with its negative, and commutativity of addition.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → AddCommGroup (Causalean.Estimation.PLR.PLRNuisance γ)","l":117,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instModuleReal","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the real vector-space structure on partially linear nuisance pairs supplies the scalar multiplication operation, the unit-scalar law, the successive-scalar law, the scalar-times-zero law, distributivity of scalar multiplication over addition of nuisance pairs, distributivity over addition of scalars, and the zero-scalar law.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Module ℝ (Causalean.Estimation.PLR.PLRNuisance γ)","l":162,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instNeg","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the negation operation on partially linear nuisance pairs assigns to each nuisance pair the pair obtained by negating both regressions pointwise.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Neg (Causalean.Estimation.PLR.PLRNuisance γ)","l":71,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instSMulReal","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the real-scalar multiplication operation on partially linear nuisance pairs assigns to every real scalar and nuisance pair the pair obtained by multiplying both regressions pointwise by that scalar.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → SMul ℝ (Causalean.Estimation.PLR.PLRNuisance γ)","l":90,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instSub","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the subtraction operation on partially linear nuisance pairs assigns to each ordered pair of nuisance pairs the pair obtained by subtracting their outcome regressions and their treatment regressions pointwise.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Sub (Causalean.Estimation.PLR.PLRNuisance γ)","l":79,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRNuisance.instZero","k":"instance","m":"Causalean.Estimation.PLR.Nuisance","a":"Estimation/PLR","t":2,"d":"For every covariate space equipped with a $\\sigma$-algebra, the zero operation on partially linear nuisance pairs assigns the pair whose outcome and treatment regressions are both identically zero.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Zero (Causalean.Estimation.PLR.PLRNuisance γ)","l":53,"f":"Causalean/Estimation/PLR/Nuisance.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem","k":"structure","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"A partially linear estimation system extends a partially linear causal model with value-space regression representatives — an outcome-regression representative and a treatment-regression representative on the covariates — subject to: each representative is measurable; the outcome representative agrees almost surely with the conditional mean of the outcome given the covariates and the treatment representative agrees almost surely with the conditional mean of the treatment given the covariates; and the treatment retains nonzero variation after partialling out the covariate, which is what makes the partialling-out Jacobian invertible.","s":"(P : Causalean.PO.POSystem) → (γ : Type u_1) → [MeasurableSpace γ] → [MeasureTheory.IsFiniteMeasure P.μ] → Type (max (max (max u_1 u_2) u_3) u_4)","l":54,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_X","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the covariate marginal measure is the population measure transported through the factual covariate map.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → MeasureTheory.Measure γ","l":125,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_X_eq","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"The covariate marginal is the image of the population measure under the observed covariate map.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), S.P_X = MeasureTheory.Measure.map S.factualX P.μ","l":139,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_Z","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the joint observed-data measure is the population measure transported through the map that records each unit's covariate, factual treatment, and factual outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → MeasureTheory.Measure (γ × ℝ × ℝ)","l":116,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.P_Z_eq","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"The joint observed-data law is the image of the population measure under the map recording the observed covariate, treatment, and outcome.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), S.P_Z = MeasureTheory.Measure.map S.factualZ P.μ","l":134,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.factualZ","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the observed-data map assigns every population unit its covariate, factual treatment, and factual outcome, in that order.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → P.Ω → γ × ℝ × ℝ","l":101,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.instIsProbabilityMeasureP_Z","k":"instance","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"For a potential-outcome system equipped with a finite probability measure, a measurable covariate space, and a partially linear regression system based on them, the system's joint observed-data law is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.IsProbabilityMeasure S.P_Z","l":85,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_P_Z","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"Integrating a measurable function under the joint observed-data law equals integrating its pullback under the population measure.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) {f : γ × ℝ × ℝ → ℝ}, Measurable f → ∫ (z : γ × ℝ × ℝ), f z ∂S.P_Z = ∫ (ω : P.Ω), f (S.factualZ ω) ∂P.μ","l":204,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_condExpZero_mul_comp_factualX","k":"theorem","m":"Causalean.Estimation.PLR.RemainderBound","a":"Estimation/PLR","t":2,"d":"Orthogonality of a σ(X)-conditionally-mean-zero variable w to any covariate function h(X): E[w·h(X)] = 0. Mirrors integral_U_resid, but the σ-algebra is σ(X) (so h(X) pulls out of the conditional expectation) instead of σ(X,D). Used to kill the three orthogonal cross terms U·Δm, Δℓ·V, V·Δm in the doubly-robust remainder.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) {w : P.Ω → ℝ} {h : γ → ℝ}, Measurable h → P.μ[w | S.sigmaX] =ᵐ[P.μ] 0 → MeasureTheory.Integrable w P.μ → MeasureTheory.Integrable (fun ω => w ω * h (S.factualX ω)) P.μ → ∫ (ω : P.Ω), w ω * h (S.factualX ω) ∂P.μ = 0","l":47,"f":"Causalean/Estimation/PLR/RemainderBound.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_plrMomentA_diff_eq","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"Bias of the empirical-Jacobian increment. The population integral of the increment Δa := mₐ(η̂, ·) − mₐ(η₀, ·) is minus the squared L²(P_X) magnitude of the treatment-regression error Δm := m_val − m̂:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η : Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (S.mVal (S.factualX ω) - η.mFn (S.factualX ω)) ^ 2) P.μ → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) * (S.mVal (S.factualX ω) - η.mFn (S.factualX ω))) P.μ → ∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentA η z - Causalean.Estimation.PLR.plrMomentA S.η₀ z ∂S.P_Z = -∫ (ω : P.Ω), (S.mVal (S.factualX ω) - η.mFn (S.factualX ω)) ^ 2 ∂P.μ","l":95,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.integral_plrMomentA_η₀_eq_J₀","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":2,"d":"The population integral of the partialling-out moment at the truth equals the Jacobian: ∫ mₐ(η₀, ·) dP_Z = J₀ = −E[(D − m_val(X))²].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), ∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentA S.η₀ z ∂S.P_Z = S.plrGeneralMoment.J₀","l":70,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.measurable_factualZ","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":2,"d":"The observed-data map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), Measurable S.factualZ","l":111,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrFeasibleEstimator","k":"def","m":"Causalean.Estimation.PLR.Feasible","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space, a partially linear estimation system built on it, an independent and identically distributed sample from its joint observed-data law, a one-shot evaluation-fold split of that sample, a sequence of estimated outcome-and-treatment regression pairs, and a sample-size index, the feasible partially linear double-machine-learning estimator maps each population state to the ratio of the evaluation-fold sum of residualized-outcome times residualized-treatment to the evaluation-fold sum of squared residualized-treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.PLR.PLRSystem P γ) → (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) → Causalean.Stat.OneShotSplit sample → (ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ) → ℕ → P.Ω → ℝ","l":69,"f":"Causalean/Estimation/PLR/Feasible.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrGeneralMoment","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the partially linear general moment system is the abstract moment system with the Robinson partialling-out score, the system's true regression pair and structural slope, and a Jacobian equal to the negative residual treatment second moment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.Estimation.PLR.PLRSystem P γ) → Causalean.Estimation.OrthogonalMoments.GeneralMoment P.Ω P.μ (γ × ℝ × ℝ) S.P_Z (Causalean.Estimation.PLR.PLRNuisance γ)","l":173,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plrMomentFunctional_factualZ","k":"theorem","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"Robinson score at the true nuisance, pulled back to the population space. For the observed data generated from a population outcome ω, the Robinson partialling-out score evaluated at the true nuisance pair, true data, and true parameter equals the true residualized outcome times the true treatment residual.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (ω : P.Ω), Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ = (S.factualY ω - S.lVal (S.factualX ω) - S.θ * (S.factualD ω - S.mVal (S.factualX ω))) * (S.factualD ω - S.mVal (S.factualX ω))","l":214,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_feasible_tendstoNormal","k":"theorem","m":"Causalean.Estimation.PLR.Feasible","a":"Estimation/PLR","t":1,"d":"Feasible partially linear DML asymptotic-normality theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. Suppose the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, the outcome- and treatment-regression errors are square-integrable in the covariate law at every fold and draw, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable and measurable as a function of the nuisance-training fold alone and jointly with the observation, and integrable and square-integrable at every fold and draw; the estimated score converges to the true score in L²(P_Z) at rate $o_p(1)$, and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; and the influence function, the one-step rescaled estimator at each n, and the normalized influence sum at each n are all measurable. Suppose in addition the empirical partialling-out Jacobian over fold B converges in probability to its population value J₀, and the rescaled feasible estimator is measurable at each n. Then the rescaled *feasible* estimator — the solved Robinson partialling-out estimator, recentered at the true slope and scaled by the square root of the fold-B sample size — converges in distribution to the same centered Gaussian as the one-step estimator: a normal law whose variance is the population second moment of the inverse-Jacobian-scaled partialling-out score at the true regressions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → (Measurable fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB n) P.μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) split.foldB n) P.μ) → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, Causalean.Estimation.PLR.plrMomentA (η_hat n ω) (sample.Z i ω)) (fun x => S.plrGeneralMoment.J₀) P.μ → ∀ (hθn_feas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.plrFeasibleEstimator sample split η_hat) S.θ₀ split.foldB n) P.μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.plrFeasibleEstimator sample split η_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (z : γ × ℝ × ℝ), (-S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) ^ 2 ∂S.P_Z)) P.μ hθn_feas","l":97,"f":"Causalean/Estimation/PLR/Feasible.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_isAsymLinear","k":"theorem","m":"Causalean.Estimation.PLR.DML","a":"Estimation/PLR","t":1,"d":"Headline partially linear DML asymptotic-linearity theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. If the model's integrability and square-integrability conditions hold — the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, at every fold and draw the outcome- and treatment-regression errors are square-integrable in the covariate law, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable, fold-A measurable, and integrable/square-integrable at every fold; the L²(P_Z) distance between the estimated and true score is $o_p(1)$; and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; then the one-step double-machine-learning estimator of the structural slope is asymptotically linear at the true slope, with influence function $-J_0^{-1}\\psi(\\eta_0,\\cdot,\\theta_0)$ — the inverse-Jacobian-scaled Robinson partialling-out score at the truth.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → Causalean.Stat.IsAsymLinear (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) sample split.foldB","l":53,"f":"Causalean/Estimation/PLR/DML.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_dml_tendstoNormal","k":"theorem","m":"Causalean.Estimation.PLR.DML","a":"Estimation/PLR","t":1,"d":"Headline partially linear DML asymptotic-normality theorem. Fix a partially linear estimation system, an i.i.d. sample of covariate-treatment-outcome triples, and a sample split whose evaluation-fold share converges to a fixed positive limit. Suppose the structural error, its product with the treatment residual, the baseline-covariate term, and the treatment are integrable, the true treatment residual is square-integrable, and the true score has finite second moment; for the estimated nuisance sequence η_hat, the outcome- and treatment-regression errors are square-integrable in the covariate law at every fold and draw, with the resulting cross terms against the structural error and the treatment residual integrable; the estimated score is jointly measurable and measurable as a function of the nuisance-training fold alone and jointly with the observation, and integrable and square-integrable at every fold and draw; the estimated score converges to the true score in L²(P_Z) at rate $o_p(1)$, and the product of the two nuisance-error seminorms is $o_p(n^{-1/2})$; and the influence function, the rescaled estimator at each n, and the normalized influence sum at each n are all measurable. Then the rescaled double-machine-learning estimator of the structural slope, recentered at the true slope and scaled by the square root of the fold-B sample size, converges in distribution to a centered Gaussian whose variance is the second moment of the inverse-Jacobian-scaled partialling-out score at the true regressions.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.U ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => ((η_hat n ω).lFn (S.factualX ω') - S.lVal (S.factualX ω')) * S.resid ω') P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => S.resid ω' * ((η_hat n ω).mFn (S.factualX ω') - S.mVal (S.factualX ω'))) P.μ) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun ω z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) → (∀ (n : ℕ), Measurable fun p => S.plrGeneralMoment.m (η_hat n p.1) p.2 S.plrGeneralMoment.θ₀) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀) S.P_Z) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ ^ 2) S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => S.plrGeneralMoment.m (η_hat n ω) z S.plrGeneralMoment.θ₀ - S.plrGeneralMoment.m S.plrGeneralMoment.η₀ z S.plrGeneralMoment.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => ↑(S.plrGeneralMoment.ρ₁ (η_hat n ω) S.plrGeneralMoment.η₀) * ↑(S.plrGeneralMoment.ρ₂ (η_hat n ω) S.plrGeneralMoment.η₀)) (fun n => ↑n ^ (-(1 / 2))) P.μ → (Measurable fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB n) P.μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum sample (fun z => -S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) split.foldB n) P.μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Estimation.OrthogonalMoments.dmlChernozhukovEstimator S.plrGeneralMoment sample split η_hat) S.θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (z : γ × ℝ × ℝ), (-S.plrGeneralMoment.J₀_inv * Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) ^ 2 ∂S.P_Z)) P.μ hθn_meas","l":171,"f":"Causalean/Estimation/PLR/DML.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_finite_var","k":"theorem","m":"Causalean.Estimation.PLR.MeanZero","a":"Estimation/PLR","t":1,"d":"If the squared Robinson partialling-out score — evaluated at the true outcome and treatment regressions and the true structural slope, pulled back to the population space — is integrable, then the squared score is integrable under the observed-data law $P_Z$, i.e. the score has finite variance at the truth.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.Integrable (fun ω => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ (S.factualZ ω) S.θ₀ ^ 2) P.μ → MeasureTheory.Integrable (fun z => Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀ ^ 2) S.P_Z","l":63,"f":"Causalean/Estimation/PLR/MeanZero.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_jacobian_consistency","k":"theorem","m":"Causalean.Estimation.PLR.JacobianConsistency","a":"Estimation/PLR","t":1,"d":"Fold-B Jacobian consistency. Assume the treatment is integrable; the true partialling-out moment at the truth is square-integrable under the observed-data law, and the squared true treatment residual is integrable; that for the estimated nuisance sequence η_hat, at every fold and draw, the treatment-regression error is square-integrable in the covariate law, both marginally and against the true treatment residual; that the treatment-regression error converges to zero in L²(P_X) at rate $o_p(1)$; and that the partialling-out-moment increment mₐ(η̂,·) − mₐ(η₀,·) is jointly measurable, fold-A measurable, square-integrable under the observed-data law, and converges to zero in L²(P_Z) at rate $o_p(1)$. Then the empirical partialling-out Jacobian at the estimated nuisance, averaged over the estimation fold, converges in probability to its population value $J_0=-E[(D-m_{val}(X))^2]$:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] [MeasureTheory.IsProbabilityMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (sample : Causalean.Stat.IIDSample P.Ω (γ × ℝ × ℝ) P.μ S.P_Z) (split : Causalean.Stat.OneShotSplit sample) (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.MemLp (Causalean.Estimation.PLR.plrMomentA S.η₀) 2 S.P_Z → MeasureTheory.Integrable (fun ω => (S.factualD ω - S.mVal (S.factualX ω)) ^ 2) P.μ → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (S.mVal (S.factualX ω') - (η_hat n ω).mFn (S.factualX ω')) ^ 2) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.Integrable (fun ω' => (S.factualD ω' - S.mVal (S.factualX ω')) * (S.mVal (S.factualX ω') - (η_hat n ω).mFn (S.factualX ω'))) P.μ) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => S.mVal x - (η_hat n ω).mFn x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => S.mVal x - (η_hat n ω).mFn x) 2 S.P_X).toReal) (fun x => 1) P.μ → (∀ (n : ℕ), Measurable (Function.uncurry fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z)) → (∀ (n : ℕ), Measurable fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) → (∀ (n : ℕ), Measurable (Function.uncurry fun ω z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z)) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) 2 S.P_Z) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.PLR.plrMomentA (η_hat n ω) z - Causalean.Estimation.PLR.plrMomentA S.η₀ z) 2 S.P_Z).toReal) (fun x => 1) P.μ → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, Causalean.Estimation.PLR.plrMomentA (η_hat n ω) (sample.Z i ω)) (fun x => S.plrGeneralMoment.J₀) P.μ","l":222,"f":"Causalean/Estimation/PLR/JacobianConsistency.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_meanZero","k":"theorem","m":"Causalean.Estimation.PLR.MeanZero","a":"Estimation/PLR","t":1,"d":"If the structural error is integrable, the product of the structural error and the true treatment residual is integrable, the baseline covariate function is integrable, and the treatment is integrable, then the Robinson partialling-out score, evaluated at the true outcome and treatment regressions and the true structural slope, has zero mean under the observed-data law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ), MeasureTheory.Integrable S.U P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.factualD P.μ → Causalean.Estimation.OrthogonalMoments.MeanZero S.plrGeneralMoment","l":38,"f":"Causalean/Estimation/PLR/MeanZero.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_remainder_bound","k":"theorem","m":"Causalean.Estimation.PLR.RemainderBound","a":"Estimation/PLR","t":1,"d":"Doubly-robust bilinear remainder. Fix any candidate nuisance pair η of outcome and treatment regressions. Assume the treatment, the baseline covariate function, and the structural error are integrable; that the outcome- and treatment-regression errors of η and the true treatment residual are square-integrable; and that the resulting orthogonal cross terms and the true structural-error/treatment-residual product are all integrable. Then the population Robinson partialling-out score at η and the true structural slope is bounded in absolute value by $(1+|\\theta_0|)$ times the product of the two L²(P_X) nuisance-error seminorms:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η : Causalean.Estimation.PLR.PLRNuisance γ), MeasureTheory.Integrable S.factualD P.μ → MeasureTheory.Integrable (fun ω => S.b (S.factualX ω)) P.μ → MeasureTheory.Integrable S.U P.μ → MeasureTheory.MemLp (fun x => η.lFn x - S.lVal x) 2 S.P_X → MeasureTheory.MemLp (fun x => η.mFn x - S.mVal x) 2 S.P_X → MeasureTheory.MemLp S.resid 2 P.μ → MeasureTheory.Integrable (fun ω => S.U ω * (η.mFn (S.factualX ω) - S.mVal (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => (η.lFn (S.factualX ω) - S.lVal (S.factualX ω)) * S.resid ω) P.μ → MeasureTheory.Integrable (fun ω => S.resid ω * (η.mFn (S.factualX ω) - S.mVal (S.factualX ω))) P.μ → MeasureTheory.Integrable (fun ω => S.U ω * S.resid ω) P.μ → |∫ (z : γ × ℝ × ℝ), Causalean.Estimation.PLR.plrMomentFunctional η z S.θ₀ ∂S.P_Z| ≤ (1 + |S.θ₀|) * ↑(S.plrGeneralMoment.ρ₁ η S.η₀) * ↑(S.plrGeneralMoment.ρ₂ η S.η₀)","l":84,"f":"Causalean/Estimation/PLR/RemainderBound.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.plr_score_diff_isLittleOp_one","k":"theorem","m":"Causalean.Estimation.PLR.ScoreL2","a":"Estimation/PLR","t":1,"d":"Score-difference L²(P_Z) o_p(1) for the partially linear model. Suppose three nonnegative bounding constants Ca, Cv, Cm satisfy: the residualized outcome $A = Y-\\ell_{val}(X)-\\theta_0(D-m_{val}(X))$ is almost surely bounded by Ca under the observed-data law; the true treatment residual $D-m_{val}(X)$ is almost surely bounded by Cv; and the treatment-regression estimation error of η_hat is uniformly bounded by Cm at every fold, draw, and covariate value. Suppose further that for the estimated nuisance sequence η_hat on the system S, the outcome- and treatment-regression estimation errors are square-integrable over the covariate law at every fold and draw, and that both regression errors converge to zero in L²(P_X) at rate $o_p(1)$. Then the L²(P_Z)-seminorm of the Robinson partialling-out score difference between the estimated and the true nuisance is itself $o_p(1)$.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.PLR.PLRSystem P γ) (η_hat : ℕ → P.Ω → Causalean.Estimation.PLR.PLRNuisance γ) {Ca Cv Cm : ℝ}, 0 ≤ Ca → 0 ≤ Cv → 0 ≤ Cm → (∀ᵐ (z : γ × ℝ × ℝ) ∂S.P_Z, |z.2.2 - S.lVal z.1 - S.θ₀ * (z.2.1 - S.mVal z.1)| ≤ Ca) → (∀ᵐ (z : γ × ℝ × ℝ) ∂S.P_Z, |z.2.1 - S.mVal z.1| ≤ Cv) → (∀ (n : ℕ) (ω : P.Ω) (x : γ), |S.mVal x - (η_hat n ω).mFn x| ≤ Cm) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X) → (∀ (n : ℕ) (ω : P.Ω), MeasureTheory.MemLp (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).lFn x - S.lVal x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun x => (η_hat n ω).mFn x - S.mVal x) 2 S.P_X).toReal) (fun x => 1) P.μ → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fun z => Causalean.Estimation.PLR.plrMomentFunctional (η_hat n ω) z S.θ₀ - Causalean.Estimation.PLR.plrMomentFunctional S.η₀ z S.θ₀) 2 S.P_Z).toReal) (fun x => 1) P.μ","l":334,"f":"Causalean/Estimation/PLR/ScoreL2.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.residSecondMoment","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the residual treatment second moment is the population integral of the squared difference between factual treatment and its value-space treatment regression evaluated at the factual covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → ℝ","l":161,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.η₀","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the true nuisance pair consists of that system's value-space outcome regression and treatment regression.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → Causalean.Estimation.PLR.PLRNuisance γ","l":144,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.PLRSystem.θ₀","k":"def","m":"Causalean.Estimation.PLR.Setup","a":"Estimation/PLR","t":1,"d":"For a partially linear potential-outcomes system with a finite population measure and a measurable covariate space and a partially linear estimation system built on it, the target parameter is the structural slope of its partially linear model.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → [inst_1 : MeasureTheory.IsFiniteMeasure P.μ] → Causalean.Estimation.PLR.PLRSystem P γ → ℝ","l":154,"f":"Causalean/Estimation/PLR/Setup.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentA","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The linear-score coefficient is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrMomentA η z","l":96,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentB","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The linear-score constant term is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrMomentB η z","l":103,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrMomentFunctional","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The Robinson partialling-out score is measurable in the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ) (θ : ℝ), Measurable fun z => Causalean.Estimation.PLR.plrMomentFunctional η z θ","l":87,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.measurable_plrResidual","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":2,"d":"The treatment residual is measurable as a function of the observed data.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ), Measurable fun z => Causalean.Estimation.PLR.plrResidual η z","l":81,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentA","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"For a measurable covariate space, a pair of outcome and treatment regression functions, and an observed covariate, treatment, and outcome triple, the coefficient of the partially linear score that is linear in the structural slope is minus the squared treatment residual.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":59,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentB","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"For a measurable covariate space, a pair of outcome and treatment regression functions, and an observed covariate, treatment, and outcome triple, the constant term in the partially linear score is the product of the treatment residual and the outcome minus its outcome-regression prediction.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":66,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMomentFunctional","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"For a measurable covariate space, a pair of outcome and treatment regression functions, an observed covariate, treatment, and outcome triple, and a real-valued structural slope, the Robinson partialling-out score is the product of the treatment residual and the outcome residual after subtracting that slope times the treatment residual.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ → ℝ","l":51,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrMoment_decomp","k":"theorem","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"Robinson score decomposition. The partialling-out score decomposes into its linear coefficient in θ times the target parameter θ plus a constant term.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (η : Causalean.Estimation.PLR.PLRNuisance γ) (z : γ × ℝ × ℝ) (θ : ℝ), Causalean.Estimation.PLR.plrMomentFunctional η z θ = Causalean.Estimation.PLR.plrMomentA η z * θ + Causalean.Estimation.PLR.plrMomentB η z","l":74,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Estimation.PLR.plrResidual","k":"def","m":"Causalean.Estimation.PLR.Moment","a":"Estimation/PLR","t":1,"d":"For a measurable covariate space, a pair of outcome and treatment regression functions, and an observed covariate, treatment, and outcome triple, the partially linear treatment residual is the observed treatment minus its treatment-regression prediction.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.Estimation.PLR.PLRNuisance γ → γ × ℝ × ℝ → ℝ","l":45,"f":"Causalean/Estimation/PLR/Moment.lean"},{"n":"Causalean.Experimentation.BettingMean.capital","k":"def","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"For any sample space, a real-valued sequence of observations, a sequence of real-valued betting fractions, and a candidate mean, the betting capital process is the process whose initial capital is one and whose capital after observation $n$ is the preceding capital multiplied by $1+\\lambda_n(X_n-m)$.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → (ℕ → Ω → ℝ) → ℝ → ℕ → Ω → ℝ","l":36,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_nonneg","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"The capital process stays nonnegative provided each bet keeps the per-step factor nonnegative (0 ≤ 1 + lamₙ(Xₙ − m)) — the admissibility constraint on the betting fractions.","s":"∀ {Ω : Type u_1} {X lam : ℕ → Ω → ℝ} {m : ℝ}, (∀ (n : ℕ) (ω : Ω), 0 ≤ 1 + lam n ω * (X n ω - m)) → ∀ (n : ℕ), 0 ≤ Causalean.Experimentation.BettingMean.capital X lam m n","l":52,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_succ","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"One step of the betting capital multiplies current wealth by the return 1 + lam n · (X n − m).","s":"∀ {Ω : Type u_1} (X lam : ℕ → Ω → ℝ) (m : ℝ) (n : ℕ), Causalean.Experimentation.BettingMean.capital X lam m (n + 1) = fun ω => Causalean.Experimentation.BettingMean.capital X lam m n ω * (1 + lam n ω * (X n ω - m))","l":47,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.capital_zero","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":2,"d":"The betting capital starts at one before any observations are processed.","s":"∀ {Ω : Type u_1} (X lam : ℕ → Ω → ℝ) (m : ℝ), Causalean.Experimentation.BettingMean.capital X lam m 0 = fun x => 1","l":44,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isAnytimeValid_betting","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"Anytime-valid test by betting. If the betting capital process built from the data stream X, betting fractions lam, and candidate mean m is a test supermartingale under μ and the target error level α is strictly positive, then the rejection region that declares significance once the capital ever reaches 1/α is an anytime-valid level-α test.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsAnytimeValid (Causalean.Experimentation.Sequential.rejectionRegion (Causalean.Experimentation.BettingMean.capital X lam m) α) μ α","l":133,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isConfidenceSequence_bettingCI","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"Betting confidence sequence (coverage). If the betting capital process built from the data stream X, betting fractions lam, and candidate mean m is a test supermartingale under μ and the target error level α is strictly positive, then the sequence of confidence sets that retains a candidate mean m only while its capital has not yet reached 1/α is a valid confidence sequence with time-uniform coverage 1 − α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsConfidenceSequence (Causalean.Experimentation.Sequential.confSeqOfWealth (Causalean.Experimentation.BettingMean.capital X lam m) α) μ α","l":123,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.BettingMean.isTestSupermartingale_capital","k":"theorem","m":"Causalean.Experimentation.BettingMean","a":"Experimentation","t":1,"d":"The capital process is a test supermartingale (the Waudby-Smith–Ramdas construction). Consider a data stream X tested against a candidate mean m via betting fractions lam, all adapted to a filtration ℱ. If the capital process itself is adapted to ℱ, each bet lam n is measurable with respect to the time-n information, the capital process, the centered increment X n − m, and the bet-scaled increment lam n · (X n − m) are all integrable at every time n, every per-step return factor 1 + lam n (X n − m) stays nonnegative, and the conditional mean of X n − m given the time-n information is zero, i.e. the bet is conditionally fair, then the betting capital process is a nonnegative test supermartingale for ℱ under μ, with E[K₀] ≤ 1.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsProbabilityMeasure μ] {X lam : ℕ → Ω → ℝ} {m : ℝ}, MeasureTheory.Adapted ℱ (Causalean.Experimentation.BettingMean.capital X lam m) → (∀ (n : ℕ), MeasureTheory.StronglyMeasurable (lam n)) → (∀ (n : ℕ), MeasureTheory.Integrable (Causalean.Experimentation.BettingMean.capital X lam m n) μ) → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => X n ω - m) μ) → (∀ (n : ℕ), MeasureTheory.Integrable (fun ω => lam n ω * (X n ω - m)) μ) → (∀ (n : ℕ) (ω : Ω), 0 ≤ 1 + lam n ω * (X n ω - m)) → (∀ (n : ℕ), μ[fun ω => X n ω - m | ↑ℱ n] =ᵐ[μ] 0) → Causalean.Experimentation.Sequential.IsTestSupermartingale (Causalean.Experimentation.BettingMean.capital X lam m) ℱ μ","l":67,"f":"Causalean/Experimentation/BettingMean.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htClusterEffect","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"Unbiasedness for the finite-population average treatment effect. Consider cluster randomization where each cluster c is assigned treatment with probability p c; suppose p c lies between 0 and 1 and, more sharply, p c lies strictly between 0 and 1, and the total number of units across clusters is positive. Then the Middleton–Aronow Horvitz–Thompson estimator of the effect — the inverse-probability-weighted treated-minus-control cluster-total contrast, normalized by the total unit count — is unbiased for the finite-population average treatment effect, the analogous unweighted contrast of the cluster-total potential outcomes normalized by the same unit count.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (n : C → ℕ) (y1 y0 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), 0 < p c) → (∀ (c : C), p c < 1) → 0 < Causalean.Experimentation.ClusterRandomizedHT.totalUnits n → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htClusterEffect p n y1 y0) = Causalean.Experimentation.ClusterRandomizedHT.totalEffect n y1 y0","l":123,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htControlTotal","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":2,"d":"The HT control total is unbiased for the control-arm population total ∑ y0 c: each cluster is a control with probability 1 − p c, undone by the weight 1 / (1 − p c).","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p y0 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), p c < 1) → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htControlTotal p y0) = ∑ c, y0 c","l":94,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.E_htTreatedTotal","k":"theorem","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":2,"d":"The HT treated total is unbiased for the treated-arm population total ∑ y1 c: each cluster is treated with probability p c, which the inverse weight 1 / p c exactly undoes.","s":"∀ {C : Type u_1} [inst : Fintype C] [inst_1 : DecidableEq C] (p y1 : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1), (∀ (c : C), 0 < p c) → (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.ClusterRandomizedHT.htTreatedTotal p y1) = ∑ c, y1 c","l":71,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htClusterEffect","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"For a finite population of clusters, cluster treatment probabilities, the number of units in each cluster, treated and control potential-outcome totals, and a realized cluster assignment, the Middleton--Aronow Horvitz--Thompson average treatment-effect estimator is the inverse-probability-weighted treated total minus control total, divided by the total number of units.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℕ) → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":55,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htControlTotal","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"For a finite population of clusters, cluster treatment probabilities, control potential-outcome totals for the clusters, and a realized cluster assignment, the Horvitz–Thompson control total is the sum, over clusters, of each untreated cluster's potential-outcome total weighted by the reciprocal of its control probability.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":43,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.htTreatedTotal","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"For a finite population of clusters, cluster treatment probabilities, treated potential-outcome totals for the clusters, and a realized cluster assignment, the Horvitz–Thompson treated total is the sum, over clusters, of each treated cluster's potential-outcome total weighted by the reciprocal of its treatment probability.","s":"{C : Type u_1} → [Fintype C] → (C → ℝ) → (C → ℝ) → (C → Bool) → ℝ","l":35,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.totalEffect","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"For a finite population of clusters, the number of units in each cluster, and treated and control potential-outcome totals, the finite-population average treatment effect is the sum of treated-minus-control cluster totals divided by the total number of units.","s":"{C : Type u_1} → [Fintype C] → (C → ℕ) → (C → ℝ) → (C → ℝ) → ℝ","l":64,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.ClusterRandomizedHT.totalUnits","k":"def","m":"Causalean.Experimentation.ClusterRandomizedHT","a":"Experimentation","t":1,"d":"For a finite population of clusters and the number of experimental units in each cluster, the total number of experimental units is the sum of the cluster sizes.","s":"{C : Type u_1} → [Fintype C] → (C → ℕ) → ℝ","l":51,"f":"Causalean/Experimentation/ClusterRandomizedHT.lean"},{"n":"Causalean.Experimentation.DesignBased.AgreeOnPrefix","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A reveal permutation, prefix length, and two assignments determine agreement through the reveal prefix: the assignments have the same values on every coordinate already revealed by this equality condition.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → Equiv.Perm (Fin N) → Fin (N + 1) → ((i : Fin N) → alpha i) → ((i : Fin N) → alpha i) → Prop","l":35,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.BlockDegreeLE","k":"def","m":"Causalean.Experimentation.DesignBased.LocalDependenceVariance","a":"Experimentation/DesignBased","t":1,"d":"For a finite population of units, a dependence block assigned to each unit, and a nonnegative integer bound, the assertion that the block family has degree at most $d$ means that, for every unit, at most $d$ blocks contain that unit. When the blocks are the in-neighbourhoods of a dependency graph, this is the out-degree bound.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → Finset U) → ℕ → Prop","l":46,"f":"Causalean/Experimentation/DesignBased/LocalDependenceVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_expoInd_cross","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Covariance of cross-exposure indicators. For a design, an exposure mapping, unit traits, units i and j, and exposure levels d and d', the design covariance of unit i's d-exposure indicator and unit j's d'-exposure indicator equals the cross joint exposure probability π_ij(d,d') minus the product of their marginal exposure probabilities.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i j : ι) (d d' : Δ), D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d') = Causalean.Experimentation.DesignBased.propPairCross D f θ i j d d' - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d'","l":103,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_expoInd_same","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Covariance of same-exposure indicators. For a design, an exposure mapping, unit traits, units i and j, and an exposure level d, the design covariance of the two units' d-exposure indicators equals the joint exposure probability π_ij(d) minus the product of their marginal exposure probabilities.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i j : ι) (d : Δ), D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d) = Causalean.Experimentation.DesignBased.propPairSame D f θ i j d - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d","l":92,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Proposition 4.4 (covariance), expanded form eq:totals_covariance. Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose the two exposures dk and dl are distinct, every unit has nonzero probability of realizing exposure dk, and every unit has nonzero probability of realizing exposure dl. Then the covariance of the two Horvitz–Thompson totals equals the off-diagonal double sum of inverse-probability-weighted joint-exposure covariance terms, minus the diagonal term ∑ᵢ y_i(dk)·y_i(dl) — a unit cannot occupy two distinct exposures at once, so the diagonal cross-indicator vanishes and leaves this unidentified term.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl) = ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl - Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl) * (y i dk / Causalean.Experimentation.DesignBased.prop D f θ i dk * (y j dl / Causalean.Experimentation.DesignBased.prop D f θ j dl)) - ∑ i, y i dk * y i dl","l":102,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Cov_htTotal_cov","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite design, outcome function y, exposure map f, and assignment θ, the covariance between two Horvitz-Thompson exposure totals equals the double sum, over ordered unit pairs, of the two inverse-probability outcome weights times the covariance of the corresponding exposure indicators.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl) = ∑ i, ∑ j, y i dk / Causalean.Experimentation.DesignBased.prop D f θ i dk * (y j dl / Causalean.Experimentation.DesignBased.prop D f θ j dl) * D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i dk) (Causalean.Experimentation.DesignBased.expoInd f θ j dl)","l":86,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.DependsOnBlock","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a population of units, a finite block of units, and a real-valued statistic of the assignment, the assertion that the statistic depends only on the block means that any two assignments agreeing on every unit in the block have the same statistic value.","s":"{U : Type u_1} → Finset U → ((U → Bool) → ℝ) → Prop","l":374,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.DesignFamily","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a finite assignment space, the design family is a set of candidate randomization designs defined on that common assignment space.","s":"(Ω : Type u_2) → [Fintype Ω] → Type u_2","l":41,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a finite assignment space, an ordered risk scale, a risk criterion for randomization designs, and two randomization designs, the assertion that the first design weakly dominates the second means that its risk is no greater than the second design's risk.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {α : Type u_2} → [Preorder α] → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":45,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates.refl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Domination is reflexive: every design dominates itself.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] (R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), Causalean.Experimentation.DesignBased.Dominates R D D","l":52,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.Dominates.trans","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Domination is transitive.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] {R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α} {D₁ D₂ D₃ : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, Causalean.Experimentation.DesignBased.Dominates R D₁ D₂ → Causalean.Experimentation.DesignBased.Dominates R D₂ D₃ → Causalean.Experimentation.DesignBased.Dominates R D₁ D₃","l":57,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.E_controlMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The control-arm mean is unbiased for the control population mean (1/N) ∑ Y0. Each unit enters the control arm with probability n₀ / N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn0 : n₁ < Fintype.card U) (Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).E (Causalean.Experimentation.DesignBased.controlMean n₁ Y0) = (∑ i, Y0 i) / ↑(Fintype.card U)","l":118,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.E_diffInMeans_eq_sate","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"Unbiasedness of difference in means. Under complete randomization in which at least one unit is treated and the treated count n₁ is strictly below the population size N, so at least one unit remains a control, the difference-in-means estimator, computed from the potential outcomes Y1 and Y0, has expectation exactly equal to the sample average treatment effect.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ), 0 < n₁ → ∀ (hn0 : n₁ < Fintype.card U) (Y1 Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).E (Causalean.Experimentation.DesignBased.diffInMeans n₁ Y1 Y0) = Causalean.Experimentation.DesignBased.sateEstimand Y1 Y0","l":186,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.E_expoInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Exposure probability as an expectation. For a design, an exposure mapping, unit traits, a unit i, and an exposure level d, the design expectation of the exposure indicator equals the generalized probability of exposure π_i(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ), D.E (Causalean.Experimentation.DesignBased.expoInd f θ i d) = Causalean.Experimentation.DesignBased.prop D f θ i d","l":84,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htEffect","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Proposition 4.4 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure dk and every unit has nonzero probability of realizing exposure dl. Then the Horvitz–Thompson effect estimator contrasting dk against dl has expectation exactly the average causal effect τ(dk,dl).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) = Causalean.Experimentation.DesignBased.tauTrue y dk dl","l":65,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson mean estimator has expectation exactly the population mean μ(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htMean D y f θ d) = Causalean.Experimentation.DesignBased.muTrue y d","l":54,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Unbiased","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson total estimator has expectation exactly the population total ∑ᵢ y i d.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, y i d","l":38,"f":"Causalean/Experimentation/DesignBased/HT/Unbiased.lean"},{"n":"Causalean.Experimentation.DesignBased.E_treatedMean","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The treated-arm mean is unbiased for the treated population mean (1/N) ∑ Y1. Each unit enters the treated arm with probability n₁ / N, so its inverse-n₁ weight averages to 1/N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U), 0 < n₁ → ∀ (Y1 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).E (Causalean.Experimentation.DesignBased.treatedMean n₁ Y1) = (∑ i, Y1 i) / ↑(Fintype.card U)","l":70,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign","k":"structure","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"A randomization design: a probability mass function p on a finite assignment space Ω whose values are nonnegative and sum to one.","s":"(Ω : Type u_1) → [Fintype Ω] → Type u_1","l":44,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb","k":"def","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"For a sequence of finite assignment spaces, a sequence of finite randomization designs, and a sequence of real-valued statistics, the assertion that the statistics are bounded in probability means that, for every positive tolerance, there is a real threshold such that, from some index onward, the probability that the statistic's absolute value is at least that threshold is at most the tolerance. This is the $O_p(1)$ counterpart of convergence in probability.","s":"{Ω : ℕ → Type u_1} → [inst : (m : ℕ) → Fintype (Ω m)] → ((m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) → ((m : ℕ) → Ω m → ℝ) → Prop","l":305,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space and two real-valued statistics of the realized assignment, the design covariance is the design expectation of the product of their deviations from their respective design expectations.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → (Ω → ℝ) → ℝ","l":68,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_comm","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is symmetric in its two arguments.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = D.Cov Y X","l":159,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal covariance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y X' Y' : Ω → ℝ}, (∀ (z : Ω), X z = X' z) → (∀ (z : Ω), Y z = Y' z) → D.Cov X Y = D.Cov X' Y'","l":128,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_const_mul_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying the left random variable by a constant multiplies covariance by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X Y : Ω → ℝ), D.Cov (fun z => c * X z) Y = c * D.Cov X Y","l":165,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_const_mul_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying the right random variable by a constant multiplies covariance by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X Y : Ω → ℝ), (D.Cov X fun z => c * Y z) = c * D.Cov X Y","l":188,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Covariance identity. For any two statistics X and Y under the design, their design covariance equals the design expectation of X·Y minus the product of their design expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = (D.E fun z => X z * Y z) - D.E X * D.E Y","l":145,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_eq_covariance","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design covariance is the measure-theoretic covariance under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), D.Cov X Y = ProbabilityTheory.covariance X Y D.toMeasure","l":85,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Bilinear expansion of covariance for linear combinations. For finite weighted sums ∑ᵢ∈s cᵢXᵢ and ∑ⱼ∈t eⱼYⱼ, built from index sets s ⊆ ι, t ⊆ κ, weights c, e, and random-variable families X, Y, their design covariance equals the double sum over s×t of the weighted covariances cᵢeⱼ·Cov(Xᵢ,Yⱼ).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} {κ : Type u_3} (s : Finset ι) (t : Finset κ) (c : ι → ℝ) (e : κ → ℝ) (X : ι → Ω → ℝ) (Y : κ → Ω → ℝ), (D.Cov (fun z => ∑ i ∈ s, c i * X i z) fun z => ∑ j ∈ t, e j * Y j z) = ∑ i ∈ s, ∑ j ∈ t, c i * e j * D.Cov (X i) (Y j)","l":217,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_apply_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Cross-coordinate independence: under the product design, functions of two distinct coordinates have zero covariance.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ∀ (g : α i → ℝ) (hfun : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Cov (fun w => g (w i)) fun w => hfun (w j)) = 0","l":75,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_block_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block covariance vanishing (complement form). For a family of independent coordinate designs D and real-valued functions f and g, suppose f depends only on the coordinates in a block A and g depends only on the coordinates outside A. Then their covariance under the product design is zero.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (A : Finset ι) (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ A, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∉ A, w i = w' i) → g w = g w') → (Causalean.Experimentation.DesignBased.prodDesign D).Cov f g = 0","l":106,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_prod_disjoint_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block covariance vanishing (two-block form). For a family of independent coordinate designs D and real-valued functions f and g, suppose the index sets S and T are disjoint, f depends only on the coordinates in S, and g depends only on the coordinates in T. Then their covariance under the product design is zero.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (S T : Finset ι), Disjoint S T → ∀ (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ S, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∈ T, w i = w' i) → g w = g w') → (Causalean.Experimentation.DesignBased.prodDesign D).Cov f g = 0","l":117,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_self","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The covariance of a random variable with itself is its variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.Cov X X = D.Var X","l":155,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sub_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is linear over differences in its left argument.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y Z : Ω → ℝ), D.Cov (fun z => X z - Y z) Z = D.Cov X Z - D.Cov Y Z","l":193,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sub_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance is linear over differences in its right argument.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y Z : Ω → ℝ), (D.Cov X fun z => Y z - Z z) = D.Cov X Y - D.Cov X Z","l":201,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sum_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance with a finite sum in the left argument is the finite sum of covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (f : ι → Ω → ℝ) (Y : Ω → ℝ), D.Cov (fun z => ∑ i ∈ s, f i z) Y = ∑ i ∈ s, D.Cov (f i) Y","l":173,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_sum_right","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Covariance with a finite sum in the right argument is the finite sum of covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (X : Ω → ℝ) (g : ι → Ω → ℝ), (D.Cov X fun z => ∑ i ∈ s, g i z) = ∑ i ∈ s, D.Cov X (g i)","l":182,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Cov_zero_left","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The covariance of the identically-zero random variable with anything is zero.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (Y : Ω → ℝ), D.Cov (fun x => 0) Y = 0","l":83,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space and a real-valued statistic of the realized assignment, the design expectation is the probability-weighted sum of that statistic over all assignments.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ","l":58,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_add","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a sum is the sum of expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.E fun z => X z + Y z) = D.E X + D.E Y","l":88,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_centered_ratio","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Centering of a plug-in ratio. In a finite design, the mean-normalized ratio X / E[X] − 1 has design mean zero whenever the mean E[X] is nonzero — the first-order (linearization) fact that a ratio statistic is exactly centered at its plug-in point.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (a : ℝ), a ≠ 0 → D.E X = a → (D.E fun z => X z / a - 1) = 0","l":74,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_centered_ratio_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Covariance of two normalized ratios (the ratio linearization kernel). In a finite design, the mean of the product of two mean-normalized ratios (X/E[X] − 1)(Y/E[Y] − 1) equals E[XY] / (E[X]·E[Y]) − 1. This is the exact second cross-moment at the heart of every Horvitz–Thompson / Hájek ratio-variance linearization.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ) (a b c : ℝ), a ≠ 0 → b ≠ 0 → D.E X = a → D.E Y = b → (D.E fun z => X z * Y z) = c → (D.E fun z => (X z / a - 1) * (Y z / b - 1)) = c / (a * b) - 1","l":87,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound","k":"theorem","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"For the stage-2 design map assigning each stage-1 outcome s a per-coordinate design D₂ s and any real-valued function X of the joint outcome, built on top of a stage-1 design D₁, the expectation of X under the compound (two-stage) design equals the double sum over stage-1 outcomes s and stage-2 profiles w of the compound probability D₁.p(s)·∏ᵢ(D₂ s i).p(wᵢ) times X(s,w).","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (X : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).E X = ∑ s, ∑ w, (D₁.p s * ∏ i, (D₂ s i).p (w i)) * X (s, w)","l":75,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound_factor","k":"theorem","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"Stage-2 collapse. For the stage-2 design map assigning each stage-1 outcome s a per-coordinate design D₂ s, the compound-design expectation of the product of a stage-1 quantity h(s) and a function g of a single group j's within-assignment wⱼ factors through the marginal expectation of group j's conditional design: E[h(s)·g(wⱼ)] = E_s[h(s)·E_{D₂ s j}[g]].","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (h : Ω₁ → ℝ) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.compound D₁ D₂).E fun sw => h sw.1 * g (sw.2 j)) = D₁.E fun s => h s * (D₂ s j).E g","l":87,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_compound_tower","k":"theorem","m":"Causalean.Experimentation.DesignBased.CompoundVariance","a":"Experimentation/DesignBased","t":2,"d":"Tower property of expectation for the compound design. The compound expectation iterates as the stage-1 expectation of the stage-2 product-design conditional expectation, E[F] = E_s[E_{D₂ s}[F(s, ·)]].","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (F : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).E F = D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).E fun w => F (s, w)","l":48,"f":"Causalean/Experimentation/DesignBased/CompoundVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ}, (∀ (z : Ω), X z = Y z) → D.E X = D.E Y","l":119,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_const","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a constant random variable is that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ), (D.E fun x => c) = c","l":84,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying a random variable by a constant on the left multiplies its expectation by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X : Ω → ℝ), (D.E fun z => c * X z) = c * D.E X","l":96,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_eq_integral","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design expectation is the integral against the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), D.E g = ∫ (x : Ω), g x ∂D.toMeasure","l":70,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_ind","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Expectation of an indicator. For any event A, the design expectation of its indicator equals the design probability of A.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.E (Causalean.Experimentation.DesignBased.FiniteDesign.ind A) = D.Pr A","l":242,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Expectation of a random variable bounded above by one is at most one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ}, (∀ (z : Ω), X z ≤ 1) → D.E X ≤ 1","l":265,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_lin_expand","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":1,"d":"Bilinear expansion of a design mean. In a finite design, suppose the mean of A·C equals AA, the mean of A·F equals AB, the mean of B·C equals BA, and the mean of B·F equals BB — the four pairwise cross-moments of the two-term linear forms (A·ai − B·bi) and (C·aj − F·bj). Then the mean of the product (A·ai − B·bi)(C·aj − F·bj) equals AA·ai·aj − AB·ai·bj − BA·bi·aj + BB·bi·bj. This is the generic bookkeeping step assembling a two-arm (e.g. treated/control) linearized variance from its component moments.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B C F : Ω → ℝ) (ai bi aj bj AA AB BA BB : ℝ), (D.E fun z => A z * C z) = AA → (D.E fun z => A z * F z) = AB → (D.E fun z => B z * C z) = BA → (D.E fun z => B z * F z) = BB → (D.E fun z => (A z * ai - B z * bi) * (C z * aj - F z * bj)) = AA * ai * aj - AB * ai * bj - BA * bi * aj + BB * bi * bj","l":107,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_map","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Transfer of expectation across a pushforward. The expectation of g under the pushforward f_* D equals the expectation of the composite g ∘ f under D.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (g : Ω' → ℝ), (D.map f).E g = D.E fun z => g (f z)","l":341,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_mul_const","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Multiplying a random variable by a constant on the right multiplies its expectation by that constant.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.E fun z => X z * c) = D.E X * c","l":101,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_neg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a negated random variable is the negated expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), (D.E fun z => -X z) = -D.E X","l":106,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Expectation of a nonnegative random variable is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ}, (∀ (z : Ω), 0 ≤ X z) → 0 ≤ D.E X","l":261,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D, the expectation under the product design of a function of a single coordinate j equals the expectation of that same function under coordinate j's own marginal design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => g (w j)) = (D j).E g","l":76,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_apply₂","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Two-coordinate factorization: under the product design, the expectation of a product of a function of one coordinate and a function of a distinct coordinate factors into the product of the two marginal expectations.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ∀ (g : α i → ℝ) (hfun : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => g (w i) * hfun (w j)) = (D i).E g * (D j).E hfun","l":49,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_block_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block factorization of expectation. For a family of independent coordinate designs D and real-valued functions f and g on the joint assignment space, suppose f is invariant under any change to the coordinates outside a block A — it depends only on the coordinates in A, and g depends only on the coordinates outside A. Then under the product design, the expectation of the product f·g factors as E[f·g] = E[f]·E[g]. The general (block-level) form of E_prod_apply₂.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (A : Finset ι) (f g : ((i : ι) → α i) → ℝ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ A, w i = w' i) → f w = f w') → (∀ (w w' : (i : ι) → α i), (∀ i ∉ A, w i = w' i) → g w = g w') → ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => f w * g w) = (Causalean.Experimentation.DesignBased.prodDesign D).E f * (Causalean.Experimentation.DesignBased.prodDesign D).E g","l":59,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_prod_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D and single-coordinate functions g, the expectation under the product design of the product ∏ᵢ g i (w i) factors into the product of the individual coordinate expectations ∏ᵢ (D i).E (g i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (g : (i : ι) → α i → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).E fun w => ∏ i, g i (w i)) = ∏ i, (D i).E (g i)","l":65,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a difference is the difference of expectations.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.E fun z => X z - Y z) = D.E X - D.E Y","l":92,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.E_sum","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Linearity of expectation over a finite sum. For a finite index set s over an index type ι and a family of random variables f, the design expectation of the finite sum ∑ᵢ∈s fᵢ equals the finite sum of the design expectations ∑ᵢ∈s E[fᵢ].","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (f : ι → Ω → ℝ), (D.E fun z => ∑ i ∈ s, f i z) = ∑ i ∈ s, D.E (f i)","l":110,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.IsConservativeVarEst","k":"def","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":1,"d":"For a randomization design, a real-valued variance estimator, and a real-valued statistic of the realized assignment, the assertion that the estimator is conservative means that its design expectation is at least the design variance of the statistic.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → (Ω → ℝ) → Prop","l":41,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space and an event with decidable membership, the design probability of the event is the design expectation of its indicator statistic.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (A : Ω → Prop) → [DecidablePred A] → ℝ","l":77,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_compound_eq_E_condPr","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Tower bridge for probabilities. Under the two-stage compound design, the unconditional probability of an event P equals the stage-1 expectation of its stage-2 conditional probability:","s":"∀ {Ω₁ : Type u_1} {ι : Type u_2} [inst : Fintype Ω₁] [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (P : Ω₁ × ((i : ι) → α i) → Prop) [inst_4 : DecidablePred P], (Causalean.Experimentation.DesignBased.compound D₁ D₂).Pr P = D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).Pr fun w => P (s, w)","l":85,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Probability respects pointwise-equivalent events.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], (∀ (z : Ω), A z ↔ B z) → D.Pr A = D.Pr B","l":290,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_eq_measureReal","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design probability of an event is the real-valued measure of the event under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_3 : DecidablePred A], D.Pr A = D.toMeasure.real {z | A z}","l":79,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_eq_zero_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.Chebyshev","a":"Experimentation/DesignBased","t":2,"d":"Zero-hitting bound for a nonzero-mean statistic. In a finite design, a statistic whose design mean is nonzero equals zero with probability at most its design variance divided by the square of its mean. This is the Chebyshev corollary that controls degeneracy of a random Horvitz–Thompson / Hájek denominator: a zero value is exactly a deviation from the mean of size equal to the mean, so it is Var / mean²-rare.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.E X ≠ 0 → (D.Pr fun z => X z = 0) ≤ D.Var X / D.E X ^ 2","l":77,"f":"Causalean/Experimentation/DesignBased/Chebyshev.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"A probability lies in [0,1]: at most one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.Pr A ≤ 1","l":286,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_map","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Transfer of probability across a pushforward. The probability of an event A under the pushforward f_* D equals the probability of its preimage under D.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (A : Ω' → Prop) [inst_2 : DecidablePred A], (D.map f).Pr A = D.Pr fun z => A (f z)","l":355,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_mono","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Monotonicity of probability: a smaller event has smaller probability.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A B : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], (∀ (z : Ω), A z → B z) → D.Pr A ≤ D.Pr B","l":295,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"A probability lies in [0,1]: nonnegativity.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], 0 ≤ D.Pr A","l":282,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_or_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Union bound for two events: Pr(P ∨ Q) ≤ Pr P + Pr Q.","s":"∀ {Ω' : Type u_2} [inst : Fintype Ω'] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω') (P Q : Ω' → Prop) [inst_1 : DecidablePred P] [inst_2 : DecidablePred Q], (D.Pr fun z => P z ∨ Q z) ≤ D.Pr P + D.Pr Q","l":115,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Pr_split","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Finite additivity: splitting an event by a second event.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (B A : Ω → Prop) [inst_1 : DecidablePred A] [inst_2 : DecidablePred B], D.Pr B = (D.Pr fun z => B z ∧ A z) + D.Pr fun z => B z ∧ ¬A z","l":306,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb","k":"def","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"For a sequence of finite assignment spaces, a sequence of finite randomization designs, a sequence of real-valued statistics, and a sequence of real-valued targets, the assertion of convergence in probability means that, for every positive $\\varepsilon$, the probability that the statistic differs from its target by at least $\\varepsilon$ tends to zero as the sequence index tends to infinity.","s":"{Ω : ℕ → Type u_1} → [inst : (m : ℕ) → Fintype (Ω m)] → ((m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) → ((m : ℕ) → Ω m → ℝ) → (ℕ → ℝ) → Prop","l":51,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.abs","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"If X m → 0 in probability then |X m| → 0 in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X : (m : ℕ) → Ω m → ℝ}, (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X fun x => 0) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => |X m z|) fun x => 0","l":193,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.add","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Convergence in probability is closed under sums.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X Y : (m : ℕ) → Ω m → ℝ} {a b : ℕ → ℝ}, Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y b → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z + Y m z) fun m => a m + b m","l":162,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Scaling a convergent sequence by a constant scales its probability limit: if X m → a m in probability then c · X m → c · a m in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X : (m : ℕ) → Ω m → ℝ} {a : ℕ → ℝ} (c : ℝ), Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => c * X m z) fun m => c * a m","l":350,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.mul_boundedInProb","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Product-tightness (o_p × O_p = o_p). If U m converges to zero in probability and V m is uniformly tight (bounded in probability), then the product U m · V m converges to zero in probability. This is the engine that turns a delta-method remainder — a vanishing factor times a bounded factor — into an o_p(1) term.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {U V : (m : ℕ) → Ω m → ℝ}, (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D U fun x => 0) → Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb D V → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => U m z * V m z) fun x => 0","l":370,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb.sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Along a sequence of finite designs, if the statistics X m converge in probability to a m and the statistics Y m converge in probability to b m, then the difference X m − Y m converges in probability to a m − b m: convergence in probability is closed under differences.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] {D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)} {X Y : (m : ℕ) → Ω m → ℝ} {a b : ℕ → ℝ}, Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y b → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z - Y m z) fun m => a m - b m","l":127,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Unbiased","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design, a real-valued estimator of the realized assignment, and a real-valued target, the assertion that the estimator is unbiased means that its design expectation equals the target.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → Prop","l":51,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space and a real-valued statistic of the realized assignment, the design variance is the design expectation of the statistic's squared deviation from its design expectation.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_compound_eq_tower","k":"theorem","m":"Causalean.Experimentation.DesignBased.CompoundVariance","a":"Experimentation/DesignBased","t":1,"d":"Law of total variance for the two-stage compound design. For a stage-1 design D₁ paired with a stage-2 design D₂ s at every stage-1 outcome s, applied to any statistic X on the joint outcome space, the total variance of X under the compound design decomposes as the stage-1 expectation of the stage-2 conditional variance of X, plus the stage-1 variance of the stage-2 conditional mean of X:","s":"∀ {Ω₁ : Type u_1} [inst : Fintype Ω₁] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {α : ι → Type u_3} [inst_3 : (i : ι) → Fintype (α i)] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (D₂ : Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (X : Ω₁ × ((i : ι) → α i) → ℝ), (Causalean.Experimentation.DesignBased.compound D₁ D₂).Var X = (D₁.E fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).Var fun w => X (s, w)) + D₁.Var fun s => (Causalean.Experimentation.DesignBased.prodDesign (D₂ s)).E fun w => X (s, w)","l":64,"f":"Causalean/Experimentation/DesignBased/CompoundVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal random variables have equal variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ}, (∀ (z : Ω), X z = Y z) → D.Var X = D.Var Y","l":123,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_const_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Variance scales by the square of a constant: Var(c·X) = c²·Var X.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (c : ℝ) (X : Ω → ℝ), (D.Var fun z => c * X z) = c ^ 2 * D.Var X","l":212,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance identity. For any statistic X, its design variance equals the design expectation of X² minus the square of its design expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), D.Var X = (D.E fun z => X z ^ 2) - D.E X ^ 2","l":137,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_eq_variance","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Reverse rewrite. The design variance is the measure-theoretic variance under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), D.Var g = ProbabilityTheory.variance g D.toMeasure","l":74,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_ind","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance of an indicator. For any event A, the design variance of its indicator equals the design probability of A times one minus that probability.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_1 : DecidablePred A], D.Var (Causalean.Experimentation.DesignBased.FiniteDesign.ind A) = D.Pr A * (1 - D.Pr A)","l":253,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"Variance of a finite linear combination. For a finite weighted sum ∑ᵢ∈s cᵢXᵢ, built from an index set s ⊆ ι, weights c, and a random-variable family X, its design variance equals the double sum of weighted covariances ∑ᵢ∈s∑ⱼ∈s cᵢcⱼ·Cov(Xᵢ,Xⱼ).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} (s : Finset ι) (c : ι → ℝ) (X : ι → Ω → ℝ), (D.Var fun z => ∑ i ∈ s, c i * X i z) = ∑ i ∈ s, ∑ j ∈ s, c i * c j * D.Cov (X i) (X j)","l":231,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Variance is nonnegative (it is the mean squared error of the centered estimator).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ), 0 ≤ D.Var X","l":83,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_prod_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":2,"d":"Single-coordinate variance: under the product design, the variance of a function of one coordinate equals the variance of that function under the coordinate's own design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (j : ι) (g : α j → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Var fun w => g (w j)) = (D j).Var g","l":63,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_prod_linear_comb","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductVariance","a":"Experimentation/DesignBased","t":1,"d":"The payload. Under the product design formed from a family of per-coordinate finite designs, the variance of a linear combination ∑ᵢ cᵢ·gᵢ(wᵢ) of single-coordinate functions gᵢ, with coefficients c, equals the sum over coordinates of the squared coefficient times the coordinate's own variance — cross-coordinate covariances vanish.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (c : ι → ℝ) (g : (i : ι) → α i → ℝ), ((Causalean.Experimentation.DesignBased.prodDesign D).Var fun w => ∑ i, c i * g i (w i)) = ∑ i, c i ^ 2 * (D i).Var (g i)","l":86,"f":"Causalean/Experimentation/DesignBased/ProductVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_sub","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"Variance of a difference: Var(X−Y) = Var X + Var Y − 2 Cov(X,Y).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), (D.Var fun z => X z - Y z) = D.Var X + D.Var Y - 2 * D.Cov X Y","l":206,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.Var_sub_const","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Shifting a random variable by a constant leaves its variance unchanged.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.Var fun z => X z - c) = D.Var X","l":63,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_Cov_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":1,"d":"Cauchy–Schwarz for the design covariance. In any finite design, the absolute covariance of two statistics X and Y is at most the product of their design standard deviations, |Cov(X,Y)| ≤ √(Var X) · √(Var Y).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X Y : Ω → ℝ), |D.Cov X Y| ≤ √(D.Var X) * √(D.Var Y)","l":92,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_Cov_le_two_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The design covariance of random variables bounded by MX and MY in absolute value pointwise is bounded by 2·MX·MY in absolute value.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X Y : Ω → ℝ} {MX MY : ℝ}, 0 ≤ MX → (∀ (z : Ω), |X z| ≤ MX) → (∀ (z : Ω), |Y z| ≤ MY) → |D.Cov X Y| ≤ 2 * MX * MY","l":62,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.abs_E_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":2,"d":"The expectation of a random variable whose absolute value is bounded by M pointwise is itself bounded by M in absolute value.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {X : Ω → ℝ} {M : ℝ}, (∀ (z : Ω), |X z| ≤ M) → |D.E X| ≤ M","l":42,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.aemeasurable_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"On a finite assignment space every statistic with a measurable codomain is a.e.-measurable under the design measure.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : MeasurableSpace β] (g : Ω → β), AEMeasurable g D.toMeasure","l":52,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.aestronglyMeasurable_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"On a finite assignment space every statistic is strongly measurable, because singletons — hence all sets — are measurable.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : MeasurableSpace β] [inst_4 : TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β] (g : Ω → β), MeasureTheory.AEStronglyMeasurable g D.toMeasure","l":43,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bernstein_abs_ge","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":1,"d":"Bernstein tail for a bounded design statistic (two-sided). For a statistic X on a finite design, suppose the bound c is nonnegative, X has design mean 0 , X is bounded in absolute value by c everywhere, the design variance of X is at most v, and the threshold ε is nonnegative. Then the design probability that X deviates from 0 by at least ε in absolute value is at most 2·exp(−ε²/(2(2v + cε))), twice the corresponding one-sided Bernstein bound.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v ε : ℝ}, 0 ≤ c → D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → D.Var X ≤ v → 0 ≤ ε → (D.Pr fun z => ε ≤ |X z|) ≤ 2 * Real.exp (-ε ^ 2 / (2 * (2 * v + c * ε)))","l":71,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bernstein_ge","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":2,"d":"Bernstein tail for a bounded design statistic (one-sided). If a statistic is bounded by c, has design mean 0, and has design variance at most v, then it exceeds a nonnegative threshold ε with probability at most exp(−ε² / (2(2v + cε))), far sharper than Chebyshev.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v ε : ℝ}, 0 ≤ c → D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → D.Var X ≤ v → 0 ≤ ε → (D.Pr fun z => ε ≤ X z) ≤ Real.exp (-ε ^ 2 / (2 * (2 * v + c * ε)))","l":58,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bias","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design, a real-valued estimator of the realized assignment, and a real-valued target, the estimator's design bias is its design expectation minus the target.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → ℝ","l":41,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.bias_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"The bias of an unbiased estimator is zero.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est : Ω → ℝ} {μ : ℝ}, D.Unbiased est μ → D.bias est μ = 0","l":74,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.boundedInProb_of_stdNormalCDF","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"For a sequence of finite-design laws and real-valued statistics, if their CDFs converge pointwise to the standard normal CDF, then the statistics are bounded in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (T : (n : ℕ) → Ω n → ℝ), (∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => T n z ≤ x) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf x))) → Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb D T","l":79,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.boundedInProb_of_var_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"An eventual variance bound together with an eventual mean bound makes a sequence bounded in probability: if the design variances Var(X m) are eventually at most V and the means E(X m) eventually lie within c of zero, then X is uniformly tight.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X : (m : ℕ) → Ω m → ℝ) {V c : ℝ}, (∀ᶠ (m : ℕ) in Filter.atTop, (D m).Var (X m) ≤ V) → (∀ᶠ (m : ℕ) in Filter.atTop, |(D m).E (X m)| ≤ c) → Causalean.Experimentation.DesignBased.FiniteDesign.BoundedInProb D X","l":314,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.chebyshev","k":"theorem","m":"Causalean.Experimentation.DesignBased.Chebyshev","a":"Experimentation/DesignBased","t":1,"d":"Finite-design Chebyshev inequality. In a finite design, for any statistic X and a strictly positive threshold ε, the design probability that X differs from its design mean by at least ε is at most the design variance of X divided by ε².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {ε : ℝ}, 0 < ε → (D.Pr fun z => ε ≤ |X z - D.E X|) ≤ D.Var X / ε ^ 2","l":37,"f":"Causalean/Experimentation/DesignBased/Chebyshev.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.chebyshev_conservative","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":1,"d":"Conservative Chebyshev bound. If the design expectation of a proposed variance estimator V̂ is at least the randomization variance of X, then for any positive threshold ε, the probability that X deviates from its mean by at least ε is bounded by that expected estimator divided by ε².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ}, D.IsConservativeVarEst Vhat X → ∀ {ε : ℝ}, 0 < ε → (D.Pr fun z => ε ≤ |X z - D.E X|) ≤ D.E Vhat / ε ^ 2","l":58,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.deterministic_tendstoInProb","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Under a sequence of finite-design laws, if a deterministic real sequence converges to a constant, then the corresponding constant-on-assignment statistics converge in finite-design probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (a : ℕ → ℝ) (c : ℝ), Filter.Tendsto a Filter.atTop (nhds c) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun n x => a n) fun x => c","l":60,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.hasSubexponentialMGF_of_bounded","k":"theorem","m":"Causalean.Experimentation.DesignBased.Concentration","a":"Experimentation/DesignBased","t":2,"d":"A bounded, mean-zero statistic on a finite design is sub-exponential under the design measure, with variance-proxy 2v and scale c. This is the finite-design instance of the Bernstein sub-exponential lemma, obtained through the measure bridge.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) {c v : ℝ} (hc : 0 ≤ c), D.E X = 0 → (∀ (z : Ω), |X z| ≤ c) → ∀ (hvar : D.Var X ≤ v), Causalean.Stat.Concentration.HasSubexponentialMGF X ⟨2 * v, mul_nonneg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl true)) (le_trans (Causalean.Experimentation.DesignBased.FiniteDesign.E_nonneg D fun x => sq_nonneg (X x - D.E X)) hvar)⟩ ⟨c, hc⟩ D.toMeasure","l":40,"f":"Causalean/Experimentation/DesignBased/Concentration.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For an event with decidable membership on an assignment space, its indicator statistic assigns one to assignments in the event and zero to all other assignments.","s":"{Ω : Type u_1} → (A : Ω → Prop) → [DecidablePred A] → Ω → ℝ","l":73,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"An event indicator is always at most one.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A] (z : Ω), Causalean.Experimentation.DesignBased.FiniteDesign.ind A z ≤ 1","l":278,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"An event indicator is always nonnegative.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A] (z : Ω), 0 ≤ Causalean.Experimentation.DesignBased.FiniteDesign.ind A z","l":273,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.ind_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"1_A ^ 2 = 1_A, the idempotence of an indicator.","s":"∀ {Ω : Type u_1} (A : Ω → Prop) [inst : DecidablePred A], (fun z => Causalean.Experimentation.DesignBased.FiniteDesign.ind A z ^ 2) = Causalean.Experimentation.DesignBased.FiniteDesign.ind A","l":248,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.instIsProbabilityMeasureToMeasure","k":"instance","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":2,"d":"For every randomization design on a finite assignment space equipped with a σ-algebra, the measure induced by that design is a probability measure: its total mass is one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), MeasureTheory.IsProbabilityMeasure D.toMeasure","l":62,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.integral_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Integral against the induced measure is the design expectation. For any statistic g, the measure-theoretic integral of g against the design's induced probability measure equals its design expectation.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), ∫ (x : Ω), g x ∂D.toMeasure = D.E g","l":73,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.isConservativeVarEst_of_E_eq_add","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":2,"d":"A variance estimator whose expectation equals the variance plus a nonnegative bias is conservative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ} {B : ℝ}, 0 ≤ B → D.E Vhat = D.Var X + B → D.IsConservativeVarEst Vhat X","l":47,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.isConservativeVarEst_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Variance.Conservative","a":"Experimentation/DesignBased/Variance","t":2,"d":"An unbiased variance estimator (its expectation equals the variance) is conservative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Vhat X : Ω → ℝ}, D.E Vhat = D.Var X → D.IsConservativeVarEst Vhat X","l":53,"f":"Causalean/Experimentation/DesignBased/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.map","k":"def","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space and a transformation from that space to another finite assignment space, the pushforward randomization design is the law of the transformed assignment: each transformed assignment receives the total probability of all original assignments mapped to it.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → {Ω' : Type u_2} → [inst : Fintype Ω'] → (Ω → Ω') → Causalean.Experimentation.DesignBased.FiniteDesign Ω'","l":317,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.map_p","k":"theorem","m":"Causalean.Experimentation.DesignBased.DesignCore","a":"Experimentation/DesignBased","t":2,"d":"The pushforward weight of y is the fiber sum ∑_{z} 1[f z = y] · D.p z.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {Ω' : Type u_2} [inst_1 : Fintype Ω'] (f : Ω → Ω') (y : Ω'), (D.map f).p y = ∑ z, if f z = y then D.p z else 0","l":335,"f":"Causalean/Experimentation/DesignBased/DesignCore.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.memLp_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.MeasureBridge","a":"Experimentation/DesignBased","t":2,"d":"Every normed-additive statistic with a suitable measurable second-countable codomain on a finite assignment space is Lᵖ under the design measure: the space is finite so every statistic is bounded, and the design measure is a probability measure, so every power is integrable.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {β : Type u_2} [inst_3 : NormedAddCommGroup β] [inst_4 : MeasurableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β] (g : Ω → β) (p : ENNReal), MeasureTheory.MemLp g p D.toMeasure","l":59,"f":"Causalean/Experimentation/DesignBased/MeasureBridge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse","k":"def","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design, a real-valued estimator of the realized assignment, and a real-valued target, the estimator's design mean squared error is the design expectation of the squared difference between the estimator and the target.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → ℝ) → ℝ → ℝ","l":46,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_congr","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Congruence: pointwise-equal estimators have equal mean squared error.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est est' : Ω → ℝ} {μ : ℝ}, (∀ (z : Ω), est z = est' z) → D.mse est μ = D.mse est' μ","l":93,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_eq_var_add_bias_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":1,"d":"Bias–variance decomposition. The mean squared error mse of estimator est for target μ equals its randomization variance plus the square of its bias: mse = Var + bias².","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), D.mse est μ = D.Var est + D.bias est μ ^ 2","l":56,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_eq_var_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"For an unbiased estimator the mean squared error equals the variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {est : Ω → ℝ} {μ : ℝ}, D.Unbiased est μ → D.mse est μ = D.Var est","l":67,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.mse_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Mean squared error is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), 0 ≤ D.mse est μ","l":79,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.nonempty_of_design","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductBlock","a":"Experimentation/DesignBased","t":2,"d":"Every coordinate type carrying a finite design is nonempty.","s":"∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (i : ι), Nonempty (α i)","l":50,"f":"Causalean/Experimentation/DesignBased/ProductBlock.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.prodDesign_Pr_reindex","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductReindex","a":"Experimentation/DesignBased","t":1,"d":"Relabeling a product design by a coordinate permutation. For a family of designs D over a common coordinate space W, indexed by ι, and a permutation σ : ι ≃ ι, the probability under prodDesign D of the predicate w ↦ P (w ∘ σ) equals the probability under the permuted product prodDesign (D ∘ σ) of P.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {W : Type u_2} [inst_2 : Fintype W] (σ : ι ≃ ι) (D : ι → Causalean.Experimentation.DesignBased.FiniteDesign W) (P : (ι → W) → Prop) [inst_3 : DecidablePred P], ((Causalean.Experimentation.DesignBased.prodDesign D).Pr fun w => P fun i => w (σ i)) = (Causalean.Experimentation.DesignBased.prodDesign fun i => D (σ i)).Pr P","l":40,"f":"Causalean/Experimentation/DesignBased/ProductReindex.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendstoInProb_continuousMap","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":2,"d":"Under a sequence of finite-design laws, if a statistic converges in probability to a constant and a real map is continuous at that constant, then applying the map preserves convergence in probability.","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (X : (n : ℕ) → Ω n → ℝ) (c : ℝ) (f : ℝ → ℝ), (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X fun x => c) → ContinuousAt f c → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun n z => f (X n z)) fun x => f c","l":82,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendstoInProb_div_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Slutsky ratio step. Along a sequence of finite designs D, fix statistics X and Y. If X m converges in probability to a m, the denominator Y m converges in probability to 1, and the limit sequence a m is uniformly bounded by a constant M, then the ratio X m / Y m converges in probability to a m. (The realized normalizer tends to one, so dividing by it does not change the probability limit.)","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X Y : (m : ℕ) → Ω m → ℝ) (a : ℕ → ℝ) (M : ℝ), (∀ (m : ℕ), |a m| ≤ M) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X a → (Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D Y fun x => 1) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D (fun m z => X m z / Y m z) a","l":200,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendstoInProb_of_var","k":"theorem","m":"Causalean.Experimentation.DesignBased.InProb","a":"Experimentation/DesignBased","t":1,"d":"Chebyshev consistency engine. Along a sequence of finite designs D, for statistics X, if the design variance of X m tends to zero as m → ∞, then X m converges in probability to its design mean E[X m].","s":"∀ {Ω : ℕ → Type u_1} [inst : (m : ℕ) → Fintype (Ω m)] (D : (m : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω m)) (X : (m : ℕ) → Ω m → ℝ), Filter.Tendsto (fun m => (D m).Var (X m)) Filter.atTop (nhds 0) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb D X fun m => (D m).E (X m)","l":101,"f":"Causalean/Experimentation/DesignBased/InProb.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendsto_E_of_uniformBound","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Mixture-lifting lemma. If a sequence of design random variables F n : Ω n → ℝ converges to a constant L *uniformly* over the assignment space — |F n s − L| ≤ B n for every s, with B n → 0 — then their expectations converge to L: (D n).E (F n) → L. Averaging a uniformly convergent family preserves the limit, since the expectation of a B n-bounded deviation is itself B n-bounded. This is the abstract content of the two-stage mixture-lifting argument: the joint law is the stage-1 average of conditional laws, and a uniform conditional limit lifts to the average.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (F : (n : ℕ) → Ω n → ℝ) (L : ℝ) (B : ℕ → ℝ), (∀ (n : ℕ) (s : Ω n), |F n s - L| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (D n).E (F n)) Filter.atTop (nhds L)","l":63,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.tendsto_E_of_uniformBound_ae","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Design expectations converge to a constant when the random variables converge uniformly on the support of each design.","s":"∀ {Ω : ℕ → Type u_2} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (F : (n : ℕ) → Ω n → ℝ) (L : ℝ) (B : ℕ → ℝ), (∀ (n : ℕ) (s : Ω n), (D n).p s ≠ 0 → |F n s - L| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (D n).E (F n)) Filter.atTop (nhds L)","l":71,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure","k":"def","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"For a randomization design on a finite assignment space equipped with a σ-algebra, the induced probability measure assigns to each assignment a point mass equal to that assignment's design probability, and sums these point masses over all assignments.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → [inst_1 : MeasurableSpace Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → MeasureTheory.Measure Ω","l":48,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_apply","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":2,"d":"The measure induced by a finite randomization design assigns each set the sum, over assignments, of each assignment's probability times the point mass of that assignment on the set.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (s : Set Ω), D.toMeasure s = ∑ z, ENNReal.ofReal (D.p z) * (MeasureTheory.Measure.dirac z) s","l":54,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_real_setOf","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Induced measure matches the design probability. For any event A, the induced measure's probability of A equals the design probability of A.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (A : Ω → Prop) [inst_3 : DecidablePred A], D.toMeasure.real {z | A z} = D.Pr A","l":88,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.toMeasure_singleton","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"The design measure of a singleton equals that singleton's design probability: (D i).toMeasure {a} = (D i).p a. (The dirac sum collapses to the single term z = a.)","s":"∀ {β : Type u_3} [inst : Fintype β] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (D : Causalean.Experimentation.DesignBased.FiniteDesign β) (a : β), D.toMeasure {a} = ENNReal.ofReal (D.p a)","l":52,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.var_edge_sum_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.EdgeVarianceBound","a":"Experimentation/DesignBased","t":1,"d":"Variance of an edge-sum over a bounded-degree dependency graph. Consider real-valued statistics b i j attached to ordered pairs of units, all defined on the same finite design. Suppose every statistic attached to an edge of a graph G is bounded in absolute value by a nonnegative constant M, G is symmetric, every unit has at most m neighbours in G, every statistic attached to a pair that is not an edge of G is identically zero, and two statistics are uncorrelated whenever no edge of G connects an endpoint of one pair to an endpoint of the other. Then the variance of the double sum ∑ᵢ∑ⱼ b i j is at most 8·M²·m³·N, where N is the number of units.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {ι : Type u_2} [inst_1 : Fintype ι] (b : ι → ι → Ω → ℝ) (G : ι → ι → Prop) [inst_2 : DecidableRel G] {M : ℝ}, 0 ≤ M → (∀ (i j : ι), G i j → ∀ (z : Ω), |b i j z| ≤ M) → (∀ (i j : ι), G i j → G j i) → ∀ {m : ℕ}, (∀ (i : ι), {j | G i j}.card ≤ m) → (∀ (i j : ι), ¬G i j → b i j = fun x => 0) → (∀ (i j k l : ι), ¬(G i k ∨ G i l ∨ G j k ∨ G j l) → D.Cov (b i j) (b k l) = 0) → (D.Var fun z => ∑ i, ∑ j, b i j z) ≤ 8 * M ^ 2 * (↑m ^ 3 * ↑(Fintype.card ι))","l":95,"f":"Causalean/Experimentation/DesignBased/EdgeVarianceBound.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.var_le_mse","k":"theorem","m":"Causalean.Experimentation.DesignBased.Risk","a":"Experimentation/DesignBased","t":2,"d":"Mean squared error is bounded below by the variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (est : Ω → ℝ) (μ : ℝ), D.Var est ≤ D.mse est μ","l":88,"f":"Causalean/Experimentation/DesignBased/Risk.lean"},{"n":"Causalean.Experimentation.DesignBased.FiniteDesign.variance_toMeasure","k":"theorem","m":"Causalean.Experimentation.DesignBased.FiniteDesignMeasure","a":"Experimentation/DesignBased","t":1,"d":"Measure-theoretic variance equals the design variance. For any statistic g, the measure-theoretic variance of g under the design's induced measure equals its design variance.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (g : Ω → ℝ), ProbabilityTheory.variance g D.toMeasure = D.Var g","l":98,"f":"Causalean/Experimentation/DesignBased/FiniteDesignMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.HeydeBrownFourthMomentPremise","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A measure (in intended use a probability measure; the premise itself does not require one), filtration, increment family, and constant define the supplied Heyde--Brown fourth-moment premise: adapted, centered, normalized increments with finite fourth moments obey the stated one-fifth-power Kolmogorov bound by the four displayed assumptions and conclusion.","s":"{N : ℕ} → {Omega : Type u_2} → [m : MeasurableSpace Omega] → MeasureTheory.Measure Omega → MeasureTheory.Filtration (Fin (N + 1)) m → (Fin N → Omega → ℝ) → ℝ → Prop","l":148,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.IsMinimaxOn","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a family of randomization designs, a finite set of states of nature that is nonempty, a risk criterion indexed by states and randomization designs, and a candidate randomization design, the minimax condition holds precisely when the candidate belongs to the family and its worst-case risk over the state set is no greater than that of every design in the family, provided risk values admit pairwise least upper bounds.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → Causalean.Experimentation.DesignBased.DesignFamily Ω → (s : Finset Y) → s.Nonempty → {α : Type u_3} → [SemilatticeSup α] → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":52,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.IsOptimalOn","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a finite assignment space, an ordered risk scale, a family of candidate randomization designs, a risk criterion, and a randomization design, the assertion that the design is optimal in the family means both that it belongs to the family and that its risk is no greater than the risk of every design in the family.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {α : Type u_2} → [Preorder α] → Causalean.Experimentation.DesignBased.DesignFamily Ω → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → Prop","l":62,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.IsOptimalOn.dominates","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"An optimal design dominates every member of its family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : Preorder α] {𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω} {R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α} {D₀ : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, Causalean.Experimentation.DesignBased.IsOptimalOn 𝒟 R D₀ → ∀ {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → Causalean.Experimentation.DesignBased.Dominates R D₀ D","l":70,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.IsPrefixMartingaleDifference","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A family of coordinate designs, reveal permutation, and increment family satisfy the prefix martingale-difference condition when each increment has zero conditional mean immediately before its reveal and is visible after that reveal.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [inst : (i : Fin N) → Fintype (alpha i)] → ((i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) → Equiv.Perm (Fin N) → (Fin N → ((i : Fin N) → alpha i) → ℝ) → Prop","l":89,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.ProperlySpecified","k":"def","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"For potential outcomes indexed by unit and exposure level, randomization potential outcomes indexed by unit and assignment, an assignment-to-exposure mapping, and unit-level attributes used by that mapping, the properly specified exposure condition states that, for every unit and every assignment, the randomization potential outcome equals the potential outcome at the exposure induced for that unit by the assignment.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (ι → Δ → ℝ) → (ι → Ω → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Prop","l":38,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.Var_htTotal","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (variance), expanded form eq:total_variance. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the diagonal sum of inverse-probability-weighted indicator variances plus the off-diagonal sum of inverse-probability-weighted indicator covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, Causalean.Experimentation.DesignBased.prop D f θ i d * (1 - Causalean.Experimentation.DesignBased.prop D f θ i d) * (y i d / Causalean.Experimentation.DesignBased.prop D f θ i d) ^ 2 + ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.DesignBased.propPairSame D f θ i j d - Causalean.Experimentation.DesignBased.prop D f θ i d * Causalean.Experimentation.DesignBased.prop D f θ j d) * (y i d / Causalean.Experimentation.DesignBased.prop D f θ i d * (y j d / Causalean.Experimentation.DesignBased.prop D f θ j d))","l":57,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Var_htTotal_cov","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Variance","a":"Experimentation/DesignBased/HT","t":1,"d":"Lemma 4.1 (variance), covariance form. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the double sum, over unit pairs, of inverse-probability-weighted indicator covariances.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) = ∑ i, ∑ j, y i d / Causalean.Experimentation.DesignBased.prop D f θ i d * (y j d / Causalean.Experimentation.DesignBased.prop D f θ j d) * D.Cov (Causalean.Experimentation.DesignBased.expoInd f θ i d) (Causalean.Experimentation.DesignBased.expoInd f θ j d)","l":42,"f":"Causalean/Experimentation/DesignBased/HT/Variance.lean"},{"n":"Causalean.Experimentation.DesignBased.Yobs","k":"def","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"For potential outcomes indexed by unit and exposure level, an assignment-to-exposure mapping, unit-level attributes used by that mapping, a unit, and an assignment, the observed outcome is that unit's potential outcome at the exposure induced by the assignment.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → ι → Ω → ℝ","l":46,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.Yobs_eq_sum","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":1,"d":"Condition 2 (consistency). The observed outcome of unit i under assignment z equals the sum, over every exposure level, of the exposure indicator times the exposure-indexed potential outcome.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] [inst_1 : Fintype Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (z : Ω), Causalean.Experimentation.DesignBased.Yobs y f θ i z = ∑ d, Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d","l":56,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.aestronglyMeasurable_reveal_of_prefixCondExp_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs, reveal permutation, prefix length, statistic, and pointwise equality to its prefix conditional expectation imply that the statistic is almost-everywhere strongly measurable for the reveal σ-algebra.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (k : Fin (N + 1)) (f : ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.prefixCondExp D pi k f = f → MeasureTheory.AEStronglyMeasurable f (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":384,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite population of units, unit-specific treatment probabilities that are each between zero and one, the Bernoulli randomization design independently assigns every unit to treatment with its own specified probability.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (p : U → ℝ) → (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":57,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_Cov_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"For distinct units i ≠ j, the two treatment indicators are uncorrelated under the Bernoulli design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) {i j : U}, i ≠ j → (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.DesignBased.treatInd i) (Causalean.Experimentation.DesignBased.treatInd j) = 0","l":104,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"A centered monomial has mean zero unless it is the constant one: the design expectation of the S-centered monomial is 1 when S is empty and 0 otherwise.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.DesignBased.centeredMonomial p S) = if S = ∅ then 1 else 0","l":218,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial_mul","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"Orthogonality of the centered monomials. Under a Bernoulli randomization in which every unit's treatment probability p i lies in [0,1], the centered monomials attached to two finite sets of units S and T are uncorrelated whenever S ≠ T, while the monomial attached to S itself has second moment ∏_{j ∈ S} p j · (1 − p j) — the product of the per-unit assignment variances — when S = T. This is the design-based analogue of the orthonormality of a Fourier basis.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.centeredMonomial p S z * Causalean.Experimentation.DesignBased.centeredMonomial p T z) = if S = T then ∏ j ∈ S, p j * (1 - p j) else 0","l":177,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_centeredMonomial_mul_treatInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Pairing a centered monomial with a raw monomial. The expectation of the S-centered monomial times the product of the treatment indicators of a set T vanishes unless every unit of S also lies in T; when it does, it equals the product of the assignment variances p j (1 − p j) over S times the treatment probabilities of the units of T outside S.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.centeredMonomial p S z * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z) = if S ⊆ T then (∏ j ∈ S, p j * (1 - p j)) * ∏ j ∈ T \\ S, p j else 0","l":235,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_ctrlInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The units in a finite set S are all left untreated together with probability ∏_{j ∈ S} (1 − p j): the expectation of the product of their control indicators is the product of their control probabilities.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) = ∏ j ∈ S, (1 - p j)","l":91,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_ctrlInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Two overlapping sets of units are simultaneously left untreated with probability ∏_{j ∈ S ∪ T} (1 − p j): the shared units are counted once, not twice.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => (∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) * ∏ j ∈ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) = ∏ j ∈ S ∪ T, (1 - p j)","l":152,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Independence across units turns the expectation of a product into a product of expectations: if every unit contributes a factor that depends only on its own coin, the design expectation of the whole product is the product over units of p i · g i (treated) + (1 − p i) · g i (control).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (g : U → Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ i, g i (z i)) = ∏ i, (p i * g i true + (1 - p i) * g i false)","l":51,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_prod_finset","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The same factorization when only the units in a finite set S contribute a factor: the units outside S are simply averaged out and leave no trace.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U) (g : U → Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ k ∈ S, g k (z k)) = ∏ k ∈ S, (p k * g k true + (1 - p k) * g k false)","l":62,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_sq_of_expansion","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Parseval's identity for the Bernoulli design. Once a statistic is written in the centered-monomial basis, its second moment is the sum over subsets of the squared coefficient times the product of the per-unit assignment variances on that subset — the basis is orthogonal, so no cross terms survive.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (F : (U → Bool) → ℝ) (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => F z ^ 2) = ∑ S ∈ Finset.univ.powerset, a S ^ 2 * ∏ j ∈ S, p j * (1 - p j)","l":328,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_sum_sq_le_blockDegree_mul_sum_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.LocalDependenceVariance","a":"Experimentation/DesignBased","t":1,"d":"Local-dependence variance bound. Under a Bernoulli design in which every unit's treatment probability lies strictly between 0 and 1, suppose each unit i contributes a term F i — attached to a dependence block N i — such that no unit's treatment is read by more than d of these blocks, every term has mean zero under the Bernoulli design, and each term F i depends only on the treatments of its own block N i. Then the second moment of the total ∑ᵢ F i is at most d times the total of the individual second moments, d · ∑ᵢ E[(F i)²]. The bound is scale free: it needs no uniform bound on the terms, no explicit decorrelation assumption, and no dependence on the population size.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 < p i) (hp1 : ∀ (i : U), p i < 1) (N : U → Finset U) (d : ℕ), Causalean.Experimentation.DesignBased.BlockDegreeLE N d → ∀ (F : U → (U → Bool) → ℝ), (∀ (i : U), (Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E (F i) = 0) → (∀ (i : U), Causalean.Experimentation.DesignBased.DependsOnBlock (N i) (F i)) → ((Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E fun z => (∑ i, F i z) ^ 2) ≤ ↑d * ∑ i, (Causalean.Experimentation.DesignBased.bernoulliDesign p (fun k => LT.lt.le (hp0 k)) fun k => LT.lt.le (hp1 k)).E fun z => F i z ^ 2","l":53,"f":"Causalean/Experimentation/DesignBased/LocalDependenceVariance.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"A quantity depending only on unit i's treatment coin has expectation equal to its value under treatment weighted by p i plus its value under control weighted by 1 - p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U) (g : Bool → ℝ), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => g (z i)) = p i * g true + (1 - p i) * g false","l":71,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability. Under the Bernoulli randomization design with per-unit treatment probabilities p i lying between 0 and 1, for two distinct units i ≠ j, the joint expectation of their treatment indicators factors as p i · p j — the units are independently assigned.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) {i j : U}, i ≠ j → ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.treatInd i z * Causalean.Experimentation.DesignBased.treatInd j z) = p i * p j","l":89,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Higher-order inclusion probability. The units in a finite set S are all treated together with probability ∏_{j ∈ S} p j; equivalently, the expectation of the product of their treatment indicators is the product of their treatment probabilities.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => ∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S, p j","l":80,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_E_treatInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Two overlapping sets of units are simultaneously treated with probability ∏_{j ∈ S ∪ T} p j: the shared units are counted once, not twice.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (S T : Finset U), ((Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E fun z => (∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S ∪ T, p j","l":142,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.bernoulliDesign_Var_treatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of unit i has variance p i (1 − p i) under the Bernoulli design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.DesignBased.treatInd i) = p i * (1 - p i)","l":81,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.bestRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a family of randomization designs that is finite and nonempty, a risk criterion indexed by states and randomization designs, and a state of nature, the best achievable risk is the least risk at that state among designs in the family, provided risk values are linearly ordered.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → {α : Type u_3} → [LinearOrder α] → (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) → Set.Finite 𝒟 → Set.Nonempty 𝒟 → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Y → α","l":74,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.bestRisk_le","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The best achievable risk is attained, hence no larger than the risk of any family member.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} {α : Type u_3} [inst_1 : LinearOrder α] (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (hfin : Set.Finite 𝒟) (hne : Set.Nonempty 𝒟) (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (y : Y) {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → Causalean.Experimentation.DesignBased.bestRisk 𝒟 hfin hne R y ≤ R y D","l":85,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.card_design_mem_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of subsets of a finite population with k + 2 units that contain two distinct specified units equals the number of ways to choose k units from the remaining population.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (i j : U), i ≠ j → ∀ (k : ℕ), Fintype.card { S // i ∈ ↑S ∧ j ∈ ↑S } = (Fintype.card U - 2).choose k","l":215,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_design_mem_succ","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of subsets of a finite population with k + 1 units that contain one specified unit equals the number of ways to choose k units from the remaining population.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (i : U) (k : ℕ), Fintype.card { S // i ∈ ↑S } = (Fintype.card U - 1).choose k","l":116,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_powersetCard_filter_mem_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Among the fixed-size subsets of a finite set that contain two distinct specified elements, the count equals the number of subsets of the remaining set after those two elements are removed. This is the finite-population counting identity behind pairwise inclusion probabilities.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (s : Finset α) (i j : α), i ∈ s → j ∈ s → i ≠ j → ∀ (k : ℕ), {t ∈ Finset.powersetCard (k + 2) s | i ∈ t ∧ j ∈ t}.card = (Finset.powersetCard k ((s.erase i).erase j)).card","l":138,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.card_powersetCard_filter_mem_succ","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Among subsets containing a specified unit and of size k + 1, there are exactly as many as there are size-k subsets after that unit is removed.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (s : Finset α), ∀ i ∈ s, ∀ (k : ℕ), {t ∈ Finset.powersetCard (k + 1) s | i ∈ t}.card = (Finset.powersetCard k (s.erase i)).card","l":74,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a population of units, unit-specific treatment probabilities, and a finite set of units, the centered monomial maps each assignment to the product, over the selected units, of that unit's treatment indicator minus its treatment probability. The empty selected set gives the constant one.","s":"{U : Type u_1} → (U → ℝ) → Finset U → (U → Bool) → ℝ","l":165,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_coef_empty_eq_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"A statistic with design mean zero has no constant term: the coefficient of the empty subset in any centered-monomial expansion of it vanishes.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (F : (U → Bool) → ℝ) (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → (Causalean.Experimentation.DesignBased.bernoulliDesign p hp0 hp1).E F = 0 → a ∅ = 0","l":357,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_coef_eq_zero_of_not_subset","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Locality of the Fourier support. If a statistic depends only on the units of a block N, every centered-monomial coefficient attached to a subset that is not contained in N is zero, so its expansion is supported on the subsets of N. (Only treatment probabilities on S must be strictly between zero and one, so those selected coordinates are nondegenerate.)","s":"∀ {U : Type u_1} [inst : Fintype U] (p : U → ℝ), (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → ∀ (N : Finset U) (F : (U → Bool) → ℝ), Causalean.Experimentation.DesignBased.DependsOnBlock N F → ∀ (a : Finset U → ℝ), (∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z) → ∀ (S : Finset U), ¬S ⊆ N → (∀ i ∈ S, 0 < p i) → (∀ i ∈ S, p i < 1) → a S = 0","l":382,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.centeredMonomial_empty","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"The centered monomial of the empty set of units is the constant function 1.","s":"∀ {U : Type u_1} (p : U → ℝ), Causalean.Experimentation.DesignBased.centeredMonomial p ∅ = fun x => 1","l":173,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite population of clusters, cluster-specific treatment probabilities that are each between zero and one, the cluster-randomization design independently assigns each cluster to treatment with its specified probability.","s":"{C : Type u_2} → [inst : Fintype C] → [inst_1 : DecidableEq C] → (p : C → ℝ) → (∀ (c : C), 0 ≤ p c) → (∀ (c : C), p c ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (C → Bool)","l":35,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. A unit i is treated with probability p (clus i), the treatment rate of its own cluster.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) (i : U), (Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E (Causalean.Experimentation.DesignBased.unitTreatInd clus i) = p (clus i)","l":50,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd_pair_diff","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"Different-cluster joint treatment. Two units in distinct clusters are jointly treated with the product of their cluster rates — distinct clusters are randomized independently.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) {i j : U}, clus i ≠ clus j → ((Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.unitTreatInd clus i z * Causalean.Experimentation.DesignBased.unitTreatInd clus j z) = p (clus i) * p (clus j)","l":78,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.clusterDesign_E_unitTreatInd_pair_same","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"Same-cluster joint treatment. For a cluster-randomization design in which each cluster's treatment probability p c lies in [0,1], if two units i and j belong to the same cluster, then they are jointly treated with probability exactly their shared cluster's rate p (clus i) — because being in the same cluster makes their treatment indicators identical.","s":"∀ {U : Type u_1} {C : Type u_2} [inst : Fintype C] [inst_1 : DecidableEq C] (p : C → ℝ) (hp0 : ∀ (c : C), 0 ≤ p c) (hp1 : ∀ (c : C), p c ≤ 1) (clus : U → C) {i j : U}, clus i = clus j → ((Causalean.Experimentation.DesignBased.clusterDesign p hp0 hp1).E fun z => Causalean.Experimentation.DesignBased.unitTreatInd clus i z * Causalean.Experimentation.DesignBased.unitTreatInd clus j z) = p (clus i)","l":60,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Coin","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a probability between zero and one, the single-unit coin randomization design assigns treatment with probability $p$ and control with probability $1-p$.","s":"(p : ℝ) → 0 ≤ p → p ≤ 1 → Causalean.Experimentation.DesignBased.FiniteDesign Bool","l":23,"f":"Causalean/Experimentation/DesignBased/Designs/Coin.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_E","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Coin","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a probability p lying in [0,1], the expectation of a real-valued function g of a single coin flip that lands heads with probability p equals the probability-weighted two-point sum p · g(true) + (1 − p) · g(false).","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (g : Bool → ℝ), (Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).E g = p * g true + (1 - p) * g false","l":38,"f":"Causalean/Experimentation/DesignBased/Designs/Coin.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_E_treat","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of a single coin, 1(true), has expectation p.","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1), ((Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).E fun b => if b = true then 1 else 0) = p","l":40,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.coinDesign_Var_treat","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":2,"d":"The treatment indicator of a single coin has variance p(1 − p).","s":"∀ (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1), ((Causalean.Experimentation.DesignBased.coinDesign p hp0 hp1).Var fun b => if b = true then 1 else 0) = p * (1 - p)","l":45,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite population of units, a nonnegative number of treated units that does not exceed the population size, the complete-randomization design assigns equal probability to every treatment allocation that treats exactly that many units.","s":"{V : Type u_2} → [inst : Fintype V] → (n₁ : ℕ) → n₁ ≤ Fintype.card V → Causalean.Experimentation.DesignBased.FiniteDesign { S // S.card = n₁ }","l":54,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_card","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"The number of possible treated sets in complete randomization is the binomial coefficient (Fintype.card U).choose n₁.","s":"∀ {V : Type u_2} [inst : Fintype V] (n₁ : ℕ), Fintype.card { S // S.card = n₁ } = (Fintype.card V).choose n₁","l":42,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_incl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. Under complete randomization, a unit is treated with probability n₁ / N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U) (i : U), ((Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).Pr fun S => i ∈ ↑S) = ↑n₁ / ↑(Fintype.card U)","l":244,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.completeRandomization_incl_pair","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.CompleteRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability. Under complete randomization treating exactly n₁ of the N units, where the treated count n₁ does not exceed the population size N, two distinct units are jointly treated with probability n₁(n₁−1) / (N(N−1)).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ) (hn : n₁ ≤ Fintype.card U) {i j : U}, i ≠ j → ((Causalean.Experimentation.DesignBased.completeRandomization n₁ hn).Pr fun S => i ∈ ↑S ∧ j ∈ ↑S) = ↑n₁ * (↑n₁ - 1) / (↑(Fintype.card U) * (↑(Fintype.card U) - 1))","l":284,"f":"Causalean/Experimentation/DesignBased/Designs/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.compound","k":"def","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"For a finite space of first-stage outcomes, a finite index set for second-stage coordinates, a finite outcome space for each coordinate, a probability design for the first stage, and a rule assigning, after each first-stage outcome, a design to each coordinate, the compound two-stage design first draws the first-stage outcome and then independently draws every coordinate from its conditional design.","s":"{Ω₁ : Type u_1} → [inst : Fintype Ω₁] → {ι : Type u_2} → [inst_1 : Fintype ι] → [inst_2 : DecidableEq ι] → {α : ι → Type u_3} → [inst_3 : (i : ι) → Fintype (α i)] → Causalean.Experimentation.DesignBased.FiniteDesign Ω₁ → (Ω₁ → (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω₁ × ((i : ι) → α i))","l":64,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.compoundCore","k":"def","m":"Causalean.Experimentation.DesignBased.TwoStage","a":"Experimentation/DesignBased","t":1,"d":"For a finite space of first-stage outcomes, a finite space of second-stage outcomes, a probability design for the first stage, and a rule assigning a probability design for the second stage after each first-stage outcome, the compound two-stage design assigns a pair of outcomes the product of its first-stage probability and its conditional second-stage probability.","s":"{Ω₁ : Type u_1} → [inst : Fintype Ω₁] → {Ω₂ : Type u_4} → [inst_1 : Fintype Ω₂] → Causalean.Experimentation.DesignBased.FiniteDesign Ω₁ → (Ω₁ → Causalean.Experimentation.DesignBased.FiniteDesign Ω₂) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω₁ × Ω₂)","l":47,"f":"Causalean/Experimentation/DesignBased/TwoStage.lean"},{"n":"Causalean.Experimentation.DesignBased.condExp_ae_eq_zero_of_prefixCondExp_eq_zero","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs, reveal permutation, prefix length, statistic, and zero explicit prefix conditional expectation imply the measure-theoretic conditional expectation is almost everywhere zero.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (k : Fin (N + 1)) (f : ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.prefixCondExp D pi k f = 0 → (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure[f | ↑(Causalean.Experimentation.DesignBased.revealFiltration pi) k] =ᵐ[(Causalean.Experimentation.DesignBased.prodDesign D).toMeasure] 0","l":395,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.conservative_wald_liminf_of_studentized_cdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.WaldCoverage","a":"Experimentation/DesignBased","t":1,"d":"Conservative Wald-interval liminf coverage. Consider a sequence of finite designs with an estimator est n of a target θ n. Suppose the normalization size m n is eventually positive, the true variance scale v n is eventually positive, and a deterministic conservative variance v̂ n eventually dominates v n. Suppose further that the studentized statistic √(m n)·(est n − θ n)/√(v n) has design-probability CDF converging to the standard-normal CDF at z and at −z, where z is nonnegative and satisfies Φ(z) = 1 − α/2. Then the two-sided interval |θ n − est n| ≤ z·√(v̂ n / m n) has asymptotic (liminf) coverage at least 1 − α.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (est : (n : ℕ) → Ω n → ℝ) (θ v vhat m : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, 0 < m n) → (∀ᶠ (n : ℕ) in Filter.atTop, 0 < v n) → (∀ᶠ (n : ℕ) in Filter.atTop, v n ≤ vhat n) → ∀ (α z : ℝ), Filter.Tendsto (fun n => (D n).Pr fun zz => √(m n) * (est n zz - θ n) / √(v n) ≤ z) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf z)) → Filter.Tendsto (fun n => (D n).Pr fun zz => √(m n) * (est n zz - θ n) / √(v n) ≤ -z) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf (-z))) → 0 ≤ z → Causalean.Experimentation.DesignBased.stdNormalCdf z = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (D n).Pr fun zz => |θ n - est n zz| ≤ z * √(vhat n / m n)) Filter.atTop","l":32,"f":"Causalean/Experimentation/DesignBased/WaldCoverage.lean"},{"n":"Causalean.Experimentation.DesignBased.continuous_stdNormalCdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"Continuity of the standard normal CDF. The standard normal cumulative distribution function Φ is continuous.","s":"Continuous Causalean.Experimentation.DesignBased.stdNormalCdf","l":68,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.controlMean","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"For a finite population of units, a number of treated units, the potential outcome under control for each unit, and a treated set containing exactly that many units, the control-arm mean is the average control potential outcome among all units outside the treated set.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":56,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.ctrlInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Control indicators take only the values zero and one, so two overlapping products of them collapse into a single product over the union of the two sets of units.","s":"∀ {U : Type u_1} [inst : DecidableEq U] (S T : Finset U) (z : U → Bool), (∏ j ∈ S, (1 - Causalean.Experimentation.DesignBased.treatInd j z)) * ∏ j ∈ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z) = ∏ j ∈ S ∪ T, (1 - Causalean.Experimentation.DesignBased.treatInd j z)","l":123,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.dependency_studentized_cdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.DependencyCLT","a":"Experimentation/DesignBased","t":1,"d":"Design-based dependency-graph CLT (studentized CDF form). Consider a sequence of finite designs D n together with a triangular array of unit contributions X n i and, for each n, a dependency-graph structure Dep n on the units. Suppose every neighbourhood in the dependency graph has size at most Dmax, the contributions are uniformly bounded by a nonnegative constant M, each contribution has design mean 0, the design second moment v n of the aggregate depSum(X n) is recorded exactly, a positive constant c eventually lower-bounds v n by c times the number of units, and the number of units diverges to infinity. Then, at every point s, the studentized aggregate depSum(X n)/√(v n) has, under the design D n, a limiting CDF equal to the standard normal CDF at s.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] [∀ (n : ℕ), MeasurableSingletonClass (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) {ι : ℕ → Type u_2} [inst_3 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (D n).toMeasure) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), (D n).E (X n i) = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ((D n).E fun ω => Causalean.SteinMethod.depSum (X n) ω ^ 2) = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (D n).Pr fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n) ≤ s) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf s))","l":43,"f":"Causalean/Experimentation/DesignBased/DependencyCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.dependency_wald_coverage","k":"theorem","m":"Causalean.Experimentation.DesignBased.WaldPipeline","a":"Experimentation/DesignBased","t":1,"d":"Design-based Wald-interval coverage from dependency-graph primitives. Consider a sequence of finite designs together with per-unit contributions X n i governed by a dependency graph Dep n on those contributions whose neighbourhoods have cardinality at most Dmax, where the contributions are uniformly bounded by a nonnegative constant M and have design mean zero. Suppose the standardizing quantity v n equals the design second moment of the contributions' sum depSum(X n), is bounded below by a positive constant c times the number of units eventually, and the number of units diverges. Suppose further that the standardized sum eventually equals √(m n)·(est n − θ n) for an estimator est n of a target θ n, the normalization m n is eventually positive, and a deterministic conservative variance v̂ n eventually dominates v n, with z the nonnegative quantile satisfying Φ(z) = 1 − α/2. Then the two-sided Wald interval |θ n − est n| ≤ z·√(v̂ n / m n) has asymptotic (liminf) coverage at least 1 − α.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] [∀ (n : ℕ), MeasurableSingletonClass (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) {ι : ℕ → Type u_2} [inst_3 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (D n).toMeasure) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), (D n).E (X n i) = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ((D n).E fun ω => Causalean.SteinMethod.depSum (X n) ω ^ 2) = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (est : (n : ℕ) → Ω n → ℝ) (θ m : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, ∀ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω = √(m n) * (est n ω - θ n)) → (∀ᶠ (n : ℕ) in Filter.atTop, 0 < m n) → ∀ (vhat : ℕ → ℝ), (∀ᶠ (n : ℕ) in Filter.atTop, v n ≤ vhat n) → ∀ (α z : ℝ), 0 ≤ z → Causalean.Experimentation.DesignBased.stdNormalCdf z = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (D n).Pr fun ω => |θ n - est n ω| ≤ z * √(vhat n / m n)) Filter.atTop","l":39,"f":"Causalean/Experimentation/DesignBased/WaldPipeline.lean"},{"n":"Causalean.Experimentation.DesignBased.diagDepGraph","k":"def","m":"Causalean.Experimentation.DesignBased.IndepSummandsCLT","a":"Experimentation/DesignBased","t":1,"d":"At each stage suppose the coordinate index set and every coordinate assignment space are finite, coordinate equality is decidable, and every singleton in every coordinate assignment space is measurable. For a family of coordinate-specific randomization designs, indexed by stage and coordinate, a family of real-valued coordinate summands, and a stage, the diagonal dependency graph is the dependency graph of the coordinate summands under the corresponding product design, in which two coordinates are adjacent exactly when they are the same coordinate.","s":"{ι : ℕ → Type u_1} → [inst : (n : ℕ) → Fintype (ι n)] → [inst_1 : (n : ℕ) → DecidableEq (ι n)] → {α : (n : ℕ) → ι n → Type u_2} → [inst_2 : (n : ℕ) → (i : ι n) → Fintype (α n i)] → [inst_3 : (n : ℕ) → (i : ι n) → MeasurableSpace (α n i)] → [∀ (n : ℕ) (i : ι n), MeasurableSingletonClass (α n i)] → (D : (n : ℕ) → (i : ι n) → Causalean.Experimentation.DesignBased.FiniteDesign (α n i)) → (g : (n : ℕ) → (i : ι n) → α n i → ℝ) → (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i w => g n i (w i)) (Causalean.Experimentation.DesignBased.prodDesign (D n)).toMeasure","l":51,"f":"Causalean/Experimentation/DesignBased/IndepSummandsCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.diffInMeans","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"For a finite population of units, a number of treated units, potential outcomes under treatment and under control, and a treated set containing exactly that many units, the difference-in-means estimator is the treated-arm mean minus the control-arm mean.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_centeredMonomial_expansion","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Completeness of the centered-monomial basis. Every real-valued statistic of a Bernoulli assignment can be written as a linear combination of the centered monomials indexed by the subsets of the population — the design-based Fourier (Walsh) expansion of the statistic.","s":"∀ {U : Type u_1} [inst : Fintype U] (p : U → ℝ) (F : (U → Bool) → ℝ), ∃ a, ∀ (z : U → Bool), F z = ∑ S ∈ Finset.univ.powerset, a S * Causalean.Experimentation.DesignBased.centeredMonomial p S z","l":279,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_isMinimaxOn","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Existence of a minimax design. For a design family 𝒟 and a risk criterion R indexed by states of nature, if the state set s is nonempty, the design family 𝒟 is finite, and 𝒟 is nonempty, then there exists a design in 𝒟 that is minimax — it minimizes the worst-case risk over s among all members of 𝒟. Immediate from exists_isOptimalOn applied to the worst-case-risk criterion.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (s : Finset Y) (hs : s.Nonempty) {α : Type u_3} [inst_1 : LinearOrder α] (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α), Set.Finite 𝒟 → Set.Nonempty 𝒟 → ∃ D₀, Causalean.Experimentation.DesignBased.IsMinimaxOn 𝒟 s hs R D₀","l":63,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.exists_isOptimalOn","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Existence of an optimal design. For a design family 𝒟 and a risk criterion R valued in a linear order, if 𝒟 is finite and 𝒟 is nonempty, then there exists a design in 𝒟 that is optimal — it dominates every other member of the family under R.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {α : Type u_2} [inst_1 : LinearOrder α] (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (R : Causalean.Experimentation.DesignBased.FiniteDesign Ω → α), Set.Finite 𝒟 → Set.Nonempty 𝒟 → ∃ D₀, Causalean.Experimentation.DesignBased.IsOptimalOn 𝒟 R D₀","l":76,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.expo","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"For an assignment-to-exposure rule, a map assigning each unit its traits, a unit, and an assignment, the unit's exposure is the exposure assigned by that rule to the assignment and the unit's traits.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → (Ω → Θ → Δ) → (ι → Θ) → ι → Ω → Δ","l":45,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"For an assignment-to-exposure rule, a map assigning each unit its traits, a unit, and an exposure condition from a set whose elements can be compared for equality, the exposure indicator maps each assignment to one when that unit receives that condition and to zero otherwise.","s":"{Ω : Type u_1} → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → (Ω → Θ → Δ) → (ι → Θ) → ι → Δ → Ω → ℝ","l":52,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_Yobs","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"On the event expo i = d, the observed outcome agrees with the potential outcome y i d; hence 1(expo i = d)·Yobs i = 1(expo i = d)·y i d.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.Yobs y f θ i z = Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d","l":68,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_Yobs_sq","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"Squared on-event substitution: 1(expo i = d)·(Yobs i)² = 1(expo i = d)·(y i d)².","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.Yobs y f θ i z ^ 2 = Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d ^ 2","l":76,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd_mul_self_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":2,"d":"A unit cannot be in two distinct exposures at once: the indicators are pointwise disjoint.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) {d d' : Δ}, d ≠ d' → ∀ (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i d z * Causalean.Experimentation.DesignBased.expoInd f θ i d' z = 0","l":115,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.expoInd₂_mul_Yobs","k":"theorem","m":"Causalean.Experimentation.DesignBased.PotentialOutcome","a":"Experimentation/DesignBased","t":2,"d":"On the event expo i = di, multiplying the observed outcome for unit i by any real quantity leaves it valid to replace that outcome by potential outcome y i di.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst : DecidableEq Δ] (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) (di : Δ) (q : Ω → ℝ) (z : Ω), Causalean.Experimentation.DesignBased.expoInd f θ i di z * q z * Causalean.Experimentation.DesignBased.Yobs y f θ i z = Causalean.Experimentation.DesignBased.expoInd f θ i di z * q z * y i di","l":83,"f":"Causalean/Experimentation/DesignBased/PotentialOutcome.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteDesign_cdf_converging_together","k":"theorem","m":"Causalean.Experimentation.DesignBased.Slutsky","a":"Experimentation/DesignBased","t":1,"d":"Finite-design CDF converging-together. Along a sequence of finite designs on two sequences of real-valued statistics Sₙ and Tₙ, suppose Sₙ is asymptotically indistinguishable from Tₙ in probability: for every η > 0, Pr(η ≤ |Sₙ − Tₙ|) → 0, the CDFs of Tₙ converge pointwise to a limit function Φ, and Φ is continuous. Then the CDFs of Sₙ converge pointwise to the same Φ.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (S T : (n : ℕ) → Ω n → ℝ) (Φ : ℝ → ℝ), (∀ (η : ℝ), 0 < η → Filter.Tendsto (fun n => (D n).Pr fun z => η ≤ |S n z - T n z|) Filter.atTop (nhds 0)) → (∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => T n z ≤ x) Filter.atTop (nhds (Φ x))) → Continuous Φ → ∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => S n z ≤ x) Filter.atTop (nhds (Φ x))","l":29,"f":"Causalean/Experimentation/DesignBased/Slutsky.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteDesign_heydeBrown_fourthMoment","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"Given coordinate designs, a reveal permutation, an increment family, and a bound constant, if the increments are prefix martingale differences, their total second moment is one, and the supplied Heyde--Brown premise holds, then their sum satisfies the finite-design one-fifth-power Kolmogorov bound.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (X : Fin N → ((i : Fin N) → alpha i) → ℝ) (C : ℝ), Causalean.Experimentation.DesignBased.IsPrefixMartingaleDifference D pi X → (∑ s, (Causalean.Experimentation.DesignBased.prodDesign D).E fun w => X s w ^ 2) = 1 → Causalean.Experimentation.DesignBased.HeydeBrownFourthMomentPremise (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure (Causalean.Experimentation.DesignBased.revealFiltration pi) X C → (Causalean.Experimentation.DesignBased.finiteKolmogorovExpr (Causalean.Experimentation.DesignBased.prodDesign D) fun w => ∑ s, X s w) ≤ C * (Causalean.Experimentation.DesignBased.finiteFourthMomentError D pi X).rpow (1 / 5)","l":452,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteDesign_symmetricBand_tendsto","k":"theorem","m":"Causalean.Experimentation.DesignBased.WaldCoverage","a":"Experimentation/DesignBased","t":2,"d":"For a sequence of finite-design laws and real-valued statistics, if their CDFs converge pointwise to a limit CDF, the limit CDF is continuous, and the symmetric-band radius is nonnegative, then the closed symmetric-band probabilities converge to the difference of the limiting CDF at the two endpoints.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (Ω n)] (D : (n : ℕ) → Causalean.Experimentation.DesignBased.FiniteDesign (Ω n)) (T : (n : ℕ) → Ω n → ℝ) (F : ℝ → ℝ), (∀ (x : ℝ), Filter.Tendsto (fun n => (D n).Pr fun z => T n z ≤ x) Filter.atTop (nhds (F x))) → Continuous F → ∀ (c : ℝ), 0 ≤ c → Filter.Tendsto (fun n => (D n).Pr fun z => -c ≤ T n z ∧ T n z ≤ c) Filter.atTop (nhds (F c - F (-c)))","l":144,"f":"Causalean/Experimentation/DesignBased/WaldCoverage.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteFourthMomentError","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A family of coordinate designs, reveal permutation, and increment family determine the finite fourth-moment error: the sum of fourth moments plus the second moment of predictable variation minus one by the displayed sum of two terms.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [inst : (i : Fin N) → Fintype (alpha i)] → ((i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) → Equiv.Perm (Fin N) → (Fin N → ((i : Fin N) → alpha i) → ℝ) → ℝ","l":114,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteFourthMomentError_eq_measureFourthMomentError","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs, reveal permutation, and increment family give an exact identification between finite-design and measure-theoretic fourth-moment errors, including the predictable-variation term.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (X : Fin N → ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.finiteFourthMomentError D pi X = Causalean.Experimentation.DesignBased.measureFourthMomentError (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure (Causalean.Experimentation.DesignBased.revealFiltration pi) X","l":432,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteKolmogorovExpr","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A finite randomization design and real statistic determine its finite-design Kolmogorov expression, the largest absolute gap between its CDF and the standard-normal CDF by taking the supremum over thresholds.","s":"{Omega : Type u_2} → [inst : Fintype Omega] → Causalean.Experimentation.DesignBased.FiniteDesign Omega → (Omega → ℝ) → ℝ","l":131,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finiteKolmogorovExpr_eq_measureKolmogorovExpr","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs and real statistic give an exact identification between the finite-design and measure-theoretic Kolmogorov expressions under the induced product-design measure.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (Y : ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.finiteKolmogorovExpr (Causalean.Experimentation.DesignBased.prodDesign D) Y = Causalean.Experimentation.DesignBased.measureKolmogorovExpr (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure Y","l":418,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finitePredictableVariation","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A family of coordinate designs, reveal permutation, and increment family determine the finite-design predictable variation, the sum of conditional second moments just before each reveal step by the displayed finite sum.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [inst : (i : Fin N) → Fintype (alpha i)] → ((i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) → Equiv.Perm (Fin N) → (Fin N → ((i : Fin N) → alpha i) → ℝ) → ((i : Fin N) → alpha i) → ℝ","l":98,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.finitePredictableVariation_ae_eq_measurePredictableVariation","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs, reveal permutation, and increment family give an almost-everywhere equality between finite-design and measure-theoretic predictable variations under the product-design measure.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (X : Fin N → ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.finitePredictableVariation D pi X =ᵐ[(Causalean.Experimentation.DesignBased.prodDesign D).toMeasure] Causalean.Experimentation.DesignBased.measurePredictableVariation (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure (Causalean.Experimentation.DesignBased.revealFiltration pi) X","l":404,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.htEffect","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, two exposure conditions, and a realized assignment define the Horvitz–Thompson estimator of the average causal effect as the estimated mean under the first condition minus the estimated mean under the second condition.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":80,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htMean","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, an exposure condition, and a realized assignment define the Horvitz–Thompson estimator of the mean potential outcome as the totalized Horvitz–Thompson estimator divided by the number of units.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":70,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htTotal","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, an exposure condition, and a realized assignment define the totalized Horvitz–Thompson estimator of the total potential outcome as the sum over units of each observed outcome multiplied by its exposure indicator and divided by its generalized exposure probability.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":45,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.htTotal_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite design D, potential outcomes y, exposure map f, assignment θ, exposure level d, and realized assignment z, the Horvitz–Thompson total equals the same sum with the observed outcome replaced termwise by the potential outcome y i d, because on each unit's exposure-indicator term the observed outcome coincides with y i d.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ) (z : Ω), Causalean.Experimentation.DesignBased.htTotal D y f θ d z = ∑ i, Causalean.Experimentation.DesignBased.expoInd f θ i d z * y i d / Causalean.Experimentation.DesignBased.prop D f θ i d","l":90,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.iIndepFun_prodDesign_eval","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"Coordinate evaluations are mutually independent under the product-design measure: viewing fun (i) (w) => w i as the family of coordinate projections, this family is iIndepFun under (prodDesign D).toMeasure. (The push-forward of iIndepFun_pi along prodDesign_toMeasure_eq_pi.)","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)), ProbabilityTheory.iIndepFun (fun i w => w i) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":86,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_apply_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"Functions of disjoint coordinate blocks are independent. Applying separate measurable function families to the coordinates in two disjoint blocks preserves their independence, even when the two blocks have different coordinatewise output spaces.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {β : ι → Type u_3} {γ : ι → Type u_4} [inst_5 : (i : ι) → MeasurableSpace (β i)] [inst_6 : (i : ι) → MeasurableSpace (γ i)] {g : (i : ι) → α i → β i} {h : (i : ι) → α i → γ i}, (∀ (i : ι), Measurable (g i)) → (∀ (i : ι), Measurable (h i)) → ∀ {A B : Finset ι}, Disjoint A B → ProbabilityTheory.IndepFun (fun w k => g (↑k) (w ↑k)) (fun w k => h (↑k) (w ↑k)) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":113,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":1,"d":"Disjoint-block independence. For a family of independent coordinate designs D, suppose the finite index sets A and B are disjoint. Then the tuple of coordinates indexed by A is probabilistically independent of the tuple of coordinates indexed by B, under the measure induced by the product design. This is the form a diagonal dependency graph G a b := a = b consumes.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {A B : Finset ι}, Disjoint A B → ProbabilityTheory.IndepFun (fun w k => w ↑k) (fun w k => w ↑k) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":101,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_eval","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"For i ≠ j, the two coordinate evaluations w ↦ w i and w ↦ w j are independent under the product-design measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {i j : ι}, i ≠ j → ProbabilityTheory.IndepFun (fun w => w i) (fun w => w j) (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":95,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.indepFun_prodDesign_of_depends_on_disjoint_blocks","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":2,"d":"General functions of disjoint coordinate blocks are independent. The strict generalization of indepFun_prodDesign_apply_blocks from coordinatewise-separable maps (g k (w k))_k to *arbitrary* measurable functions that each depend only on a block: if F is unchanged by any assignment that agrees on S, and G is unchanged by any assignment that agrees on the disjoint set T, then F and G are independent under the product-design measure. This is the hypothesis a dependency-graph construction actually consumes, where each summand depends jointly (not coordinatewise) on a neighbourhood block.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] {β : Type u_3} {γ : Type u_4} [inst_5 : MeasurableSpace β] [inst_6 : MeasurableSpace γ] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) {S T : Finset ι}, Disjoint S T → ∀ (F : ((i : ι) → α i) → β) (G : ((i : ι) → α i) → γ), (∀ (w w' : (i : ι) → α i), (∀ i ∈ S, w i = w' i) → F w = F w') → (∀ (w w' : (i : ι) → α i), (∀ i ∈ T, w i = w' i) → G w = G w') → ProbabilityTheory.IndepFun F G (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure","l":128,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.le_worstRisk","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The worst-case risk dominates the risk at every state in the family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} {α : Type u_3} [inst_1 : SemilatticeSup α] (s : Finset Y) (hs : s.Nonempty) (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) {y : Y}, y ∈ s → R y D ≤ Causalean.Experimentation.DesignBased.worstRisk s hs R D","l":46,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.measureFourthMomentError","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A measure (in intended use a probability measure; none is required), filtration, and increment family determine the measure-theoretic fourth-moment error: the sum of fourth moments plus the second moment of predictable variation minus one by the displayed sum of two terms.","s":"{N : ℕ} → {Omega : Type u_2} → [m : MeasurableSpace Omega] → MeasureTheory.Measure Omega → MeasureTheory.Filtration (Fin (N + 1)) m → (Fin N → Omega → ℝ) → ℝ","l":122,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.measureKolmogorovExpr","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A measure and real random variable determine its measure-theoretic Kolmogorov expression, the largest absolute gap between its CDF and the standard-normal CDF by taking the supremum over thresholds. No probability-measure assumption is made: the CDF term is the real-valued measure of the lower set (zero where that measure is infinite), and an unbounded family of gaps has supremum zero by convention; for a probability measure the gaps are bounded and this is the usual Kolmogorov distance.","s":"{Omega : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → ℝ) → ℝ","l":138,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.measurePredictableVariation","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A measure (in intended use a probability measure; none is required), filtration, and increment family determine the measure-theoretic predictable variation, the sum of conditional second moments at the preceding filtration times by the displayed finite sum.","s":"{N : ℕ} → {Omega : Type u_2} → [m : MeasurableSpace Omega] → MeasureTheory.Measure Omega → MeasureTheory.Filtration (Fin (N + 1)) m → (Fin N → Omega → ℝ) → Omega → ℝ","l":106,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.monotone_stdNormalCdf","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative distribution function is monotone in its threshold.","s":"Monotone Causalean.Experimentation.DesignBased.stdNormalCdf","l":58,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a finite assignment space, an estimator specified separately for each randomization design, and a real-valued target, the mean-squared-error risk function assigns to each design the mean squared error of that design's estimator about the target, evaluated under that design.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → (Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ) → ℝ → Causalean.Experimentation.DesignBased.FiniteDesign Ω → ℝ","l":95,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk_eq_var_of_unbiased","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"For an estimator that is unbiased under every design in the family, the mean-squared-error risk coincides with the variance there.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {est : Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ} {μ : ℝ} {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D.Unbiased (est D) μ → Causalean.Experimentation.DesignBased.mseRisk est μ D = D.Var (est D)","l":107,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.mseRisk_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Mean-squared-error risk is nonnegative.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (est : Causalean.Experimentation.DesignBased.FiniteDesign Ω → Ω → ℝ) (μ : ℝ) (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω), 0 ≤ Causalean.Experimentation.DesignBased.mseRisk est μ D","l":102,"f":"Causalean/Experimentation/DesignBased/Optimality.lean"},{"n":"Causalean.Experimentation.DesignBased.muTrue","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite population, a schedule of potential outcomes for every unit and exposure condition and an exposure condition define the finite-population mean potential outcome as the sum of the units' potential outcomes under that condition divided by the number of units.","s":"{ι : Type u_2} → {Δ : Type u_4} → [Fintype ι] → (ι → Δ → ℝ) → Δ → ℝ","l":56,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanFraction","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a first real-valued input and a second real-valued input, the Neyman allocation fraction is $\\sqrt{A}/(\\sqrt{A}+\\sqrt{B})$. When the inputs are nonnegative arm variances, this is the ratio of the treated arm's standard deviation to the sum of the two arms' standard deviations.","s":"ℝ → ℝ → ℝ","l":37,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanFraction_mem_Ioo","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The Neyman fraction lies in (0,1).","s":"∀ {A B : ℝ}, 0 < A → 0 < B → 0 < Causalean.Experimentation.DesignBased.neymanFraction A B ∧ Causalean.Experimentation.DesignBased.neymanFraction A B < 1","l":68,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neymanOptimalValue","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a first real-valued input and a second real-valued input, the Neyman optimal-value formula is $(\\sqrt{A}+\\sqrt{B})^2$. When the inputs are nonnegative arm variances, this is the squared sum of their standard deviations.","s":"ℝ → ℝ → ℝ","l":43,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_eq_at_fraction","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":1,"d":"Neyman allocation optimum. For nonnegative treated-arm and control-arm outcome variances A and B, evaluating the two-arm variance A/x + B/(1−x) at the Neyman fraction x* = √A/(√A+√B) yields exactly its lower bound (√A + √B)²; hence the Neyman fraction minimizes the variance over (0,1).","s":"∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → A / Causalean.Experimentation.DesignBased.neymanFraction A B + B / (1 - Causalean.Experimentation.DesignBased.neymanFraction A B) = Causalean.Experimentation.DesignBased.neymanOptimalValue A B","l":79,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_isMinimizer","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"The Neyman fraction is a minimizer: its two-arm variance is no larger than the variance at any treatment fraction in (0,1).","s":"∀ {A B : ℝ}, 0 ≤ A → 0 ≤ B → ∀ (x : ℝ), 0 < x → x < 1 → A / Causalean.Experimentation.DesignBased.neymanFraction A B + B / (1 - Causalean.Experimentation.DesignBased.neymanFraction A B) ≤ A / x + B / (1 - x)","l":106,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.neyman_allocation_lower_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Neyman","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Neyman allocation lower bound. For nonnegative arm variances A, B and any treatment fraction x ∈ (0,1), the two-arm variance is at least (√A + √B)².","s":"∀ {A B x : ℝ}, 0 ≤ A → 0 ≤ B → 0 < x → x < 1 → Causalean.Experimentation.DesignBased.neymanOptimalValue A B ≤ A / x + B / (1 - x)","l":48,"f":"Causalean/Experimentation/DesignBased/Optimality/Neyman.lean"},{"n":"Causalean.Experimentation.DesignBased.prefixCondExp","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A family of coordinate designs, reveal permutation, prefix length, and statistic determine the explicit prefix conditional expectation, obtained by averaging over unrevealed coordinates while holding revealed coordinates fixed by the stated finite sum.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [inst : (i : Fin N) → Fintype (alpha i)] → ((i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) → Equiv.Perm (Fin N) → Fin (N + 1) → (((i : Fin N) → alpha i) → ℝ) → ((i : Fin N) → alpha i) → ℝ","l":75,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.prefixCondExp_ae_eq_condExp","k":"theorem","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":2,"d":"A family of coordinate designs, reveal permutation, prefix length, and statistic give an almost-everywhere identification of the explicit prefix average with conditional expectation under the product-design measure and its reveal σ-algebra.","s":"∀ {N : ℕ} {alpha : Fin N → Type u_1} [inst : (i : Fin N) → Fintype (alpha i)] [inst_1 : (i : Fin N) → MeasurableSpace (alpha i)] [∀ (i : Fin N), MeasurableSingletonClass (alpha i)] (D : (i : Fin N) → Causalean.Experimentation.DesignBased.FiniteDesign (alpha i)) (pi : Equiv.Perm (Fin N)) (k : Fin (N + 1)) (f : ((i : Fin N) → alpha i) → ℝ), Causalean.Experimentation.DesignBased.prefixCondExp D pi k f =ᵐ[(Causalean.Experimentation.DesignBased.prodDesign D).toMeasure] (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure[f | ↑(Causalean.Experimentation.DesignBased.revealFiltration pi) k]","l":165,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.prefixRank","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A reveal permutation and coordinate determine that coordinate's reveal rank, namely the step at which the permutation reveals it.","s":"{N : ℕ} → Equiv.Perm (Fin N) → Fin N → Fin N","l":31,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a finite, equality-comparable index set whose coordinate assignment spaces are finite, a family of coordinate-specific randomization designs defines the product design, which randomizes all coordinates independently: the probability of an assignment vector is the product of the probabilities assigned to its coordinate values by their respective designs.","s":"{ι : Type u_1} → [inst : Fintype ι] → [inst_1 : DecidableEq ι] → {α : ι → Type u_2} → [inst_2 : (i : ι) → Fintype (α i)] → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) → Causalean.Experimentation.DesignBased.FiniteDesign ((i : ι) → α i)","l":45,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_clt","k":"theorem","m":"Causalean.Experimentation.DesignBased.IndepSummandsCLT","a":"Experimentation/DesignBased","t":1,"d":"Independent-summands CLT over product designs. Fix a family of coordinate designs D n i and real-valued per-coordinate summands g n i, one pair per stage n and coordinate i. Suppose there is a sequence of nonnegative bounds B n tending to zero, every summand g n i a is bounded in absolute value by B n, the number of coordinates at stage n times B n cubed tends to zero, each summand has mean zero under its own coordinate design, and the total sum ∑ᵢ g n i (w i) has design variance exactly one under the product design at every stage n. Then the design probability that the sum is at most any fixed threshold s converges, as n → ∞, to the standard normal cumulative distribution function Φ(s).","s":"∀ {ι : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (ι n)] [inst_1 : (n : ℕ) → DecidableEq (ι n)] {α : (n : ℕ) → ι n → Type u_2} [inst_2 : (n : ℕ) → (i : ι n) → Fintype (α n i)] [inst_3 : (n : ℕ) → (i : ι n) → MeasurableSpace (α n i)] [∀ (n : ℕ) (i : ι n), MeasurableSingletonClass (α n i)] (D : (n : ℕ) → (i : ι n) → Causalean.Experimentation.DesignBased.FiniteDesign (α n i)) (g : (n : ℕ) → (i : ι n) → α n i → ℝ) (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n) (a : α n i), |g n i a| ≤ B n) → Filter.Tendsto (fun n => ↑(Fintype.card (ι n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n), (D n i).E (g n i) = 0) → (∀ (n : ℕ), ((Causalean.Experimentation.DesignBased.prodDesign (D n)).Var fun w => ∑ i, g n i (w i)) = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (Causalean.Experimentation.DesignBased.prodDesign (D n)).Pr fun w => ∑ i, g n i (w i) ≤ s) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf s))","l":78,"f":"Causalean/Experimentation/DesignBased/IndepSummandsCLT.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_p","k":"theorem","m":"Causalean.Experimentation.DesignBased.Product","a":"Experimentation/DesignBased","t":1,"d":"For a family of independent coordinate designs D and an assignment w, the product-design probability of w equals the product of the coordinate probabilities ∏ᵢ (D i).p (w i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)) (w : (i : ι) → α i), (Causalean.Experimentation.DesignBased.prodDesign D).p w = ∏ i, (D i).p (w i)","l":57,"f":"Causalean/Experimentation/DesignBased/Product.lean"},{"n":"Causalean.Experimentation.DesignBased.prodDesign_toMeasure_eq_pi","k":"theorem","m":"Causalean.Experimentation.DesignBased.ProductMeasure","a":"Experimentation/DesignBased","t":1,"d":"The product design IS Mathlib's product measure. For a family of independent coordinate designs D, the probability measure induced by the product design prodDesign D coincides exactly with Mathlib's product measure of the coordinate design measures. Both are finite measures on the Fintype ∀ i, α i, so they agree iff they agree on every singleton {w}; on a singleton both sides evaluate to ∏ i, (D i).p (w i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {α : ι → Type u_2} [inst_2 : (i : ι) → Fintype (α i)] [inst_3 : (i : ι) → MeasurableSpace (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (D : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (α i)), (Causalean.Experimentation.DesignBased.prodDesign D).toMeasure = MeasureTheory.Measure.pi fun i => (D i).toMeasure","l":69,"f":"Causalean/Experimentation/DesignBased/ProductMeasure.lean"},{"n":"Causalean.Experimentation.DesignBased.prop","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"On a finite assignment space, for a finite randomization design, an assignment-to-exposure rule, a map assigning each unit its traits, a unit, and an exposure condition from a set whose elements can be compared for equality, the generalized probability of exposure is the design probability that the unit receives that condition.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → Δ → ℝ","l":59,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairCross","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"On a finite assignment space, for a finite randomization design, an assignment-to-exposure rule, a map assigning each unit its traits, two units, and two exposure conditions from a set whose elements can be compared for equality, the cross-condition joint exposure probability is the design expectation of the product of the first unit's indicator for the first condition and the second unit's indicator for the second condition.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → ι → Δ → Δ → ℝ","l":75,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairCross_self_of_ne","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":2,"d":"The self cross-exposure joint probability vanishes for distinct exposures.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι) {d d' : Δ}, d ≠ d' → Causalean.Experimentation.DesignBased.propPairCross D f θ i i d d' = 0","l":124,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.propPairSame","k":"def","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"On a finite assignment space, for a finite randomization design, an assignment-to-exposure rule, a map assigning each unit its traits, two units, and an exposure condition from a set whose elements can be compared for equality, the same-condition joint exposure probability is the design expectation of the product of the two units' indicators for that condition.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → Θ → Δ) → (ι → Θ) → ι → ι → Δ → ℝ","l":67,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.ratio_remainder_capped_bound","k":"theorem","m":"Causalean.Experimentation.DesignBased.RatioLinearization","a":"Experimentation/DesignBased","t":2,"d":"Capped ratio-remainder bound. Consider a ratio estimator A / D of a target μ, scaled by √n (here sn with sn² = n), and let G = A − μ·D be the centered numerator. The scaled second-order remainder √n·(A/D − μ) − √n⁻¹·G equals √n⁻¹·G·(n/D − 1), and on the event where the denominator is at least half its target (equivalently n/D ≤ 2) it is bounded by 2·|√n⁻¹·G|·|n⁻¹·D − 1|. This is the per-arm remainder control a design-based ratio / Hájek CLT uses to reduce the studentized estimator to its linear score.","s":"∀ {n sn A D μ G : ℝ}, 0 < n → sn ^ 2 = n → 0 < D → n / D ≤ 2 → G = A - μ * D → |sn * (A / D - μ) - sn⁻¹ * G| ≤ 2 * |sn⁻¹ * G| * |n⁻¹ * D - 1|","l":34,"f":"Causalean/Experimentation/DesignBased/RatioLinearization.lean"},{"n":"Causalean.Experimentation.DesignBased.regret","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a family of randomization designs that is finite and nonempty, a risk criterion indexed by states and randomization designs, a state of nature, and a randomization design, the design's regret is its risk at that state minus the least risk achievable there within the family, provided risk values form a linearly ordered additive group whose order is preserved by right addition.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) → Set.Finite 𝒟 → Set.Nonempty 𝒟 → {α : Type u_3} → [inst_1 : AddGroup α] → [inst_2 : LinearOrder α] → [AddRightMono α] → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α","l":92,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.regret_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":2,"d":"Regret is nonnegative for every member of the family.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {Y : Type u_2} (𝒟 : Causalean.Experimentation.DesignBased.DesignFamily Ω) (hfin : Set.Finite 𝒟) (hne : Set.Nonempty 𝒟) {α : Type u_3} [inst_1 : AddGroup α] [inst_2 : LinearOrder α] [inst_3 : AddRightMono α] (R : Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) (y : Y) {D : Causalean.Experimentation.DesignBased.FiniteDesign Ω}, D ∈ 𝒟 → 0 ≤ Causalean.Experimentation.DesignBased.regret 𝒟 hfin hne R y D","l":102,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.DesignBased.revealFiltration","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A reveal permutation determines the reveal filtration, whose time-k σ-algebra is the prefix σ-algebra, is monotone in reveal time, and lies inside the full product σ-algebra.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [inst : (i : Fin N) → MeasurableSpace (alpha i)] → Equiv.Perm (Fin N) → MeasureTheory.Filtration (Fin (N + 1)) inferInstance","l":51,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.revealMeasurableSpace","k":"def","m":"Causalean.Experimentation.DesignBased.HeydeBrown","a":"Experimentation/DesignBased","t":1,"d":"A reveal permutation and prefix length determine the σ-algebra generated by the revealed coordinates, which records exactly the information available after that many reveal steps by generating from those coordinate evaluations.","s":"{N : ℕ} → {alpha : Fin N → Type u_1} → [(i : Fin N) → MeasurableSpace (alpha i)] → Equiv.Perm (Fin N) → Fin (N + 1) → MeasurableSpace ((i : Fin N) → alpha i)","l":42,"f":"Causalean/Experimentation/DesignBased/HeydeBrown.lean"},{"n":"Causalean.Experimentation.DesignBased.sateEstimand","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"For a finite population of units with potential outcomes under treatment and under control, the sample average treatment effect is the population mean of the unit-level difference between those two potential outcomes.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → ℝ) → ℝ","l":43,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf","k":"def","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"For a real threshold, the standard normal cumulative distribution function is the probability that a standard normal random variable is no greater than that threshold.","s":"ℝ → ℝ","l":38,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_eq","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The design-based .real (Iic) presentation agrees with the canonical stdNormalCDF.","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf t = Causalean.Mathlib.stdNormalCDF t","l":46,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_le_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative probability is at most one.","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf t ≤ 1","l":54,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_neg","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":1,"d":"Symmetry of the standard normal CDF. For any threshold t, the standard normal CDF satisfies Φ(−t) = 1 − Φ(t).","s":"∀ (t : ℝ), Causalean.Experimentation.DesignBased.stdNormalCdf (-t) = 1 - Causalean.Experimentation.DesignBased.stdNormalCdf t","l":63,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stdNormalCdf_nonneg","k":"theorem","m":"Causalean.Experimentation.DesignBased.GaussianCDF","a":"Experimentation/DesignBased","t":2,"d":"The standard-normal cumulative probability is nonnegative.","s":"∀ (t : ℝ), 0 ≤ Causalean.Experimentation.DesignBased.stdNormalCdf t","l":50,"f":"Causalean/Experimentation/DesignBased/GaussianCDF.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite collection of strata, each with a finite population of units, and a specified number of treated units in every stratum that does not exceed that stratum's population size, the stratified randomization design independently selects exactly the specified number of units in each stratum.","s":"{K : Type u_1} → [inst : Fintype K] → [inst_1 : DecidableEq K] → {V : K → Type u_2} → [inst_2 : (k : K) → Fintype (V k)] → (n₁ : K → ℕ) → (∀ (k : K), n₁ k ≤ Fintype.card (V k)) → Causalean.Experimentation.DesignBased.FiniteDesign ((k : K) → { S // S.card = n₁ k })","l":39,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":2,"d":"First-order inclusion probability. A unit i in stratum k is treated with probability n₁ k / N k, the complete-randomization rate of its own stratum.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) (k : K) (i : V k), ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k)) = ↑(n₁ k) / ↑(Fintype.card (V k))","l":49,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl_pair_across","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":1,"d":"Second-order inclusion probability across strata. For a stratified design in which each stratum k treats exactly n₁ k of its N k units, with n₁ k never exceeding N k, if k and k' are distinct strata, then a unit i in stratum k and a unit i' in stratum k' are jointly treated with probability the product of their two strata's treatment rates, (n₁ k / N k) · (n₁ k' / N k') — the strata are randomized independently.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) {k k' : K}, k ≠ k' → ∀ (i : V k) (i' : V k'), ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k) ∧ i' ∈ ↑(z k')) = ↑(n₁ k) / ↑(Fintype.card (V k)) * (↑(n₁ k') / ↑(Fintype.card (V k')))","l":89,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.stratifiedDesign_incl_pair_within","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.Stratified","a":"Experimentation/DesignBased/Designs","t":2,"d":"Second-order inclusion probability within one stratum. Two distinct units in the same stratum k are jointly treated with the complete-randomization second-order inclusion probability for that stratum.","s":"∀ {K : Type u_1} [inst : Fintype K] [inst_1 : DecidableEq K] {V : K → Type u_2} [inst_2 : (k : K) → Fintype (V k)] [inst_3 : (k : K) → DecidableEq (V k)] (n₁ : K → ℕ) (hn : ∀ (k : K), n₁ k ≤ Fintype.card (V k)) (k : K) {i j : V k}, i ≠ j → ((Causalean.Experimentation.DesignBased.stratifiedDesign n₁ hn).Pr fun z => i ∈ ↑(z k) ∧ j ∈ ↑(z k)) = ↑(n₁ k) * (↑(n₁ k) - 1) / (↑(Fintype.card (V k)) * (↑(Fintype.card (V k)) - 1))","l":67,"f":"Causalean/Experimentation/DesignBased/Designs/Stratified.lean"},{"n":"Causalean.Experimentation.DesignBased.sum_prop_eq_one","k":"theorem","m":"Causalean.Experimentation.DesignBased.Exposure","a":"Experimentation/DesignBased","t":1,"d":"Exposure probabilities sum to one. For a design, an exposure mapping, unit traits, and a unit i, the generalized exposure probabilities of unit i, summed over all exposure levels, equal one.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq Δ] [inst_2 : Fintype Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (f : Ω → Θ → Δ) (θ : ι → Θ) (i : ι), ∑ d, Causalean.Experimentation.DesignBased.prop D f θ i d = 1","l":134,"f":"Causalean/Experimentation/DesignBased/Exposure.lean"},{"n":"Causalean.Experimentation.DesignBased.tauTrue","k":"def","m":"Causalean.Experimentation.DesignBased.HT.Estimator","a":"Experimentation/DesignBased/HT","t":1,"d":"For a finite population, a schedule of potential outcomes for every unit and exposure condition and two exposure conditions define the finite-population average causal effect as the mean potential outcome under the first condition minus that under the second condition.","s":"{ι : Type u_2} → {Δ : Type u_4} → [Fintype ι] → (ι → Δ → ℝ) → Δ → Δ → ℝ","l":63,"f":"Causalean/Experimentation/DesignBased/HT/Estimator.lean"},{"n":"Causalean.Experimentation.DesignBased.treatInd","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.Bernoulli","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite population of units, a unit, and a treatment assignment for that population, the treatment indicator for that unit equals one when the unit is assigned treatment and zero otherwise.","s":"{U : Type u_1} → U → (U → Bool) → ℝ","l":66,"f":"Causalean/Experimentation/DesignBased/Designs/Bernoulli.lean"},{"n":"Causalean.Experimentation.DesignBased.treatInd_prod_mul_prod","k":"theorem","m":"Causalean.Experimentation.DesignBased.Designs.BernoulliMoments","a":"Experimentation/DesignBased/Designs","t":2,"d":"Treatment indicators take only the values zero and one, so two overlapping products of them collapse into a single product over the union of the two sets of units.","s":"∀ {U : Type u_1} [inst : DecidableEq U] (S T : Finset U) (z : U → Bool), (∏ j ∈ S, Causalean.Experimentation.DesignBased.treatInd j z) * ∏ j ∈ T, Causalean.Experimentation.DesignBased.treatInd j z = ∏ j ∈ S ∪ T, Causalean.Experimentation.DesignBased.treatInd j z","l":105,"f":"Causalean/Experimentation/DesignBased/Designs/BernoulliMoments.lean"},{"n":"Causalean.Experimentation.DesignBased.treatedMean","k":"def","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":1,"d":"For a finite population of units, a number of treated units, the potential outcome under treatment for each unit, and a treated set containing exactly that many units, the treated-arm mean is the average treatment potential outcome among the treated units.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n₁ : ℕ) → (U → ℝ) → { S // S.card = n₁ } → ℝ","l":49,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.unbiased_diffInMeans","k":"theorem","m":"Causalean.Experimentation.DesignBased.Estimators.DifferenceInMeans","a":"Experimentation/DesignBased/Estimators","t":2,"d":"The difference in means is unbiased for the SATE, in the Unbiased predicate form.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n₁ : ℕ), 0 < n₁ → ∀ (hn0 : n₁ < Fintype.card U) (Y1 Y0 : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n₁ (Nat.le_of_lt hn0)).Unbiased (Causalean.Experimentation.DesignBased.diffInMeans n₁ Y1 Y0) (Causalean.Experimentation.DesignBased.sateEstimand Y1 Y0)","l":202,"f":"Causalean/Experimentation/DesignBased/Estimators/DifferenceInMeans.lean"},{"n":"Causalean.Experimentation.DesignBased.unitTreatInd","k":"def","m":"Causalean.Experimentation.DesignBased.Designs.ClusterRandomization","a":"Experimentation/DesignBased/Designs","t":1,"d":"For a finite population of units, a finite population of clusters, a map assigning every unit to a cluster, a unit, and a treatment assignment for the clusters, the unit-level treatment indicator equals one if that unit's cluster is treated and zero otherwise.","s":"{U : Type u_1} → {C : Type u_2} → (U → C) → U → (C → Bool) → ℝ","l":44,"f":"Causalean/Experimentation/DesignBased/Designs/ClusterRandomization.lean"},{"n":"Causalean.Experimentation.DesignBased.worstRisk","k":"def","m":"Causalean.Experimentation.DesignBased.Optimality.Minimax","a":"Experimentation/DesignBased/Optimality","t":1,"d":"For a finite set of states of nature that is nonempty, a risk criterion indexed by states and randomization designs, and a randomization design, the worst-case risk is the largest risk incurred by that design as the state ranges over the given set, provided risk values admit pairwise least upper bounds.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {Y : Type u_2} → {α : Type u_3} → [SemilatticeSup α] → (s : Finset Y) → s.Nonempty → (Y → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α) → Causalean.Experimentation.DesignBased.FiniteDesign Ω → α","l":38,"f":"Causalean/Experimentation/DesignBased/Optimality/Minimax.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEstA_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.6 (cova). Suppose only the two treatment arms dk, dl are distinct and every unit has nonzero exposure propensity under dk and under dl — with NO positive-cross-joint assumption. Then the general Horvitz–Thompson covariance estimator Ĉov_A is nonpositively biased for the true design covariance of the two HT totals: zero-cross-joint off-diagonal pairs drop from the first sum, while the Young correction (summing over every j with π_{ij}(dk,dl)=0, including the diagonal) dominates the corresponding −y_i(dk)y_j(dl) covariance contributions termwise via y_i(dk)y_j(dl) ≤ y_i(dk)²/2 + y_j(dl)²/2.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEstA D y f θ dk dl) ≤ D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":601,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEst_eq_of_noEffect","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.5 (no_bias_cov). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, every off-diagonal pair has nonzero cross-arm joint exposure propensity — the positive marginal and positive cross-joint regime — and the two exposures share the same potential outcomes: y_i(dk) = y_i(dl) for every unit i. Then the Horvitz–Thompson covariance estimator is exactly unbiased for the true design covariance of the two HT totals: the Young diagonal correction (y_i²/2 + y_i²/2) = y_i² = y_i(dk)y_i(dl) is exact, so the nonpositive bias of Proposition 5.4 vanishes.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → (∀ (i : ι), y i dk = y i dl) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl) = D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":478,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htCovEst_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.4 (ncov). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero cross-arm joint exposure propensity — the positive marginal and positive cross-joint regime. Then the Horvitz–Thompson covariance estimator is nonpositively biased for the true design covariance of the two HT totals.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl) ≤ D.Cov (Causalean.Experimentation.DesignBased.htTotal D y f θ dk) (Causalean.Experimentation.DesignBased.htTotal D y f θ dl)","l":166,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htEffectVarEstA_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.7, general form (consvar). Suppose only the two treatment arms dk, dl are distinct and every unit has nonzero exposure propensity under dk and under dl — without any positive-joint assumption. Then the assembled general Horvitz–Thompson effect-variance estimator has nonnegative bias: its expectation is at least the true design variance of the effect estimator τ̂.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) ≤ D.E (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEstA D y f θ dk dl)","l":795,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htEffectVarEst_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.7 (consvar). Suppose the two treatment arms dk, dl are distinct, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm joint exposure propensity under dk and under dl, as well as nonzero cross-arm joint exposure propensity — the positive-joint regime. Then the assembled Horvitz–Thompson effect-variance estimator has nonnegative bias: its expectation is at least the true design variance of the effect estimator τ̂.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dk ≠ 0) → (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htEffect D y f θ dk dl) ≤ D.E (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst D y f θ dk dl)","l":269,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Lemma 5.1 (varun). In the positive-joint regime — every unit has nonzero exposure propensity under d and every off-diagonal pair has nonzero same-arm joint exposure propensity under d — the Horvitz–Thompson variance estimator is exactly unbiased for the true design variance of the HT total.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → (∀ (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j d ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d) = D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d)","l":97,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst_add_htA2_ge","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.3 (a2). Assuming only every unit has nonzero exposure propensity under d, adding the Young correction Â₂ to the Horvitz–Thompson variance estimator makes it conservative for the true design variance of the HT total: Var[ŷᵀ(d)] ≤ E[V̂ + Â₂].","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) ≤ D.E fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d z + Causalean.Experimentation.ExposureMappingInterference.htA2 D y f θ d z","l":418,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.E_htVarEst_eq_addBias","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"Proposition 5.2 (varbias). Assuming only every unit has nonzero exposure propensity under d — without the positive-joint assumption — the expectation of the Horvitz–Thompson variance estimator equals the true design variance of the HT total plus the signed zero-joint correction A = ∑_{π_{ij}(d)=0} y_i(d)·y_j(d).","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (d : Δ), (∀ (i : ι), Causalean.Experimentation.DesignBased.prop D f θ i d ≠ 0) → D.E (Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ d) = D.Var (Causalean.Experimentation.DesignBased.htTotal D y f θ d) + ∑ i, ∑ j ∈ Finset.univ.erase i, (if Causalean.Experimentation.DesignBased.propPairSame D f θ i j d = 0 then 1 else 0) * (y i d * y j d)","l":314,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment","k":"structure","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"A design-based experiment on a finite population: a finite assignment space Ω equipped with a randomization design D over it, a finite population of units ι each assigned a trait in a trait space Θ, exposure-indexed potential outcomes y, one real value per unit and exposure in an exposure space Δ, and an exposure mapping f sending each assignment and unit trait to a realized exposure.","s":"Type 1","l":37,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.N","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Given an experiment, the population size is the number of units in its finite population.","s":"Causalean.Experimentation.ExposureMappingInterference.Experiment → ℕ","l":78,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.Var_htMean_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Variance bound. Under a uniform bound c on |y_i(d)/π_i(d)|, the variance of the HT mean estimator is O((N + ∑_{i,j} g_{ij})/N²).","s":"∀ (E : Causalean.Experimentation.ExposureMappingInterference.Experiment) (d : E.Δ) {c : ℝ}, 0 ≤ c → (∀ (i : E.ι), |E.y i d / Causalean.Experimentation.DesignBased.prop E.D E.f E.θ i d| ≤ c) → E.D.Var (Causalean.Experimentation.DesignBased.htMean E.D E.y E.f E.θ d) ≤ c ^ 2 * (↑E.N + ∑ i, ∑ j ∈ Finset.univ.erase i, E.gdep d i j) / ↑E.N ^ 2","l":99,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.effRaw","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Given an experiment, two target exposure levels, a unit, and a realized assignment, the raw effect contribution is that unit's inverse-propensity-weighted observed outcome under the first exposure level minus its analogous weighted observed outcome under the second exposure level.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.ι → E.Ω → ℝ","l":49,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.effSummand","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Given an experiment, two target exposure levels, a unit, and a realized assignment, the standardized effect summand is the unit's raw effect contribution minus its design expectation, divided by the population size times the true standard deviation of the Horvitz--Thompson effect estimator.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.ι → E.Ω → ℝ","l":57,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.gdep","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Given an experiment, an exposure level, and two units, the pairwise dependency indicator equals zero when their indicators for that exposure have zero design covariance, and equals one otherwise.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.ι → E.ι → ℝ","l":71,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.instMeasurableSingletonΩ","k":"instance","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"For every exposure-mapping experiment, every singleton subset of its assignment space is measurable under the experiment's measurable structure.","s":"∀ (E : Causalean.Experimentation.ExposureMappingInterference.Experiment), MeasurableSingletonClass E.Ω","l":174,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.instMeasurableSpaceΩ","k":"instance","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"For every exposure-mapping experiment, the measurable structure on its finite assignment space is the top σ-algebra, under which every subset is measurable.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → MeasurableSpace E.Ω","l":167,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.Experiment.studentizedEffect","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinCLT","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Given an experiment, two target exposure levels, and a realized assignment, the studentized Horvitz--Thompson effect statistic is the estimated effect minus the corresponding finite-population effect, divided by the true standard deviation of the estimator under the randomization design.","s":"(E : Causalean.Experimentation.ExposureMappingInterference.Experiment) → E.Δ → E.Δ → E.Ω → ℝ","l":36,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinCLT.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT","k":"structure","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinCLT","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Chen-Shao local-dependence CLT interface, bounded-neighborhood case. Along a sequence of experiments Exp with a chosen pair of exposure levels dk, dl at each sample size, this records that the design probability that the studentized Horvitz–Thompson effect statistic falls below any fixed threshold converges, as the sample size grows, to the standard normal CDF at that threshold.","s":"(Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) → ((n : ℕ) → (Exp n).Δ) → ((n : ℕ) → (Exp n).Δ) → Prop","l":46,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinCLT.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.abs_vb_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Boundedness of the edge-function (Conditions 1 + 1'): |vb i j z| ≤ vbBound c₁ c₂ c₃.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (i : ι) (d : Δ), |y i d| ≤ c₁) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dk) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dl) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dk ≤ c₂) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dl ≤ c₂) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≤ c₃) → ∀ (i j : ι) (z : Ω), |Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j z| ≤ Causalean.Experimentation.ExposureMappingInterference.vbBound c₁ c₂ c₃","l":343,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.diagCov","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, a unit, and an assignment, the diagonal covariance correction summand is one half of the inverse-propensity-weighted squared observed outcome for each target exposure, added together.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → Ω → ℝ","l":84,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.diagVar","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, an exposure level, a unit, and an assignment, the diagonal variance summand is the observed exposure indicator times one minus that unit's exposure probability times the square of its inverse-propensity-weighted observed outcome.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → ι → Ω → ℝ","l":47,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htA2","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, an exposure level, and an assignment, the Young-inequality correction adds, over every ordered pair of distinct units with zero same-exposure joint probability, the two corresponding half-weighted squared observed-outcome terms.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":400,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htCovEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, and an assignment, the Horvitz--Thompson covariance estimator is its ordered-pair cross-exposure term over distinct units minus its single-unit Young correction.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":65,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htCovEstA","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, and an assignment, the general Horvitz--Thompson covariance estimator is its distinct-unit cross-exposure sum minus a Young correction over every ordered pair with zero cross-exposure joint probability.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":578,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, and an assignment, the conservative effect-variance estimator is the sum of the two variance estimators minus twice the covariance estimator, divided by the square of the population size.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":84,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEstA","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, and an assignment, the general conservative effect-variance estimator is the two variance estimators and their Young corrections, minus twice the general covariance estimator, divided by the square of the population size.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → Ω → ℝ","l":780,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst_undershoot_tendsto_zero","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Reduction of feasible-interval variance consistency to L²-relative consistency. For a pair of treatment sequences dk, dl with dk n ≠ dl n for every n, suppose every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm and nonzero cross-arm joint exposure propensities — the standing overlap conditions giving the conservativeness Var[τ̂] ≤ E[V̂]. If the true effect-estimator variance is everywhere positive and the L²-relative-consistency limit Var[V̂]/Var[τ̂]² → 0 holds, then the conservative variance estimator undershoots the true variance only with vanishing probability: for every ε > 0, Pr[Vh < (1−ε)·Var[τ̂]] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ), dk n ≠ dl n) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≠ 0) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z < (1 - ε) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)","l":48,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htMean_consistent_of_var","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Consistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Chebyshev consistency. Along a sequence of experiments with a sequence of treatment assignments d such that every unit's exposure probability under d is nonzero, if the design variance of the Horvitz–Thompson mean estimator tends to 0, then for any positive threshold ε, the estimator is consistent: Pr[|μ̂ − μ| ≥ ε] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (d : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (d n) ≠ 0) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.DesignBased.htMean (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (d n))) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => ε ≤ |Causalean.Experimentation.DesignBased.htMean (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (d n) z - Causalean.Experimentation.DesignBased.muTrue (Exp n).y (d n)|) Filter.atTop (nhds 0)","l":186,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Consistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htVarEst","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Variance.Conservative","a":"Experimentation/ExposureMappingInterference/Variance","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, an exposure level, and an assignment, the Horvitz--Thompson variance estimator is the sum of its single-unit inverse-propensity terms and its ordered-pair terms over distinct units.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [Fintype ι] → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Ω → ℝ","l":47,"f":"Causalean/Experimentation/ExposureMappingInterference/Variance/Conservative.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.htVarEst_add_sub_eq_edgeSum","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Edge-sum identity. V̂_raw = ∑_{i,j} vb i j: the three single-index sums of ŷVar(dk), ŷVar(dl), Ĉov fold into the diagonal i = j, and the three pairwise sums into the off-diagonal i ≠ j.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), (fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dk z + Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dl z - 2 * Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl z) = fun z => ∑ i, ∑ j, Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j z","l":135,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from a bounded-degree dependency graph. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a dependency graph Dg whose neighbourhoods have cardinality at most m, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the summands are pointwise bounded by a nonnegative sequence B n with B n → 0 and population size times B n cubed tending to 0. Then the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * B n ^ 3) Filter.atTop (nhds 0) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":309,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_paper_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from the literal Aronow–Samii conditions. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a bounded-degree dependency graph Dg, potential outcomes under dk and under dl are uniformly bounded by a constant c1, and the exposure propensities under dk and under dl are bounded away from 0 by a positive constant c2. If the population size diverges, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, and population size times that variance converges to a positive limit cVar, then the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → ∀ (c1 : ℝ), (∀ (n : ℕ) (i : (Exp n).ι), |(Exp n).y i (dk n)| ≤ c1) → (∀ (n : ℕ) (i : (Exp n).ι), |(Exp n).y i (dl n)| ≤ c1) → ∀ (c2 : ℝ), 0 < c2 → (∀ (n : ℕ) (i : (Exp n).ι), 1 / c2 ≤ Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / c2 ≤ Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ (cVar : ℝ), 0 < cVar → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds cVar) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":459,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.localDependenceCLT_of_stein","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Local-dependence CLT from the abstract Stein conditions. For a pair of treatment sequences dk, dl and a choice of dependency neighbourhoods N n i, suppose the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the per-unit effect summands are pointwise bounded by a nonnegative sequence B n. If each summand is independent of the sum of summands outside its neighbourhood, and the two Stein negligibility limits hold — the design variance of the neighbourhood cross-term sum tends to 0 and the summed third-moment-type error term tends to 0 — then the studentized Horvitz–Thompson effect statistic converges in distribution to a standard normal.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (N : (n : ℕ) → (Exp n).ι → Finset (Exp n).ι), (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι), ProbabilityTheory.IndepFun ((Exp n).effSummand (dk n) (dl n) i) (fun z => ∑ j ∈ Finset.univ \\ N n i, (Exp n).effSummand (dk n) (dl n) j z) (Exp n).D.toMeasure) → Filter.Tendsto (fun n => (Exp n).D.Var fun z => ∑ i, (Exp n).effSummand (dk n) (dl n) i z * ∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, (Exp n).D.E fun z => |(Exp n).effSummand (dk n) (dl n) i z| * (∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) ^ 2) Filter.atTop (nhds 0) → Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl","l":181,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.offCov","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, two units, and an assignment, the off-diagonal covariance summand is the product of the corresponding two exposure indicators, their centered cross-exposure probability divided by their cross-exposure probability, and their two inverse-propensity-weighted observed outcomes.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":70,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.offVar","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, an exposure level, two units, and an assignment, the off-diagonal variance summand is the product of their observed exposure indicators, their centered joint-exposure probability divided by their joint-exposure probability, and their two inverse-propensity-weighted observed outcomes.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → ι → ι → Ω → ℝ","l":57,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.relVar_of_NsqVar_tendsto","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Bridge from Var[N·V̂] → 0 to L²-relative consistency. The paper's appendix establishes Var[N·V̂_n] → 0 (here as N²·Var[V̂_n] → 0); together with Condition 4 (N·Var[τ̂_n] → c > 0) this yields the relative limit Var[V̂_n]/Var[τ̂_n]² → 0, via the identity Var[V̂]/Var² = (N²·Var[V̂])/(N·Var)² and Tendsto.div (numerator → 0, denominator → c² > 0).","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds c) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) ^ 2 * (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0)","l":171,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.var_NsqVhat_tendsto_zero_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstConsistencyConditions","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Scaled variance-estimator variance vanishes (Aronow–Samii Prop 6.6 core). Along a sequence of experiments with distinct treatment arms dk ≠ dl whose outcomes are uniformly bounded by a nonnegative constant c₁ (Condition 1), suppose the marginal exposure propensities under dk and dl are positive with inverses uniformly bounded by a nonnegative constant c₂, and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃ (Condition 1'). Suppose further a symmetric, reflexive dependency relation G of degree at most m such that every non-adjacent pair's joint exposure propensities factor as if independent and every quadruple with no adjacent index pair has zero covariance between the corresponding variance-estimator kernel terms (Condition 3), and the population size diverges. Then the scaled variance of the conservative variance estimator vanishes: N²·Var[V̂_n] → 0.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), (∀ (n : ℕ), dk n ≠ dl n) → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (n : ℕ) (i : (Exp n).ι) (d : (Exp n).Δ), |(Exp n).y i d| ≤ c₁) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≤ c₂) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≤ c₂) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≤ c₃) → ∀ (G : (n : ℕ) → (Exp n).ι → (Exp n).ι → Prop) (decG : (n : ℕ) → DecidableRel (G n)), (∀ (n : ℕ) (i : (Exp n).ι), G n i i) → (∀ (n : ℕ) (i j : (Exp n).ι), G n i j → G n j i) → ∀ {m : ℕ}, (∀ (n : ℕ) (i : (Exp n).ι), {j | G n i j}.card ≤ m) → (∀ (n : ℕ) (i j : (Exp n).ι), ¬G n i j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dk n) ∧ Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n) ∧ Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n)) → (∀ (n : ℕ) (i j k l : (Exp n).ι), ¬(G n i k ∨ G n i l ∨ G n j k ∨ G n j l) → (Exp n).D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) i j) (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) k l) = 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) ^ 2 * (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)","l":94,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstConsistencyConditions.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.var_htEdgeStat_le","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Per-population quadruple-sum bound (Aronow–Samii appendix, Prop 6.6). Suppose the two treatment arms are distinct and outcomes are uniformly bounded by a nonnegative constant c₁ (Condition 1). Suppose the marginal exposure propensities under each arm are positive with inverses uniformly bounded by a nonnegative constant c₂, and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃ (Condition 1'). Suppose further a symmetric, reflexive dependency relation G of degree at most m makes every non-adjacent pair's joint exposure propensities factor as if independent and gives every quadruple with no adjacent index pair zero covariance between the variance-estimator kernel terms (Condition 3). Then the variance of the raw conservative variance estimator is linear in the population size: Var[ŷVar(dk)+ŷVar(dl)−2Ĉov] ≤ 8·(vbBound c₁ c₂ c₃)²·m³·N.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ), dk ≠ dl → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → (∀ (i : ι) (d : Δ), |y i d| ≤ c₁) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dk) → (∀ (i : ι), 0 < Causalean.Experimentation.DesignBased.prop D f θ i dl) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dk ≤ c₂) → (∀ (i : ι), 1 / Causalean.Experimentation.DesignBased.prop D f θ i dl ≤ c₂) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl ≤ c₃) → (∀ (i j : ι), 1 / Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl ≤ c₃) → ∀ (G : ι → ι → Prop) [inst_4 : DecidableRel G], (∀ (i : ι), G i i) → (∀ (i j : ι), G i j → G j i) → ∀ {m : ℕ}, (∀ (i : ι), {j | G i j}.card ≤ m) → (∀ (i j : ι), ¬G i j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dk ∧ Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl = Causalean.Experimentation.DesignBased.prop D f θ i dl * Causalean.Experimentation.DesignBased.prop D f θ j dl ∧ Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl) → (∀ (i j k l : ι), ¬(G i k ∨ G i l ∨ G j k ∨ G j l) → D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j) (Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl k l) = 0) → (D.Var fun z => Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dk z + Causalean.Experimentation.ExposureMappingInterference.htVarEst D y f θ dl z - 2 * Causalean.Experimentation.ExposureMappingInterference.htCovEst D y f θ dk dl z) ≤ 8 * Causalean.Experimentation.ExposureMappingInterference.vbBound c₁ c₂ c₃ ^ 2 * (↑m ^ 3 * ↑(Fintype.card ι))","l":426,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vb","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, two units, and an assignment, the edge function for the unscaled effect-variance estimator equals the diagonal edge term when the two units coincide and the off-diagonal edge term otherwise.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq ι] → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":115,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbBound","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For three real constants, the edge-function bound is $4(1+c_3)c_1^2(c_2^2+c_2)$.","s":"ℝ → ℝ → ℝ → ℝ","l":125,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbDiag","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, a unit, and an assignment, the diagonal edge term is the sum of the two diagonal variance summands plus twice the diagonal covariance correction.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → Ω → ℝ","l":95,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vbOff","k":"def","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"For a finite assignment space, a finite unit population, a unit-trait space, an exposure space, a randomization design, exposure-indexed potential outcomes, an exposure mapping, unit traits, two exposure levels, two units, and an assignment, the off-diagonal edge term is the sum of the two off-diagonal variance summands minus twice the off-diagonal covariance summand.","s":"{Ω : Type u_1} → [inst : Fintype Ω] → {ι : Type u_2} → {Θ : Type u_3} → {Δ : Type u_4} → [DecidableEq Δ] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (ι → Δ → ℝ) → (Ω → Θ → Δ) → (ι → Θ) → Δ → Δ → ι → ι → Ω → ℝ","l":105,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.vb_eq_zero_of_indep","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstQuadBound","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":2,"d":"Off-edge vanishing. If the same-exposure and cross-exposure pair propensities factor as products for an off-diagonal pair i ≠ j, then the centered-cross factors vanish and vb i j = 0; Condition 3 supplies these factorization hypotheses for non-adjacent pairs.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] {ι : Type u_2} {Θ : Type u_3} {Δ : Type u_4} [inst_1 : DecidableEq ι] [inst_2 : DecidableEq Δ] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (y : ι → Δ → ℝ) (f : Ω → Θ → Δ) (θ : ι → Θ) (dk dl : Δ) (i j : ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dk = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dk → Causalean.Experimentation.DesignBased.propPairSame D f θ i j dl = Causalean.Experimentation.DesignBased.prop D f θ i dl * Causalean.Experimentation.DesignBased.prop D f θ j dl → Causalean.Experimentation.DesignBased.propPairCross D f θ i j dk dl = Causalean.Experimentation.DesignBased.prop D f θ i dk * Causalean.Experimentation.DesignBased.prop D f θ j dl → Causalean.Experimentation.ExposureMappingInterference.vb D y f θ dk dl i j = fun x => 0","l":413,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstQuadBound.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Intervals","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Proposition 6.5 (asymptotic coverage, true standard error). For a pair of treatment sequences dk, dl, suppose the studentized Horvitz–Thompson effect statistic satisfies the local-dependence central limit theorem and the design variance of the effect estimator is everywhere positive — the operative form of Condition 4 (nonzero limiting variance). Then with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, the Wald interval centered at the Horvitz–Thompson effect estimator, using its true standard error, attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":35,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Intervals.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.Intervals","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Proposition 6.5 (asymptotic coverage, feasible/estimated standard error — the paper's actual interval). For a pair of treatment sequences dk, dl, suppose the studentized effect statistic satisfies the local-dependence central limit theorem, the design variance of the effect estimator is everywhere positive, and the conservative variance estimator V̂ undershoots the true variance only with vanishing probability. Then with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, the Wald interval τ̂ ± zq·√(V̂[τ̂]), using the estimated variance htEffectVarEst exactly as in Aronow–Samii Prop 6.5, attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).D.Pr fun z => Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z < (1 - ε) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds 0)) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":176,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/Intervals.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarEstConsistencyConditions","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Feasible Wald coverage from primitive boundedness, overlap, dependency-graph, and variance-growth conditions. Along a sequence of experiments with distinct treatment arms dk ≠ dl whose outcomes are uniformly bounded by a nonnegative constant c₁, suppose the marginal exposure propensities under dk and dl are positive with inverses uniformly bounded by a nonnegative constant c₂ — taken strictly positive — and the same-arm and cross-arm pairwise joint exposure propensities have inverses uniformly bounded by a nonnegative constant c₃, with all off-diagonal joint propensities strictly positive. Suppose a symmetric, reflexive dependency relation G of degree at most m makes every non-adjacent pair's joint exposure propensities factor as if independent and gives every quadruple with no adjacent index pair zero covariance between the variance-estimator kernel terms; suppose also the per-unit effect summands admit a Stein dependency graph Dg of degree at most m, that the population size diverges while population size times the true effect-estimator variance converges to a positive constant cVar and that variance is everywhere positive, and that zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2. Then the feasible Wald interval τ̂ ± zq·√V̂ attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) {m : ℕ} (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), dk n ≠ dl n) → ∀ {c₁ c₂ c₃ : ℝ}, 0 ≤ c₁ → 0 ≤ c₂ → 0 ≤ c₃ → 0 < c₂ → (∀ (n : ℕ) (i : (Exp n).ι) (d : (Exp n).Δ), |(Exp n).y i d| ≤ c₁) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n)) → (∀ (n : ℕ) (i : (Exp n).ι), 0 < Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n)) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≤ c₂) → (∀ (n : ℕ) (i : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≤ c₂) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≤ c₃) → (∀ (n : ℕ) (i j : (Exp n).ι), 1 / Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≤ c₃) → ∀ (G : (n : ℕ) → (Exp n).ι → (Exp n).ι → Prop) (decG : (n : ℕ) → DecidableRel (G n)), (∀ (n : ℕ) (i : (Exp n).ι), G n i i) → (∀ (n : ℕ) (i j : (Exp n).ι), G n i j → G n j i) → (∀ (n : ℕ) (i : (Exp n).ι), {j | G n i j}.card ≤ m) → (∀ (n : ℕ) (i j : (Exp n).ι), ¬G n i j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dk n) ∧ Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n) ∧ Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) = Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) * Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ j (dl n)) → (∀ (n : ℕ) (i j k l : (Exp n).ι), ¬(G n i k ∨ G n i l ∨ G n j k ∨ G n j l) → (Exp n).D.Cov (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) i j) (Causalean.Experimentation.ExposureMappingInterference.vb (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) k l) = 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι)) Filter.atTop Filter.atTop → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n)) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n)) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → 0 < Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n)) → ∀ {cVar : ℝ}, 0 < cVar → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) Filter.atTop (nhds cVar) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":151,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarEstConsistencyConditions.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_feasible_of_relVar","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.VarianceConsistency","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Feasible Wald coverage from L²-relative variance consistency (capstone). For a pair of treatment sequences dk, dl with dk n ≠ dl n for every n, suppose the studentized effect statistic satisfies the local-dependence central limit theorem, every unit has nonzero exposure propensity under dk and under dl, and every off-diagonal pair has nonzero same-arm and nonzero cross-arm joint exposure propensities. If the true effect-estimator variance is everywhere positive and the L²-relative-consistency limit Var[V̂]/Var[τ̂]² → 0 holds, with zq a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the paper's actual interval τ̂ ± zq·√V̂ attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ), Causalean.Experimentation.ExposureMappingInterference.LocalDependenceCLT Exp dk dl → (∀ (n : ℕ), dk n ≠ dl n) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dk n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairSame (Exp n).D (Exp n).f (Exp n).θ i j (dl n) ≠ 0) → (∀ (n : ℕ) (i j : (Exp n).ι), i ≠ j → Causalean.Experimentation.DesignBased.propPairCross (Exp n).D (Exp n).f (Exp n).θ i j (dk n) (dl n) ≠ 0) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → Filter.Tendsto (fun n => (Exp n).D.Var (Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) / (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)) ^ 2) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √(Causalean.Experimentation.ExposureMappingInterference.htEffectVarEst (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z)) Filter.atTop","l":126,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/VarianceConsistency.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_of_conditions","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Aronow–Samii oracle Wald coverage from bounded-degree primitive conditions. For a pair of treatment sequences dk, dl, suppose the per-unit effect summands admit a dependency graph Dg whose neighbourhoods have cardinality at most m, the design variance of the Horvitz–Thompson effect estimator is everywhere positive, and every unit has nonzero exposure probability under dk and under dl. If the summands are pointwise bounded by a nonnegative sequence B n with B n → 0 and population size times B n cubed tending to 0, and zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the oracle Wald interval τ̂ ± zq·√Var[τ̂] attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (Dg : (n : ℕ) → Causalean.SteinMethod.DepGraph (fun i => (Exp n).effSummand (dk n) (dl n) i) (Exp n).D.toMeasure) (m : ℕ), (∀ (n : ℕ) (i : (Exp n).ι), ((Dg n).nbhd i).card ≤ m) → (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (Exp n).ι) * B n ^ 3) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":370,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.ExposureMappingInterference.wald_coverage_of_stein","k":"theorem","m":"Causalean.Experimentation.ExposureMappingInterference.Asymptotics.SteinInstance","a":"Experimentation/ExposureMappingInterference/Asymptotics","t":1,"d":"Aronow–Samii oracle Wald coverage from primitive Stein-discharge conditions. For a pair of treatment sequences dk, dl and a choice of dependency neighbourhoods N n i, suppose the design variance of the Horvitz–Thompson effect estimator is everywhere positive, every unit has nonzero exposure probability under dk and under dl, and the per-unit effect summands are pointwise bounded by a nonnegative sequence B n. If each summand is independent of the sum of summands outside its neighbourhood and the two Stein negligibility limits hold — the design variance of the neighbourhood cross-term sum tends to 0 and the summed third-moment-type error term tends to 0 — and zq is a nonnegative quantile satisfying Φ(zq) = 1 − α/2, then the oracle Wald interval τ̂ ± zq·√Var[τ̂] attains asymptotic (liminf) coverage at least 1 − α.","s":"∀ (Exp : ℕ → Causalean.Experimentation.ExposureMappingInterference.Experiment) (dk dl : (n : ℕ) → (Exp n).Δ) (N : (n : ℕ) → (Exp n).ι → Finset (Exp n).ι), (∀ (n : ℕ), 0 < (Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n))) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dk n) ≠ 0) → (∀ (n : ℕ) (i : (Exp n).ι), Causalean.Experimentation.DesignBased.prop (Exp n).D (Exp n).f (Exp n).θ i (dl n) ≠ 0) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι) (z : (Exp n).Ω), |(Exp n).effSummand (dk n) (dl n) i z| ≤ B n) → (∀ (n : ℕ) (i : (Exp n).ι), ProbabilityTheory.IndepFun ((Exp n).effSummand (dk n) (dl n) i) (fun z => ∑ j ∈ Finset.univ \\ N n i, (Exp n).effSummand (dk n) (dl n) j z) (Exp n).D.toMeasure) → Filter.Tendsto (fun n => (Exp n).D.Var fun z => ∑ i, (Exp n).effSummand (dk n) (dl n) i z * ∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, (Exp n).D.E fun z => |(Exp n).effSummand (dk n) (dl n) i z| * (∑ j ∈ N n i, (Exp n).effSummand (dk n) (dl n) j z) ^ 2) Filter.atTop (nhds 0) → ∀ {α : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - α / 2 → 1 - α ≤ Filter.liminf (fun n => (Exp n).D.Pr fun z => |Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n) z - Causalean.Experimentation.DesignBased.tauTrue (Exp n).y (dk n) (dl n)| ≤ zq * √((Exp n).D.Var (Causalean.Experimentation.DesignBased.htEffect (Exp n).D (Exp n).y (Exp n).f (Exp n).θ (dk n) (dl n)))) Filter.atTop","l":264,"f":"Causalean/Experimentation/ExposureMappingInterference/Asymptotics/SteinInstance.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.E_sampleMean","k":"theorem","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"Unbiasedness of the sample mean (Li & Ding 2017, Thm 1 moments). For a sample size that is positive and at most the population size, the mean of a simple random sample of that size, drawn without replacement from the finite population of outcomes y, is unbiased for the population mean: each unit is sampled with probability n/N, which the 1/n weight averages to 1/N.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n : ℕ) (hn : n ≤ Fintype.card U), 0 < n → ∀ (y : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n hn).E (Causalean.Experimentation.FinitePopulationMoments.sampleMean n y) = Causalean.Experimentation.FinitePopulationMoments.popMean y","l":73,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.Var_sampleMean","k":"theorem","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"Variance of the sample mean (Li & Ding 2017, Thm 1 / Cochran). For a sample size that is positive and at most the population size, when the population contains at least two units, the randomization variance of the simple-random-sample mean equals (1/n − 1/N)·v_N, the sampling-without-replacement variance.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (n : ℕ) (hn : n ≤ Fintype.card U), 0 < n → 2 ≤ Fintype.card U → ∀ (y : U → ℝ), (Causalean.Experimentation.DesignBased.completeRandomization n hn).Var (Causalean.Experimentation.FinitePopulationMoments.sampleMean n y) = (1 / ↑n - 1 / ↑(Fintype.card U)) * Causalean.Experimentation.FinitePopulationMoments.popVar y","l":83,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popMaxSqDev","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"For a nonempty finite population of units and a real-valued outcome for each unit, the maximum squared deviation is the largest squared deviation of an outcome from the finite-population mean.","s":"{U : Type u_1} → [Fintype U] → [Nonempty U] → (U → ℝ) → ℝ","l":56,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popMean","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"For a finite population of units and a real-valued outcome for each unit, the finite-population mean is the sum of all outcomes divided by the number of units, with a zero denominator understood to yield zero.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → ℝ","l":44,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.popVar","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"For a finite population of units and a real-valued outcome for each unit, the finite-population variance is the sum of squared deviations from the finite-population mean divided by one fewer than the number of units, with a zero denominator understood to yield zero.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → ℝ","l":49,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.FinitePopulationMoments.sampleMean","k":"def","m":"Causalean.Experimentation.FinitePopulationMoments","a":"Experimentation","t":1,"d":"For a finite population with decidable unit identity, a nonnegative integer sample size, a real-valued outcome for each unit, and a selected sample containing exactly that many units, the simple-random-sample mean is the sum of the selected outcomes divided by the sample size, with a zero denominator understood to yield zero.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (n : ℕ) → (U → ℝ) → { S // S.card = n } → ℝ","l":65,"f":"Causalean/Experimentation/FinitePopulationMoments.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.E_matchedPairEstimator","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"Unbiasedness of the matched-pair estimator. Provided there is at least one pair, under the matched-pair design the difference-in-means estimator is unbiased for the sample average treatment effect.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ), 0 < Fintype.card P → Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator y1 y0) = Causalean.Experimentation.MatchedPairDesign.sate y1 y0","l":87,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.E_pairContribution","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":2,"d":"The expected per-pair contribution is pair p's average treatment effect ½ ∑_b (y1 p b − y0 p b) — the fair coin gives each position probability ½ of being treated.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ) (p : P), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.pairContribution y1 y0 p) = (∑ b, (y1 p b - y0 p b)) / 2","l":68,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.Var_matchedPairEstimator","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":1,"d":"Variance of the matched-pair estimator. Under the matched-pair design, the difference-in-means estimator built from potential outcomes y1 and y0 has randomization variance equal to 1/(4N²) times the sum of squared within-pair imbalances.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.Var (Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator y1 y0) = (∑ p, Causalean.Experimentation.MatchedPairDesign.pairImbalance y1 y0 p ^ 2) / (4 * ↑(Fintype.card P) ^ 2)","l":76,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.Var_pairContribution","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":2,"d":"The variance of a single pair's contribution is ¼ times its squared within-pair imbalance — the variance of the fair two-point random variable y1 p (z p) − y0 p (¬ z p).","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (y1 y0 : P → Bool → ℝ) (p : P), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.Var (Causalean.Experimentation.MatchedPairDesign.pairContribution y1 y0 p) = Causalean.Experimentation.MatchedPairDesign.pairImbalance y1 y0 p ^ 2 / 4","l":55,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"For a finite collection of pair labels with decidable identity, the matched-pair randomization design independently draws a fair coin for every pair and treats the position selected by that coin, leaving the other position as control. This is a size-two stratified design with one treated unit in each pair rather than independent assignment over all units.","s":"{P : Type u_1} → [inst : Fintype P] → [inst_1 : DecidableEq P] → Causalean.Experimentation.DesignBased.FiniteDesign (P → Bool)","l":54,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"First-order inclusion probability. Under the matched-pair design, the unit occupying position b of pair p is treated with probability 1/2.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p : P) (b : Bool), Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E (Causalean.Experimentation.MatchedPairDesign.mpTreatInd p b) = 1 / 2","l":79,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd_cross","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"Cross-pair independence. For pairs p and p' that are distinct, the probability that position b of p and position b' of p' are treated simultaneously equals ¼ — units in distinct pairs are treated independently, and each position is treated with probability ½, so the joint probability factors as ½ · ½.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p p' : P), p ≠ p' → ∀ (b b' : Bool), (Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E fun z => Causalean.Experimentation.MatchedPairDesign.mpTreatInd p b z * Causalean.Experimentation.MatchedPairDesign.mpTreatInd p' b' z) = 1 / 4","l":103,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_E_mpTreatInd_within","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"Within-pair negative dependence. Under the matched-pair design, the two units of pair p are never treated together, so the expectation of the product of their treatment indicators is zero — perfect negative dependence.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P] (p : P), (Causalean.Experimentation.MatchedPairDesign.matchedPairDesign.E fun z => Causalean.Experimentation.MatchedPairDesign.mpTreatInd p true z * Causalean.Experimentation.MatchedPairDesign.mpTreatInd p false z) = 0","l":91,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairDesign_eq_prod_pairCoin","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":2,"d":"The matched-pair design is the product of the independent pair-level fair coins.","s":"∀ {P : Type u_1} [inst : Fintype P] [inst_1 : DecidableEq P], Causalean.Experimentation.MatchedPairDesign.matchedPairDesign = Causalean.Experimentation.DesignBased.prodDesign fun x => Causalean.Experimentation.MatchedPairDesign.pairCoinDesign","l":62,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.matchedPairEstimator","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"For a finite collection of pairs, the treated and control potential outcomes of every position in every pair, and an assignment selecting a treated position in every pair, the matched-pair difference-in-means estimator is the sum of the observed treated-minus-control contributions across pairs divided by the number of pairs, with a zero denominator understood to yield zero.","s":"{P : Type u_1} → [Fintype P] → (P → Bool → ℝ) → (P → Bool → ℝ) → (P → Bool) → ℝ","l":60,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.mpTreatInd","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"For a pair label, one of its two positions, and an assignment selecting a treated position in every pair, the treatment indicator for that unit equals one when the assignment selects that position and zero otherwise.","s":"{P : Type u_1} → P → Bool → (P → Bool) → ℝ","l":66,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.mpTreatInd_within","k":"theorem","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":2,"d":"Within-pair exclusivity. Exactly one position of each pair is treated: the two units' indicators sum to one on every assignment.","s":"∀ {P : Type u_1} (p : P) (z : P → Bool), Causalean.Experimentation.MatchedPairDesign.mpTreatInd p true z + Causalean.Experimentation.MatchedPairDesign.mpTreatInd p false z = 1","l":72,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairCoinDesign","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.MatchedPair","a":"Experimentation/MatchedPairDesign","t":1,"d":"The per-pair fair-coin randomization design assigns probability one half to each of the two positions: one outcome selects the first position for treatment and the other selects the second position.","s":"Causalean.Experimentation.DesignBased.FiniteDesign Bool","l":48,"f":"Causalean/Experimentation/MatchedPairDesign/MatchedPair.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairContribution","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"For the treated and control potential outcomes of every position in every pair, a particular pair, and an assignment selecting a treated position in every pair, the observed treated-minus-control contribution of that pair is its treated potential outcome at the selected position minus its control potential outcome at the other position.","s":"{P : Type u_1} → (P → Bool → ℝ) → (P → Bool → ℝ) → P → (P → Bool) → ℝ","l":53,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.pairImbalance","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Variance","a":"Experimentation/MatchedPairDesign","t":1,"d":"For the treated and control potential outcomes of every position in every pair and a particular pair, the within-pair imbalance is the difference between the two positions in the sum of their treated and control potential outcomes.","s":"{P : Type u_1} → (P → Bool → ℝ) → (P → Bool → ℝ) → P → ℝ","l":47,"f":"Causalean/Experimentation/MatchedPairDesign/Variance.lean"},{"n":"Causalean.Experimentation.MatchedPairDesign.sate","k":"def","m":"Causalean.Experimentation.MatchedPairDesign.Estimator","a":"Experimentation/MatchedPairDesign","t":1,"d":"For a finite collection of pairs and the treated and control potential outcomes of every position in every pair, the sample average treatment effect is the sum, over both positions of all pairs, of the treated potential outcome minus the control potential outcome, divided by twice the number of pairs, with a zero denominator understood to yield zero.","s":"{P : Type u_1} → [Fintype P] → (P → Bool → ℝ) → (P → Bool → ℝ) → ℝ","l":46,"f":"Causalean/Experimentation/MatchedPairDesign/Estimator.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment","k":"structure","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"An adaptive sequential experiment: a time-indexed information flow ℱ on the outcome space together with a propensity process propensity recording the treatment probability of the unit arriving at each time, subject to three conditions — the process is predictable: the time-0 propensity is measurable with respect to the initial information, and the time-(t+1) propensity depends only on the information available at time t, every propensity is nonnegative, and every propensity is at most one.","s":"(Ω : Type u_1) → MeasurableSpace Ω → Type u_1","l":41,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment.HasOverlap","k":"def","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"For a measurable outcome space, an adaptive experiment on that space, and a real number serving as an overlap margin, the overlap condition with that margin holds precisely when the margin is strictly positive and at every time and on every outcome history, the experiment's treatment probability lies between the margin and one minus the margin.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → Causalean.Experimentation.Sequential.AdaptiveExperiment Ω m0 → ℝ → Prop","l":67,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.AdaptiveExperiment.propensity_pos_of_overlap","k":"theorem","m":"Causalean.Experimentation.Sequential.AdaptiveDesign","a":"Experimentation/Sequential","t":1,"d":"Under overlap with margin δ, every propensity is at least δ, hence strictly positive — so inverse-propensity weights are finite.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {E : Causalean.Experimentation.Sequential.AdaptiveExperiment Ω m0} {δ : ℝ}, E.HasOverlap δ → ∀ (t : ℕ) (ω : Ω), 0 < E.propensity t ω","l":78,"f":"Causalean/Experimentation/Sequential/AdaptiveDesign.lean"},{"n":"Causalean.Experimentation.Sequential.IsAnytimeValid","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"For a measurable sample space, an event on that space, a measure on the sample space, and a real-valued test level, the property of being anytime-valid at that level means that the measure of the event is at most $\\max\\{α,0\\}$.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → Set Ω → MeasureTheory.Measure Ω → ℝ → Prop","l":47,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.IsConfidenceSequence","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"For a measurable sample space, a time-indexed coverage predicate on that space, a measure on the sample space, and a real-valued level, the property of being a confidence sequence at that level means that the measure of outcomes on which coverage fails at at least one time is at most $\\max\\{α,0\\}$.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → (ℕ → Ω → Prop) → MeasureTheory.Measure Ω → ℝ → Prop","l":63,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.IsTestSupermartingale","k":"def","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":1,"d":"Given a sample space, a σ-algebra on that space, a real-valued process indexed by nonnegative integer times, a filtration on that σ-algebra, and a measure on that measurable space, a test supermartingale is a supermartingale adapted to that filtration whose value is nonnegative at every time and sample point and whose expected initial value is at most one.","s":"{Ω : Type u_1} → {m0 : MeasurableSpace Ω} → (ℕ → Ω → ℝ) → MeasureTheory.Filtration ℕ m0 → MeasureTheory.Measure Ω → Prop","l":45,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.confSeqOfWealth","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"For a sample space and a real-valued wealth process indexed by time and outcome and a real-valued level, the wealth-inverted coverage predicate holds at a given time and outcome exactly when the wealth at that time is strictly less than $1/α$, with this quotient understood as zero when $α=0$.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → ℕ → Ω → Prop","l":70,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.isAnytimeValid_rejectionRegion","k":"theorem","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"Anytime-valid type-I error control. If the wealth process M is a test supermartingale for the filtration ℱ under μ and the level α is positive, then the event that M ever reaches 1/α has probability at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsAnytimeValid (Causalean.Experimentation.Sequential.rejectionRegion M α) μ α","l":53,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.isConfidenceSequence_confSeqOfWealth","k":"theorem","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"Confidence-sequence coverage. If M is a test supermartingale for the filtration ℱ under μ and the level α is positive, then the cover obtained by requiring M's wealth to stay below 1/α fails at some time with probability at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → Causalean.Experimentation.Sequential.IsConfidenceSequence (Causalean.Experimentation.Sequential.confSeqOfWealth M α) μ α","l":76,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.rejectionRegion","k":"def","m":"Causalean.Experimentation.Sequential.AnytimeValid","a":"Experimentation/Sequential","t":1,"d":"For a sample space and a real-valued wealth process indexed by time and outcome, and a real-valued test level, the sequential-test rejection region is the event that the wealth reaches or exceeds $1/α$ at at least one time, with this quotient understood as zero when $α=0$.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → Set Ω","l":41,"f":"Causalean/Experimentation/Sequential/AnytimeValid.lean"},{"n":"Causalean.Experimentation.Sequential.supermartingale_maximal_ineq","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":2,"d":"Finite-horizon supermartingale maximal inequality. For a nonnegative supermartingale M and level λ > 0, the probability that M reaches λ by time n is at most E[M₀] / λ.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, MeasureTheory.Supermartingale M ℱ μ → (∀ (n : ℕ), 0 ≤ M n) → ∀ {lam : ℝ}, 0 < lam → ∀ (n : ℕ), μ {ω | lam ≤ (Finset.range (n + 1)).sup' Finset.nonempty_range_add_one fun k => M k ω} ≤ ENNReal.ofReal ((∫ (x : Ω), M 0 x ∂μ) / lam)","l":55,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.ville_inequality","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":1,"d":"Ville's inequality (time-uniform maximal inequality). If M is a supermartingale adapted to the filtration ℱ under the finite measure μ, M is everywhere nonnegative, and the level λ is positive, then the probability that M ever reaches λ is at most E[M₀] / λ, the bound taken over the event of reaching the boundary at some finite time.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, MeasureTheory.Supermartingale M ℱ μ → (∀ (n : ℕ), 0 ≤ M n) → ∀ {lam : ℝ}, 0 < lam → μ {ω | ∃ n, lam ≤ M n ω} ≤ ENNReal.ofReal ((∫ (x : Ω), M 0 x ∂μ) / lam)","l":132,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.Sequential.ville_test","k":"theorem","m":"Causalean.Experimentation.Sequential.Ville","a":"Experimentation/Sequential","t":2,"d":"Ville's inequality for a test supermartingale: the chance of ever reaching 1/α is at most α.","s":"∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {M : ℕ → Ω → ℝ}, Causalean.Experimentation.Sequential.IsTestSupermartingale M ℱ μ → ∀ {α : ℝ}, 0 < α → μ {ω | ∃ n, 1 / α ≤ M n ω} ≤ ENNReal.ofReal α","l":163,"f":"Causalean/Experimentation/Sequential/Ville.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_eq_locProd","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"The network-HAC estimator equals the localized double sum ∑ᵢ Xᵢ · (∑_{j ∈ N i} Xⱼ): the pointwise identity that lets the dependency-graph variance bound apply to V̂ verbatim.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) (ω : Ω), F.netHACVarEst ω = ∑ i, F.X i ω * ∑ k ∈ F.toDepGraph.nbhd i, F.X k ω","l":44,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_memLp","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.Consistency","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"The network-HAC estimator is in L². With summands bounded by B and degree ≤ m, the estimator V̂ is pointwise bounded by card(V)·m·B², hence square-integrable; this is the moment hypothesis Chebyshev's inequality needs.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ] {B : ℝ}, 0 ≤ B → (∀ (i : V) (ω : Ω), |F.X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : V), (F.toDepGraph.nbhd i).card ≤ m) → MeasureTheory.MemLp (fun ω => F.netHACVarEst ω) 2 μ","l":42,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/Consistency.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_variance_le","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":1,"d":"Estimator-variance bound. For a super-population network field F, given a nonnegative bound B such that every summand is bounded in absolute value by B and every network neighborhood has size at most m, the variance of the network-HAC estimator V̂ = ∑ᵢ ∑_{j∈N i} Xᵢ Xⱼ is at most 2·m⁵·card(V)·B⁴.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ] {B : ℝ}, 0 ≤ B → (∀ (i : V) (ω : Ω), |F.X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : V), (F.toDepGraph.nbhd i).card ≤ m) → ProbabilityTheory.variance (fun ω => F.netHACVarEst ω) μ ≤ 2 * ↑m ^ 5 * ↑(Fintype.card V) * B ^ 4","l":50,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHACVarEst_variance_tendsto_zero","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.VarianceBound","a":"Experimentation/SuperPopulation/HACConsistency","t":2,"d":"Estimator variance tends to zero. Along a sequence of super-population network fields F n in the CLT regime — bounded degree m, summands bounded by B n with B n → 0, and card(V n)·(B n)³ → 0 — the variance of the network-HAC estimator tends to zero.","s":"∀ {V : ℕ → Type u_3} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_4} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => (F n).netHACVarEst ω) (μ n)) Filter.atTop (nhds 0)","l":70,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/VarianceBound.lean"},{"n":"Causalean.Experimentation.SuperPopulation.HACConsistency.netHAC_consistent","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HACConsistency.Consistency","a":"Experimentation/SuperPopulation/HACConsistency","t":1,"d":"HAC consistency (convergence in probability). Consider a sequence of super-population network fields F n over probability spaces with measures μ n, each in the CLT regime — dependency-graph degree at most m, summands uniformly bounded by a sequence B n tending to zero with card(Vₙ)·(Bₙ)³ → 0, and square-integrable, mean-zero summands — and an arbitrary error tolerance ε > 0. Then the network-HAC variance estimator converges in probability to the variance of the network sum: the probability that the estimator deviates from that variance by at least ε tends to zero as n → ∞.","s":"∀ {V : ℕ → Type u_3} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_4} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : V n), MeasureTheory.MemLp ((F n).X i) 2 (μ n)) → (∀ (n : ℕ) (i : V n), ∫ (ω : Ω n), (F n).X i ω ∂μ n = 0) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (μ n).real {ω | ε ≤ |(F n).netHACVarEst ω - ProbabilityTheory.variance (Causalean.SteinMethod.depSum (F n).X) (μ n)|}) Filter.atTop (nhds 0)","l":85,"f":"Causalean/Experimentation/SuperPopulation/HACConsistency/Consistency.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField","k":"def","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"Given a finite population of units, a measurable sample space, a measure on that space, a real-valued outcome for every unit and sample point, a decidable relation between units, the assumption that every unit is related to itself, the assumption that this relation is symmetric, the assumption that every outcome is measurable, the assumption that outcome vectors for any two finite sets with no relation joining them are independent, and a real normalizing constant, the centered and normalized network field has, for each unit, its outcome minus its expectation under the given measure, divided by that constant; it retains the supplied relation and has independent vectors on finite sets with no relation joining them.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (Y : V → Ω → ℝ) → (adj : V → V → Prop) → [DecidableRel adj] → (∀ (i : V), adj i i) → (∀ (i j : V), adj i j → adj j i) → (∀ (i : V), Measurable (Y i)) → (∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) → ℝ → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ","l":41,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_X","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"The standardized summand for unit i at sample point ω is the centered outcome Yᵢ ω − E[Yᵢ] divided by the normalizing constant s.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ) (i : V) (ω : Ω), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω = (Y i ω - ∫ (x : Ω), Y i x ∂μ) / s","l":83,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_abs_le","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":2,"d":"Uniform summand bound. If the centered outcomes are bounded, |Yᵢ − E[Yᵢ]| ≤ c, then each standardized summand satisfies |Xᵢ| ≤ 2c/s (with s > 0). The tight bound is c/s, weakened to 2c/s to match the engine's card·Bₙ³ → 0 smallness with Bₙ = 2c/s.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ), 0 < s → ∀ (c : ℝ), (∀ (i : V) (ω : Ω), |Y i ω - ∫ (x : Ω), Y i x ∂μ| ≤ c) → ∀ (i : V) (ω : Ω), |(Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω| ≤ 2 * c / s","l":124,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_adj","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Field","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"The interference network underlying the standardized field is exactly the supplied network adj.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_3 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ) (s : ℝ), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).adj = adj","l":89,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Field.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_integral_eq_zero","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":2,"d":"Mean-zero summands. Each standardized summand has integral zero: E[Xᵢ] = (E[Yᵢ] − E[Yᵢ]) / s = 0. (Uses integrability of Yᵢ, from MemLp Yᵢ 2.)","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_4 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ), (∀ (i : V), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (s : ℝ) (i : V), ∫ (ω : Ω), (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X i ω ∂μ = 0","l":39,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField_sq_integral","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.Hypotheses","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"Unit total variance. If each outcome is square-integrable, the normalizing constant s is positive, and s² equals the variance of the network sum of outcomes, s² = Var(∑ᵢ Yᵢ), then the standardized network sum ∑ᵢ Xᵢ = (∑ᵢ Yᵢ − ∑ᵢ E[Yᵢ]) / s has unit total variance: ∫ (∑ᵢ Xᵢ)² = 1. This is the field-variance hypothesis of networkSum_clt (∫ (depSum X)² = 1), derived from the outcome sum-variance.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : V → Ω → ℝ) (adj : V → V → Prop) [inst_4 : DecidableRel adj] (hrefl : ∀ (i : V), adj i i) (hsymm : ∀ (i j : V), adj i j → adj j i) (hmeasY : ∀ (i : V), Measurable (Y i)) (hindepY : ∀ (A B : Finset V), (∀ a ∈ A, ∀ b ∈ B, ¬adj a b) → ProbabilityTheory.IndepFun (fun ω k => Y (↑k) ω) (fun ω k => Y (↑k) ω) μ), (∀ (i : V), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (s : ℝ), 0 < s → s ^ 2 = ProbabilityTheory.variance (fun ω => ∑ i, Y i ω) μ → ∫ (ω : Ω), Causalean.SteinMethod.depSum (Causalean.Experimentation.SuperPopulation.MeanCLT.centeredNormalizedField Y adj hrefl hsymm hmeasY hindepY s).X ω ^ 2 ∂μ = 1","l":49,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/Hypotheses.lean"},{"n":"Causalean.Experimentation.SuperPopulation.MeanCLT.networkMean_clt","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.MeanCLT.MeanCLT","a":"Experimentation/SuperPopulation/MeanCLT","t":1,"d":"Asymptotic normality of a centered super-population network sum. Fix a family of outcomes Y n over probability spaces with measures μ n and a reflexive, symmetric adjacency relation adj n recording which units interfere, with every outcome measurable, outcome tuples on non-adjacent unit sets independent — i.e. m-dependence — and adjacency degree bounded by m. Assume every outcome is square-integrable, a positive normalizing constant s n with (s n)² = Var(∑ᵢ Yₙᵢ), outcomes uniformly bounded around their means by a sequence c n, and the negligibility rate card(Vₙ)·(cₙ/sₙ)³ → 0. Then the network sum, centered by subtracting the sum of the individual outcome means and divided by s n, converges in distribution to the standard normal.","s":"∀ {V : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (V n)] {Ω : ℕ → Type u_2} [inst_1 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (Y : (n : ℕ) → V n → Ω n → ℝ) (adj : (n : ℕ) → V n → V n → Prop) [inst_3 : (n : ℕ) → DecidableRel (adj n)], (∀ (n : ℕ) (i : V n), adj n i i) → (∀ (n : ℕ) (i j : V n), adj n i j → adj n j i) → (∀ (n : ℕ) (i : V n), Measurable (Y n i)) → (∀ (n : ℕ) (A B : Finset (V n)), (∀ a ∈ A, ∀ b ∈ B, ¬adj n a b) → ProbabilityTheory.IndepFun (fun ω k => Y n (↑k) ω) (fun ω k => Y n (↑k) ω) (μ n)) → ∀ (m : ℕ), (∀ (n : ℕ) (i : V n), {j | adj n i j}.card ≤ m) → (∀ (n : ℕ) (i : V n), MeasureTheory.MemLp (Y n i) 2 (μ n)) → ∀ (s : ℕ → ℝ), (∀ (n : ℕ), 0 < s n) → (∀ (n : ℕ), s n ^ 2 = ProbabilityTheory.variance (fun ω => ∑ i, Y n i ω) (μ n)) → ∀ (c : ℕ → ℝ), (∀ (n : ℕ) (i : V n) (ω : Ω n), |Y n i ω - ∫ (x : Ω n), Y n i x ∂μ n| ≤ c n) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * (c n / s n) ^ 3) Filter.atTop (nhds 0) → ∀ (t : ℝ), Filter.Tendsto (fun n => (μ n).real {ω | (∑ i, Y n i ω - ∑ i, ∫ (x : Ω n), Y n i x ∂μ n) / s n ≤ t}) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic t)))","l":36,"f":"Causalean/Experimentation/SuperPopulation/MeanCLT/MeanCLT.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence","k":"structure","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"A super-population locally-dependent network field: bundles node-level random summands on a common ambient probability space together with a network relation between units that is reflexive and symmetric, requires every summand to be measurable, and requires any two collections of units joined by no edge to carry independent summand tuples (exact m-dependence beyond the network). This is the model-based counterpart of the finite design — the randomness is the population draw, not the assignment — and the m-dependence sibling of an i.i.d. sample.","s":"(V : Type u_3) → (Ω : Type u_4) → [Fintype V] → [DecidableEq V] → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type (max u_3 u_4)","l":39,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.mem_nbhd_iff","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Unit j lies in unit i's network neighborhood if and only if i and j are adjacent in the underlying interference network.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) {i j : V}, j ∈ F.nbhd i ↔ F.adj i j","l":93,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.nbhd","k":"def","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Given a finite population of units, a measurable sample space, a measure on that space, a super-population locally dependent network field, and a unit, the closed network neighborhood of that unit is the finite set of all units related to it, including the unit itself by reflexivity.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ → V → Finset V","l":86,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.netHACVarEst","k":"def","m":"Causalean.Experimentation.SuperPopulation.HAC","a":"Experimentation/SuperPopulation","t":1,"d":"Given a finite population of units, a measurable sample space, a measure on that space, a super-population locally dependent network field, and a sample point, the network-HAC variance estimator is the sum, over every unit and every unit in its closed network neighborhood, of the product of their realized summands. It targets the variance of the network sum, whose cross-terms vanish outside the network neighborhoods.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ → Ω → ℝ","l":33,"f":"Causalean/Experimentation/SuperPopulation/HAC.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.netHACVarEst_integral_eq_variance","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.HAC","a":"Experimentation/SuperPopulation","t":1,"d":"Unbiasedness of the network-HAC estimator. Under a probability measure with square- integrable summands that are mean zero, the expectation of the network-HAC estimator equals the variance of the network sum: E[V̂] = Var(∑ᵢ Xᵢ).","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : V), MeasureTheory.MemLp (F.X i) 2 μ) → (∀ (i : V), ∫ (ω : Ω), F.X i ω ∂μ = 0) → ∫ (ω : Ω), F.netHACVarEst ω ∂μ = ProbabilityTheory.variance (Causalean.SteinMethod.depSum F.X) μ","l":41,"f":"Causalean/Experimentation/SuperPopulation/HAC.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.self_mem_nbhd","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Every unit lies in its own network neighborhood.","s":"∀ {V : Type u_1} {Ω : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] [inst_2 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) (i : V), i ∈ F.nbhd i","l":98,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.NetworkDependence.toDepGraph","k":"def","m":"Causalean.Experimentation.SuperPopulation.Basic","a":"Experimentation/SuperPopulation","t":1,"d":"Given a finite population of units, a measurable sample space, a measure on that space, and a super-population locally dependent network field, its dependency graph representation is the graph with the same random summands, unit relation, reflexivity, symmetry, measurability, and finite-set independence condition.","s":"{V : Type u_1} → {Ω : Type u_2} → [inst : Fintype V] → [inst_1 : DecidableEq V] → [inst_2 : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → (F : Causalean.Experimentation.SuperPopulation.NetworkDependence V Ω μ) → Causalean.SteinMethod.DepGraph F.X μ","l":72,"f":"Causalean/Experimentation/SuperPopulation/Basic.lean"},{"n":"Causalean.Experimentation.SuperPopulation.networkSum_clt","k":"theorem","m":"Causalean.Experimentation.SuperPopulation.CLT","a":"Experimentation/SuperPopulation","t":1,"d":"m-dependent network CLT (super-population). Consider a sequence of super-population network fields F n, each on a probability space with measure μ n, whose dependency graph has degree at most m, whose summands are uniformly bounded in absolute value by a sequence B n tending to zero fast enough that card(Vₙ)·(Bₙ)³ → 0, and whose summands are mean zero with the network sum having unit total variance for every n. Then the network sum's cumulative distribution function converges pointwise to the standard normal CDF: P[∑ᵢ (F n).X i ≤ s] → Φ(s).","s":"∀ {V : ℕ → Type u_1} [inst : (n : ℕ) → Fintype (V n)] [inst_1 : (n : ℕ) → DecidableEq (V n)] {Ω : ℕ → Type u_2} [inst_2 : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (F : (n : ℕ) → Causalean.Experimentation.SuperPopulation.NetworkDependence (V n) (Ω n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : V n), ((F n).toDepGraph.nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : V n) (ω : Ω n), |(F n).X i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (V n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : V n), ∫ (ω : Ω n), (F n).X i ω ∂μ n = 0) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (F n).X ω ^ 2 ∂μ n = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (F n).X) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":29,"f":"Causalean/Experimentation/SuperPopulation/CLT.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_direct","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a within-group randomization design for every group, and the potential outcome of each unit under every within-group assignment, the Hudgens--Halloran direct-effect contrast is the population average potential outcome under treatment minus that under control, both evaluated under the supplied design.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":109,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_indirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a first within-group randomization design for every group, a second such design for every group, and the potential outcome of each unit under every within-group assignment, the indirect, or spillover, causal effect is the population average potential outcome under control using the second design minus that using the first design.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_overall","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a first within-group randomization design for every group, a second such design for every group, and the potential outcome of each unit under every within-group assignment, the overall causal effect is the population marginal average potential outcome using the second design minus that using the first design.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":139,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_total","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a first within-group randomization design for every group, a second such design for every group, and the potential outcome of each unit under every within-group assignment, the total causal effect is the population average potential outcome under control using the second design minus that under treatment using the first design.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":127,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.CE_total_decomp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Mixed-orientation decomposition identity. For per-group two-stage designs ψ (treatment strategy) and φ (control strategy) governing each group's within-group assignment and an outcome recorded for every group, unit, and realized within-group assignment, the total contrast — the population control-state mean under φ minus the population treated-state mean under ψ — equals the indirect contrast — the population control-state mean under φ minus the population control-state mean under ψ — minus the direct contrast — the population treated-state mean under ψ minus the population control-state mean under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {n : ι → ℕ} (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ), Causalean.Experimentation.TwoStageInterference.CE_total ψ φ Y = Causalean.Experimentation.TwoStageInterference.CE_indirect ψ φ Y - Causalean.Experimentation.TwoStageInterference.CE_direct ψ Y","l":136,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.DEbar_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Estimand reduction. Under the homogeneity and regularity bundle, the population average treatment-minus-control direct-effect contrast collapses to the common group-level contrast δ.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), (Exp n).DEbar = δ","l":177,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_Shat","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceMoments","a":"Experimentation/TwoStageInterference","t":1,"d":"Expectation of an observed sample variance. Let U be a {0,1}-valued selection family over a design ρ on length-n binary assignments, so that every Uⱼ is idempotent, taking only the values 0 and 1, and suppose the group size n and n − 1 are both nonzero, as are the real-valued selection count M and M − 1. If each Uⱼ has first moment M/n, every two distinct units j and k have second moment M(M−1)/(n(n−1)) for the product Uⱼ·Uₖ, and exactly M units are selected on every assignment the design gives positive weight, then the expectation of the realized sample variance of x over the selected units — (1/(M−1))∑ⱼ Uⱼ(xⱼ − x̄_U)² with x̄_U = (∑ Uⱼxⱼ)/M — equals the population sample variance (1/(n−1))∑ⱼ(xⱼ − x̄)².","s":"∀ {n : ℕ} (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)) (M : ℝ) (x : Fin n → ℝ) (U : Fin n → (Fin n → Bool) → ℝ), ↑n ≠ 0 → ↑n - 1 ≠ 0 → M ≠ 0 → M - 1 ≠ 0 → (∀ (j : Fin n) (w : Fin n → Bool), U j w * U j w = U j w) → (∀ (j : Fin n), ρ.E (U j) = M / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => U j w * U k w) = M * (M - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, U j w = M) → (ρ.E fun w => (∑ j, U j w * (x j - (∑ i, U i w * x i) / M) ^ 2) / (M - 1)) = (∑ j, (x j - (∑ i, x i) / ↑n) ^ 2) / (↑n - 1)","l":64,"f":"Causalean/Experimentation/TwoStageInterference/VarianceMoments.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_ShatControl","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Expected observed control sample variance. The control analogue: for any design satisfying the same completely-randomized moment hypotheses, the expectation of the realized untreated-state sample variance among the n−K control units equals the population untreated-state sample variance S₀: E[Ŝ₀] = S₀.","s":"∀ {n : ℕ} (K : ℕ) (b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.E (Causalean.Experimentation.TwoStageInterference.ShatControl K b) = Causalean.Experimentation.TwoStageInterference.S0 b","l":165,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_ShatTreated","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Expected observed treated sample variance. For any within-group design whose treatment indicators satisfy the completely-randomized moment hypotheses — first moment K/n (hmean), pairwise second moment K(K−1)/(n(n−1)) (hpair), and a deterministic treated count K on the support (hsupp) — the expectation of the realized treated-state sample variance equals the population treated-state sample variance S₁: E[Ŝ₁] = S₁.","s":"∀ {n : ℕ} (K : ℕ) (a : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.E (Causalean.Experimentation.TwoStageInterference.ShatTreated K a) = Causalean.Experimentation.TwoStageInterference.S1 a","l":143,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_congr_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceMoments","a":"Experimentation/TwoStageInterference","t":2,"d":"Support congruence for expectation. Two random variables that agree on every assignment the design gives positive weight have equal expectation, since the off-support assignments contribute 0 = p w · _ to the finite sum either way.","s":"∀ {n : ℕ} (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)) {f g : (Fin n → Bool) → ℝ}, (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → f w = g w) → ρ.E f = ρ.E g","l":53,"f":"Causalean/Experimentation/TwoStageInterference/VarianceMoments.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estDirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Direct-contrast unbiasedness (Theorem 1 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size C of ψ-selected groups is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by ψ each unit's control propensity is m0/n and each unit's treatment propensity is m1/n, and that the stage-1 design selects each group into the ψ arm with probability C/N. Then the Horvitz–Thompson estimator built from the ψ-selected groups is unbiased for the direct-effect contrast — the population average outcome under treatment minus under control, both evaluated under strategy ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0 m1 : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m0 i ≠ 0) → (∀ (i : ι), m1 i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0 i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1 i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estDirect Y m0 m1 C) = Causalean.Experimentation.TwoStageInterference.CE_direct ψ Y","l":143,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estIndirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Indirect-effect unbiasedness (Theorem 2 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm control unit count m0ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's control propensity is m0ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the indirect-effect estimator is unbiased for the spillover contrast: the population average control outcome under φ minus the population average control outcome under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0φ m0ψ : ι → ℝ) (dφ dψ : ℝ), dφ ≠ 0 → dψ ≠ 0 → (∀ (i : ι), m0φ i ≠ 0) → (∀ (i : ι), m0ψ i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((φ i).Pr fun w => w j = false) = m0φ i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0ψ i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = false) = dφ / ↑(Fintype.card ι)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = dψ / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estIndirect Y m0φ m0ψ dφ dψ) = Causalean.Experimentation.TwoStageInterference.CE_indirect ψ φ Y","l":168,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_estTotal","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Total-effect unbiasedness (Theorem 3 contrast). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, assume the target sample size dφ of φ-selected groups is nonzero, the target sample size dψ of ψ-selected groups is nonzero, every group's φ-arm control unit count m0φ is nonzero, every group's ψ-arm treatment unit count m1ψ is nonzero, and every group's size n is nonzero. Suppose that within each group randomized by φ each unit's control propensity is m0φ/n, that within each group randomized by ψ each unit's treatment propensity is m1ψ/n, that the stage-1 design selects each group into the φ arm with probability dφ/N, and that it selects each group into the ψ arm with probability dψ/N. Then the total-effect estimator is unbiased for the contrast between the population average control outcome under φ and the population average treatment outcome under ψ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0φ m1ψ : ι → ℝ) (dφ dψ : ℝ), dφ ≠ 0 → dψ ≠ 0 → (∀ (i : ι), m0φ i ≠ 0) → (∀ (i : ι), m1ψ i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((φ i).Pr fun w => w j = false) = m0φ i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1ψ i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = false) = dφ / ↑(Fintype.card ι)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = dψ / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.estTotal Y m0φ m1ψ dφ dψ) = Causalean.Experimentation.TwoStageInterference.CE_total ψ φ Y","l":197,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_groupDiff_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The conditional mean of the per-group contrast estimator under the ψ-design is the common group-level contrast: (ψ i).E (dᵢ) = δ. Immediate from E_groupEst (twice) and homogeneity.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) (i : (Exp n).ι), ((Exp n).ψ i).E (Causalean.Experimentation.TwoStageInterference.groupDiff (Exp n) i) = δ","l":268,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_groupEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Within-group unbiasedness. Fix a per-group design ρ governing the within-group treatment randomization and an outcome recorded for every group, unit, and realized within-group assignment, together with a group i, a treatment state z, and a nonzero real number m used as the treated-count denominator. Assume group i has a nonzero number of units and that every unit of group i receives treatment state z with the same probability m divided by the group's size. Then the expected value, under ρ, of the empirical mean outcome among the z-treated units of group i equals the group's average potential outcome under z.","s":"∀ {ι : Type u_1} {n : ι → ℕ} (ρ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (i : ι) (z : Bool) (m : ℝ), m ≠ 0 → ↑(n i) ≠ 0 → (∀ (j : Fin (n i)), ((ρ i).Pr fun w => w j = z) = m / ↑(n i)) → (ρ i).E (Causalean.Experimentation.TwoStageInterference.groupEst Y i z m) = Causalean.Experimentation.TwoStageInterference.groupMean ρ Y i z","l":45,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_popEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Unbiased","a":"Experimentation/TwoStageInterference","t":1,"d":"Population unbiasedness (Theorem 1). Consider the two per-group designs ψ and φ governing the within-group randomization when a group is respectively assigned the ψ-strategy or the φ-strategy at stage 1 and an outcome recorded for every group, unit, and realized within-group assignment. Fix a treatment state z, a nonzero real number C used as the population-estimator denominator, and a family m with every group's value m i nonzero; suppose every group has a nonzero number of units, within every ψ-assigned group every unit receives treatment state z with the same probability m i divided by the group's size, and the marginal probability of each group being assigned the ψ-strategy at stage 1 equals C divided by the number of groups. Then the expected value, under the compound two-stage design, of the population estimator restricted to the ψ-assigned groups equals the population average potential outcome under z computed from the ψ-design.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.popEst Y z true m C) = Causalean.Experimentation.TwoStageInterference.popMean ψ Y z","l":83,"f":"Causalean/Experimentation/TwoStageInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_popEst_pick","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"Population unbiasedness, either selection (generalizing Theorem 1). For the two-stage design that allocates groups to strategy ψ or strategy φ and records outcomes via the potential-outcome function Y, fix an arbitrary selection flag pick together with a within-group design ρ meant to govern every group whose stage-1 flag equals pick, where on the event that a group's stage-1 flag equals pick, its conditional within-group design (ψ if flagged true, φ if flagged false) actually equals ρ. Assume the normalizing group count denom is nonzero, every group's unit count m at treatment level z is nonzero, and every group's size n is nonzero. Suppose that within each group governed by ρ each unit's propensity of being assigned treatment level z is m/n, and that the stage-1 design selects each group flagged pick with probability denom/N. Then the population estimator on the groups selected by pick is unbiased for the population average potential outcome at level z computed under design ρ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (pick : Bool) (denom : ℝ) (ρ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)), denom ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (s : Causalean.Experimentation.TwoStageInterference.StratAssign ι) (i : ι), s i = pick → (if s i = true then ψ i else φ i) = ρ i) → (∀ (i : ι) (j : Fin (n i)), ((ρ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = pick) = denom / ↑(Fintype.card ι)) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).E (Causalean.Experimentation.TwoStageInterference.popEst Y z pick m denom) = Causalean.Experimentation.TwoStageInterference.popMean ρ Y z","l":64,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_varHat_conservative","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Conservativeness of the within-group variance estimator (Hudgens–Halloran 2008, Eq. 9). For any design satisfying the completely-randomized moment hypotheses (hmean/hpair/hsupp) and treating exactly K of n units (with 2 ≤ K and K + 2 ≤ n, so both sample variances are well defined), the conservative estimator overstates the randomization variance, Var(τ̂) ≤ E[v̂ar].","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 2 ≤ K → K + 2 ≤ n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → (∀ (w : Fin n → Bool), ρ.p w ≠ 0 → ∑ j, Causalean.Experimentation.TwoStageInterference.T j w = ↑K) → ρ.Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) ≤ ρ.E (Causalean.Experimentation.TwoStageInterference.varHat K a b)","l":225,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.E_varHat_conservative_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Conservativeness of the within-group variance estimator, for the completely randomized design (Hudgens–Halloran 2008, Eq. 9). For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where at least two units are treated and at least two units remain in control (K + 2 ≤ n) — so both within-group sample variances are well defined. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design is at most the expectation, under the same design, of the conservative variance estimator v̂ar.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 2 ≤ K → ∀ (hKn2 : K + 2 ≤ n), (Causalean.Experimentation.TwoStageInterference.crd K (le_trans (Nat.le_add_right K 2) hKn2)).Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) ≤ (Causalean.Experimentation.TwoStageInterference.crd K (le_trans (Nat.le_add_right K 2) hKn2)).E (Causalean.Experimentation.TwoStageInterference.varHat K a b)","l":268,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.FiniteDesign.Var_add_const","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"Variance is invariant under adding a constant: Var(X + c) = Var X.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (X : Ω → ℝ) (c : ℝ), (D.Var fun z => X z + c) = D.Var X","l":57,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Homogeneous","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Homogeneity and regularity bundle for a sequence of Liu–Hudgens experiments. Faithfully encodes the hypotheses of Proposition 5.1 for the Hudgens-Halloran orientation: the studentized statistic is the standardized contrast estimator; every group-level direct-effect contrast equals a common value δ (homogeneity); every within-group contrast-estimator variance equals a common value v(n) that is positive; the centered per-group contrast estimator is uniformly bounded; every stage-1 selection supported by the design flags exactly C groups; the resulting rate sequence tends to zero together with its cubed Lyapunov rate (the many-groups asymptotic regime); and the conditional distribution of the studentized statistic does not depend on which stage-1 selection occurred, the analytic form of homogeneity that lifts the conditional CLT to the average.","s":"(Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) → ℝ → ((n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ) → ℝ → ℝ → (ℕ → ℝ) → Prop","l":134,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Reference data for one identical-groups experiment. All groups in the population share a common size K, a common pair of within-group allocation strategies ψ₀ and φ₀, common potential outcomes Y₀, and common control and treatment unit counts, each assumed nonzero, as is the group size K. A stage-1 design D₁ assigns each group a strategy, selecting a nonzero number C of groups for ψ, out of a population of at least two groups; every unit's within-group control propensity equals m0₀/K and its treatment propensity equals m1₀/K, every group's stage-1 selection propensity equals C/N, and every pair's joint selection propensity equals C(C−1)/(N(N−1)). Together these assemble the constant LHExperiment toExp.","s":"Type 1","l":107,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.condDesign_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The conditional design of the constant experiment is the genuine product, over the non-dependent space R.ι → (Fin K → Bool), of the per-group strategy designs.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι), Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s = Causalean.Experimentation.DesignBased.prodDesign fun i => if s i = true then R.ψ₀ else R.φ₀","l":207,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.estD_equivariant","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Estimator equivariance. Because all groups are identical, the aggregate contrast estimator sees a selection only through its pattern: if σ aligns the patterns of s' and s (s' i = s (σ i)), then evaluating at s' and the relabeled assignment w ∘ σ equals the value at s and w.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι) (σ : R.ι ≃ R.ι), (∀ (i : R.ι), s' i = s (σ i)) → ∀ (w : R.ι → Fin R.K → Bool), R.toExp.estD (s', fun i => w (σ i)) = R.toExp.estD (s, w)","l":220,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.groupDiff_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment, every group's contrast estimator equals the common groupDiff₀.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), Causalean.Experimentation.TwoStageInterference.groupDiff R.toExp i = R.groupDiff₀","l":203,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.groupDiff₀","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment and a realized assignment within its common-size group, the common per-group treatment-minus-control contrast estimator is the sum of potential outcomes for treated units divided by the common treated count minus the analogous sum for control units divided by the common control count.","s":"(R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) → (Fin R.K → Bool) → ℝ","l":195,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hhom_of_identical","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Derived homogeneity hypothesis under identical groups. For two stage-1 selections each flagging exactly C groups, the conditional studentized CDF is the same. This is exactly the analytic homogeneity hypothesis hhom, now a theorem: the selection-symmetry permutation σ (exists_equiv_selection) relabels the conditional product design (prodDesign_Pr_reindex) while carrying the studentized statistic (studId_equivariant), so the two CDFs coincide.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t : ℝ) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι), (∑ i, if s i = true then 1 else 0) = R.C → (∑ i, if s' i = true then 1 else 0) = R.C → ((Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s).Pr fun w => R.studId (s, w) ≤ t) = (Causalean.Experimentation.TwoStageInterference.condDesign R.toExp s').Pr fun w => R.studId (s', w) ≤ t","l":244,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hv_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment every group's within-group variance equals the common refVar.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), (R.toExp.ψ i).Var (Causalean.Experimentation.TwoStageInterference.groupDiff R.toExp i) = R.refVar","l":311,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.hδ_toExp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"In the constant experiment every group's level contrast equals the common refDelta.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (i : R.ι), Causalean.Experimentation.TwoStageInterference.groupMean R.toExp.ψ R.toExp.Y i true - Causalean.Experimentation.TwoStageInterference.groupMean R.toExp.ψ R.toExp.Y i false = R.refDelta","l":306,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refDelta","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment, the common group-level direct-effect contrast is its common group-average potential outcome under treatment minus that under control.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → ℝ","l":296,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refGroupMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment and a treatment status, the common group-average potential outcome is the arithmetic mean over units in the common group of their expected potential outcomes under the reference allocation strategy, conditional on their own treatment having that status.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → Bool → ℝ","l":288,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.refVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment, the common within-group contrast-estimator variance is the variance, under the reference first allocation strategy, of the common per-group treatment-minus-control contrast estimator.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → ℝ","l":301,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.studId","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment and a realized first-stage strategy assignment together with a realized within-group assignment for every group, the studentized treatment-minus-control contrast statistic is the aggregate direct-effect estimator minus its population direct effect, divided by the square root of its direct-effect variance.","s":"(R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) → Causalean.Experimentation.TwoStageInterference.StratAssign R.ι × (R.ι → Fin R.K → Bool) → ℝ","l":212,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.studId_equivariant","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Studentized equivariance. The studentized statistic is likewise selection-pattern equivariant, since DEbar and directVar are scalars and the estimator is equivariant.","s":"∀ (R : Causalean.Experimentation.TwoStageInterference.IdenticalRef) (s s' : Causalean.Experimentation.TwoStageInterference.StratAssign R.ι) (σ : R.ι ≃ R.ι), (∀ (i : R.ι), s' i = s (σ i)) → ∀ (w : R.ι → Fin R.K → Bool), R.studId (s', fun i => w (σ i)) = R.studId (s, w)","l":234,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.IdenticalRef.toExp","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given reference data for one identical-groups experiment, the constant Liu--Hudgens experiment assigns every group the common size, two common within-group randomization designs, common potential outcomes, and common treated and control counts in that reference data. Thus its conditional within-group randomization is a product across groups.","s":"Causalean.Experimentation.TwoStageInterference.IdenticalRef → Causalean.Experimentation.TwoStageInterference.LHExperiment","l":168,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment","k":"structure","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"A single Liu–Hudgens (2014) two-stage experiment, packaged so that a sequence of them can be studied for large-sample inference. Carries a finite population of groups with per-group sizes, a stage-1 design assigning each group a strategy, the per-group allocation strategies ψ and φ, partial-interference potential outcomes, and design-fixed control and treatment unit counts per group, together with the regularity conditions that the number C of ψ-selected groups is nonzero, every group has nonzero control and treatment counts and nonzero size, the population has at least one group and at least two groups, every unit's within-group control propensity equals m0 i / nᵢ and its treatment propensity equals m1 i / nᵢ, every group's stage-1 selection propensity equals C/N, and every pair's joint selection propensity equals C(C−1)/(N(N−1)) — exactly the hypothesis lists of E_estDirect and Var_estDirect.","s":"Type 1","l":52,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.DEbar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment, the population-average treatment-minus-control direct effect is the mean potential outcome under treatment minus the mean potential outcome under control, with both means evaluated under the experiment's treatment strategy.","s":"Causalean.Experimentation.TwoStageInterference.LHExperiment → ℝ","l":126,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.E_estD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Unbiasedness bridge. The Horvitz–Thompson estimator is unbiased for the population average treatment-minus-control direct-effect contrast.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), E.jointD.E E.estD = E.DEbar","l":137,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.directVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment, the closed-form two-stage design variance of the treatment-minus-control direct-effect estimator equals $(1-C/N)S_\\mu^2/C+(CN)^{-1}\\sum_i V_i$, where $N$ is the number of groups, $C$ is the number assigned the treatment strategy, $S_\\mu^2$ is the population sample variance of the group-level direct-effect contrasts, and $V_i$ is the variance of group $i$'s within-group contrast estimator under the treatment strategy.","s":"Causalean.Experimentation.TwoStageInterference.LHExperiment → ℝ","l":129,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.estD","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment, the Horvitz--Thompson estimator of its treatment-minus-control direct effect assigns a real-valued estimate to every joint realization of the group-level strategy assignment and all within-group assignments.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.TwoStageInterference.StratAssign E.ι × ((i : E.ι) → Fin (E.gsize i) → Bool) → ℝ","l":122,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.jointD","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment, the joint two-stage randomization design first draws the group-level strategy assignment and then draws each group's within-group assignment from the design corresponding to its assigned strategy.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign E.ι × ((i : E.ι) → Fin (E.gsize i) → Bool))","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.LHExperiment.var_estD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Setup","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Variance bridge. The design variance of the treatment-minus-control direct-effect contrast estimator equals the closed-form two-stage variance directVar.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), E.jointD.Var E.estD = E.directVar","l":145,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Setup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.M_nonneg_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Nonnegativity of the bound constant M, available whenever the experiment has at least one group.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) [Nonempty (Exp n).ι], 0 ≤ M","l":280,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.S0","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units and their control potential outcomes, the population sample variance of the control potential outcomes is $(n-1)^{-1}\\sum_j(b_j-\\bar b)^2$.","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":124,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.S1","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units and their treated potential outcomes, the population sample variance of the treated potential outcomes is $(n-1)^{-1}\\sum_j(a_j-\\bar a)^2$.","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":121,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatControl","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, their control potential outcomes, and a realized assignment, the observed sample variance among control units is $(n-K-1)^{-1}\\sum_j(1-T_j)(b_j-\\bar b_{\\mathrm{obs}})^2$.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":68,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatControl_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The observed control sample variance is nonnegative, provided K + 1 ≤ n (so the denominator n−K−1 ≥ 0).","s":"∀ {n : ℕ} (K : ℕ) (b : Fin n → ℝ), K + 1 ≤ n → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.ShatControl K b w","l":95,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatTreated","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, their treated potential outcomes, and a realized assignment, the observed sample variance among treated units is $(K-1)^{-1}\\sum_jT_j(a_j-\\bar a_{\\mathrm{obs}})^2$.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":64,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.ShatTreated_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The observed treated sample variance is nonnegative, provided 1 ≤ K (so the denominator K−1 ≥ 0): it is a nonnegative-weighted sum of squares divided by a nonnegative number.","s":"∀ {n : ℕ} (K : ℕ) (a : Fin n → ℝ), 1 ≤ K → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.ShatTreated K a w","l":86,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.SmuVar","k":"def","m":"Causalean.Experimentation.TwoStageInterference.StageOne","a":"Experimentation/TwoStageInterference","t":1,"d":"For a finite collection of groups and a real-valued group-level quantity $\\mu_i$, the population sample variance of the group-level quantity is $(N-1)^{-1}\\sum_i(\\mu_i-\\bar\\mu)^2$, where $N$ is the number of groups and $\\bar\\mu=N^{-1}\\sum_i\\mu_i$.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ℝ","l":46,"f":"Causalean/Experimentation/TwoStageInterference/StageOne.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Stau","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, their treated potential outcomes, and their control potential outcomes, the population sample variance of the unit-level treatment effects is $(n-1)^{-1}\\sum_j[(a_j-b_j)-(\\bar a-\\bar b)]^2$.","s":"{n : ℕ} → (Fin n → ℝ) → (Fin n → ℝ) → ℝ","l":127,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratAssign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the first-stage strategy-assignment space consists of all assignments that give each group one of two allocation strategies.","s":"Type u_2 → Type u_2","l":45,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratifiedInterference","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"For a collection of groups, their group sizes, and a potential-outcome schedule, the stratified-interference condition holds exactly when, for every group, every unit in that group, and every two within-group treatment assignments, equal own treatment and equal numbers of other treated units imply equal potential outcomes for that unit.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Prop","l":59,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.StratifiedInterference.elim","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":2,"d":"Restatement of stratified interference in terms of the two raw summaries: equal own treatment and equal count of treated others force equal outcomes.","s":"∀ {ι : Type u_1} {n : ι → ℕ} {Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ}, Causalean.Experimentation.TwoStageInterference.StratifiedInterference Y → ∀ (i : ι) (j : Fin (n i)) (w w' : Causalean.Experimentation.TwoStageInterference.WAssign n i), w j = w' j → Causalean.Experimentation.TwoStageInterference.numTreatedOthers i j w = Causalean.Experimentation.TwoStageInterference.numTreatedOthers i j w' → Y i j w = Y i j w'","l":65,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.T","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units and a unit $j$, the treatment indicator of unit $j$ equals one for each within-group assignment that treats $j$ and zero otherwise.","s":"{n : ℕ} → Fin n → (Fin n → Bool) → ℝ","l":104,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.T_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"The treatment indicator is nonnegative.","s":"∀ {n : ℕ} (j : Fin n) (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.T j w","l":78,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_estDirect","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroupEffect","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 6 (two-stage variance of the direct-effect contrast estimator). Consider the two-stage design that first allocates each group to strategy ψ or strategy φ and then randomizes the group's units accordingly, with outcomes recorded by the potential-outcome function Y. Assume the target sample size C of selected groups is nonzero, the number N of groups is nonzero, N minus one is nonzero, every group's control-arm unit count m0 is nonzero, every group's treatment-arm unit count m1 is nonzero, and every group's size n is nonzero. Suppose the stage-1 design draws a simple random sample of C of the N groups, so that each group is selected with probability C/N and each pair of distinct groups is jointly selected with probability C(C−1)/(N(N−1)), and that within a selected ψ-group each unit's control propensity is m0/n and each unit's treatment propensity is m1/n. Then the randomization variance of the treatment-minus-control estimator decomposes into a between-group term — the finite-population-corrected sample variance of the group-level treatment-minus-control contrasts, scaled by (1 − C/N)/C — plus a within-group term averaging, over the N groups and scaled by 1/(C·N), the conditional variance of each group's within-group contrast estimator.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (m0 m1 : ι → ℝ) (C : ℝ), C ≠ 0 → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι), m0 i ≠ 0) → (∀ (i : ι), m1 i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = false) = m0 i / ↑(n i)) → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = true) = m1 i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var (Causalean.Experimentation.TwoStageInterference.estDirect Y m0 m1 C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i true - Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i false) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var fun w => Causalean.Experimentation.TwoStageInterference.groupEst Y i true (m1 i) w - Causalean.Experimentation.TwoStageInterference.groupEst Y i false (m0 i) w","l":49,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroupEffect.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_estDirect_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroupEffect","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 6, for the mixed two-stage design. Consider the completely randomized two-stage design in which stage 1 draws a simple random sample of C groups out of the population of N groups, and each drawn group is completely randomized by treating K of its n units, with outcomes recorded by the potential-outcome function Y. Assume the sample size C is strictly positive, C is strictly less than the number N of groups, every group's treated-unit count K is strictly positive, and every group's treated count K is strictly less than its size n. Then the randomization variance of the treatment-minus-control estimator on this design equals the same between-group/within-group decomposition as Var_estDirect, with control count n−K and treatment count K in each group.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (K : ι → ℕ) (C : ℕ), 0 < C → ∀ (hCN : C < Fintype.card ι), (∀ (i : ι), 0 < K i) → ∀ (hKn : ∀ (i : ι), K i < n i), (Causalean.Experimentation.TwoStageInterference.jointDesign (Causalean.Experimentation.TwoStageInterference.crdOn C (LT.lt.le hCN)) (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.estDirect Y (fun i => ↑(n i) - ↑(K i)) (fun i => ↑(K i)) ↑C) = ((1 - ↑C / ↑(Fintype.card ι)) / ↑C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i true - Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i false) + 1 / (↑C * ↑(Fintype.card ι)) * ∑ i, (Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var fun w => Causalean.Experimentation.TwoStageInterference.groupEst Y i true (↑(K i)) w - Causalean.Experimentation.TwoStageInterference.groupEst Y i false (↑(n i) - ↑(K i)) w","l":112,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroupEffect.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_groupAgg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Two-stage variance decomposition, abstract per-group statistic. For an arbitrary within-group allocation design ψ and comparison design φ and an arbitrary per-group statistic g, assume the selected-group count C is nonzero, the population has nonzero size and at least two groups, and that the stage-1 selection satisfies first-order inclusion probability C/N for every group and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups. Then the randomization variance of the aggregate (∑ᵢ 1(Sᵢ=ψ)·g i(wᵢ))/C under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the conditional means (ψ i).E (g i), plus a within-group term (1/(C·N)) times the sum of the conditional variances (ψ i).Var (g i).","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (g : (i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (C : ℝ), C ≠ 0 → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → ((Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var fun sw => (∑ i, if sw.1 i = true then g i (sw.2 i) else 0) / C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => (ψ i).E (g i)) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var (g i)","l":145,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_popEst","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 4 (two-stage variance decomposition of Ŷ(z;ψ)). For arbitrary within-group allocation designs ψ and comparison designs φ and potential outcomes Y, assume the treated/control unit counts m i are nonzero, the group sizes n i are nonzero, the population has nonzero size and at least two groups, the selected-group count C is nonzero, that the within-group z-propensity of every unit equals m i / n i, and that the stage-1 selection satisfies first-order inclusion probability C/N and second-order (pairwise) inclusion probability C(C−1)/(N(N−1)) — the moments of a simple random sample of C of the N groups, each then randomized by its within-group design. Then the randomization variance of the population estimator Ŷ(z;ψ) under the two-stage design jointDesign D₁ ψ φ splits into a between-group SRS term with finite-population correction (1 − C/N)/C applied to the population sample variance of the group-level potential outcomes ȳ_i(z;ψ), plus a within-group term (1/(C·N)) times the sum of the per-group conditional variances of the within-group estimator.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι)) (ψ φ : (i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (m : ι → ℝ) (C : ℝ), C ≠ 0 → (∀ (i : ι), m i ≠ 0) → (∀ (i : ι), ↑(n i) ≠ 0) → ↑(Fintype.card ι) ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → (∀ (i : ι) (j : Fin (n i)), ((ψ i).Pr fun w => w j = z) = m i / ↑(n i)) → (∀ (i : ι), (D₁.Pr fun s => s i = true) = C / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s i = true) s * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun s => s j = true) s) = C * (C - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (Causalean.Experimentation.TwoStageInterference.jointDesign D₁ ψ φ).Var (Causalean.Experimentation.TwoStageInterference.popEst Y z true m C) = ((1 - C / ↑(Fintype.card ι)) / C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean ψ Y i z) + 1 / (C * ↑(Fintype.card ι)) * ∑ i, (ψ i).Var (Causalean.Experimentation.TwoStageInterference.groupEst Y i z (m i))","l":187,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_popEst_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.BetweenGroup","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 4, for the mixed two-stage design. For arbitrary potential outcomes Y, assume the selected-group count C is positive and strictly less than the number of groups, and that every group's treated-unit count K i is positive and strictly less than the group's size n i. Then, under the mixed two-stage design — stage 1 a simple random sample of C of the groups, each selected group i completely randomized to treat K i of its n i units — Var_popEst's between/within variance decomposition holds for the population estimator Ŷ(z; ·), with all of Var_popEst's design-moment hypotheses discharged by the completely-randomized design's inclusion and propensity facts.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {n : ι → ℕ} (Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) (z : Bool) (K : ι → ℕ) (C : ℕ), 0 < C → ∀ (hCN : C < Fintype.card ι), (∀ (i : ι), 0 < K i) → ∀ (hKn : ∀ (i : ι), K i < n i), (Causalean.Experimentation.TwoStageInterference.jointDesign (Causalean.Experimentation.TwoStageInterference.crdOn C (LT.lt.le hCN)) (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.popEst Y z true (fun i => bif z then ↑(K i) else ↑(n i) - ↑(K i)) ↑C) = ((1 - ↑C / ↑(Fintype.card ι)) / ↑C * Causalean.Experimentation.TwoStageInterference.SmuVar fun i => Causalean.Experimentation.TwoStageInterference.groupMean (fun i => Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))) Y i z) + 1 / (↑C * ↑(Fintype.card ι)) * ∑ i, (Causalean.Experimentation.TwoStageInterference.crd (K i) (LT.lt.le (hKn i))).Var (Causalean.Experimentation.TwoStageInterference.groupEst Y i z (bif z then ↑(K i) else ↑(n i) - ↑(K i)))","l":225,"f":"Causalean/Experimentation/TwoStageInterference/BetweenGroup.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_srs_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.StageOne","a":"Experimentation/TwoStageInterference","t":1,"d":"Stage-1 / between-group SRS variance term (Hudgens–Halloran 2008, the between-group term of Theorems 4 and 6). Under simple random sampling of m of the N := card ι groups, with {0,1} selection indicators U satisfying the SRS first- and second-order selection moments (hmean, hpair) and the Bernoulli diagonal variance (hvar), the sampling variance of the sample mean (∑ᵢ Uᵢ·μᵢ)/m of the group-level quantities μ equals (1 − m/N)/m times the population sample variance SmuVar μ.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {Ω₁ : Type u_2} [inst_2 : Fintype Ω₁] (D₁ : Causalean.Experimentation.DesignBased.FiniteDesign Ω₁) (U : ι → Ω₁ → ℝ) (μ : ι → ℝ) (m : ℝ), (∀ (i : ι), D₁.E (U i) = m / ↑(Fintype.card ι)) → (∀ (i j : ι), i ≠ j → (D₁.E fun s => U i s * U j s) = m * (m - 1) / (↑(Fintype.card ι) * (↑(Fintype.card ι) - 1))) → (∀ (i : ι), D₁.Var (U i) = m / ↑(Fintype.card ι) * (1 - m / ↑(Fintype.card ι))) → m ≠ 0 → ↑(Fintype.card ι) - 1 ≠ 0 → ↑(Fintype.card ι) ≠ 0 → (D₁.Var fun s => (∑ i, U i s * μ i) / m) = (1 - m / ↑(Fintype.card ι)) / m * Causalean.Experimentation.TwoStageInterference.SmuVar μ","l":61,"f":"Causalean/Experimentation/TwoStageInterference/StageOne.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_tauHat","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 5 (within-group / Neyman form). For any within-group design whose treatment indicators have first moment K/n (hmean) and pairwise second moment K(K−1)/(n(n−1)) (hpair) — the moments of the completely randomized (mixed) design of Assumption 1, which treats exactly K of n units — with the two-valued potential outcomes a (treated state) and b (control state), the randomization variance of the difference-in-means estimator is S₁/K + S₀/(n−K) − Sτ/n.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), 0 < K → K < n → (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → ρ.Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) = Causalean.Experimentation.TwoStageInterference.S1 a / ↑K + Causalean.Experimentation.TwoStageInterference.S0 b / (↑n - ↑K) - Causalean.Experimentation.TwoStageInterference.Stau a b / ↑n","l":170,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.Var_tauHat_CRD","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"Hudgens–Halloran (2008), Theorem 5, for the completely randomized design. For a group of n units with potential outcomes a (treated state) and b (untreated state), consider the completely randomized within-group design that treats exactly K units uniformly at random, where K is positive and strictly less than the group size n. Then the randomization variance of the control-minus-treatment difference-in-means estimator under this design equals S₁/K + S₀/(n−K) − Sτ/n, where S₁ and S₀ are the population sample variances of the treated-state and untreated-state outcomes and Sτ is the population sample variance of the unit-level treatment effects.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 0 < K → ∀ (hKn : K < n), (Causalean.Experimentation.TwoStageInterference.crd K (LT.lt.le hKn)).Var (Causalean.Experimentation.TwoStageInterference.tauHat K a b) = Causalean.Experimentation.TwoStageInterference.S1 a / ↑K + Causalean.Experimentation.TwoStageInterference.S0 b / (↑n - ↑K) - Causalean.Experimentation.TwoStageInterference.Stau a b / ↑n","l":258,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.WAssign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the number of units in every group and a group, the within-group assignment space consists of all assignments that give each unit in that group a treated or untreated indicator.","s":"{ι : Type u_1} → (ι → ℕ) → ι → Type","l":40,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cltSummand","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a sequence of Liu--Hudgens experiments, an experiment index $n$, a common group-level contrast $\\delta$, a sequence of within-group variances, a stage-one strategy assignment, a group, and that group's within-group treatment assignment, the scaled per-coordinate summand of the conditional studentized statistic is the selected-group indicator times the centered group contrast estimator, divided by $\\sqrt{C_n v_n}$.","s":"{Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} → (n : ℕ) → ℝ → (ℕ → ℝ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι → (i : (Exp n).ι) → (Fin ((Exp n).gsize i) → Bool) → ℝ","l":220,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.condCLT_ref","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischargeMain","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Conditional CLT for the reference selection. Applying the independent-summands CLT prodDesign_clt to the per-coordinate summands gₛ₀,ᵢ over the conditional product design gives the conditional studentized CDF at the reference selection refSel Exp n converging to Φ(t).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → Filter.Tendsto (fun n => (Causalean.Experimentation.TwoStageInterference.condDesign (Exp n) (Causalean.Experimentation.TwoStageInterference.refSel Exp n)).Pr fun w => stud n (Causalean.Experimentation.TwoStageInterference.refSel Exp n, w) ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":45,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischargeMain.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.condDesign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment and a stage-one strategy assignment, the conditional stage-two design independently assigns each group according to its treatment design when that group is assigned treatment and according to its control design otherwise.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → Causalean.Experimentation.TwoStageInterference.StratAssign E.ι → Causalean.Experimentation.DesignBased.FiniteDesign ((i : E.ι) → Fin (E.gsize i) → Bool)","l":113,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cov_diag","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The diagonal covariance Cov(Tⱼ, Tⱼ) = Var(Tⱼ) = (K/n)(1 − K/n).","s":"∀ {n : ℕ} (K : ℕ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → ∀ (j : Fin n), ρ.Cov (Causalean.Experimentation.TwoStageInterference.T j) (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n * (1 - ↑K / ↑n)","l":141,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.cov_offdiag","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The off-diagonal covariance Cov(Tⱼ, Tₖ) = K(K−1)/(n(n−1)) − (K/n)² for j ≠ k.","s":"∀ {n : ℕ} (K : ℕ) (ρ : Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)), (∀ (j : Fin n), ρ.E (Causalean.Experimentation.TwoStageInterference.T j) = ↑K / ↑n) → (∀ (j k : Fin n), j ≠ k → (ρ.E fun w => Causalean.Experimentation.TwoStageInterference.T j w * Causalean.Experimentation.TwoStageInterference.T k w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))) → ∀ (j k : Fin n), j ≠ k → ρ.Cov (Causalean.Experimentation.TwoStageInterference.T j) (Causalean.Experimentation.TwoStageInterference.T k) = ↑K * (↑K - 1) / (↑n * (↑n - 1)) - ↑K / ↑n * (↑K / ↑n)","l":151,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"For a group containing $n$ units, a target treated count $K$ no greater than the group size, the within-group completely randomized assignment design assigns positive probability only to assignments with exactly $K$ treated members and makes every such treated subset equally likely. This is the finite-group specialization of the general completely randomized assignment design.","s":"{n : ℕ} → (K : ℕ) → K ≤ n → Causalean.Experimentation.DesignBased.FiniteDesign (Fin n → Bool)","l":147,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"For a finite population of units, a target treated count $K$ no greater than the population size, the completely randomized assignment design assigns positive probability only to assignments with exactly $K$ treated units and makes every such treated subset equally likely. It is realized as the pushforward of the uniform design on size-$K$ treated subsets along their indicator assignments.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (K : ℕ) → K ≤ Fintype.card U → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":65,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"First-order inclusion probability: each unit i is treated with probability K/N, i.e. E[Tᵢ] = K/N. Derived from completeRandomization_incl.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i : U), (Causalean.Experimentation.TwoStageInterference.crdOn K hK).E (Causalean.Experimentation.DesignBased.FiniteDesign.ind fun w => w i = true) = ↑K / ↑(Fintype.card U)","l":73,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_mean_compl","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Complementary first moment: each unit i is untreated with probability (N−K)/N, i.e. E[1−Tᵢ] = (N−K)/N. The control-arm propensity, from the indicator complement 1 − Tᵢ.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i : U), ((Causalean.Experimentation.TwoStageInterference.crdOn K hK).E fun w => 1 - Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w) = (↑(Fintype.card U) - ↑K) / ↑(Fintype.card U)","l":83,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_pair","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"Second-order inclusion probability of the completely randomized design. Under the completely randomized design that treats exactly K of the N units in the population uniformly over all size-K treated subsets, if i and j are two distinct units then the probability that both i and j are treated equals K(K−1)/(N(N−1)).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (i j : U), i ≠ j → ((Causalean.Experimentation.TwoStageInterference.crdOn K hK).E fun w => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w) = ↑K * (↑K - 1) / (↑(Fintype.card U) * (↑(Fintype.card U) - 1))","l":93,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdOn_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Deterministic treated count on the design's support: any assignment with positive design weight treats exactly K units, i.e. ∑ᵢ Tᵢ = K.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (K : ℕ) (hK : K ≤ Fintype.card U) (w : U → Bool), (Causalean.Experimentation.TwoStageInterference.crdOn K hK).p w ≠ 0 → ∑ i, Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w i = true) w = ↑K","l":113,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdToBool","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"For a group containing $n$ units and a target treated count $K$, the within-group treated-set indicator assignment maps a subset of exactly $K$ group members to the assignment that marks precisely those members as treated.","s":"{n : ℕ} → (K : ℕ) → { S // S.card = K } → Fin n → Bool","l":142,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crdToBoolOn","k":"def","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":1,"d":"For a finite population of units and a target treated count $K$, the treated-set indicator assignment maps a treated subset containing exactly $K$ units to the assignment that marks precisely its members as treated.","s":"{U : Type u_1} → [DecidableEq U] → (K : ℕ) → { S // S.card = K } → U → Bool","l":59,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_mean","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"First-order inclusion probability of the within-group design: E[Tⱼ] = K/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), (Causalean.Experimentation.TwoStageInterference.crd K hK).E (Causalean.Experimentation.DesignBased.FiniteDesign.ind fun w => w j = true) = ↑K / ↑n","l":155,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_pair","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Second-order inclusion probability of the within-group design: E[Tⱼ Tₖ] = K(K−1)/(n(n−1)) for j ≠ k.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j k : Fin n), j ≠ k → ((Causalean.Experimentation.TwoStageInterference.crd K hK).E fun w => Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w * Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w k = true) w) = ↑K * (↑K - 1) / (↑n * (↑n - 1))","l":160,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_prop_false","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Control propensity of the within-group design: each unit j is in control with probability (n−K)/n, i.e. Pr[wⱼ = false] = (n−K)/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), ((Causalean.Experimentation.TwoStageInterference.crd K hK).Pr fun w => w j = false) = (↑n - ↑K) / ↑n","l":180,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_prop_true","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Treatment propensity of the within-group design: each unit j is treated with probability K/n, i.e. Pr[wⱼ = true] = K/n.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (j : Fin n), ((Causalean.Experimentation.TwoStageInterference.crd K hK).Pr fun w => w j = true) = ↑K / ↑n","l":174,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.crd_supp","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.CompleteRandomization","a":"Experimentation/TwoStageInterference","t":2,"d":"Deterministic treated count on the within-group design's support: ∑ⱼ Tⱼ = K.","s":"∀ {n : ℕ} (K : ℕ) (hK : K ≤ n) (w : Fin n → Bool), (Causalean.Experimentation.TwoStageInterference.crd K hK).p w ≠ 0 → ∑ j, Causalean.Experimentation.DesignBased.FiniteDesign.ind (fun w => w j = true) w = ↑K","l":168,"f":"Causalean/Experimentation/TwoStageInterference/CompleteRandomization.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLT","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 (Liu–Hudgens 2014), asymptotic normality of the treatment-minus-control direct-effect contrast — conditional on the uniform conditional CLT. Along a sequence of two-stage Hudgens–Halloran experiments Exp : ℕ → LHExperiment, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast and cond n s the within-selection (stage-2 product) design when stage 1 selects the strategy assignment s. Assume the conditional studentized CDFs at t converge to the standard normal CDF Φ(t) uniformly over selections s, with a vanishing uniform bound. Then the unconditional (joint-design) studentized CDF at t converges to Φ(t):","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (t : ℝ) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → ∀ (cond : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι → Causalean.Experimentation.DesignBased.FiniteDesign ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), cond n s = Causalean.Experimentation.DesignBased.prodDesign fun i => if s i = true then (Exp n).ψ i else (Exp n).φ i) → (∃ B, Filter.Tendsto B Filter.atTop (nhds 0) ∧ ∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), |((cond n s).Pr fun w => stud n (s, w) ≤ t) - Causalean.Experimentation.DesignBased.stdNormalCdf t| ≤ B n) → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLT.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischargeMain","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 (Liu–Hudgens 2014), fully primitive. Under the homogeneity and regularity bundle, the studentized treatment-minus-control direct-effect contrast is asymptotically standard normal: its joint-design CDF at t converges to Φ(t).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":146,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischargeMain.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directEffect_clt_identical","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Proposition 5.1 under literally identical groups. Along a sequence of identical-groups experiments R — common size, allocation strategies, potential outcomes, and unit counts — sharing one group-level treatment-minus-control direct-effect contrast δ with a positive common within-group variance, a uniform bound M on the centered per-group contrast estimator, every supported stage-1 selection flagging exactly C groups, and the many-groups rate M/√(C·v) → 0 together with its Lyapunov cube card·(M/√(C·v))³ → 0 — the studentized contrast statistic is asymptotically standard normal. No analytic homogeneity hypothesis is assumed: it is derived from the concrete identical-groups structure via hhom_of_identical.","s":"∀ (R : ℕ → Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t δ M : ℝ), (∀ (n : ℕ), (R n).refDelta = δ) → (∀ (n : ℕ), 0 < (R n).refVar) → (∀ (n : ℕ) (i : (R n).toExp.ι) (w : Fin ((R n).toExp.gsize i) → Bool), |Causalean.Experimentation.TwoStageInterference.groupDiff (R n).toExp i w - δ| ≤ M) → (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (R n).toExp.ι), (R n).toExp.D₁.p s ≠ 0 → (∑ i, if s i = true then 1 else 0) = (R n).toExp.C) → Filter.Tendsto (fun n => M / √((R n).toExp.C * (R n).refVar)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (R n).toExp.ι) * (M / √((R n).toExp.C * (R n).refVar)) ^ 3) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => (R n).toExp.jointD.Pr fun sw => (R n).studId sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))","l":359,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.directVar_eq_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Variance reduction. Under the homogeneity and regularity bundle, the two-stage design variance of the direct-effect contrast collapses to v n / C: the between-group term vanishes since the population variance of a constant is zero, and the within-group term averages to v n / C.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), (Exp n).directVar = v n / (Exp n).C","l":187,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estD_eq_agg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The experiment-level contrast estimator is the mean of the selected groups' per-group treatment-minus-control contrast estimators.","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment) (s : Causalean.Experimentation.TwoStageInterference.StratAssign E.ι) (w : (i : E.ι) → Fin (E.gsize i) → Bool), E.estD (s, w) = (∑ i, (if s i = true then 1 else 0) * Causalean.Experimentation.TwoStageInterference.groupDiff E i (w i)) / E.C","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estDirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, the potential outcome of each unit under every within-group assignment, a control denominator for every group, a treatment denominator for every group, a real population denominator, and a realized joint assignment, the direct-effect estimator is the estimated treatment mean minus the estimated control mean among groups assigned the first allocation strategy.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":175,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estDirect_consistent","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Consistency","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Consistency of the direct-effect contrast estimator (Liu–Hudgens 2014). Along a sequence of two-stage experiments, if the closed-form design variance directVar tends to zero as the number of groups grows, then for every positive tolerance ε, the Horvitz-Thompson estimator is consistent for the population average treatment-minus-control direct-effect contrast: the probability that the deviation exceeds ε tends to zero.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment), Filter.Tendsto (fun n => (Exp n).directVar) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => ε ≤ |(Exp n).estD sw - (Exp n).DEbar|) Filter.atTop (nhds 0)","l":41,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Consistency.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estIndirect","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"For a finite collection of groups with their respective unit counts, a potential-outcome schedule, the control-arm counts for the $\\phi$ and $\\psi$ strategies, the corresponding stage-one normalizing counts, and a realized two-stage assignment, the Horvitz--Thompson estimator of the indirect (spillover) effect is the control-outcome population estimator among $\\phi$-assigned groups minus that among $\\psi$-assigned groups.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":121,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.estTotal","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Effects","a":"Experimentation/TwoStageInterference","t":1,"d":"For a finite collection of groups with their respective unit counts, a potential-outcome schedule, the $\\phi$-control and $\\psi$-treatment arm counts, the corresponding stage-one normalizing counts, and a realized two-stage assignment, the Horvitz--Thompson estimator of the total effect is the control-outcome population estimator among $\\phi$-assigned groups minus the treatment-outcome population estimator among $\\psi$-assigned groups.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":131,"f":"Causalean/Experimentation/TwoStageInterference/Effects.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_equiv_selection","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Selection-symmetry permutation. Given two Boolean selections s, s' on a finite type ι that flag the same number of indices (card {i | s i} = card {i | s' i}), there is a permutation σ : ι ≃ ι aligning their patterns: s' i = s (σ i) for every i. Built by gluing a bijection of the selected sets (equal cardinality) with a bijection of their complements (equal cardinality, since the total is fixed) through Equiv.sumCompl.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (s s' : ι → Bool), {i | s i = true}.card = {i | s' i = true}.card → ∃ σ, ∀ (i : ι), s' i = s (σ i)","l":63,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_strat_factor","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"Factorization through the exposure. Under stratified interference of the potential outcomes Y, there is a family g i j : Bool × ℕ → ℝ of exposure-indexed potential outcomes such that every outcome factors as Y i j w = g i j (stratExpo i j w).","s":"∀ {ι : Type u_1} {n : ι → ℕ} {Y : (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ}, Causalean.Experimentation.TwoStageInterference.StratifiedInterference Y → ∃ g, ∀ (i : ι) (j : Fin (n i)) (w : Causalean.Experimentation.TwoStageInterference.WAssign n i), Y i j w = g i j (Causalean.Experimentation.TwoStageInterference.stratExpo i j w)","l":81,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.exists_support_selection","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"Some stage-1 selection lies in the support of D₁ (its probabilities sum to one).","s":"∀ (E : Causalean.Experimentation.TwoStageInterference.LHExperiment), ∃ s, E.D₁.p s ≠ 0","l":287,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupDiff","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a Liu--Hudgens experiment, one of its groups, and a within-group treatment assignment, the per-group treatment-minus-control direct-effect estimator is the treated-strategy group estimator minus the control-strategy group estimator.","s":"(E : Causalean.Experimentation.TwoStageInterference.LHExperiment) → (i : E.ι) → (Fin (E.gsize i) → Bool) → ℝ","l":105,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupEst","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the number of units in every group, the potential outcome of each unit under every within-group assignment, a group, a treatment status, a real denominator, and a realized within-group assignment, the within-group estimator is the sum of realized outcomes of units in that group with the specified status, divided by the supplied denominator. The denominator is intended to be the design-fixed number of such units.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Bool → ℝ → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ","l":153,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.groupMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the number of units in every group, a within-group randomization design for every group, the potential outcome of each unit under every within-group assignment, a group, and a treatment status, the group average potential outcome is the arithmetic mean of the individual average potential outcomes of all units in that group at that status.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ι → Bool → ℝ","l":73,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.homogeneous_of_identical","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Identical","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Given a sequence of identical-groups reference experiments, a real evaluation threshold, a real common direct-effect contrast, a real uniform bound, the assumption that every reference direct-effect contrast equals that common contrast, the assumption that every reference within-group contrast variance is positive, the assumption that, for every experiment, group, and within-group assignment, the absolute difference between the realized group contrast and the common contrast is at most the uniform bound, the assumption that every first-stage assignment with positive probability selects exactly its prescribed number of groups, the assumption that, as the sequence index grows, $M/\\sqrt{C_n v_n}$ converges to zero, where $C_n$ is the prescribed selected-group count and $v_n$ the reference within-group variance, and the assumption that the number of groups times $(M/\\sqrt{C_n v_n})^3$ converges to zero, the homogeneity bundle for this sequence has the stated threshold, studentized statistics, common contrast, bound, and reference within-group variances. Its conditional-distribution homogeneity follows from literal identity of the groups.","s":"∀ (R : ℕ → Causalean.Experimentation.TwoStageInterference.IdenticalRef) (t δ M : ℝ), (∀ (n : ℕ), (R n).refDelta = δ) → (∀ (n : ℕ), 0 < (R n).refVar) → (∀ (n : ℕ) (i : (R n).toExp.ι) (w : Fin ((R n).toExp.gsize i) → Bool), |Causalean.Experimentation.TwoStageInterference.groupDiff (R n).toExp i w - δ| ≤ M) → (∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (R n).toExp.ι), (R n).toExp.D₁.p s ≠ 0 → (∑ i, if s i = true then 1 else 0) = (R n).toExp.C) → Filter.Tendsto (fun n => M / √((R n).toExp.C * (R n).refVar)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (R n).toExp.ι) * (M / √((R n).toExp.C * (R n).refVar)) ^ 3) Filter.atTop (nhds 0) → Causalean.Experimentation.TwoStageInterference.Homogeneous (fun n => (R n).toExp) t (fun n => (R n).studId) δ M fun n => (R n).refVar","l":320,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Identical.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.indMarg","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the number of units in every group, a within-group randomization design for every group, the potential outcome of each unit under every within-group assignment, a group, and a unit in that group, the marginal individual average potential outcome is that unit's expected outcome under its group's design, averaging over its own treatment status.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Fin (n i) → ℝ","l":91,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.indMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a population of groups, the number of units in every group, a within-group randomization design for every group, the potential outcome of each unit under every within-group assignment, a group, a unit in that group, and a treatment status, the individual average potential outcome is that unit's expected outcome under the group's design conditional on its own treatment having that status, expressed as the corresponding weighted expectation divided by the probability of that status.","s":"{ι : Type u_1} → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → (i : ι) → Fin (n i) → Bool → ℝ","l":62,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.jointDesign","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a first-stage finite randomization design over the two allocation strategies and two within-group finite randomization designs for every group, the joint two-stage randomization design first draws the strategy assignment and then, conditionally for each group, draws its within-group assignment from the first design when that group receives the first strategy and from the second design otherwise.","s":"{ι : Type u_1} → [inst : Fintype ι] → [inst_1 : DecidableEq ι] → {n : ι → ℕ} → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i))","l":49,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.nonempty_of_refSel","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The reference selection flags at least one group, so the group index type is nonempty.","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ), Nonempty (Exp n).ι","l":305,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.numTreatedOthers","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"For a collection of groups, their group sizes, a group $i$, a unit $j$ in that group, and a within-group treatment assignment, the number of other treated units is the number of units in group $i$ other than $j$ that the assignment treats.","s":"{ι : Type u_1} → {n : ι → ℕ} → (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℕ","l":49,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.obsMeanControl","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, their control potential outcomes, and a realized assignment, the observed mean among control units is $(n-K)^{-1}\\sum_j(1-T_j)b_j$.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":60,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.obsMeanTreated","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, their treated potential outcomes, and a realized assignment, the observed mean among treated units is $K^{-1}\\sum_j T_j a_j$.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":56,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.one_sub_T_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":2,"d":"1 − Tⱼ (the control indicator) is nonnegative.","s":"∀ {n : ℕ} (j : Fin n) (w : Fin n → Bool), 0 ≤ 1 - Causalean.Experimentation.TwoStageInterference.T j w","l":82,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popEst","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, the potential outcome of each unit under every within-group assignment, a treatment status, a selected allocation strategy, a real within-group denominator for every group, a real population denominator, and a realized joint assignment, the population estimator is the sum of within-group estimators over groups assigned the selected strategy, divided by the population denominator. Each within-group denominator is intended to be the design-fixed number of units with the specified status.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Bool → Bool → (ι → ℝ) → ℝ → Causalean.Experimentation.TwoStageInterference.StratAssign ι × ((i : ι) → Causalean.Experimentation.TwoStageInterference.WAssign n i) → ℝ","l":163,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMarg","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a within-group randomization design for every group and the potential outcome of each unit under every within-group assignment, the population marginal average potential outcome is the arithmetic mean, over all groups, of their averages of marginal individual outcomes.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → ℝ","l":100,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMean","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Basic","a":"Experimentation/TwoStageInterference","t":1,"d":"Given a finite population of groups, the number of units in every group, a within-group randomization design for every group, the potential outcome of each unit under every within-group assignment, and a treatment status, the population average potential outcome is the arithmetic mean, over all groups, of their group average potential outcomes at that status.","s":"{ι : Type u_1} → [Fintype ι] → {n : ι → ℕ} → ((i : ι) → Causalean.Experimentation.DesignBased.FiniteDesign (Causalean.Experimentation.TwoStageInterference.WAssign n i)) → ((i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → ℝ) → Bool → ℝ","l":82,"f":"Causalean/Experimentation/TwoStageInterference/Basic.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.popMeanV","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units and a real-valued quantity $x_j$ for each unit, the population mean of that quantity is $n^{-1}\\sum_j x_j$.","s":"{n : ℕ} → (Fin n → ℝ) → ℝ","l":118,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.refSel","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"For a sequence of Liu--Hudgens experiments and an experiment index $n$, the reference stage-one selection is a fixed strategy assignment having positive probability under that experiment's stage-one design.","s":"(Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) → (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι","l":296,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.refSel_mem","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":2,"d":"The fixed reference selection has positive stage-one design probability.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (n : ℕ), (Exp n).D₁.p (Causalean.Experimentation.TwoStageInterference.refSel Exp n) ≠ 0","l":300,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stratExpo","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":1,"d":"For a collection of groups, their group sizes, a group $i$, a unit $j$ in that group, and a within-group treatment assignment, the stratified-interference exposure of unit $j$ is the pair consisting of $j$'s own treatment and the number of other treated units in group $i$.","s":"{ι : Type u_1} → {n : ι → ℕ} → (i : ι) → Fin (n i) → Causalean.Experimentation.TwoStageInterference.WAssign n i → Bool × ℕ","l":53,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stratExpo_exists","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Stratified","a":"Experimentation/TwoStageInterference","t":2,"d":"The exposure summary of unit (i,j) is always realized — namely by w itself; so every stratified exposure value reachable from some assignment has a witnessing assignment.","s":"∀ {ι : Type u_1} {n : ι → ℕ} (i : ι) (j : Fin (n i)) (w : Causalean.Experimentation.TwoStageInterference.WAssign n i), ∃ w', Causalean.Experimentation.TwoStageInterference.stratExpo i j w' = Causalean.Experimentation.TwoStageInterference.stratExpo i j w","l":74,"f":"Causalean/Experimentation/TwoStageInterference/Stratified.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.stud_eq_sum_of_homogeneous","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.CLTDischarge","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Studentized = independent sum. For a stage-1 stratified assignment s that the design supports with positive probability, then, under homogeneity, for every within-group assignment pattern w, the conditional studentized statistic decomposes as the normalized independent sum stud(s,w) = ∑ᵢ cltSummand n δ v s i (w i).","s":"∀ {Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment} {t δ M : ℝ} {v : ℕ → ℝ} {stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ}, Causalean.Experimentation.TwoStageInterference.Homogeneous Exp t stud δ M v → ∀ (n : ℕ) (s : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι), (Exp n).D₁.p s ≠ 0 → ∀ (w : (i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool), stud n (s, w) = ∑ i, Causalean.Experimentation.TwoStageInterference.cltSummand n δ v s i (w i)","l":226,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/CLTDischarge.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.sum_sub_mean_sq","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"The sum of squared deviations equals the raw second moment minus the squared first moment divided by m: ∑ⱼ (xⱼ − x̄)² = ∑ⱼ xⱼ² − (∑ⱼ xⱼ)²/m, where x̄ = (∑ x)/m.","s":"∀ {m : ℕ}, 0 < m → ∀ (x : Fin m → ℝ), ∑ j, (x j - (∑ i, x i) / ↑m) ^ 2 = ∑ j, x j ^ 2 - (∑ i, x i) ^ 2 / ↑m","l":85,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.sum_sum_ite_quadratic","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":2,"d":"A double sum of cⱼ cₖ weighted by a two-valued kernel (vd on the diagonal, vo off it) collapses to vo·(∑ c)² + (vd − vo)·∑ c². This is the algebraic core that turns the Var_linear_comb double sum into the Neyman split form.","s":"∀ {α : Type u_1} [inst : DecidableEq α] (s : Finset α) (c : α → ℝ) (vd vo : ℝ), (∑ j ∈ s, ∑ k ∈ s, c j * c k * if j = k then vd else vo) = vo * (∑ j ∈ s, c j) ^ 2 + (vd - vo) * ∑ j ∈ s, c j ^ 2","l":67,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.tauHat","k":"def","m":"Causalean.Experimentation.TwoStageInterference.Variance","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, unit-level treated potential outcomes, and unit-level control potential outcomes, the difference-in-means estimator maps each realized assignment to the mean control potential outcome among the $n-K$ control units minus the mean treated potential outcome among the $K$ treated units.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":110,"f":"Causalean/Experimentation/TwoStageInterference/Variance.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.varHat","k":"def","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"For a population of $n$ units, a treated-unit count $K$, their treated potential outcomes, their control potential outcomes, and a realized assignment, the conservative variance estimator is the treated observed sample variance divided by $K$ plus the control observed sample variance divided by $n-K$.","s":"{n : ℕ} → ℕ → (Fin n → ℝ) → (Fin n → ℝ) → (Fin n → Bool) → ℝ","l":72,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.varHat_nonneg","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.VarianceConservative","a":"Experimentation/TwoStageInterference","t":1,"d":"Pointwise nonnegativity of the conservative variance estimator. For a group of n units with potential outcomes a and b, provided at least one unit is treated and at least one unit remains in control (K + 1 ≤ n) — so both within-group sample-variance denominators K and n − K are positive — then for every realized assignment w, the conservative variance estimator v̂ar is nonnegative, being the sum of two nonnegative sample variances each divided by a positive count.","s":"∀ {n : ℕ} (K : ℕ) (a b : Fin n → ℝ), 1 ≤ K → K + 1 ≤ n → ∀ (w : Fin n → Bool), 0 ≤ Causalean.Experimentation.TwoStageInterference.varHat K a b w","l":104,"f":"Causalean/Experimentation/TwoStageInterference/VarianceConservative.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.wald_coverage_feasible","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.WaldFeasible","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Asymptotic feasible Wald coverage (Liu–Hudgens 2014). Along a sequence of two-stage Hudgens–Halloran experiments Exp, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast, and assume its per-threshold CDF converges to the standard normal CDF at every threshold and the design variance is everywhere positive. Let Vh n be an arbitrary variance estimator that is conservative-consistent — for every slack ε > 0 the probability it undershoots (1−ε) times the true variance tends to zero, and let zq ≥ 0 be the standard-normal upper quantile at level γ, Φ(zq) = 1 − γ/2. Then the feasible Wald interval D̂E ± zq·√(Vh) attains asymptotic coverage of DE̅ at least 1 − γ.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → ∀ (Vh : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (t : ℝ), Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ t) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf t))) → (∀ (n : ℕ), 0 < (Exp n).directVar) → (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => Vh n sw < (1 - ε) * (Exp n).directVar) Filter.atTop (nhds 0)) → ∀ {γ : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - γ / 2 → 1 - γ ≤ Filter.liminf (fun n => (Exp n).jointD.Pr fun sw => |(Exp n).estD sw - (Exp n).DEbar| ≤ zq * √(Vh n sw)) Filter.atTop","l":123,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/WaldFeasible.lean"},{"n":"Causalean.Experimentation.TwoStageInterference.wald_coverage_oracle","k":"theorem","m":"Causalean.Experimentation.TwoStageInterference.Asymptotic.Wald","a":"Experimentation/TwoStageInterference/Asymptotic","t":1,"d":"Asymptotic oracle Wald coverage (Liu–Hudgens 2014). Along a sequence of two-stage Hudgens–Halloran experiments Exp, let stud n be the studentized statistic (D̂E − DE̅)/√directVar for the treatment-minus-control direct-effect contrast, and assume the design variance is everywhere positive. Let zq ≥ 0 be the standard-normal upper quantile at level γ, i.e. Φ(zq) = 1 − γ/2, and assume the two studentized-CDF limits of stud at zq and −zq converge to Φ(zq) and Φ(−zq) — exactly the conclusion of the direct-contrast CLT. Then the oracle Wald interval D̂E ± zq·√directVar attains asymptotic coverage of DE̅ at least 1 − γ.","s":"∀ (Exp : ℕ → Causalean.Experimentation.TwoStageInterference.LHExperiment) (stud : (n : ℕ) → Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool) → ℝ), (∀ (n : ℕ) (sw : Causalean.Experimentation.TwoStageInterference.StratAssign (Exp n).ι × ((i : (Exp n).ι) → Fin ((Exp n).gsize i) → Bool)), stud n sw = ((Exp n).estD sw - (Exp n).DEbar) / √(Exp n).directVar) → (∀ (n : ℕ), 0 < (Exp n).directVar) → ∀ {γ : ℝ} (zq : ℝ), 0 ≤ zq → Causalean.Experimentation.DesignBased.stdNormalCdf zq = 1 - γ / 2 → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ zq) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf zq)) → Filter.Tendsto (fun n => (Exp n).jointD.Pr fun sw => stud n sw ≤ -zq) Filter.atTop (nhds (Causalean.Experimentation.DesignBased.stdNormalCdf (-zq))) → 1 - γ ≤ Filter.liminf (fun n => (Exp n).jointD.Pr fun sw => |(Exp n).estD sw - (Exp n).DEbar| ≤ zq * √(Exp n).directVar) Filter.atTop","l":54,"f":"Causalean/Experimentation/TwoStageInterference/Asymptotic/Wald.lean"},{"n":"Causalean.Experimentation.UnknownInterference.ACATE","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, a potential-outcome schedule, and an assignment of treatments to all units, the assignment-conditional average treatment effect is the average, over all units, of their own-treatment effects with the remaining assignments held fixed.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":88,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.AhatCtrl","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, a potential-outcome schedule, and a realized treatment assignment, the control numerator average is the population average of the control Horvitz--Thompson summands.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":88,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.AhatTreat","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, a potential-outcome schedule, and a realized treatment assignment, the treated numerator average is the population average of the treated Horvitz--Thompson summands.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":82,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.BhatCtrl","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, and a realized treatment assignment, the control weight average is the population average of the control inverse-probability weight summands.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → Bool) → ℝ","l":100,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.BhatTreat","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, and a realized treatment assignment, the treated weight average is the population average of the treated inverse-probability weight summands.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → Bool) → ℝ","l":94,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.EATE","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, a randomization design over treatment assignments, and a potential-outcome schedule, the expected average treatment effect is the design expectation of the assignment-conditional average treatment effect. Under no interference this quantity equals the conventional average treatment effect because the assignment-conditional effect is then constant across assignments.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool) → (U → (U → Bool) → ℝ) → ℝ","l":95,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_AhatTreat_sub_E_AhatCtrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The difference of the numerator means is EATE: E[Â₁] − E[Â₀] = EATE (the HT estimator is Â₁ − Â₀, and it is unbiased).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatTreat p y) - (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatCtrl p y) = Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y","l":683,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_BhatCtrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"With nonzero control propensities and a nonempty population, the control weight average has mean exactly one: E[B̂₀] = 1.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), 1 - p i ≠ 0) → 1 ≤ Fintype.card U → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.BhatCtrl p) = 1","l":140,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_BhatTreat","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"With nonzero treatment propensities and a nonempty population, the treated weight average has mean exactly one: E[B̂₁] = 1.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → 1 ≤ Fintype.card U → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.BhatTreat p) = 1","l":115,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_VhatBer_bias","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"The anti-conservativeness mechanism (Sävje–Aronow–Hudgens 2021). Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the conventional variance estimator's expected value equals Var(ĤT) + n⁻²·(∑ᵢ(E ĤTᵢ)² − ∑ᵢ∑_{j≠i} Cov(ĤTᵢ,ĤTⱼ)): the true sampling variance of the Horvitz–Thompson estimator, plus the average squared per-unit mean, minus the off-diagonal covariances between units' Horvitz–Thompson summands.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.VhatBer p y) = (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) + (∑ i, (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htSummand p y i) ^ 2 - ∑ i, ∑ j ∈ Finset.univ.erase i, (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.UnknownInterference.htSummand p y i) (Causalean.Experimentation.UnknownInterference.htSummand p y j)) / ↑(Fintype.card U) ^ 2","l":111,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.E_htSummand","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Unbiased","a":"Experimentation/UnknownInterference","t":1,"d":"Per-unit unbiasedness. Under a Bernoulli design in which every unit's treatment probability lies in [0,1] and is neither exactly zero nor exactly one, so both the treatment and control propensities are nonzero, the i-th Horvitz–Thompson summand has the same expectation as unit i's assignment-conditional treatment effect τ_i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (i : U), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htSummand p y i) = (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.tau y i)","l":49,"f":"Causalean/Experimentation/UnknownInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.UnknownInterference.InterfDep","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, a potential-outcome schedule, and two units, the interference-dependence relation holds exactly when some unit's treatment interferes with both units' outcomes; equivalently, the two outcomes may be affected by a common treatment.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → U → Prop","l":57,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.Interferes","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, a potential-outcome schedule, and two units, the potential source unit and the affected unit, the interference indicator holds exactly when the units are identical or there is an assignment under which reversing the source unit's treatment changes the affected unit's outcome.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → U → Prop","l":50,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment","k":"structure","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"A single Sävje–Aronow–Hudgens Bernoulli experiment, packaged so that a sequence of them models the growing-sample regime. Carries a finite population of units, marginal treatment probabilities, and potential outcomes indexed by the full assignment vector, together with an overlap regularity constant k at least one; the conditions that there is at least one unit, every treatment probability lies between 0 and 1, is bounded below by 1/k and above by 1 − 1/k (overlap); and every unit's potential outcome has second moment at most k² under the resulting Bernoulli design.","s":"Type 1","l":52,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D","k":"def","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"For a packaged Sävje--Aronow--Hudgens experiment, the experiment's Bernoulli randomization design independently assigns each unit treatment according to that experiment's stated marginal treatment probabilities.","s":"(E : Causalean.Experimentation.UnknownInterference.SAHExperiment) → Causalean.Experimentation.DesignBased.FiniteDesign (E.U → Bool)","l":92,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D_E_htEst","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Unbiasedness (bundle form). The Horvitz–Thompson estimator's expectation under the experiment's Bernoulli design equals the experiment's EATE estimand.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment), E.D.E (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) = E.eate","l":117,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.D_Var_htEst_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Variance bound (bundle form). The Horvitz–Thompson estimator's variance is at most the fourth power of the regularity constant times the average interference degree, divided by the population size.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment), E.D.Var (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) ≤ E.k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar E.y / ↑(Fintype.card E.U)","l":124,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.chebyshev_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Chebyshev tail bound. For any deviation threshold ε > 0, the probability that the Horvitz–Thompson estimator deviates from the EATE by at least ε is at most (k⁴·d̄/n)/ε², where k is the experiment's regularity constant, d̄ is the average interference degree, and n is the population size.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) {ε : ℝ}, 0 < ε → (E.D.Pr fun z => ε ≤ |Causalean.Experimentation.UnknownInterference.htEst E.p E.y z - E.eate|) ≤ E.k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar E.y / ↑(Fintype.card E.U) / ε ^ 2","l":132,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.eate","k":"def","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"For a packaged Sävje--Aronow--Hudgens experiment, the experiment's expected average treatment-effect estimand is the expectation, under its Bernoulli randomization design, of the assignment-conditional average treatment effect determined by its potential outcomes.","s":"Causalean.Experimentation.UnknownInterference.SAHExperiment → ℝ","l":97,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.one_sub_p_ne_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":2,"d":"One minus the treatment probability is nonzero (from upper overlap).","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (i : E.U), 1 - E.p i ≠ 0","l":108,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.SAHExperiment.p_ne_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":2,"d":"The treatment probabilities are nonzero (from lower overlap).","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (i : E.U), E.p i ≠ 0","l":102,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.VhatBer","k":"def","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, a potential-outcome schedule, and a realized treatment assignment, the conventional Horvitz--Thompson variance estimator is the population-size-squared-normalized sum of squared unit-level Horvitz--Thompson summands. Pointwise it equals the sum of the treated and control squared inverse-probability-weighted outcome terms because their cross product is zero.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":63,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.abs_E_AhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The control numerator mean is the average control potential outcome, uniformly bounded by k.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => |y i (Function.update z i false)|) ≤ k) → |(Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatCtrl p y)| ≤ k","l":652,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.abs_E_AhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"The treated numerator mean is the average treated potential outcome, uniformly bounded by k via the potential-outcome moment bound (Assumption C).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => |y i (Function.update z i true)|) ≤ k) → |(Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.AhatTreat p y)| ≤ k","l":619,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign","k":"def","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"For every finite population of units whose members can be compared for equality, every unit-specific treatment-probability function, and every such function whose values are nonnegative and at most one, the Bernoulli randomization design is the probability design on all binary treatment assignments in which each unit is independently treated with its specified probability.","s":"{U : Type u_1} → [inst : Fintype U] → [inst_1 : DecidableEq U] → (p : U → ℝ) → (∀ (i : U), 0 ≤ p i) → (∀ (i : U), p i ≤ 1) → Causalean.Experimentation.DesignBased.FiniteDesign (U → Bool)","l":44,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_ctrl","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginal control-indicator expectation under the Bernoulli design. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of the indicator that unit i is untreated equals 1 - p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => if z i = true then 0 else 1) = 1 - p i","l":74,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_eval","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginalizing the Bernoulli design to a single coordinate. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of any function g of a single unit i's treatment status coincides with its expectation under that unit's own coin design with success probability p i — the Zᵢ ⊥ Z₋ᵢ marginalization.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U) (g : Bool → ℝ), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => g (z i)) = (Causalean.Experimentation.DesignBased.coinDesign (p i) (hp0 i) (hp1 i)).E g","l":55,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.bernoulliDesign_E_treat","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Bernoulli","a":"Experimentation/UnknownInterference","t":1,"d":"Marginal treatment-indicator expectation under the Bernoulli design. Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], the expected value of the indicator that unit i is treated equals p i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => if z i = true then 1 else 0) = p i","l":66,"f":"Causalean/Experimentation/UnknownInterference/Bernoulli.lean"},{"n":"Causalean.Experimentation.UnknownInterference.chebyshev_ci_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"Chebyshev confidence interval for EATE. For the Bernoulli design with per-unit treatment probabilities p that take values in [0, 1] and are never exactly zero or one, for any value V that is positive and bounds the Horvitz–Thompson estimator's true sampling variance from above, and for any positive significance level α, the interval ĤT ± √(V/α) covers the EATE estimand with probability at least 1 − α.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) {V α : ℝ}, 0 < V → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ V → 0 < α → 1 - α ≤ (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Pr fun z => |Causalean.Experimentation.UnknownInterference.htEst p y z - Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y| ≤ √(V / α)","l":265,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.cov_htSummand_zero","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"No covariance off the interference-dependence graph. Under a Bernoulli design in which each unit i is treated independently with probability p i, where every p i lies in the unit interval, if units i and j are not interference dependent — no unit's treatment affects both units' outcomes — then their Horvitz–Thompson summands have zero covariance under this design.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) {i j : U}, ¬Causalean.Experimentation.UnknownInterference.InterfDep y i j → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Cov (Causalean.Experimentation.UnknownInterference.htSummand p y i) (Causalean.Experimentation.UnknownInterference.htSummand p y j) = 0","l":93,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.dbar","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units and a potential-outcome schedule, the average interference dependence is the unnormalized count of interference-dependent ordered pairs divided by the population size. For a nonempty population it equals one under no interference and the population size when every ordered pair is interference dependent; restricted interference is the condition that it is of smaller order than the population size.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → ℝ","l":71,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.dbarCount","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units and a potential-outcome schedule, the unnormalized interference-dependence count is the number of ordered pairs of units that are interference dependent.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → ℝ","l":65,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.degDep","k":"def","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, a potential-outcome schedule, and a unit, the interference degree is the number of units that are interference dependent with that unit.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → ℝ","l":72,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.disjoint_interferers_of_not_interfDep","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":2,"d":"If i and j are not interference dependent, their interferer blocks are disjoint.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (y : U → (U → Bool) → ℝ) {i j : U}, ¬Causalean.Experimentation.UnknownInterference.InterfDep y i j → Disjoint (Causalean.Experimentation.UnknownInterference.interferers y i) (Causalean.Experimentation.UnknownInterference.interferers y j)","l":82,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.eate_ci_kbound","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"A concrete finite-sample confidence interval for EATE (Sävje–Aronow–Hudgens 2021). Suppose the Bernoulli design has treatment probabilities p that lie in [0, 1], are never exactly zero or one, and in fact stay within [1/k, 1 - 1/k] for some regularity constant k ≥ 1; suppose also that the population is nonempty, every unit's outcome has second moment at most k², and the significance level α is positive. Then the Chebyshev interval ĤT ± √(k⁴·d̄/(n·α)), where d̄ is the average interference degree and n the population size, covers the EATE estimand with probability at least 1 − α — a valid (conservative) interval that needs only the regularity constant k and the interference measure d̄.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → ∀ {α : ℝ}, 0 < α → 1 - α ≤ (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Pr fun z => |Causalean.Experimentation.UnknownInterference.htEst p y z - Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y| ≤ √(k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / (↑(Fintype.card U) * α))","l":342,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.hajekEst","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, a potential-outcome schedule, and a realized treatment assignment, the Hájek estimator is the treated numerator average divided by the treated weight average, minus the control numerator average divided by the control weight average.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":106,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.hajek_consistent_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"Consistency of the Hájek estimator for EATE (Sävje–Aronow–Hudgens 2021). Along a sequence of Bernoulli experiments for which the regularity constants are uniformly bounded above by a constant M, k⁴·d̄/n → 0 along the sequence — restricted interference, and the mean absolute treated and control potential outcomes are each bounded by that experiment's own regularity constant k (the paper's Assumption C), the Hájek estimator converges in probability to the EATE.","s":"∀ (Exp : ℕ → Causalean.Experimentation.UnknownInterference.SAHExperiment) (M : ℝ), (∀ (m : ℕ), (Exp m).k ≤ M) → Filter.Tendsto (fun m => (Exp m).k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar (Exp m).y / ↑(Fintype.card (Exp m).U)) Filter.atTop (nhds 0) → (∀ (m : ℕ) (i : (Exp m).U), ((Exp m).D.E fun z => |(Exp m).y i (Function.update z i true)|) ≤ (Exp m).k) → (∀ (m : ℕ) (i : (Exp m).U), ((Exp m).D.E fun z => |(Exp m).y i (Function.update z i false)|) ≤ (Exp m).k) → Causalean.Experimentation.DesignBased.FiniteDesign.TendstoInProb (fun m => (Exp m).D) (fun m => Causalean.Experimentation.UnknownInterference.hajekEst (Exp m).p (Exp m).y) fun m => (Exp m).eate","l":726,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htCtrlSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, marginal treatment probabilities, a potential-outcome schedule, a unit, and a realized treatment assignment, the control Horvitz--Thompson summand is that unit's observed outcome multiplied by its control indicator and divided by its control probability.","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":63,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a finite population of units, marginal treatment probabilities, a potential-outcome schedule, and a realized treatment assignment, the Horvitz--Thompson estimator is the average of the units' treated-minus-control inverse-probability-weighted outcome summands.","s":"{U : Type u_1} → [Fintype U] → (U → ℝ) → (U → (U → Bool) → ℝ) → (U → Bool) → ℝ","l":114,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst_consistent_eate","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Consistency of Horvitz–Thompson for EATE under unknown interference (Sävje–Aronow–Hudgens 2021). Along a sequence of Bernoulli experiments along which k⁴·d̄/n → 0 — restricted interference with a controlled regularity constant, for every fixed deviation threshold ε > 0, the probability that the Horvitz–Thompson estimator deviates from the EATE by at least ε tends to zero along the sequence, i.e. the HT estimator converges in probability to the EATE.","s":"∀ (Exp : ℕ → Causalean.Experimentation.UnknownInterference.SAHExperiment), Filter.Tendsto (fun m => (Exp m).k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar (Exp m).y / ↑(Fintype.card (Exp m).U)) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun m => (Exp m).D.Pr fun z => ε ≤ |Causalean.Experimentation.UnknownInterference.htEst (Exp m).p (Exp m).y z - (Exp m).eate|) Filter.atTop (nhds 0)","l":147,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htEst_unbiased","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Unbiased","a":"Experimentation/UnknownInterference","t":1,"d":"Horvitz–Thompson unbiasedness for EATE (Sävje–Aronow–Hudgens 2021). Under a Bernoulli design in which every unit's treatment probability lies in [0,1] and is neither exactly zero nor exactly one, so both the treatment and control propensities are nonzero, the Horvitz–Thompson estimator is exactly unbiased for the expected average treatment effect.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1), (∀ (i : U), p i ≠ 0) → (∀ (i : U), 1 - p i ≠ 0) → ∀ (y : U → (U → Bool) → ℝ), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.htEst p y) = Causalean.Experimentation.UnknownInterference.EATE (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1) y","l":150,"f":"Causalean/Experimentation/UnknownInterference/Unbiased.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, marginal treatment probabilities, a potential-outcome schedule, a unit, and a realized treatment assignment, the unit-level Horvitz--Thompson summand is its observed outcome weighted by the reciprocal of its treatment probability if treated, minus the same outcome weighted by the reciprocal of its control probability if untreated.","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":105,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htSummand_depends_on_interferers","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":2,"d":"The iᵗʰ HT summand depends only on the treatments of the units interfering with i.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (y : U → (U → Bool) → ℝ) (i : U) (z z' : U → Bool), (∀ ℓ ∈ Causalean.Experimentation.UnknownInterference.interferers y i, z ℓ = z' ℓ) → Causalean.Experimentation.UnknownInterference.htSummand p y i z = Causalean.Experimentation.UnknownInterference.htSummand p y i z'","l":66,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.htTreatSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, marginal treatment probabilities, a potential-outcome schedule, a unit, and a realized treatment assignment, the treated Horvitz--Thompson summand is that unit's observed outcome multiplied by its treatment indicator and divided by its treatment probability.","s":"{U : Type u_1} → (U → ℝ) → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":56,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.interferers","k":"def","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"For every finite population of units whose members can be compared for equality, every schedule of potential outcomes indexed by a unit and a complete binary treatment assignment, and every unit in that population, the interferer set is the finite set consisting of that unit and every other unit whose treatment can affect that unit's potential outcome.","s":"{U : Type u_1} → [Fintype U] → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → Finset U","l":57,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.root_n_var","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Consistency","a":"Experimentation/UnknownInterference","t":1,"d":"Root-n variance scaling under bounded interference (Sävje–Aronow–Hudgens 2021). In one bundled experiment, if the average interference degree d̄ is bounded above by a constant C, then the sample size n times the variance of the Horvitz–Thompson estimator is at most k⁴·C, where k is the experiment's regularity constant — the finite-sample variance inequality that supports a root-n rate in bounded-interference sequences.","s":"∀ (E : Causalean.Experimentation.UnknownInterference.SAHExperiment) (C : ℝ), Causalean.Experimentation.UnknownInterference.dbar E.y ≤ C → ↑(Fintype.card E.U) * E.D.Var (Causalean.Experimentation.UnknownInterference.htEst E.p E.y) ≤ E.k ^ 4 * C","l":167,"f":"Causalean/Experimentation/UnknownInterference/Consistency.lean"},{"n":"Causalean.Experimentation.UnknownInterference.tau","k":"def","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, a potential-outcome schedule, a unit, and an assignment of treatments to all units, the assignment-conditional unit-level treatment effect is that unit's outcome when its own treatment is set to treated minus its outcome when its own treatment is set to control, with every other unit's assignment held fixed.","s":"{U : Type u_1} → [DecidableEq U] → (U → (U → Bool) → ℝ) → U → (U → Bool) → ℝ","l":81,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_AhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the control numerator average: Var(Â₀) ≤ k⁴·d̄/n.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.AhatCtrl p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":473,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_AhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the treated numerator average: Var(Â₁) ≤ k⁴·d̄/n (same disjoint-block argument as the HT estimator).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.AhatTreat p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":419,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_BhatCtrl_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the control weight average: Var(B̂₀) ≤ k²/n.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), p i ≤ 1 - k⁻¹) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.BhatCtrl p) ≤ k ^ 2 / ↑(Fintype.card U)","l":574,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_BhatTreat_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":2,"d":"Variance bound for the treated weight average: Var(B̂₁) ≤ k²/n (single-coordinate independence — the weight summands Zᵢ/pᵢ depend on disjoint singletons).","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.BhatTreat p) ≤ k ^ 2 / ↑(Fintype.card U)","l":525,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htEst_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"The Horvitz–Thompson variance bound (Sävje–Aronow–Hudgens 2021). Fix a regularity constant k at least 1 over a nonempty finite unit population, with a Bernoulli design whose per-unit treatment probabilities lie in the unit interval and further satisfy the overlap bounds k⁻¹ ≤ p i ≤ 1 − k⁻¹ for every unit. If the outcome function's second moment E[(y i)²] is at most k² for every unit under this design, then the Horvitz–Thompson estimator's variance is at most k⁴ · d̄ / n, where d̄ is the average interference-dependence degree and n the number of units.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → 1 ≤ Fintype.card U → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ k ^ 4 * Causalean.Experimentation.UnknownInterference.dbar y / ↑(Fintype.card U)","l":182,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htEst_le_inflated","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Confidence","a":"Experimentation/UnknownInterference","t":1,"d":"Conservative inflation (Sävje–Aronow–Hudgens 2021). Under the Bernoulli design with per-unit treatment probabilities p taking values in [0, 1], if every unit's interference degree — the number of units it is interference-dependent with — is at most a bound D, then inflating the conventional variance estimator's expectation by a factor 1 + D gives a conservative bound on the Horvitz–Thompson estimator's true variance: Var(ĤT) ≤ (1 + D)·E[V̂_Ber].","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (D : ℝ), (∀ (i : U), Causalean.Experimentation.UnknownInterference.degDep y i ≤ D) → (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htEst p y) ≤ (1 + D) * (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E (Causalean.Experimentation.UnknownInterference.VhatBer p y)","l":151,"f":"Causalean/Experimentation/UnknownInterference/Confidence.lean"},{"n":"Causalean.Experimentation.UnknownInterference.var_htSummand_le","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.VarianceBound","a":"Experimentation/UnknownInterference","t":1,"d":"Per-summand variance bound. Fix a regularity constant k at least 1 and a Bernoulli design with per-unit treatment probabilities lying in the unit interval and further satisfying the overlap bounds k⁻¹ ≤ p i ≤ 1 − k⁻¹ for every unit. If the outcome function's second moment E[(y i)²] is at most k² for every unit under this design, then the variance of the iᵗʰ Horvitz–Thompson summand is at most k⁴.","s":"∀ {U : Type u_1} [inst : Fintype U] [inst_1 : DecidableEq U] (p : U → ℝ) (hp0 : ∀ (i : U), 0 ≤ p i) (hp1 : ∀ (i : U), p i ≤ 1) (y : U → (U → Bool) → ℝ) (k : ℝ), 1 ≤ k → (∀ (i : U), k⁻¹ ≤ p i) → (∀ (i : U), p i ≤ 1 - k⁻¹) → (∀ (i : U), ((Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).E fun z => y i z ^ 2) ≤ k ^ 2) → ∀ (i : U), (Causalean.Experimentation.UnknownInterference.bernoulliDesign p hp0 hp1).Var (Causalean.Experimentation.UnknownInterference.htSummand p y i) ≤ k ^ 4","l":120,"f":"Causalean/Experimentation/UnknownInterference/VarianceBound.lean"},{"n":"Causalean.Experimentation.UnknownInterference.weightCtrlSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, marginal treatment probabilities, a unit, and a realized treatment assignment, the control inverse-probability weight summand is the unit's control indicator divided by its control probability.","s":"{U : Type u_1} → (U → ℝ) → U → (U → Bool) → ℝ","l":76,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.weightTreatSummand","k":"def","m":"Causalean.Experimentation.UnknownInterference.Hajek","a":"Experimentation/UnknownInterference","t":1,"d":"For a population of units, marginal treatment probabilities, a unit, and a realized treatment assignment, the treated inverse-probability weight summand is the unit's treatment indicator divided by its treatment probability.","s":"{U : Type u_1} → (U → ℝ) → U → (U → Bool) → ℝ","l":70,"f":"Causalean/Experimentation/UnknownInterference/Hajek.lean"},{"n":"Causalean.Experimentation.UnknownInterference.y_eq_of_agree_on_interferers","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"A unit's outcome depends only on the units that interfere with it. If two treatment assignments z and z' agree on every unit that interferes with unit i, then unit i's outcome is the same under both assignments: y i z = y i z'.","s":"∀ {U : Type u_1} [inst : DecidableEq U] [Finite U] (y : U → (U → Bool) → ℝ) (i : U) (z z' : U → Bool), (∀ (ℓ : U), Causalean.Experimentation.UnknownInterference.Interferes y ℓ i → z ℓ = z' ℓ) → y i z = y i z'","l":145,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.Experimentation.UnknownInterference.y_update_eq_of_not_interferes","k":"theorem","m":"Causalean.Experimentation.UnknownInterference.Basic","a":"Experimentation/UnknownInterference","t":1,"d":"When unit ℓ does not interfere with unit i, flipping ℓ's treatment while holding every other unit's assignment fixed never changes i's realized outcome.","s":"∀ {U : Type u_1} [inst : DecidableEq U] {y : U → (U → Bool) → ℝ} {ℓ i : U}, ¬Causalean.Experimentation.UnknownInterference.Interferes y ℓ i → ∀ (z : U → Bool), y i z = y i (Function.update z ℓ !z ℓ)","l":127,"f":"Causalean/Experimentation/UnknownInterference/Basic.lean"},{"n":"Causalean.GaussMarkov.SphericalErrors","k":"def","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"For an observation index set in which equality can be decided, a real square matrix indexed by those observations, and a real scale parameter, the spherical-errors condition holds exactly when the matrix equals $σ^2$ times the identity matrix.","s":"{Obs : Type u_1} → [DecidableEq Obs] → Matrix Obs Obs ℝ → ℝ → Prop","l":68,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.SphericalFamily","k":"def","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"For a sample space equipped with a σ-algebra, a finite observation index set, a real-valued random variable for each observation, a measure on the sample space, and a real scale parameter, the spherical-family condition holds exactly when every random variable has variance $σ^2$ and every two indexed by distinct observations have covariance zero.","s":"{Ω : Type u_1} → {Obs : Type u_2} → {mΩ : MeasurableSpace Ω} → (Obs → Ω → ℝ) → MeasureTheory.Measure Ω → ℝ → Prop","l":75,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.colSpan_dotProduct_leftNull","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":2,"d":"Column span is Euclidean-orthogonal to the left null space: if z lies in the left null space of X (z ᵥ* X = 0), then any column-span vector X *ᵥ g is orthogonal to z.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] {X : Matrix Obs Param ℝ} {g : Param → ℝ} {z : Obs → ℝ}, Matrix.vecMul z X = 0 → X.mulVec g ⬝ᵥ z = 0","l":51,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.covMatrix","k":"def","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"For a sample space equipped with a σ-algebra, a finite observation index set, a real-valued random variable for each observation, and a measure on the sample space, the covariance matrix has entry $(i,j)$ equal to the covariance of the $i$th and $j$th random variables under that measure.","s":"{Ω : Type u_1} → {Obs : Type u_2} → {mΩ : MeasurableSpace Ω} → (Obs → Ω → ℝ) → MeasureTheory.Measure Ω → Matrix Obs Obs ℝ","l":37,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.covMatrix_posSemidef","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":2,"d":"A covariance matrix of an L² random family is positive semidefinite: it is symmetric (cov is symmetric) and its quadratic form is a genuine variance, hence nonnegative. (Fintype Obs appears only under the PosSemidef definition, which the unusedFintypeInType linter cannot see; it is genuinely required.)","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → (Causalean.GaussMarkov.covMatrix Y μ).PosSemidef","l":37,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.dotProduct_self_nonneg'","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":2,"d":"Nonnegativity of the Euclidean self dot product.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] (v : Obs → ℝ), 0 ≤ v ⬝ᵥ v","l":58,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.gauss_markov_gls","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":1,"d":"Finite Gauss-Markov theorem, general covariance (GLS). Fix a design matrix X, a target combination c, and a positive-semidefinite covariance matrix S. Among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, if wStar is a GLS weight — meaning S *ᵥ wStar lies in the column span of X, then the quadratic form wStarᵀ S wStar is no larger than wᵀ S w. Specializes to gauss_markov_spherical when S = σ² I.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] {X : Matrix Obs Param ℝ} {c : Param → ℝ} {S : Matrix Obs Obs ℝ}, S.PosSemidef → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, S.mulVec wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S wStar ≤ Causalean.GaussMarkov.quadVar S w","l":93,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.gauss_markov_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.LeastNorm","a":"Estimation/GaussMarkov","t":1,"d":"Finite Gauss-Markov theorem, spherical case. Fix a design matrix X, a target combination c, and suppose the covariance matrix S is spherical: S = σ² I for some scale σ. Among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, if wStar lies in the column span of X, wStar = X *ᵥ g, then the quadratic form wStarᵀ S wStar is no larger than wᵀ S w.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Obs] {X : Matrix Obs Param ℝ} {c : Param → ℝ} {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S wStar ≤ Causalean.GaussMarkov.quadVar S w","l":63,"f":"Causalean/Estimation/GaussMarkov/LeastNorm.lean"},{"n":"Causalean.GaussMarkov.olsWeight","k":"def","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":1,"d":"For finite observation and parameter index sets, with parameter indices that can be distinguished, a real design matrix, and a real vector specifying a linear combination of the parameters, the ordinary-least-squares weight vector is $X(X^\\mathsf{T}X)^{-1}c$.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Param → ℝ) → Obs → ℝ","l":36,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_blue_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":1,"d":"OLS is BLUE under spherical errors. For an invertible normal-equations matrix Xᵀ X — full column rank — the ordinary-least-squares weight olsWeight X c estimating the target combination c is well defined. Under spherical errors S = σ² I, among all weight vectors w satisfying the same unbiasedness constraint w ᵥ* X = c, the quadratic form at the OLS weight is no larger than at any other unbiased weight w.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] [inst_3 : DecidableEq Obs] {X : Matrix Obs Param ℝ} (c : Param → ℝ), IsUnit (X.transpose * X).det → ∀ {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ {w : Obs → ℝ}, Matrix.vecMul w X = c → Causalean.GaussMarkov.quadVar S (Causalean.GaussMarkov.olsWeight X c) ≤ Causalean.GaussMarkov.quadVar S w","l":64,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_mem_colSpan","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":2,"d":"The OLS weight lies in the column span of X.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (c : Param → ℝ), Causalean.GaussMarkov.olsWeight X c = X.mulVec ((X.transpose * X)⁻¹.mulVec c)","l":47,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.olsWeight_unbiased","k":"theorem","m":"Causalean.Estimation.GaussMarkov.OLS","a":"Estimation/GaussMarkov","t":2,"d":"The OLS weight satisfies the unbiasedness constraint wStar ᵥ* X = c, provided XᵀX is invertible (full column rank).","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] {X : Matrix Obs Param ℝ} (c : Param → ℝ), IsUnit (X.transpose * X).det → Matrix.vecMul (Causalean.GaussMarkov.olsWeight X c) X = c","l":51,"f":"Causalean/Estimation/GaussMarkov/OLS.lean"},{"n":"Causalean.GaussMarkov.quadVar","k":"def","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"For a finite observation index set, a real covariance matrix indexed by that set, and a real weight assigned to each observation, the quadratic variance form is $w^\\mathsf{T} S w$.","s":"{Obs : Type u_1} → [Fintype Obs] → Matrix Obs Obs ℝ → (Obs → ℝ) → ℝ","l":46,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_def","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":2,"d":"The quadratic variance form unfolds to the weight-covariance-weight product.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] (S : Matrix Obs Obs ℝ) (w : Obs → ℝ), Causalean.GaussMarkov.quadVar S w = w ⬝ᵥ S.mulVec w","l":55,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_nonneg","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":2,"d":"A positive-semidefinite covariance matrix gives a nonnegative variance for every linear weight.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] {S : Matrix Obs Obs ℝ}, S.PosSemidef → ∀ (w : Obs → ℝ), 0 ≤ Causalean.GaussMarkov.quadVar S w","l":59,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.quadVar_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.QuadForm","a":"Estimation/GaussMarkov","t":1,"d":"Under spherical errors, i.e. when the error covariance matrix S equals σ² times the identity, the linear-estimator variance functional at weight vector w reduces to the common variance σ² times the squared Euclidean length of w.","s":"∀ {Obs : Type u_1} [inst : Fintype Obs] [inst_1 : DecidableEq Obs] {S : Matrix Obs Obs ℝ} {σ : ℝ}, Causalean.GaussMarkov.SphericalErrors S σ → ∀ (w : Obs → ℝ), Causalean.GaussMarkov.quadVar S w = σ ^ 2 * w ⬝ᵥ w","l":75,"f":"Causalean/Estimation/GaussMarkov/QuadForm.lean"},{"n":"Causalean.GaussMarkov.sphericalFamily_covMatrix","k":"theorem","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":2,"d":"A spherical random family has a spherical (scalar-identity) covariance matrix.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : DecidableEq Obs] {Y : Obs → Ω → ℝ} {σ : ℝ}, (∀ (i : Obs), AEMeasurable (Y i) μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → Causalean.GaussMarkov.SphericalErrors (Causalean.GaussMarkov.covMatrix Y μ) σ","l":84,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.GaussMarkov.variance_blue_gls","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":1,"d":"BLUE for known covariance (GLS, variance form). For a finite family of random variables Y i, each square-integrable, consider linear combinations ∑ i, w i * Y i whose weights satisfy the same unbiasedness constraint w ᵥ* X = c relative to a design X and target combination c. If the weight wStar is a generalized-least-squares (GLS) weight — i.e. the family's covariance matrix applied to wStar lies in the column span of X, then the linear combination built from wStar has variance no larger than that of any other unbiased linear combination.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Param : Type u_3} [inst_1 : Fintype Param] [MeasureTheory.IsProbabilityMeasure μ] {X : Matrix Obs Param ℝ} {c : Param → ℝ} (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, (Causalean.GaussMarkov.covMatrix Y μ).mulVec wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → ProbabilityTheory.variance (fun ω => ∑ i, wStar i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ","l":79,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.variance_blue_spherical","k":"theorem","m":"Causalean.Estimation.GaussMarkov.BLUE","a":"Estimation/GaussMarkov","t":1,"d":"BLUE under spherical errors (variance form). For a finite family of random variables Y i, each square-integrable, suppose the family is spherical: distinct cells are uncorrelated and every cell has the same variance σ². Among all linear combinations ∑ i, w i * Y i whose weights satisfy the same unbiasedness constraint w ᵥ* X = c relative to a design X and target combination c, if wStar lies in the column span of X, wStar = X *ᵥ g, then the linear combination built from wStar (the OLS weight) has variance no larger than that of any other unbiased linear combination.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Param : Type u_3} [inst_1 : Fintype Param] [MeasureTheory.IsProbabilityMeasure μ] {X : Matrix Obs Param ℝ} {c : Param → ℝ} (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {σ : ℝ}, Causalean.GaussMarkov.SphericalFamily Y μ σ → ∀ {w wStar : Obs → ℝ} {g : Param → ℝ}, wStar = X.mulVec g → Matrix.vecMul wStar X = c → Matrix.vecMul w X = c → ProbabilityTheory.variance (fun ω => ∑ i, wStar i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ","l":57,"f":"Causalean/Estimation/GaussMarkov/BLUE.lean"},{"n":"Causalean.GaussMarkov.variance_linearCombination","k":"theorem","m":"Causalean.Estimation.GaussMarkov.Variance","a":"Estimation/GaussMarkov","t":1,"d":"Variance of a linear combination = covariance quadratic form. For a finite family of random variables Y i, each square-integrable, and any weight vector w, the variance of the random linear combination ∑ i, w i * Y i equals the quadratic form wᵀ Σ w, where Σ is the family's covariance matrix.","s":"∀ {Ω : Type u_1} {Obs : Type u_2} [inst : Fintype Obs] {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Obs → Ω → ℝ), (∀ (i : Obs), MeasureTheory.MemLp (Y i) 2 μ) → ∀ (w : Obs → ℝ), ProbabilityTheory.variance (fun ω => ∑ i, w i * Y i ω) μ = Causalean.GaussMarkov.quadVar (Causalean.GaussMarkov.covMatrix Y μ) w","l":45,"f":"Causalean/Estimation/GaussMarkov/Variance.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization","k":"structure","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"A positive DAG density is a normalized parent-local factorization whose local densities are finite and continuous and all share one explicit strictly positive lower bound. The structure itself does not require the coordinate spaces to be compact; the results about it in this and the companion modules assume compact, nonempty, standard Borel coordinate spaces with open-positive reference measures.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → Causalean.DAG V → (X : V → Type uX) → [inst : (i : V) → MeasurableSpace (X i)] → [(i : V) → TopologicalSpace (X i)] → (μ : (i : V) → MeasureTheory.Measure (X i)) → [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → Type (max uV uX)","l":225,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.AEFactorIndependentOf","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"A child's local factor is almost-everywhere independent of coordinate j when the update identity holds for product-reference-almost every assignment and parent value.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → V → V → Prop","l":270,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.CondIndepCoordinates","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"Coordinate i is conditionally independent of coordinate j given C under the observational product-density law.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → [∀ (i : V), StandardBorelSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → V → V → Finset V → Prop","l":255,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.EdgeWitness","k":"structure","m":"Causalean.Graph.FiniteDensity.Positive.Witness","a":"Graph/FiniteDensity/Positive","t":1,"d":"A compact-density edge witness records four domain points at which the child's local cross-product contrast is nonzero.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → V → V → Type (max uV uX)","l":34,"f":"Causalean/Graph/FiniteDensity/Positive/Witness.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.EdgeWitness.eventually_not_condIndep","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Witness","a":"Graph/FiniteDensity/Positive","t":2,"d":"A fixed nonzero compact-domain contrast remains a nonzero witness, and hence continues to rule out edge conditional independence, throughout a sufficiently small uniform neighborhood.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → ∀ (w : M.EdgeWitness i j), ∃ ε > 0, ∀ (N : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ), M.FactorSupClose N ε → ¬N.CondIndepCoordinates i j ((G.parents i).erase j)","l":66,"f":"Causalean/Graph/FiniteDensity/Positive/Witness.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.EdgeWitness.not_condIndep","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Witness","a":"Graph/FiniteDensity/Positive","t":2,"d":"A nonzero local factor contrast on an edge rules out conditional independence of the edge endpoints given the other parents.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → ∀ (w : M.EdgeWitness i j), ¬M.CondIndepCoordinates i j ((G.parents i).erase j)","l":50,"f":"Causalean/Graph/FiniteDensity/Positive/Witness.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.FactorIndependentOf","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"A child's local conditional density is independent of coordinate j when updating only that coordinate leaves the factor unchanged at every point of the compact product domain.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → V → V → Prop","l":264,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.FactorSupClose","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Witness","a":"Graph/FiniteDensity/Positive","t":1,"d":"Two compact positive mechanisms are uniformly close when the real values of all local density factors differ by less than one common radius over every node and domain point.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → ℝ → Prop","l":59,"f":"Causalean/Graph/FiniteDensity/Positive/Witness.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.aeFactorIndependent_iff_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"Continuity and full support upgrade almost-everywhere independence of a local density factor from one coordinate to pointwise independence on the entire compact product domain.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, M.AEFactorIndependentOf i j ↔ M.FactorIndependentOf i j","l":431,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.aeFactorIndependent_of_edge_condIndep","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i, conditional independence of the endpoints given the other parents forces reference-almost-everywhere independence of the child's local factor from j.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → M.CondIndepCoordinates i j ((G.parents i).erase j) → M.AEFactorIndependentOf i j","l":114,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.aeFactorIndependent_of_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"Pointwise independence of a local factor from one coordinate implies the corresponding iterated product-reference almost-everywhere update identity.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, M.FactorIndependentOf i j → M.AEFactorIndependentOf i j","l":392,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.all_edge_witnesses_open","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Witness","a":"Graph/FiniteDensity/Positive","t":1,"d":"A compact positive factorization and a nonzero local-contrast witness for each directed edge have one positive uniform factor neighborhood in which every edge remains conditionally dependent given its other parents.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) (w : (i j : V) → G.edge j i → M.EdgeWitness i j), ∃ ε > 0, ∀ (N : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ), M.FactorSupClose N ε → ∀ (i j : V), G.edge j i → ¬N.CondIndepCoordinates i j ((G.parents i).erase j)","l":124,"f":"Causalean/Graph/FiniteDensity/Positive/Witness.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.condDistrib_child_given_parents","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":2,"d":"A compact positive factorization and node have a regular conditional law of that node given its parents equal to the reference measure weighted by the node's normalized local factor.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [inst_7 : ∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) (i : V), have P := G.parents i; have x₀ := Classical.choice inferInstance; have d := fun p z => M.toFactorization.factor i (Function.update (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateExtension P x₀ p) i z); ⇑(ProbabilityTheory.condDistrib (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection P) M.observationalMeasure) =ᵐ[MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection P) M.observationalMeasure] ⇑((ProbabilityTheory.Kernel.const ((k : ↥P) → X ↑k) (μ i)).withDensity d)","l":276,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.edge_condIndep_iff_aeFactorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i in a compact positive DAG density, conditional independence of i and j given the other parents is equivalent to reference-almost-everywhere independence of the child's local factor from j.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → (M.CondIndepCoordinates i j ((G.parents i).erase j) ↔ M.AEFactorIndependentOf i j)","l":554,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.edge_condIndep_iff_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":1,"d":"A compact positive factorization and directed edge have conditional independence of its endpoints given the child's other parents exactly when the child's local factor is pointwise unchanged by changing the parent coordinate.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → (M.CondIndepCoordinates i j ((G.parents i).erase j) ↔ M.FactorIndependentOf i j)","l":563,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.edge_condIndep_iff_localContrast_zero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i, conditional independence given the other parents is equivalent to vanishing of every local factor cross-product contrast.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → (M.CondIndepCoordinates i j ((G.parents i).erase j) ↔ ∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0)","l":663,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.edge_condIndep_of_aeFactorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i, reference-almost-everywhere independence of the child's local factor from j implies conditional independence of the endpoints given the other parents.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → M.AEFactorIndependentOf i j → M.CondIndepCoordinates i j ((G.parents i).erase j)","l":545,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.edge_condIndep_of_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i, pointwise independence of the child's local factor from j implies conditional independence of the endpoints given the other parents.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [inst_8 : ∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → M.FactorIndependentOf i j → M.CondIndepCoordinates i j ((G.parents i).erase j)","l":438,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.factorIndependent_iff_localContrast_zero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"For an edge j → i, pointwise independence of the child's factor from j is equivalent to vanishing of all its local four-point cross-product contrasts.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → (M.FactorIndependentOf i j ↔ ∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0)","l":654,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.factorIndependent_of_aeFactorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"Continuity and full support upgrade the iterated product-reference almost-everywhere update identity for a local factor to pointwise independence on the whole compact product domain.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, M.AEFactorIndependentOf i j → M.FactorIndependentOf i j","l":400,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.factorIndependent_of_localContrast_zero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"If every four-point local cross-product contrast for an edge vanishes, normalization and strict positivity force the child's factor to be independent of that parent coordinate.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → (∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0) → M.FactorIndependentOf i j","l":595,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.localContrast","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":1,"d":"The real four-point cross-product contrast of a child's local density varies its own coordinate and one other coordinate while holding all remaining coordinates fixed. The other coordinate is arbitrary here; the edge characterization applies it to a parent of the child, and if it equals the child's own coordinate the contrast is identically zero.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → (i j : V) → ((k : V) → X k) → X i → X i → X j → X j → ℝ","l":99,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.localContrast_zero_of_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Edge","a":"Graph/FiniteDensity/Positive","t":2,"d":"Pointwise independence of a child's factor from a parent coordinate forces every associated four-point local cross-product contrast to vanish.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} [inst_4 : (i : V) → TopologicalSpace (X i)] [∀ (i : V), BorelSpace (X i)] [∀ (i : V), CompactSpace (X i)] [∀ (i : V), Nonempty (X i)] [∀ (i : V), StandardBorelSpace (X i)] [∀ (i : V), (μ i).IsOpenPosMeasure] (M : Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ) {i j : V}, G.edge j i → M.FactorIndependentOf i j → ∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0","l":571,"f":"Causalean/Graph/FiniteDensity/Positive/Edge.lean"},{"n":"Causalean.Graph.FiniteDensity.CompactPositiveFactorization.observationalMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"The observational law of a compact positive factorization is the product reference measure weighted by the product of its local densities.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] → {G : Causalean.DAG V} → [inst_4 : (i : V) → TopologicalSpace (X i)] → Causalean.Graph.FiniteDensity.CompactPositiveFactorization G X μ → MeasureTheory.Measure ((i : V) → X i)","l":251,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.CubeDependsOn","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A node set and an extended-nonnegative function determine cube-local dependence, by requiring agreement on those coordinates to determine the function value at any two cube points.","s":"{V : Type u_1} → Finset V → ((V → ℝ) → ENNReal) → Prop","l":213,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization","k":"structure","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A finite DAG, coordinate value spaces, and coordinate reference measures determine a factorization interface of measurable, parent-local conditional densities normalized in their own coordinates.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → (X : V → Type u_3) → [inst : (i : V) → MeasurableSpace (X i)] → ((i : V) → MeasureTheory.Measure (X i)) → Type (max u_1 u_3)","l":26,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.ancestralMarginal_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, distinct intervention and queried nodes, evidence that the intervention target is not an ancestor of the queried node, and a normalized replacement density give identical observational and interventional laws on the queried node's ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i j : V}, j ≠ i → ¬G.isAncestor j i → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ), MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) B.observationalMeasure = MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) (B.interventionMeasure j q)","l":86,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.factor_update_eq_of_ne_of_not_parent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Leaf","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, two distinct nodes, evidence that the updated node is not a parent of the other, a full assignment, and a new coordinate value leave the other node's local factor unchanged.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {k l : V}, k ≠ l → k ∉ G.parents l → ∀ (v : (i : V) → X i) (x : X k), B.factor l (Function.update v k x) = B.factor l v","l":29,"f":"Causalean/Graph/FiniteDensity/Leaf.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.instIsFiniteMeasureObservationalMeasure","k":"instance","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"The observational measure induced by a normalized finite DAG factorization is a finite measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ), MeasureTheory.IsFiniteMeasure B.observationalMeasure","l":210,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.intervene","k":"def","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, intervention target, and normalized replacement density determine the factorization obtained by replacing that target's factor.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → (j : V) → Causalean.Graph.FiniteDensity.InterventionDensity j X μ → Causalean.Graph.FiniteDensity.Factorization G X μ","l":91,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.interventionDensity","k":"def","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, intervention target, replacement density, and full assignment determine the truncated product density for that intervention.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → (j : V) → Causalean.Graph.FiniteDensity.InterventionDensity j X μ → ((i : V) → X i) → ENNReal","l":122,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.interventionMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, an intervention target, and a normalized replacement density determine the corresponding single-target interventional product measure.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → (j : V) → Causalean.Graph.FiniteDensity.InterventionDensity j X μ → MeasureTheory.Measure ((i : V) → X i)","l":33,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lintegral_node_given_parents","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Basic","a":"Graph/FiniteDensity/Positive","t":1,"d":"A normalized finite-DAG density factorization, node, parent-coordinate test, node-value test, and measurability of those two tests ensure that integrating their product under the observational law equals integrating the parent test against the node's local conditional density.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (i : V) (h : ((k : ↥(G.parents i)) → X ↑k) → ENNReal) (g : X i → ENNReal), Measurable h → Measurable g → ∫⁻ (x : (i : V) → X i), h (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (G.parents i) x) * g (x i) ∂B.observationalMeasure = ∫⁻ (x : (i : V) → X i), h (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (G.parents i) x) * ∫⁻ (z : X i), g z * B.factor i (Function.update x i z) ∂μ i ∂B.observationalMeasure","l":32,"f":"Causalean/Graph/FiniteDensity/Positive/Basic.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lintegral_observationalDensity_lt_top","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"The observational density of a normalized finite DAG factorization has finite integral against the product reference measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ), ∫⁻ (x : (i : V) → X i), B.observationalDensity x ∂MeasureTheory.Measure.pi μ < ⊤","l":183,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lintegral_partialDensity_leaf","k":"theorem","m":"Causalean.Graph.FiniteDensity.Leaf","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, a leaf within a finite node set, and a fixed assignment of the other coordinates show that integrating the normalized leaf factor removes exactly that factor from the product.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {S : Finset V} {k : V}, Causalean.Graph.FiniteDensity.IsLeafIn G k S → ∀ (v : (i : V) → X i), ∫⁻ (x : X k), B.partialDensity S (Function.update v k x) ∂μ k = B.partialDensity (S.erase k) v","l":62,"f":"Causalean/Graph/FiniteDensity/Leaf.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lmarginal_compl_interventionDensity_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Elimination","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, a parent-closed retained set, an intervention target outside that set, and a normalized replacement density show that integrating the intervention density outside the set leaves the same retained factor product.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {A : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G A → ∀ {j : V}, j ∉ A → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ), ∫⋯∫⁻_Finset.univ \\ A, B.interventionDensity j q ∂μ = B.partialDensity A","l":95,"f":"Causalean/Graph/FiniteDensity/Elimination.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lmarginal_compl_observationalDensity_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Elimination","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization and a parent-closed retained set show that integrating the observational density outside the set leaves precisely its retained factor product.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {A : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G A → ∫⋯∫⁻_Finset.univ \\ A, B.observationalDensity ∂μ = B.partialDensity A","l":78,"f":"Causalean/Graph/FiniteDensity/Elimination.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.lmarginal_partialDensity_union_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Elimination","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, a parent-closed retained set, and its disjointness from an eliminable set show that integrating every eliminable coordinate removes all of their factors.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {A T : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G A → Disjoint A T → ∫⋯∫⁻_T, B.partialDensity (A ∪ T) ∂μ = B.partialDensity A","l":43,"f":"Causalean/Graph/FiniteDensity/Elimination.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.localMarkovParents_of_parentClosed","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Local","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"For a finite DAG density factorization, a parent-closed coordinate block, and a vertex omitted from that block, the vertex coordinate is conditionally independent of the block's non-parent coordinates given its parent coordinates.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] [inst_3 : ∀ (i : V), StandardBorelSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_4 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i : V} {P : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G P → i ∉ P → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (G.parents i)) inferInstance) (Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le (G.parents i)) (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (P \\ G.parents i)) B.observationalMeasure","l":164,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Local.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.localMarkovSuperset_of_parentClosed","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Main","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"For a finite DAG density factorization, a parent-closed block omitting a vertex, and a conditioning subset of that block containing every parent, the vertex coordinate is conditionally independent of all remaining block coordinates given the chosen conditioning coordinates.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [mX : (i : V) → MeasurableSpace (X i)] [inst_2 : ∀ (i : V), StandardBorelSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i : V} {P A : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G P → i ∉ P → A ⊆ P → G.parents i ⊆ A → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) inferInstance) (Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le A) (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (P \\ A)) B.observationalMeasure","l":86,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.map_ancestralProjection_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, distinct intervention and queried nodes, evidence that the intervention target is not an ancestor of the queried node, a normalized replacement density, and a measurable map of the ancestral subproduct give the same observational and interventional law after that map.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i j : V}, j ≠ i → ¬G.isAncestor j i → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ) {Y : Type u_3} [inst_4 : MeasurableSpace Y] (mix : ((k : ↥(Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) → X ↑k) → Y), Measurable mix → MeasureTheory.Measure.map (mix ∘ Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) B.observationalMeasure = MeasureTheory.Measure.map (mix ∘ Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) (B.interventionMeasure j q)","l":141,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.map_eq_of_dependsOn_nodeAncestralClosure","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, distinct intervention and queried nodes, evidence that the intervention target is not an ancestor of the queried node, a normalized replacement density, a measurable outcome map, its dependence only on the queried node's ancestral closure, and an anchor assignment give the same observational and interventional outcome law.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i j : V}, j ≠ i → ¬G.isAncestor j i → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ) {Y : Type u_3} [inst_4 : MeasurableSpace Y] (f : ((k : V) → X k) → Y), Measurable f → Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i) f → ∀ (x₀ : (k : V) → X k), MeasureTheory.Measure.map f B.observationalMeasure = MeasureTheory.Measure.map f (B.interventionMeasure j q)","l":103,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.measurable_interventionDensity","k":"theorem","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, intervention target, and replacement density give a measurable truncated intervention density.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (j : V) (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ), Measurable (B.interventionDensity j q)","l":152,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.measurable_observationalDensity","k":"theorem","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization has a measurable observational product density.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ), Measurable B.observationalDensity","l":62,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.measurable_partialDensity","k":"theorem","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization and a finite node set have a measurable partial product density.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (S : Finset V), Measurable (B.partialDensity S)","l":53,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.observationalDensity","k":"def","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization and a full assignment determine the observational product density over all nodes.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → ((i : V) → X i) → ENNReal","l":47,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.observationalDensity_intervene","k":"theorem","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, intervention target, and replacement density give the same product density whether intervention is represented by factor replacement or by the explicit truncated product.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (j : V) (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ), (B.intervene j q).observationalDensity = B.interventionDensity j q","l":129,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.observationalMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization determines its observational product measure.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → MeasureTheory.Measure ((i : V) → X i)","l":29,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.orderedLocalMarkov","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Main","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"For a finite DAG density factorization, an arbitrary topological ranking, a vertex, a conditioning predecessor set, and proof that it lies among the predecessors and contains every parent, the vertex coordinate is conditionally independent of all other predecessors given that set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [mX : (i : V) → MeasurableSpace (X i)] [inst_2 : ∀ (i : V), StandardBorelSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V), ∀ A ⊆ Causalean.Graph.FiniteDensity.predecessors τ i, G.parents i ⊆ A → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) inferInstance) (Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le A) (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.predecessors τ i \\ A)) B.observationalMeasure","l":179,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.orderedLocalMarkov_canonical","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Main","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"For a finite DAG density factorization, a vertex, a conditioning set in the canonical predecessor block, and proof that it is a predecessor subset containing every parent, the vertex coordinate is conditionally independent of all other canonical predecessors given that set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [mX : (i : V) → MeasurableSpace (X i)] [inst_2 : ∀ (i : V), StandardBorelSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [inst_3 : ∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (i : V), ∀ A ⊆ Causalean.Graph.FiniteDensity.predecessors (Causalean.Graph.FiniteDensity.canonicalTopologicalRanking G) i, G.parents i ⊆ A → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) inferInstance) (Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le A) (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.predecessors (Causalean.Graph.FiniteDensity.canonicalTopologicalRanking G) i \\ A)) B.observationalMeasure","l":195,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.parentClosed_marginal_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, a parent-closed retained set, an intervention target outside that set, and a normalized replacement density give identical observational and interventional marginals on the retained set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {A : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G A → ∀ {j : V}, j ∉ A → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ), MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) B.observationalMeasure = MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) (B.interventionMeasure j q)","l":72,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.partialDensity","k":"def","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"A DAG factorization, a finite node set, and a full assignment determine the product density over that node set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {X : V → Type u_2} → [inst_2 : (i : V) → MeasurableSpace (X i)] → {μ : (i : V) → MeasureTheory.Measure (X i)} → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.Factorization G X μ → Finset V → ((i : V) → X i) → ENNReal","l":41,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.partialDensity_erase_update_leaf","k":"theorem","m":"Causalean.Graph.FiniteDensity.Leaf","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, a leaf of a finite node set, a full assignment, and a new leaf-coordinate value leave the product of all remaining factors unchanged.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {S : Finset V} {k : V}, Causalean.Graph.FiniteDensity.IsLeafIn G k S → ∀ (v : (i : V) → X i) (x : X k), B.partialDensity (S.erase k) (Function.update v k x) = B.partialDensity (S.erase k) v","l":45,"f":"Causalean/Graph/FiniteDensity/Leaf.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.targetRatio_dependsOn","k":"theorem","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, a target node, and a numerator depending on that node's value give a density ratio determined only by the target and its parents.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) (i : V) (numerator : X i → ENNReal), Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn (insert i (G.parents i)) fun v => numerator (v i) / B.factor i v","l":69,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.Factorization.targetRatio_map_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A DAG factorization, distinct intervention and ratio nodes, evidence that the intervention target is not an ancestor of the ratio node, a normalized replacement density, a node-coordinate numerator, measurability of the resulting ratio, and an anchor assignment give the same observational and interventional law for the canonical density ratio.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.Factorization G X μ) {i j : V}, j ≠ i → ¬G.isAncestor j i → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j X μ) (numerator : X i → ENNReal), (Measurable fun v => numerator (v i) / B.factor i v) → ∀ (x₀ : (k : V) → X k), MeasureTheory.Measure.map (fun v => numerator (v i) / B.factor i v) B.observationalMeasure = MeasureTheory.Measure.map (fun v => numerator (v i) / B.factor i v) (B.interventionMeasure j q)","l":120,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.InterventionDensity","k":"structure","m":"Causalean.Graph.FiniteDensity.Factorization","a":"Graph/FiniteDensity","t":1,"d":"An intervention target, coordinate value spaces, and coordinate reference measures determine a measurable unit-mass replacement density for that target.","s":"{V : Type u_1} → V → (X : V → Type u_3) → [inst : (i : V) → MeasurableSpace (X i)] → ((i : V) → MeasureTheory.Measure (X i)) → Type u_3","l":80,"f":"Causalean/Graph/FiniteDensity/Factorization.lean"},{"n":"Causalean.Graph.FiniteDensity.IsLeafIn","k":"def","m":"Causalean.Graph.FiniteDensity.Leaf","a":"Graph/FiniteDensity","t":1,"d":"A finite DAG, a node, and a finite node set determine the property that the node is a leaf within that set.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V → Prop","l":24,"f":"Causalean/Graph/FiniteDensity/Leaf.lean"},{"n":"Causalean.Graph.FiniteDensity.MeasurableOnSet","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A function and a set determine the proposition that the function is measurable on that set, by testing the restriction to the set's subtype.","s":"{X : Type u_2} → {Y : Type u_3} → [MeasurableSpace X] → [MeasurableSpace Y] → (X → Y) → Set X → Prop","l":207,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.ParentClosed","k":"def","m":"Causalean.Graph.FiniteDensity.Elimination","a":"Graph/FiniteDensity","t":1,"d":"A finite DAG and a finite node set determine the property that the set contains every parent of each of its nodes.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Finset V → Prop","l":24,"f":"Causalean/Graph/FiniteDensity/Elimination.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism","k":"structure","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"A positive finite-state DAG mechanism consists of normalized, strictly positive local factors, each depending only on its own coordinate and its graph parents.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → Causalean.DAG V → (X : V → Type uX) → [(i : V) → Fintype (X i)] → Type (max uV uX)","l":32,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.CondIndepCoordinates","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"The cross-product conditional-independence identity for two coordinates given a set C: for every assignment and every values of the two coordinates, the cylinder masses satisfy the usual cross-multiplied finite conditional-probability identity. It expresses conditional independence when the two coordinates are distinct and neither lies in C, as in every use here (a child and one of its parents, given the remaining parents); for overlapping arguments the identity is not conditional independence.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → [(i : V) → DecidableEq (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → V → V → Finset V → Prop","l":305,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.EdgeWitness","k":"structure","m":"Causalean.Graph.FiniteDensity.Positive.FiniteWitness","a":"Graph/FiniteDensity/Positive","t":1,"d":"A local edge witness records four coordinate values at which the child's cross-product contrast is nonzero.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → V → V → Type (max uV uX)","l":30,"f":"Causalean/Graph/FiniteDensity/Positive/FiniteWitness.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.EdgeWitness.eventually_nonzero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.FiniteWitness","a":"Graph/FiniteDensity/Positive","t":2,"d":"A fixed nonzero local contrast remains nonzero for every sufficiently small uniform perturbation of all local factors.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [(i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) {i j : V} (w : M.EdgeWitness i j), ∃ ε > 0, ∀ (N : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X), M.FactorSupClose N ε → N.localContrast i j w.base w.child₀ w.child₁ w.parent₀ w.parent₁ ≠ 0","l":60,"f":"Causalean/Graph/FiniteDensity/Positive/FiniteWitness.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.EdgeWitness.not_condIndep","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.FiniteWitness","a":"Graph/FiniteDensity/Positive","t":2,"d":"A nonzero local factor contrast on an edge proves failure of conditional independence of the edge endpoints given the child's other parents.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) {i j : V}, G.edge j i → ∀ (w : M.EdgeWitness i j), ¬M.CondIndepCoordinates i j ((G.parents i).erase j)","l":46,"f":"Causalean/Graph/FiniteDensity/Positive/FiniteWitness.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.FactorIndependentOf","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"A local factor is independent of coordinate j when changing only that coordinate never changes the factor.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → V → V → Prop","l":318,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.FactorSupClose","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.FiniteWitness","a":"Graph/FiniteDensity/Positive","t":1,"d":"Two finite DAG mechanisms are uniformly factor-close when every local factor differs by less than the prescribed radius at every full assignment.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → ℝ → Prop","l":55,"f":"Causalean/Graph/FiniteDensity/Positive/FiniteWitness.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.all_edge_witnesses_open","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.FiniteWitness","a":"Graph/FiniteDensity/Positive","t":1,"d":"A positive finite-state DAG mechanism and a nonzero local-contrast witness for each directed edge have one positive uniform factor neighborhood in which all edge conditional dependences persist.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) (w : (i j : V) → G.edge j i → M.EdgeWitness i j), ∃ ε > 0, ∀ (N : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X), M.FactorSupClose N ε → ∀ (i j : V), G.edge j i → ¬N.CondIndepCoordinates i j ((G.parents i).erase j)","l":102,"f":"Causalean/Graph/FiniteDensity/Positive/FiniteWitness.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.conditionalMass","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"The pointwise conditional mass of coordinate i taking a given value, given the coordinates in C, is the mass of the cylinder fixing i and C divided by the mass of the cylinder fixing C (zero if that mass is zero). It is the elementary conditional probability when i is not in C, which is how it is used; if i belongs to C, the numerator overrides that conditioning value.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → [(i : V) → DecidableEq (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → (i : V) → Finset V → ((i : V) → X i) → X i → ℝ","l":59,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.conditionalMass_given_parents","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":2,"d":"Conditioning a node on all of its parents recovers exactly its normalized local factor.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) (i : V) (x : (i : V) → X i) (z : X i), M.conditionalMass i (G.parents i) x z = M.factor i (Function.update x i z)","l":178,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.edge_condIndep_iff_factorIndependent","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":2,"d":"For a directed edge j → i, conditional independence of its endpoints given the other parents is equivalent to pointwise independence of the child's local factor from j.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) {i j : V}, G.edge j i → (M.CondIndepCoordinates i j ((G.parents i).erase j) ↔ M.FactorIndependentOf i j)","l":428,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.edge_condIndep_iff_localContrast_zero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"A positive finite-state DAG mechanism and directed edge have conditional independence of the endpoints given the other parents exactly when every local four-point factor contrast vanishes.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) {i j : V}, G.edge j i → (M.CondIndepCoordinates i j ((G.parents i).erase j) ↔ ∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0)","l":546,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.factorIndependent_iff_localContrast_zero","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":2,"d":"For a directed edge j → i, independence of the child's factor from j is equivalent to vanishing of every local four-point cross-product contrast.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [(i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) {i j : V}, G.edge j i → (M.FactorIndependentOf i j ↔ ∀ (x : (k : V) → X k) (xi xi' : X i) (xj xj' : X j), M.localContrast i j x xi xi' xj xj' = 0)","l":485,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.jointMass","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"The joint mass of an assignment is the product of all local factors.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → ((i : V) → X i) → ℝ","l":49,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.localContrast","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"The four-point local contrast compares the two-by-two cross products of a child's factor as its own value and one candidate parent value vary, with all other coordinates held fixed.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → (i j : V) → ((k : V) → X k) → X i → X i → X j → X j → ℝ","l":323,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.marginalMass","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":1,"d":"The mass of the cylinder fixing the coordinates in S to their values in x is obtained by summing the joint product mass over all compatible full assignments.","s":"{V : Type uV} → [inst : Fintype V] → [inst_1 : DecidableEq V] → {X : V → Type uX} → [inst_2 : (i : V) → Fintype (X i)] → [(i : V) → DecidableEq (X i)] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X → Finset V → ((i : V) → X i) → ℝ","l":52,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.marginalMass_pos","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":2,"d":"Every cylinder marginal of a positive finite DAG mechanism is strictly positive.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [inst_3 : (i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X) (S : Finset V) (x : (i : V) → X i), 0 < M.marginalMass S x","l":115,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism.sum_jointMass","k":"theorem","m":"Causalean.Graph.FiniteDensity.Positive.Finite","a":"Graph/FiniteDensity/Positive","t":2,"d":"The product density of a normalized positive finite DAG mechanism has total mass one.","s":"∀ {V : Type uV} [inst : Fintype V] [inst_1 : DecidableEq V] {X : V → Type uX} [inst_2 : (i : V) → Fintype (X i)] [(i : V) → DecidableEq (X i)] [∀ (i : V), Nonempty (X i)] {G : Causalean.DAG V} (M : Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism G X), ∑ x, M.jointMass x = 1","l":66,"f":"Causalean/Graph/FiniteDensity/Positive/Finite.lean"},{"n":"Causalean.Graph.FiniteDensity.TopologicalRanking","k":"structure","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"A finite DAG is equipped with a numeric position for each vertex, distinct positions for distinct vertices, and strictly increasing positions along every directed edge.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Type u_1","l":21,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.UnitCubeFactorization","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A finite DAG determines the factorization interface whose coordinates use unit-interval-restricted Lebesgue reference measure.","s":"(V : Type u_1) → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Type (max u_1 0)","l":63,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.UnitCubeFactorization.ancestralMarginal_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A unit-cube DAG factorization, distinct intervention and queried nodes, evidence that the intervention target is not an ancestor of the queried node, and a normalized replacement density give identical observational and interventional laws on the queried node's ancestral closure.","s":"∀ (V : Type u_1) [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.UnitCubeFactorization V G) {i j : V}, j ≠ i → ¬G.isAncestor j i → ∀ (q : Causalean.Graph.FiniteDensity.InterventionDensity j (fun x => ℝ) fun x => Causalean.Graph.FiniteDensity.unitIntervalReference), MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) (Causalean.Graph.FiniteDensity.Factorization.observationalMeasure B) = MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)) (Causalean.Graph.FiniteDensity.Factorization.interventionMeasure B j q)","l":68,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.UnitCubeFactorization.instIsFiniteMeasureUnitCubeObservational","k":"instance","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Main","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"The unit-cube observational density associated with a unit-cube factorization induces a finite measure when combined with the unit-cube reference measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.UnitCubeFactorization V G), MeasureTheory.IsFiniteMeasure ((Causalean.Graph.FiniteDensity.unitCubeReference V).withDensity (Causalean.Graph.FiniteDensity.Factorization.observationalDensity B))","l":212,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.UnitCubeFactorization.orderedLocalMarkov_unitCubeReference","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Main","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"For a unit-cube DAG density factorization, an arbitrary topological ranking, a vertex, a conditioning predecessor set, and proof that it lies among the predecessors and contains every parent, the vertex is conditionally independent of all remaining predecessors given that set under the unit-cube reference measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (B : Causalean.Graph.FiniteDensity.UnitCubeFactorization V G) (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V), ∀ A ⊆ Causalean.Graph.FiniteDensity.predecessors τ i, G.parents i ⊆ A → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) inferInstance) (Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le A) (fun x => x i) (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.predecessors τ i \\ A)) ((Causalean.Graph.FiniteDensity.unitCubeReference V).withDensity (Causalean.Graph.FiniteDensity.Factorization.observationalDensity B))","l":223,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.binaryEdgeDAG","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Examples","a":"Graph/FiniteDensity/Positive","t":1,"d":"The two-node DAG has the single directed edge 0 → 1.","s":"Causalean.DAG (Fin 2)","l":20,"f":"Causalean/Graph/FiniteDensity/Positive/Examples.lean"},{"n":"Causalean.Graph.FiniteDensity.binaryMechanism","k":"def","m":"Causalean.Graph.FiniteDensity.Positive.Examples","a":"Graph/FiniteDensity/Positive","t":1,"d":"The binary example has a fair root and a child which matches its parent with probability three quarters.","s":"Causalean.Graph.FiniteDensity.PositiveFiniteDAGMechanism Causalean.Graph.FiniteDensity.binaryEdgeDAG fun x => Bool","l":29,"f":"Causalean/Graph/FiniteDensity/Positive/Examples.lean"},{"n":"Causalean.Graph.FiniteDensity.canonicalTopologicalRanking","k":"def","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"A finite DAG has the canonical topological ranking supplied by its DAG API.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G : Causalean.DAG V) → Causalean.Graph.FiniteDensity.TopologicalRanking G","l":32,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.clampCube","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A coordinate type and a real assignment determine the coordinatewise clamp to the closed unit cube, by clamping each coordinate between zero and one.","s":"(V : Type u_1) → (V → ℝ) → V → ℝ","l":103,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampCube_eq_self","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"A finite coordinate type and a point in its unit cube are such that the coordinatewise clamp fixes that point.","s":"∀ (V : Type u_1) [DecidableEq V] [Fintype V] {v : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → Causalean.Graph.FiniteDensity.clampCube V v = v","l":129,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampCube_mem","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Every real assignment over a finite coordinate type is sent by the coordinatewise clamp into the finite unit cube.","s":"∀ (V : Type u_1) [DecidableEq V] [Fintype V] (v : V → ℝ), Causalean.Graph.FiniteDensity.clampCube V v ∈ Causalean.Graph.FiniteDensity.unitCube V","l":117,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampCube_update","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"An assignment, a coordinate, and a replacement value satisfy the update-and-clamp identity.","s":"∀ (V : Type u_1) [inst : DecidableEq V] [Fintype V] (v : V → ℝ) (i : V) (x : ℝ), Causalean.Graph.FiniteDensity.clampCube V (Function.update v i x) = Function.update (Causalean.Graph.FiniteDensity.clampCube V v) i (max 0 (min 1 x))","l":141,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampCube_update_of_mem","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"An assignment, a coordinate, and a replacement value in the unit interval satisfy the update identity in which only the unchanged context is clamped.","s":"∀ (V : Type u_1) [inst : DecidableEq V] [Fintype V] (v : V → ℝ) (i : V) {x : ℝ}, x ∈ Set.Icc 0 1 → Causalean.Graph.FiniteDensity.clampCube V (Function.update v i x) = Function.update (Causalean.Graph.FiniteDensity.clampCube V v) i x","l":152,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampedFactor","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A cube-local extended-nonnegative function determines its globally defined clamped extension, by evaluating it after coordinatewise clamping.","s":"{V : Type u_1} → ((V → ℝ) → ENNReal) → (V → ℝ) → ENNReal","l":222,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.clampedOfRealFactor","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A cube-local real function determines its globally defined nonnegative clamped extension, by applying the nonnegative-real conversion after coordinatewise clamping.","s":"{V : Type u_1} → ((V → ℝ) → ℝ) → (V → ℝ) → ENNReal","l":227,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.coordinateConditioning_comap_le","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"The σ-algebra generated by projection onto a conditioning coordinate block is no finer than the ambient product σ-algebra.","s":"∀ {V : Type u_1} [DecidableEq V] [Fintype V] {X : V → Type u_2} [inst : (i : V) → MeasurableSpace (X i)] (A : Finset V), MeasurableSpace.comap (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection A) inferInstance ≤ inferInstance","l":81,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.cubeInterventionDensityMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Cube-local extended-nonnegative factors, a target coordinate, and a replacement density determine the corresponding cube-restricted single-target product-density measure, by replacing that coordinate's factor in the joint product.","s":"{V : Type u_1} → [DecidableEq V] → [Fintype V] → (V → (V → ℝ) → ENNReal) → V → (ℝ → ENNReal) → MeasureTheory.Measure (V → ℝ)","l":308,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.cubeProductDensityMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Cube-local extended-nonnegative factors determine the corresponding cube-restricted observational product-density measure, by weighting restricted Lebesgue volume with their joint product.","s":"{V : Type u_1} → [Fintype V] → (V → (V → ℝ) → ENNReal) → MeasureTheory.Measure (V → ℝ)","l":301,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.exists_leaf_in_union_of_parentClosed","k":"theorem","m":"Causalean.Graph.FiniteDensity.Elimination","a":"Graph/FiniteDensity","t":2,"d":"A parent-closed retained set, its disjointness from an eliminable set, and a nonempty eliminable set produce an eliminable node that is a leaf of their union.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {A T : Finset V}, Causalean.Graph.FiniteDensity.ParentClosed G A → Disjoint A T → T.Nonempty → ∃ k ∈ T, Causalean.Graph.FiniteDensity.IsLeafIn G k (A ∪ T)","l":29,"f":"Causalean/Graph/FiniteDensity/Elimination.lean"},{"n":"Causalean.Graph.FiniteDensity.interventionDensityOfCube","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A target coordinate and a cube-local replacement density that is measurable on the interval and normalized there determine a global intervention density, by scalar clamping.","s":"{V : Type u_1} → (j : V) → (q : ℝ → ENNReal) → Causalean.Graph.FiniteDensity.MeasurableOnSet q (Set.Icc 0 1) → ∫⁻ (x : ℝ) in Set.Icc 0 1, q x = 1 → Causalean.Graph.FiniteDensity.InterventionDensity j (fun x => ℝ) fun x => Causalean.Graph.FiniteDensity.unitIntervalReference","l":342,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.interventionDensityOfRealCube","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A target coordinate and a real replacement density that is measurable on the unit interval, nonnegative there, and Lebesgue-normalized determine a global intervention density, by nonnegative-real conversion followed by scalar clamping.","s":"{V : Type u_1} → (j : V) → (q : ℝ → ℝ) → Causalean.Graph.FiniteDensity.MeasurableOnSet q (Set.Icc 0 1) → (∀ x ∈ Set.Icc 0 1, 0 ≤ q x) → ∫ (x : ℝ) in Set.Icc 0 1, q x = 1 → Causalean.Graph.FiniteDensity.InterventionDensity j (fun x => ℝ) fun x => Causalean.Graph.FiniteDensity.unitIntervalReference","l":513,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.interventionMeasure_unitCubeFactorizationOfCubeFactors","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Cube-local factors with cube measurability, node-and-parent locality, and coordinate-wise normalization, together with a target coordinate, a replacement density, its interval measurability, and its normalization, give an intervention law equal to the cube-restricted replacement product-density measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ENNReal) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hlocal : ∀ (i : V), Causalean.Graph.FiniteDensity.CubeDependsOn (insert i (G.parents i)) (p i)) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫⁻ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) (j : V) (q : ℝ → ENNReal) (hqmeas : Causalean.Graph.FiniteDensity.MeasurableOnSet q (Set.Icc 0 1)) (hqnorm : ∫⁻ (x : ℝ) in Set.Icc 0 1, q x = 1), Causalean.Graph.FiniteDensity.Factorization.interventionMeasure (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfCubeFactors p hmeas hlocal hnorm) j (Causalean.Graph.FiniteDensity.interventionDensityOfCube j q hqmeas hqnorm) = Causalean.Graph.FiniteDensity.cubeInterventionDensityMeasure p j q","l":367,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.interventionMeasure_unitCubeFactorizationOfRealCubeFactors","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Real cube factors with cube measurability, nonnegativity, node-and-parent locality, and coordinate-wise normalization, together with a target coordinate, a real replacement density, its interval measurability, its nonnegativity, and its normalization, give an intervention law equal to the usual cube-restricted real replacement product-density measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ℝ) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hnonneg : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, 0 ≤ p i v) (hlocal : ∀ (i : V) {v w : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → w ∈ Causalean.Graph.FiniteDensity.unitCube V → (∀ k ∈ insert i (G.parents i), v k = w k) → p i v = p i w) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) (j : V) (q : ℝ → ℝ) (hqmeas : Causalean.Graph.FiniteDensity.MeasurableOnSet q (Set.Icc 0 1)) (hqnonneg : ∀ x ∈ Set.Icc 0 1, 0 ≤ q x) (hqnorm : ∫ (x : ℝ) in Set.Icc 0 1, q x = 1), Causalean.Graph.FiniteDensity.Factorization.interventionMeasure (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfRealCubeFactors p hmeas hnonneg hlocal hnorm) j (Causalean.Graph.FiniteDensity.interventionDensityOfRealCube j q hqmeas hqnonneg hqnorm) = Causalean.Graph.FiniteDensity.realCubeInterventionDensityMeasure p j q","l":534,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.measurableOnSet_of_continuousOn_cube","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Real cube factors with continuity on the finite unit cube have the cube-local measurability required by the real-valued factorization constructor.","s":"∀ {V : Type u_1} [DecidableEq V] [Fintype V] (p : V → (V → ℝ) → ℝ), (∀ (i : V), ContinuousOn (p i) (Causalean.Graph.FiniteDensity.unitCube V)) → ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)","l":578,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.measurable_clampCube","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"The coordinatewise clamp for a finite coordinate type is measurable.","s":"∀ (V : Type u_1) [DecidableEq V] [Fintype V], Measurable (Causalean.Graph.FiniteDensity.clampCube V)","l":109,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.measurable_clampedFactor","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"A cube-local extended-nonnegative function with cube-local measurability has a globally measurable clamped extension.","s":"∀ {V : Type u_1} [DecidableEq V] [Fintype V] {p : (V → ℝ) → ENNReal}, Causalean.Graph.FiniteDensity.MeasurableOnSet p (Causalean.Graph.FiniteDensity.unitCube V) → Measurable (Causalean.Graph.FiniteDensity.clampedFactor p)","l":233,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.measurable_clampedOfRealFactor","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"A cube-local real function with cube-local measurability has a globally measurable nonnegative clamped extension.","s":"∀ {V : Type u_1} [DecidableEq V] [Fintype V] {p : (V → ℝ) → ℝ}, Causalean.Graph.FiniteDensity.MeasurableOnSet p (Causalean.Graph.FiniteDensity.unitCube V) → Measurable (Causalean.Graph.FiniteDensity.clampedOfRealFactor p)","l":243,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.measurable_orderedCoordinate","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"Reading a single coordinate from a finite product is measurable.","s":"∀ {V : Type u_1} [DecidableEq V] [Fintype V] {X : V → Type u_2} [inst : (i : V) → MeasurableSpace (X i)] (i : V), Measurable fun x => x i","l":67,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.measurable_predecessorResidualProjection","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"Projecting onto the predecessors outside a conditioning set is measurable.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {X : V → Type u_2} [inst_2 : (i : V) → MeasurableSpace (X i)] {G : Causalean.DAG V} (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V) (A : Finset V), Measurable (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection (Causalean.Graph.FiniteDensity.predecessors τ i \\ A))","l":73,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.mem_nodeAncestralClosure_iff","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A queried node and a candidate node satisfy ancestral-closure membership exactly when the candidate is the queried node or an ancestor of it.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} {i j : V}, j ∈ Causalean.Graph.FiniteDensity.nodeAncestralClosure G i ↔ j = i ∨ G.isAncestor j i","l":44,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.nodeAncestralClosure","k":"def","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":1,"d":"A finite DAG and a node determine the node's ancestral closure, including the node and all strict ancestors.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → V → Finset V","l":39,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.not_mem_predecessors","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"A vertex does not belong to its strict predecessor set.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V), i ∉ Causalean.Graph.FiniteDensity.predecessors τ i","l":52,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.observationalMeasure_unitCubeFactorizationOfCubeFactors","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Cube-local factors with cube measurability, node-and-parent locality, and coordinate-wise normalization give an observational law of the clamped factorization equal to the cube-restricted product-density measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ENNReal) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hlocal : ∀ (i : V), Causalean.Graph.FiniteDensity.CubeDependsOn (insert i (G.parents i)) (p i)) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫⁻ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1), Causalean.Graph.FiniteDensity.Factorization.observationalMeasure (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfCubeFactors p hmeas hlocal hnorm) = Causalean.Graph.FiniteDensity.cubeProductDensityMeasure p","l":316,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.observationalMeasure_unitCubeFactorizationOfRealCubeFactors","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Real cube factors with cube measurability, nonnegativity, node-and-parent locality, and coordinate-wise normalization give an observational law equal to the usual cube-restricted real product-density measure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ℝ) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hnonneg : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, 0 ≤ p i v) (hlocal : ∀ (i : V) {v w : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → w ∈ Causalean.Graph.FiniteDensity.unitCube V → (∀ k ∈ insert i (G.parents i), v k = w k) → p i v = p i w) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1), Causalean.Graph.FiniteDensity.Factorization.observationalMeasure (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfRealCubeFactors p hmeas hnonneg hlocal hnorm) = Causalean.Graph.FiniteDensity.realCubeProductDensityMeasure p","l":481,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.parentClosed_nodeAncestralClosure","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A node has an ancestral closure that contains every parent of each of its nodes.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (i : V), Causalean.Graph.FiniteDensity.ParentClosed G (Causalean.Graph.FiniteDensity.nodeAncestralClosure G i)","l":51,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.parentClosed_predecessors","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":2,"d":"The predecessor set of a vertex in a topologically ranked DAG contains all parents of each of its members.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V), Causalean.Graph.FiniteDensity.ParentClosed G (Causalean.Graph.FiniteDensity.predecessors τ i)","l":58,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.parents_subset_predecessors","k":"theorem","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"A topological ranking places every parent of a vertex among its predecessors.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (τ : Causalean.Graph.FiniteDensity.TopologicalRanking G) (i : V), G.parents i ⊆ Causalean.Graph.FiniteDensity.predecessors τ i","l":43,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.predecessors","k":"def","m":"Causalean.Graph.FiniteDensity.OrderedLocalMarkov.Coordinates","a":"Graph/FiniteDensity/OrderedLocalMarkov","t":1,"d":"A topological ranking and vertex determine the finite set of vertices strictly preceding that vertex.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → Causalean.Graph.FiniteDensity.TopologicalRanking G → V → Finset V","l":38,"f":"Causalean/Graph/FiniteDensity/OrderedLocalMarkov/Coordinates.lean"},{"n":"Causalean.Graph.FiniteDensity.realCubeInterventionDensityMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Real-valued cube factors, a target coordinate, and a real-valued replacement function for that coordinate, none sign-restricted, determine their cube-restricted single-target product-density measure: Lebesgue measure restricted to the unit cube, weighted by the product in which the target factor is replaced by the replacement evaluated at the target coordinate, with a negative product truncated to zero.","s":"{V : Type u_1} → [DecidableEq V] → [Fintype V] → (V → (V → ℝ) → ℝ) → V → (ℝ → ℝ) → MeasureTheory.Measure (V → ℝ)","l":471,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.realCubeProductDensityMeasure","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Real-valued cube factors, with no sign restriction, determine their cube-restricted observational product-density measure: Lebesgue measure restricted to the unit cube, weighted by the joint product of the factors, where a negative product is truncated to zero.","s":"{V : Type u_1} → [Fintype V] → (V → (V → ℝ) → ℝ) → MeasureTheory.Measure (V → ℝ)","l":464,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.selfParents_subset_nodeAncestralClosure","k":"theorem","m":"Causalean.Graph.FiniteDensity.Main","a":"Graph/FiniteDensity","t":2,"d":"A node and all of its parents belong to that node's ancestral closure.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (i : V), insert i (G.parents i) ⊆ Causalean.Graph.FiniteDensity.nodeAncestralClosure G i","l":62,"f":"Causalean/Graph/FiniteDensity/Main.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCube","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A vertex population determines the unit cube of real assignments: the set of assignments whose every coordinate lies in the closed interval from zero to one.","s":"(V : Type u_1) → Set (V → ℝ)","l":28,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeFactorizationOfCubeFactors","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Cube-local extended-nonnegative factors that are measurable on the cube, local to each node and its parents on the cube, and normalized in their own coordinates on the cube determine a global unit-cube DAG factorization, by using their clamped extensions as factors.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → (p : V → (V → ℝ) → ENNReal) → (∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) → (∀ (i : V), Causalean.Graph.FiniteDensity.CubeDependsOn (insert i (G.parents i)) (p i)) → (∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫⁻ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) → Causalean.Graph.FiniteDensity.UnitCubeFactorization V G","l":253,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeFactorizationOfCubeFactors_factor_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Cube-local factors with cube measurability, node-and-parent locality, and coordinate-wise normalization, evaluated at a node and a cube point, give a constructed factor equal to the original factor.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ENNReal) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hlocal : ∀ (i : V), Causalean.Graph.FiniteDensity.CubeDependsOn (insert i (G.parents i)) (p i)) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫⁻ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) (i : V) {v : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfCubeFactors p hmeas hlocal hnorm).factor i v = p i v","l":286,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeFactorizationOfRealCubeFactors","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"Real-valued cube factors that are measurable on the cube, nonnegative there, local to each node and its parents, and normalized by Lebesgue integration in their own coordinate determine a global unit-cube DAG factorization, by conversion to extended nonnegative factors.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → {G : Causalean.DAG V} → (p : V → (V → ℝ) → ℝ) → (∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) → (∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, 0 ≤ p i v) → (∀ (i : V) {v w : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → w ∈ Causalean.Graph.FiniteDensity.unitCube V → (∀ k ∈ insert i (G.parents i), v k = w k) → p i v = p i w) → (∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) → Causalean.Graph.FiniteDensity.UnitCubeFactorization V G","l":402,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeFactorizationOfRealCubeFactors_factor_eq","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"Real cube factors with cube measurability, nonnegativity, node-and-parent locality, and coordinate-wise normalization, evaluated at a node and a cube point, give a constructed factor equal to the nonnegative-real conversion of the original factor.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G : Causalean.DAG V} (p : V → (V → ℝ) → ℝ) (hmeas : ∀ (i : V), Causalean.Graph.FiniteDensity.MeasurableOnSet (p i) (Causalean.Graph.FiniteDensity.unitCube V)) (hnonneg : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, 0 ≤ p i v) (hlocal : ∀ (i : V) {v w : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → w ∈ Causalean.Graph.FiniteDensity.unitCube V → (∀ k ∈ insert i (G.parents i), v k = w k) → p i v = p i w) (hnorm : ∀ (i : V), ∀ v ∈ Causalean.Graph.FiniteDensity.unitCube V, ∫ (x : ℝ) in Set.Icc 0 1, p i (Function.update v i x) = 1) (i : V) {v : V → ℝ}, v ∈ Causalean.Graph.FiniteDensity.unitCube V → (Causalean.Graph.FiniteDensity.unitCubeFactorizationOfRealCubeFactors p hmeas hnonneg hlocal hnorm).factor i v = ENNReal.ofReal (p i v)","l":442,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeReference","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"A finite vertex population determines the product reference measure on its unit cube.","s":"(V : Type u_1) → [Fintype V] → MeasureTheory.Measure (V → ℝ)","l":38,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeReference_compl","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"A finite vertex population has a unit-cube product reference measure concentrated on the ambient finite unit cube.","s":"∀ (V : Type u_2) [DecidableEq V] [inst : Fintype V], (Causalean.Graph.FiniteDensity.unitCubeReference V) (Causalean.Graph.FiniteDensity.unitCube V)ᶜ = 0","l":43,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitCubeReference_eq_volume_restrict","k":"theorem","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":2,"d":"For a finite coordinate type, the product of unit-interval Lebesgue restrictions is Lebesgue volume restricted to the finite unit cube.","s":"∀ (V : Type u_1) [DecidableEq V] [inst : Fintype V], Causalean.Graph.FiniteDensity.unitCubeReference V = MeasureTheory.volume.restrict (Causalean.Graph.FiniteDensity.unitCube V)","l":175,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.Graph.FiniteDensity.unitIntervalReference","k":"def","m":"Causalean.Graph.FiniteDensity.Cube","a":"Graph/FiniteDensity","t":1,"d":"The one-coordinate reference measure on the unit interval is Lebesgue measure restricted to the closed interval from zero to one.","s":"MeasureTheory.Measure ℝ","l":33,"f":"Causalean/Graph/FiniteDensity/Cube.lean"},{"n":"Causalean.InducedFrom","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"For a single-world intervention graph and a selected finite set of nodes, the induced subgraph is the graph obtained by restricting the original graph to those selected nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N","l":64,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.ML.AchievesL2Rate","k":"def","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":1,"d":"For a measurable sample space, a measurable covariate space, a sequence of estimated regression functions indexed by sample size and experiment outcome, a target regression function, a joint covariate--response measure, a real-valued rate sequence, and an experiment measure, the predicate that the estimators achieve the stated L² rate holds precisely when for every sample size and experiment outcome, the L² seminorm of the estimation error under the covariate marginal of the joint measure is finite, and the resulting real-valued sequence of L² seminorms is stochastically bounded at the supplied rate under the experiment measure.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {X : Type u_2} → [inst_1 : MeasurableSpace X] → (ℕ → Ω → X → ℝ) → (X → ℝ) → MeasureTheory.Measure (X × ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":196,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.Activation","k":"structure","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"An activation function bundles a scalar map together with a Lipschitz constant and the certificate that the map is Lipschitz with that constant (e.g. ReLU, sigmoid, and tanh are all 1-Lipschitz).","s":"Type","l":33,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.Activation.applyVec","k":"def","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"For a vector dimension, an activation-function bundle, and an input vector, the coordinatewise activation is the vector obtained by applying the bundle's scalar activation to each coordinate of the input.","s":"{n : ℕ} → Causalean.ML.Activation → (Fin n → ℝ) → Fin n → ℝ","l":44,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.Causal.aipw_mlNuisance_meanZero_of_wellSpecified","k":"theorem","m":"Causalean.ML.CausalApplication.AIPWExample","a":"ML/CausalApplication","t":1,"d":"End-to-end identification with ML nuisances. In a backdoor estimation system S, suppose the propensity is bounded away from 0 and 1 by ε (strict overlap), the backdoor identification assumptions hold, the squared factual outcome is integrable, and the squared potential outcome under each treatment arm is integrable. If the ML-learned outcome-regression functions and propensity function are measurable and agree pointwise with the true outcome regression and true propensity, respectively (correct specification), then the AIPW moment functional built from the ML-learned nuisance vector integrates to zero at the true average treatment effect S.θ₀ — the doubly-robust estimating equation identifies the ATE with machine-learned nuisances.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.Estimation.ATE.BackdoorEstimationSystem P γ) {ε : ℝ}, S.StrictOverlap ε → S.Assumptions → MeasureTheory.Integrable (fun ω => S.factualY ω ^ 2) P.μ → (∀ (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω ^ 2) P.μ) → ∀ {mhat : Bool → γ → ℝ} {ehat : γ → ℝ} (hmhat : ∀ (b : Bool), Measurable (mhat b)) (hehat : Measurable ehat), (∀ (b : Bool) (x : γ), mhat b x = S.μ_val b x) → (∀ (x : γ), ehat x = S.e_val x) → ∫ (z : γ × Bool × ℝ), Causalean.Estimation.ATE.BackdoorEstimationSystem.aipwMomentFunctional (Causalean.ML.Causal.mlNuisanceVec mhat ehat hmhat hehat) z S.θ₀ ∂S.P_Z = 0","l":27,"f":"Causalean/ML/CausalApplication/AIPWExample.lean"},{"n":"Causalean.ML.Causal.condExp_of_isL2Projection","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"On a finite joint covariate–response law P, if the candidate function m is measurable, the response coordinate is integrable, the composed function z ↦ m(z.1) is integrable, and m is an L² projection of the response onto functions of the covariate — its residual is uncorrelated with every measurable, integrable function of the covariate, then m agrees P-almost everywhere with the conditional expectation of the response given the covariate σ-algebra.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure (X × ℝ)) [MeasureTheory.IsFiniteMeasure P] {m : X → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) P → MeasureTheory.Integrable (fun z => m z.1) P → Causalean.ML.IsL2Projection P m → (fun z => m z.1) =ᵐ[P] P[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":116,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.covarSigma","k":"def","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"For a covariate space equipped with a measurable structure, the covariate σ-algebra on the joint covariate--response sample space is the σ-algebra generated by observing only the covariate coordinate.","s":"{X : Type u_1} → [MeasurableSpace X] → MeasurableSpace (X × ℝ)","l":32,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.covarSigma_le","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":2,"d":"The covariate sigma-algebra is a sub-sigma-algebra of the full joint observation sigma-algebra.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Causalean.ML.Causal.covarSigma ≤ inferInstance","l":35,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.dml_rate_conditions_of_quarter_rates","k":"theorem","m":"Causalean.ML.CausalApplication.RateAssembly","a":"ML/CausalApplication","t":1,"d":"Assembly. If, for every treatment arm, the outcome-regression estimation error is o_p(n^{-1/4}), and the propensity estimation error is o_p(n^{-1/4}), then each error is separately o_p(1), and for every arm the pointwise product of the outcome and propensity errors is o_p(n^{-1/2}) — together these are DML's three nuisance-rate conditions.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {μErr : Bool → ℕ → Ω → ℝ} {eErr : ℕ → Ω → ℝ}, (∀ (a : Bool), Causalean.Stat.IsLittleOp (μErr a) (fun n => ↑n ^ (-(1 / 4))) μ) → Causalean.Stat.IsLittleOp eErr (fun n => ↑n ^ (-(1 / 4))) μ → (∀ (a : Bool), Causalean.Stat.IsLittleOp (μErr a) (fun x => 1) μ) ∧ Causalean.Stat.IsLittleOp eErr (fun x => 1) μ ∧ ∀ (a : Bool), Causalean.Stat.IsLittleOp (fun n ω => μErr a n ω * eErr n ω) (fun n => ↑n ^ (-(1 / 2))) μ","l":36,"f":"Causalean/ML/CausalApplication/RateAssembly.lean"},{"n":"Causalean.ML.Causal.isL2Projection_of_condExp","k":"theorem","m":"Causalean.ML.CausalApplication.RegressionBridge","a":"ML/CausalApplication","t":1,"d":"On a finite joint covariate–response law P, if the candidate function m is measurable, the response coordinate is integrable, and m agrees P-almost everywhere with the conditional expectation of the response given the covariate σ-algebra, then m is an L² projection of the response onto functions of the covariate: its residual is uncorrelated with every measurable, integrable function of the covariate.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure (X × ℝ)) [MeasureTheory.IsFiniteMeasure P] {m : X → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) P → (fun z => m z.1) =ᵐ[P] P[fun z => z.2 | Causalean.ML.Causal.covarSigma] → Causalean.ML.IsL2Projection P m","l":40,"f":"Causalean/ML/CausalApplication/RegressionBridge.lean"},{"n":"Causalean.ML.Causal.mlNuisanceVec","k":"def","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"For a measurable covariate space, two outcome-regression functions indexed by a Boolean treatment level, a propensity function, the hypothesis that each outcome-regression function is measurable, and the hypothesis that the propensity function is measurable, the machine-learning nuisance vector consists of the supplied outcome-regression functions, the supplied propensity function, their outcome-regression measurability certificates, and the propensity measurability certificate.","s":"{γ : Type u_1} → [inst : MeasurableSpace γ] → (μ_fn : Bool → γ → ℝ) → (e_fn : γ → ℝ) → (∀ (b : Bool), Measurable (μ_fn b)) → Measurable e_fn → Causalean.Estimation.ATE.NuisanceVec γ","l":55,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.mlOutcomeRegression_ae_eq","k":"theorem","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"Outcome-regression recovery. On a finite covariate–outcome law Pd, if the candidate regression function m is measurable, the outcome coordinate is integrable, the composed function z ↦ m(z.1) is integrable, and m is the squared-loss population target — its residual Y − m(X) is uncorrelated with every measurable, integrable function of the covariate, then m agrees Pd-almost everywhere with the conditional expectation of the outcome given the covariate σ-algebra, i.e. m is the outcome regression μ(d, x) = E[Y ∣ X = x, D = d].","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (Pd : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure Pd] {m : γ → ℝ}, Measurable m → MeasureTheory.Integrable (fun z => z.2) Pd → MeasureTheory.Integrable (fun z => m z.1) Pd → Causalean.ML.IsL2Projection Pd m → (fun z => m z.1) =ᵐ[Pd] Pd[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":26,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.mlPropensity_ae_eq","k":"theorem","m":"Causalean.ML.CausalApplication.Nuisance","a":"ML/CausalApplication","t":1,"d":"Propensity recovery. On a finite covariate–treatment-indicator law Pe, if the candidate propensity function e is measurable, the treatment indicator is integrable, the composed function z ↦ e(z.1) is integrable, and e is the squared-loss population target — its residual is uncorrelated with every measurable, integrable function of the covariate, then e agrees Pe-almost everywhere with the conditional expectation of the treatment indicator given the covariate σ-algebra, i.e. e is the propensity P(D = 1 ∣ X).","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (Pe : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure Pe] {e : γ → ℝ}, Measurable e → MeasureTheory.Integrable (fun z => z.2) Pe → MeasureTheory.Integrable (fun z => e z.1) Pe → Causalean.ML.IsL2Projection Pe e → (fun z => e z.1) =ᵐ[Pe] Pe[fun z => z.2 | Causalean.ML.Causal.covarSigma]","l":41,"f":"Causalean/ML/CausalApplication/Nuisance.lean"},{"n":"Causalean.ML.Causal.rpow_quarter_le_one","k":"theorem","m":"Causalean.ML.CausalApplication.RateAssembly","a":"ML/CausalApplication","t":2,"d":"The n^{-1/4} rate is bounded by 1 (so it weakens to the o_p(1) rate).","s":"∀ (n : ℕ), ↑n ^ (-(1 / 4)) ≤ 1","l":28,"f":"Causalean/ML/CausalApplication/RateAssembly.lean"},{"n":"Causalean.ML.DenseLayer","k":"structure","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"A dense affine layer bundles a weight matrix and a bias vector determining the map $x \\mapsto Wx + b$ from Fin m inputs to Fin n outputs.","s":"ℕ → ℕ → Type","l":21,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.DenseLayer.eval","k":"def","m":"Causalean.ML.NeuralNet.Layer","a":"ML/NeuralNet","t":1,"d":"For an input dimension, an output dimension, a dense affine layer, and an input vector, the layer evaluation is the output vector whose $j$th coordinate is the weighted sum of the input coordinates plus the $j$th bias.","s":"{m n : ℕ} → Causalean.ML.DenseLayer m n → (Fin m → ℝ) → Fin n → ℝ","l":29,"f":"Causalean/ML/NeuralNet/Layer.lean"},{"n":"Causalean.ML.FeatureMap","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A finite feature map bundles a transform sending each input to its vector of K feature values. Linear-in-features predictors use x ↦ ⟪β, φ x⟫; the identity feature map recovers ordinary linear regression, and other choices of the transform recover polynomial, spline, or Fourier (sieve) regression.","s":"Type u_1 → (K : Type u_2) → [Fintype K] → Type (max u_1 u_2)","l":45,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.FinitePartitionPredictor","k":"structure","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"A piecewise-constant predictor on a finite partition: a finite index type cell of partition cells, a map chooseCell assigning each input to its cell, and a constant predicted value value on each cell.","s":"Type u_1 → Type (max 1 u_1)","l":18,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.FinitePartitionPredictor.eval","k":"def","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"For an input space, a finite-partition predictor, and an input, the tree prediction is the constant value assigned to the partition cell containing that input.","s":"{X : Type u_1} → Causalean.ML.FinitePartitionPredictor X → X → ℝ","l":33,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.FinitePartitionPredictor.eval_eqOn_cell","k":"theorem","m":"Causalean.ML.Tree.FinitePartitionPredictor","a":"ML/Tree","t":1,"d":"For a finite-partition predictor T, on the set of inputs mapped to a given cell c, the predictor's output equals the constant value assigned to that cell.","s":"∀ {X : Type u_1} (T : Causalean.ML.FinitePartitionPredictor X) (c : T.cell), Set.EqOn T.eval (fun x => T.value c) {x | T.chooseCell x = c}","l":37,"f":"Causalean/ML/Tree/FinitePartitionPredictor.lean"},{"n":"Causalean.ML.HasFinitePopulationRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a measurable input space, a measurable label space, a pointwise loss, a joint input--label measure, and a prediction rule, the finite-population-risk condition holds exactly when the loss comparing the rule's prediction with the observed label is integrable under the joint measure.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → Prop","l":58,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.HypothesisClass","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A hypothesis class packaged extensionally as a set of admissible prediction functions from the covariates to the outcome, subject to the requirement that every function admitted to the class is measurable.","s":"(X : Type u_1) → (Y : Type u_2) → [MeasurableSpace X] → [MeasurableSpace Y] → Type (max u_1 u_2)","l":36,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.IsApproxERMP","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat minimizes a parametric objective up to slack ε ≥ 0.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ → Θ → ℝ → Prop","l":34,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsERM","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"An empirical-risk minimizer is an admissible prediction rule whose finite-sample risk criterion is no larger than that of any other admissible rule. The criterion is the inverse-cardinality-scaled loss sum over a nonempty finite sample.","s":"{ι : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [Fintype ι] → [Nonempty ι] → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.HypothesisClass X Y → Causalean.ML.Loss Y → (ι → X × Y) → (X → Y) → Prop","l":55,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsERMP","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat exactly minimizes a parametric objective over the admissible set.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ → Θ → Prop","l":27,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsL2Projection","k":"def","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":1,"d":"For a measurable covariate space, a joint measure of a covariate and a real-valued response, and a candidate regression function, the L²-projection condition holds precisely when, for every measurable real-valued function of the covariate whose product with the residual is integrable, the integral of that product is zero.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → MeasureTheory.Measure (X × ℝ) → (X → ℝ) → Prop","l":47,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.IsPSDkernel","k":"def","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"For an input space and a real-valued kernel on that space, the positive-semidefinite-kernel property holds exactly when, for every finite sample, its Gram matrix is positive semidefinite.","s":"{X : Type u_1} → (X → X → ℝ) → Prop","l":29,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.IsPopulationOLS","k":"def","m":"Causalean.ML.Linear.Population","a":"ML/Linear","t":1,"d":"For a measurable covariate space, a finite feature index set, a joint covariate–response measure, a feature map, and a coefficient vector, the population ordinary-least-squares condition holds exactly when the integral of the product of the corresponding linear-predictor residual and each feature coordinate is zero.","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → (K → ℝ) → Prop","l":25,"f":"Causalean/ML/Linear/Population.lean"},{"n":"Causalean.ML.IsPopulationRegLogistic","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"For a measurable covariate space, a finite feature index set, a joint covariate–response measure, a feature map, a penalty level, and a coefficient vector, the penalized population logistic first-order condition holds exactly when, for every feature coordinate, the integral of the logistic score times that coordinate plus twice the penalty level times its coefficient is zero.","s":"{γ : Type u_2} → {K : Type u_3} → [inst : MeasurableSpace γ] → [inst_1 : Fintype K] → MeasureTheory.Measure (γ × ℝ) → Causalean.ML.FeatureMap γ K → ℝ → (K → ℝ) → Prop","l":47,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.IsPopulationRidge","k":"def","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"For a measurable covariate space, a finite feature index set, a joint covariate–response measure, a feature map, a penalty level, and a coefficient vector, the regularized population normal-equation condition holds exactly when, for every feature coordinate, the integral of the product of the linear-predictor residual and that coordinate equals the penalty level times its coefficient.","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → ℝ → (K → ℝ) → Prop","l":37,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.IsPopulationRiskMinimizer","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"A population-risk minimizer bundles the claim that a prediction rule belongs to the hypothesis class, that it has finite expected loss under the population measure, that every other rule admissible in the class also has finite expected loss, and that its population risk is no larger than that of any other rule in the class.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.HypothesisClass X Y → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → Prop","l":67,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.IsRKHS","k":"structure","m":"Causalean.ML.Kernel.RKHS","a":"ML/Kernel","t":1,"d":"The candidate reproducing-kernel Hilbert space structure on a real inner-product space H of functions on X, given an evaluation map and a representer map, satisfies the reproducing property: evaluating any function of H at a point equals its inner product with the representer of that point.","s":"(X : Type u_1) → (H : Type u_2) → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → (H → X → ℝ) → (X → H) → Prop","l":18,"f":"Causalean/ML/Kernel/RKHS.lean"},{"n":"Causalean.ML.KFeat","k":"def","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"For a feature dimension, a feature-radius bound, and a response-radius bound, the feature–response data space consists of pairs whose feature component lies in the closed Euclidean ball of the stated feature radius centered at zero and whose real-valued response lies in the closed interval centered at zero with the stated response radius.","s":"ℕ → ℝ → ℝ → Type","l":34,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.KWeight","k":"def","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"For a feature dimension and a weight-radius bound, the weight-vector space is the closed Euclidean ball centered at zero with the stated weight radius.","s":"ℕ → ℝ → Type","l":41,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.LassoFeat","k":"def","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"For a dimension $d$, a coordinate bound $X_{\\infty}$, and a response bound $Y_b$, the lasso feature--response space consists of pairs whose $d$ feature coordinates have absolute value at most $X_{\\infty}$ and whose response lies in the closed interval $[-Y_b,Y_b]$.","s":"ℕ → ℝ → ℝ → Type","l":34,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.LassoWeight","k":"def","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"For a dimension $d$ and a radius $W$, the lasso weight space is the set of $d$-dimensional real weight vectors whose coordinate $\\ell^1$ norm is at most $W$.","s":"ℕ → ℝ → Type","l":40,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.Loss","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a label space, a pointwise loss is a real-valued function that compares a predicted label with an observed label.","s":"Type u_1 → Type u_1","l":25,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.Predictor","k":"structure","m":"Causalean.ML.Core.Hypothesis","a":"ML/Core","t":1,"d":"A parametrized family of predictors bundles an admissible set of parameter values together with a map sending each parameter to a prediction function from the covariates to the outcome.","s":"Type u_1 → Type u_2 → Type u_3 → Type (max (max u_1 u_2) u_3)","l":27,"f":"Causalean/ML/Core/Hypothesis.lean"},{"n":"Causalean.ML.ProperBinaryLoss","k":"def","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"For a real-valued loss of a predicted probability and a binary outcome, the property of being a proper binary loss means that, for every true probability $\\eta$ in $[0,1]$ and every candidate probability $q$ in $[0,1]$, the conditional expected loss at $\\eta$ is no greater than that at $q$: $\\eta L(\\eta,1)+(1-\\eta)L(\\eta,0) \\le \\eta L(q,1)+(1-\\eta)L(q,0)$.","s":"(ℝ → Bool → ℝ) → Prop","l":28,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.RandomForest","k":"structure","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"A random forest: one finite-partition regression tree tree for each index in Fin T.","s":"Type u_1 → ℕ → Type (max 1 u_1)","l":25,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RandomForest.eval","k":"def","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"Given an input domain, a nonnegative number of trees, a random forest on that domain, and an input point, the forest prediction is the arithmetic average of the predictions made at that point by all trees in the forest.","s":"{X : Type u_1} → {T : ℕ} → Causalean.ML.RandomForest X T → X → ℝ","l":31,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RandomForest.eval_mem_Icc","k":"theorem","m":"Causalean.ML.Tree.Forest","a":"ML/Tree","t":1,"d":"For a random forest F of T finite-partition regression trees, if the forest is nonempty (T > 0) and every tree predicts within [a, b] at every input, then the forest's averaged prediction also lies in [a, b] at every input x.","s":"∀ {X : Type u_1} {T : ℕ} (F : Causalean.ML.RandomForest X T), 0 < T → ∀ {a b : ℝ}, (∀ (t : Fin T) (x : X), (F.tree t).eval x ∈ Set.Icc a b) → ∀ (x : X), F.eval x ∈ Set.Icc a b","l":37,"f":"Causalean/ML/Tree/Forest.lean"},{"n":"Causalean.ML.RegularizedERM","k":"structure","m":"Causalean.ML.Core.ERM","a":"ML/Core","t":1,"d":"θhat minimizes a penalized objective objective θ + lam · penalty θ with a nonnegative regularization weight lam.","s":"{Θ : Type u_1} → (Θ → ℝ) → (Θ → ℝ) → ℝ → Set Θ → Θ → Prop","l":44,"f":"Causalean/ML/Core/ERM.lean"},{"n":"Causalean.ML.StrictProperBinaryLoss","k":"def","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"For a real-valued loss of a predicted probability and a binary outcome, the property of being a strictly proper binary loss means both that the loss is proper and that, for every true probability $\\eta$ and every candidate probability $q$ in $[0,1]$, equality between their conditional expected losses implies $q=\\eta$.","s":"(ℝ → Bool → ℝ) → Prop","l":35,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.SubgradientAt","k":"def","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":1,"d":"In a real inner-product space, for a real-valued function, a subset of the space, a vector, and a point, the statement that the vector is a subgradient of the function at the point relative to the subset requires the point to belong to the subset and for every point $y$ in that subset, $f(x)+\\langle g,y-x\\rangle\\le f(y)$.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (E → ℝ) → Set E → E → E → Prop","l":121,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.bernoulliCE","k":"def","m":"Causalean.ML.Binary.FisherConsistency","a":"ML/Binary","t":1,"d":"For a true success probability $\\eta$ and a predicted success probability $q$, the Bernoulli cross-entropy is $-\\eta\\log q-(1-\\eta)\\log(1-q)$.","s":"ℝ → ℝ → ℝ","l":19,"f":"Causalean/ML/Binary/FisherConsistency.lean"},{"n":"Causalean.ML.bernoulliCE_isMinOn","k":"theorem","m":"Causalean.ML.Binary.FisherConsistency","a":"ML/Binary","t":1,"d":"For a true probability η strictly between 0 and 1, the Bernoulli cross-entropy q ↦ bernoulliCE η q attains its minimum over the open interval (0,1) exactly at the truth q = η.","s":"∀ {η : ℝ}, η ∈ Set.Ioo 0 1 → IsMinOn (fun q => Causalean.ML.bernoulliCE η q) (Set.Ioo 0 1) η","l":25,"f":"Causalean/ML/Binary/FisherConsistency.lean"},{"n":"Causalean.ML.bestLinearPredictor_minimizes_populationRisk","k":"theorem","m":"Causalean.ML.Linear.Population","a":"ML/Linear","t":1,"d":"For a probability measure P on features and outcome and a finite feature map φ, if the residual of the linear predictor with coefficients βstar is uncorrelated in expectation with every feature, the population squared-loss risks of the βstar- and β-predictors are both finite, and each feature is integrable against that residual, then the population squared risk of the βstar-predictor is at most that of any other linear-in-features predictor with coefficients β.","s":"∀ {X' : Type u_1} {K : Type u_2} [inst : MeasurableSpace X'] [inst_1 : Fintype K] (P : MeasureTheory.Measure (X' × ℝ)) (φ : Causalean.ML.FeatureMap X' K) {βstar : K → ℝ}, Causalean.ML.IsPopulationOLS P φ βstar → ∀ (β : K → ℝ), (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) → (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k) → (∀ (k : K), MeasureTheory.Integrable (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) P) → (Causalean.ML.populationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) ≤ Causalean.ML.populationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k","l":32,"f":"Causalean/ML/Linear/Population.lean"},{"n":"Causalean.ML.bool01","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"For a binary label, the zero--one real-valued encoding of that label is one when the label is true and zero when it is false.","s":"Bool → ℝ","l":24,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_false","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"A false Boolean label is encoded as zero.","s":"Causalean.ML.bool01 false = 0","l":48,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_le_one","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The zero-one encoding of a Boolean label is always at most one.","s":"∀ (b : Bool), Causalean.ML.bool01 b ≤ 1","l":55,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_nonneg","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The zero-one encoding of a Boolean label is always nonnegative.","s":"∀ (b : Bool), 0 ≤ Causalean.ML.bool01 b","l":51,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.bool01_true","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"A true Boolean label is encoded as the real number one.","s":"Causalean.ML.bool01 true = 1","l":45,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.centered_score_mean_isBigOp","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The centered empirical score mean Ŝₙ − λβ⋆ = (Ĉₙ − Ĝₙβ⋆) − λβ⋆ = n⁻¹ Σ (φᵢ(yᵢ−⟨β⋆,φᵢ⟩) − λβ⋆) is O_p(n^{-1/2}). The population normal equations make the summands centered.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ} (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → Causalean.Stat.IsBigOp (fun n ω => ‖Causalean.ML.empiricalCross φ S.Z n ω - (Causalean.ML.empiricalGram φ S.Z n ω).mulVec βstar - lam • βstar‖) (fun n => (√↑n)⁻¹) μ","l":254,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.clampedSq","k":"def","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"For a real clipping bound and a real input, the clamped-square surrogate is the square of the input after applying the interval-clipping rule with endpoints $-c$ and $c$. This definition applies to every pair of real numbers, including when the two endpoints are not ordered.","s":"ℝ → ℝ → ℝ","l":31,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_eq_sq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"On the band where t lies within c in absolute value, the clamped square clampedSq c t equals the genuine square t².","s":"∀ {c t : ℝ}, |t| ≤ c → Causalean.ML.clampedSq c t = t ^ 2","l":45,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_le_sq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"For a nonnegative bound c, the clamped square clampedSq c t never exceeds c², for every real t.","s":"∀ {c : ℝ}, 0 ≤ c → ∀ (t : ℝ), Causalean.ML.clampedSq c t ≤ c ^ 2","l":53,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.clampedSq_nonneg","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":2,"d":"The clamped square is nonnegative.","s":"∀ (c t : ℝ), 0 ≤ Causalean.ML.clampedSq c t","l":50,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.continuousAt_matrixInv_entry","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"Continuity of an inverse-matrix entry. At any matrix with nonzero determinant, the map M ↦ (M⁻¹)ₖₗ is continuous.","s":"∀ {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] {G₀ : Matrix K K ℝ}, G₀.det ≠ 0 → ∀ (k l : K), ContinuousAt (fun M => M⁻¹ k l) G₀","l":395,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.continuous_clampedSq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":2,"d":"The clamped square is continuous.","s":"∀ (c : ℝ), Continuous (Causalean.ML.clampedSq c)","l":40,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.continuous_logisticEmpRisk","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"The empirical logistic risk is continuous.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool), Continuous (Causalean.ML.logisticEmpRisk Z)","l":107,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.continuous_softplus","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"The softplus function x ↦ log(1 + eˣ) is continuous on the whole real line.","s":"Continuous Causalean.ML.softplus","l":44,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_comp_inner","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"A convex scalar function precomposed with the linear score θ ↦ ⟪θ, x⟫ is convex. This is the workhorse turning convexity of a margin loss into convexity of the parameter objective.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {φ : ℝ → ℝ}, ConvexOn ℝ Set.univ φ → ∀ (x : E), ConvexOn ℝ Set.univ fun θ => φ (inner ℝ θ x)","l":64,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.convexOn_l1penalty","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"The L1 penalty is convex as a function of the coefficient vector.","s":"∀ {Param : Type u_2} [inst : Fintype Param], ConvexOn ℝ Set.univ Causalean.ML.l1penalty","l":50,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_lassoObjective","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The lasso objective is convex for λ ≥ 0.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 ≤ lam → ConvexOn ℝ Set.univ (Causalean.ML.lassoObjective X y lam)","l":116,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_logisticEmpRisk","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"For any finite sample of feature–label pairs Z, the empirical logistic risk, as a function of the coefficient, is convex.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool), ConvexOn ℝ Set.univ (Causalean.ML.logisticEmpRisk Z)","l":82,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_logisticScoreLoss","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"The score-space logistic loss is convex in the score.","s":"∀ (y : Bool), ConvexOn ℝ Set.univ fun t => Causalean.ML.logisticScoreLoss y t","l":63,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.convexOn_olsObjective","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The least-squares objective is convex in the coefficients.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), ConvexOn ℝ Set.univ (Causalean.ML.olsObjective X y)","l":74,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.convexOn_softplus","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":2,"d":"Softplus is convex.","s":"ConvexOn ℝ Set.univ Causalean.ML.softplus","l":56,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.eLpNorm_predictor_sub_le","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"Deterministic predictor Lipschitz bound. The prediction L²(P_X) error is bounded by a finite constant times the coefficient error: ‖∑ₖ δₖ φ·ₖ‖_{L²} ≤ C ‖δ‖.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → ∀ (βstar : K → ℝ), ∃ C, 0 ≤ C ∧ ∀ (β : K → ℝ), (MeasureTheory.eLpNorm (fun x => ∑ k, β k * φ.φ x k - ∑ k, βstar k * φ.φ x k) 2 (MeasureTheory.Measure.map Prod.fst P)).toReal ≤ C * ‖β - βstar‖","l":641,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.effectiveDimension","k":"def","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":1,"d":"Given an arbitrary set of eigenvalue indices, a family of real eigenvalues indexed by that set, and a real regularization level, the effective dimension is the infinite sum $\\sum_i \\mu_i/(\\mu_i+\\lambda)$. This definition imposes no positivity, summability, or nonzero-denominator condition; a summand with zero denominator, and the total when the summand family is not summable, are assigned the value zero.","s":"{ι : Type u_1} → (ι → ℝ) → ℝ → ℝ","l":44,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_antitone","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Monotonicity in the regularization level. More regularization shrinks the effective dimension: λ₁ ≤ λ₂ ⇒ N(λ₂) ≤ N(λ₁).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam₁ lam₂ : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam₁ → lam₁ ≤ lam₂ → Summable μ → Causalean.ML.effectiveDimension μ lam₂ ≤ Causalean.ML.effectiveDimension μ lam₁","l":107,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_le_card","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Over finitely many eigenvalues, N(λ) never exceeds the ambient dimension #ι.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Causalean.ML.effectiveDimension μ lam ≤ ↑(Fintype.card ι)","l":97,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_le_trace_div","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":1,"d":"Dimension-free bound. For an eigenvalue family μ and regularization level lam, if every eigenvalue is nonnegative, the regularization level is strictly positive, and the eigenvalues are summable, i.e. the operator is trace-class, then the effective dimension N(lam) = ∑ᵢ μᵢ/(μᵢ+lam) is at most the trace ∑ᵢ μᵢ divided by lam. This is the bound that, with eigenvalue decay μᵢ ≍ i^{-b}, yields N(λ) = O(λ^{-1/b}).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Summable μ → Causalean.ML.effectiveDimension μ lam ≤ (∑' (i : ι), μ i) / lam","l":83,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_nonneg","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"The effective dimension is nonnegative.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → 0 ≤ Causalean.ML.effectiveDimension μ lam","l":78,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_le_div","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand is dominated by μᵢ/λ.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), μ i / (μ i + lam) ≤ μ i / lam","l":59,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_le_one","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand is at most 1.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), μ i / (μ i + lam) ≤ 1","l":64,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.effectiveDimension_term_nonneg","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"Each summand μᵢ/(μᵢ+λ) is nonnegative (for nonnegative eigenvalues and λ > 0).","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → ∀ (i : ι), 0 ≤ μ i / (μ i + lam)","l":54,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.ML.empiricalCross","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Given an arbitrary sample space, an arbitrary covariate space, a finite set of feature coordinates, a feature map from covariates into those coordinates, a sequence of data samples indexed by sample-space outcomes, a nonnegative sample size, and a sample-space outcome selecting one realized sequence, the empirical feature--response cross moment is $n^{-1}\\sum_{i<n}y_i\\phi(x_i)$, where $(x_i,y_i)$ is the $i$th realized observation.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℕ → Ω → K → ℝ","l":53,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalGram","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Given an arbitrary sample space, an arbitrary covariate space, a finite set of feature coordinates, a feature map from covariates into those coordinates, a sequence of data samples indexed by sample-space outcomes, a nonnegative sample size, and a sample-space outcome selecting one realized sequence, the empirical feature Gram matrix is $n^{-1}\\sum_{i<n}\\phi(x_i)\\phi(x_i)^\\mathsf{T}$, where $(x_i,y_i)$ is the $i$th realized observation.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℕ → Ω → Matrix K K ℝ","l":42,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalGram_posSemidef","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The empirical Gram is positive semidefinite (average of rank-one φφᵀ).","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) (n : ℕ) (ω : Ω), (Causalean.ML.empiricalGram φ Z n ω).PosSemidef","l":206,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.empiricalRademacherComplexity_innerBall_le","k":"theorem","m":"Causalean.ML.Kernel.RKHSRademacher","a":"ML/Kernel","t":2,"d":"For a sample of vectors bounded by κ and linear weights bounded by r, the empirical Rademacher complexity of the resulting linear class is at most κ times r divided by the square root of the sample size.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} {n : ℕ} (κ r : ℝ), 0 ≤ κ → 0 ≤ r → ∀ (Y : Fin n → E), (∀ (k : Fin n), ‖Y k‖ ≤ κ) → ∀ (w : ι → E), (∀ (i : ι), ‖w i‖ ≤ r) → empiricalRademacherComplexity n (fun i a => inner ℝ (w i) a) Y ≤ κ * r / √↑n","l":442,"f":"Causalean/ML/Kernel/RKHSRademacher.lean"},{"n":"Causalean.ML.empiricalRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a nonempty finite sample index set, an input space, a label space, a pointwise loss, a sample of input--label pairs, and a prediction rule, the empirical risk is the average, over all sample indices, of the loss comparing the rule's prediction at that observation's input with that observation's label.","s":"{ι : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [Fintype ι] → [Nonempty ι] → Causalean.ML.Loss Y → (ι → X × Y) → (X → Y) → ℝ","l":28,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.empiricalRiskP","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a nonempty finite sample index set, a parameter space, an input space, a label space, a parametric predictor, a pointwise loss, a sample of input--label pairs, and a parameter value, the parametric empirical risk is the empirical average loss of the prediction rule selected by that parameter value.","s":"{ι : Type u_1} → {Θ : Type u_2} → {X : Type u_3} → {Y : Type u_4} → [Fintype ι] → [Nonempty ι] → Causalean.ML.Predictor Θ X Y → Causalean.ML.Loss Y → (ι → X × Y) → Θ → ℝ","l":33,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.empiricalRiskP_convexOn_of_loss_convex","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"If every per-sample parameter loss is convex on Θset, so is the empirical risk of the predictor.","s":"∀ {ι : Type u_1} {Θ : Type u_2} {X : Type u_3} {Y : Type u_4} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : AddCommGroup Θ] [inst_3 : Module ℝ Θ] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (S : ι → X × Y) (Θset : Set Θ), Convex ℝ Θset → (∀ (i : ι), ConvexOn ℝ Θset fun θ => loss (M.predict θ (S i).1) (S i).2) → ConvexOn ℝ Θset (Causalean.ML.empiricalRiskP M loss S)","l":27,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.empiricalRisk_squaredLoss_linear","k":"theorem","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":2,"d":"Bridge to the spine: the empirical squared-loss risk of the linear-in-features predictor x ↦ ⟪β, φ x⟫ equals (card ι)⁻¹ times the OLS objective of the design matrix Xᵢₖ = (φ xᵢ)ₖ.","s":"∀ {ι : Type u_3} {K : Type u_4} {X' : Type u_5} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : Fintype K] (φ : Causalean.ML.FeatureMap X' K) (S : ι → X' × ℝ) (β : K → ℝ), (Causalean.ML.empiricalRisk Causalean.ML.squaredLoss S fun x => ∑ k, β k * φ.φ x k) = (↑(Fintype.card ι))⁻¹ * Causalean.ML.olsObjective (fun i k => φ.φ (S i).1 k) (fun i => (S i).2) β","l":89,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.evalLayers","k":"def","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"For a nonnegative network width and an activation function, the evaluation of a uniform-width feedforward network maps a finite ordered list of equal-width dense layers and an input vector to its output vector. For an empty list, the output is the input itself; for a list whose first layer is followed by further layers, the output applies the first layer and then evaluates the remaining layers.","s":"{n : ℕ} → Causalean.ML.Activation → List (Causalean.ML.DenseLayer n n) → (Fin n → ℝ) → Fin n → ℝ","l":29,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.evalLayers_append","k":"theorem","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":2,"d":"Structure — composition. Evaluating concatenated layer lists is the composition of the two evaluations.","s":"∀ {n : ℕ} (σ : Causalean.ML.Activation) (Ls Ms : List (Causalean.ML.DenseLayer n n)) (x : Fin n → ℝ), Causalean.ML.evalLayers σ (Ls ++ Ms) x = Causalean.ML.evalLayers σ Ms (Causalean.ML.evalLayers σ Ls x)","l":39,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.evalLayers_lipschitz","k":"theorem","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"Structure — Lipschitz. For a uniform-width feedforward network with activation σ and layer list Ls, if each layer's affine-then-activation map is Lipschitz with the constant assigned to it by k, then the whole network evaluation is Lipschitz with constant equal to the product of the per-layer constants.","s":"∀ {n : ℕ} (σ : Causalean.ML.Activation) (Ls : List (Causalean.ML.DenseLayer n n)) (k : Causalean.ML.DenseLayer n n → NNReal), (∀ L ∈ Ls, LipschitzWith (k L) (Causalean.ML.layerMap σ L)) → LipschitzWith (List.map k Ls).prod (Causalean.ML.evalLayers σ Ls)","l":48,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.exists_isMinOn_of_isCompact","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"Existence of a minimizer on a nonempty compact set (Weierstrass).","s":"∀ {Θ : Type u_1} [inst : TopologicalSpace Θ] {objective : Θ → ℝ} {Θset : Set Θ}, Θset.Nonempty → IsCompact Θset → ContinuousOn objective Θset → ∃ θhat ∈ Θset, IsMinOn objective Θset θhat","l":78,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.exists_isMinOn_univ_of_coercive","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":1,"d":"Existence via coercivity. On a proper normed real vector space, if the objective function is continuous and it tends to infinity along the cocompact filter, i.e. it is coercive — it grows without bound as the argument leaves every compact set, then a global minimizer of the objective over the whole space exists.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [ProperSpace E] {objective : E → ℝ}, Continuous objective → Filter.Tendsto objective (Filter.cocompact E) Filter.atTop → ∃ a, IsMinOn objective Set.univ a","l":91,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.fderiv_eq_zero_of_isMinOn_univ","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"First-order optimality on the whole space: at an unconstrained minimizer the Fréchet derivative vanishes.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {a : E}, IsMinOn f Set.univ a → HasFDerivAt f f' a → f' = 0","l":109,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.gram","k":"def","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"For an input space, a sample size, a real-valued kernel on that space, and a sample of that size, the Gram matrix is the matrix whose row-$i$, column-$j$ entry is the kernel evaluated at the $i$th and $j$th sample points.","s":"{X : Type u_1} → {n : ℕ} → (X → X → ℝ) → (Fin n → X) → Matrix (Fin n) (Fin n) ℝ","l":22,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.gram_add_lambda_posDef","k":"theorem","m":"Causalean.ML.Kernel.Gram","a":"ML/Kernel","t":1,"d":"For a finite sample x and kernel k, if k is positive semidefinite, i.e. every Gram matrix it produces is PSD and the regularization level lam is strictly positive, then the regularized Gram matrix gram k x + lam·I is positive definite, hence invertible.","s":"∀ {X : Type u_1} {n : ℕ} {k : X → X → ℝ}, Causalean.ML.IsPSDkernel k → ∀ (x : Fin n → X) {lam : ℝ}, 0 < lam → (Causalean.ML.gram k x + lam • 1).PosDef","l":35,"f":"Causalean/ML/Kernel/Gram.lean"},{"n":"Causalean.ML.iidEmpiricalRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a measurable sample space, an input space and a label space equipped jointly with a measurable structure, an experiment measure, a joint input--label measure, an independent and identically distributed sample with those measures, a pointwise loss, a prediction rule, and a sample size, the i.i.d. empirical risk is the random sample average of the loss comparing the rule's prediction with the observed label over the first $n$ sample observations.","s":"{Ω : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace (X × Y)] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure (X × Y)} → Causalean.Stat.IIDSample Ω (X × Y) μ P → Causalean.ML.Loss Y → (X → Y) → ℕ → Ω → ℝ","l":50,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.imageClass","k":"def","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":1,"d":"Given an arbitrary parameter space, an arbitrary covariate space equipped with a σ-algebra, an arbitrary outcome space equipped with a σ-algebra, a parametric predictor together with its admissible parameter set and evidence that the prediction function is measurable for every admissible parameter, the realized hypothesis class consists exactly of the prediction functions obtained by letting the parameter range over that admissible set; its members are equipped with the stated measurability guarantee.","s":"{Θ : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (M : Causalean.ML.Predictor Θ X Y) → (∀ θ ∈ M.paramSet, Measurable (M.predict θ)) → Causalean.ML.HypothesisClass X Y","l":22,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.isERMP_to_extensional","k":"theorem","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":2,"d":"A parametric empirical-risk minimizer pushes forward to an extensional empirical-risk minimizer over the realized hypothesis class.","s":"∀ {ι : Type u_1} {Θ : Type u_2} {X : Type u_3} {Y : Type u_4} [inst : Fintype ι] [inst_1 : Nonempty ι] [inst_2 : MeasurableSpace X] [inst_3 : MeasurableSpace Y] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (S : ι → X × Y) (hmeas : ∀ θ ∈ M.paramSet, Measurable (M.predict θ)) {θhat : Θ}, Causalean.ML.IsERMP (Causalean.ML.empiricalRiskP M loss S) M.paramSet θhat → Causalean.ML.IsERM (Causalean.ML.imageClass M hmeas) loss S (M.predict θhat)","l":37,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.isLittleOp_mul_quarter","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":1,"d":"The n^{-1/4} product rule. If the sequence Xn is o_p(n^{-1/4}) and the sequence Yn is o_p(n^{-1/4}) under the probability law μ, then their pointwise product Xn·Yn is o_p(n^{-1/2}) — the DML product-rate condition from per-nuisance n^{-1/4} rates.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp Xn (fun n => ↑n ^ (-(1 / 4))) μ → Causalean.Stat.IsLittleOp Yn (fun n => ↑n ^ (-(1 / 4))) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun n => ↑n ^ (-(1 / 2))) μ","l":43,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.isLittleOp_one_of_le_one","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":2,"d":"Rate weakening: an o_p(rn) sequence with rn ≤ 1 is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), rn n ≤ 1) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ","l":29,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.isLittleOp_quarter_of_isBigOp_sqrt","k":"theorem","m":"Causalean.ML.Core.Rate","a":"ML/Core","t":2,"d":"An O_p(n^{-1/2}) (root-n) rate is o_p(n^{-1/4}): a method's root-n estimation rate clears the DML o_p(n^{-1/4}) bar.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ}, Causalean.Stat.IsBigOp Xn (fun n => (√↑n)⁻¹) μ → Causalean.Stat.IsLittleOp Xn (fun n => ↑n ^ (-(1 / 4))) μ","l":106,"f":"Causalean/ML/Core/Rate.lean"},{"n":"Causalean.ML.kernel_erm_excess_rate","k":"theorem","m":"Causalean.ML.Kernel.Rate","a":"ML/Kernel","t":1,"d":"Kernel ERM excess-risk rate over the L² ball. For features valued in the closed Xb-ball and linear predictors indexed by the closed W-ball, if the feature bound Xb is nonnegative, the weight bound W is nonnegative, the feature map X is measurable, the constant t satisfies the calibration t·(Xb·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical risk no larger than that of the comparator wstar, then the probability that the excess population risk of ŵ over wstar exceeds 4·Xb·W/√n + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb W : ℝ}, 0 ≤ Xb → 0 ≤ W → ∀ (X : Ω → ↑(Metric.closedBall 0 Xb)), Measurable X → ∀ {t : ℝ}, t * (Xb * W) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → ↑(Metric.closedBall 0 W)) (wstar : ↑(Metric.closedBall 0 W)), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, inner ℝ ↑(ŵ ω) ↑(X (ω k)) ≤ (↑n)⁻¹ * ∑ k, inner ℝ ↑wstar ↑(X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * (Xb * W / √↑n) + 2 * ε < ∫ (x : Ω), (fun ω' => inner ℝ ↑(ŵ ω) ↑(X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => inner ℝ ↑wstar ↑(X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":82,"f":"Causalean/ML/Kernel/Rate.lean"},{"n":"Causalean.ML.kernel_erm_squaredLoss_excess_rate","k":"theorem","m":"Causalean.ML.Kernel.SquaredLoss","a":"ML/Kernel","t":1,"d":"Kernel ERM squared-loss excess-risk rate over the L² ball. For data pairing features in the closed Xb-ball with responses in [-Yb, Yb], and linear predictors indexed by the closed W-ball, if Xb is nonnegative, Yb is nonnegative, W is nonnegative, the data map X is measurable, the constant t satisfies the calibration t·((Xb·W)² + 2·Yb·Xb·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical squared loss no larger than that of the comparator wstar, then for the squared loss (⟪w,x⟩ − y)², the probability that the excess population risk of ŵ over wstar exceeds 4·(4(XbW)² + 2·Yb·Xb·W)/√n + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb Yb W : ℝ}, 0 ≤ Xb → 0 ≤ Yb → 0 ≤ W → ∀ (X : Ω → Causalean.ML.KFeat d Xb Yb), Measurable X → ∀ {t : ℝ}, t * ((Xb * W) ^ 2 + 2 * Yb * (Xb * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.KWeight d W) (wstar : Causalean.ML.KWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, (inner ℝ ↑(ŵ ω) ↑(X (ω k)).1 - ↑(X (ω k)).2) ^ 2 ≤ (↑n)⁻¹ * ∑ k, (inner ℝ ↑wstar ↑(X (ω k)).1 - ↑(X (ω k)).2) ^ 2) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * ((4 * (Xb * W) ^ 2 + 2 * Yb * Xb * W) / √↑n) + 2 * ε < ∫ (x : Ω), (fun ω' => (inner ℝ ↑(ŵ ω) ↑(X ω').1 - ↑(X ω').2) ^ 2) x ∂μ - ∫ (x : Ω), (fun ω' => (inner ℝ ↑wstar ↑(X ω').1 - ↑(X ω').2) ^ 2) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":46,"f":"Causalean/ML/Kernel/SquaredLoss.lean"},{"n":"Causalean.ML.krrRisk","k":"def","m":"Causalean.ML.Kernel.Ridge","a":"ML/Kernel","t":1,"d":"Given an arbitrary covariate space, a real inner-product space, a rule for evaluating a candidate function at a covariate value, a nonnegative sample size, a finite sample of covariates, the corresponding real-valued responses, a real regularization level, and a candidate function in that inner-product space, the kernel-ridge regularized empirical risk is the average squared prediction error plus $\\lambda$ times the squared norm of the candidate function.","s":"{X : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → (H → X → ℝ) → {n : ℕ} → (Fin n → X) → (Fin n → ℝ) → ℝ → H → ℝ","l":23,"f":"Causalean/ML/Kernel/Ridge.lean"},{"n":"Causalean.ML.l1penalty","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"For a finite coefficient index set and a coefficient vector, the L1 penalty is the sum, over all coefficient indices, of the absolute values of its coordinates.","s":"{Param : Type u_2} → [Fintype Param] → (Param → ℝ) → ℝ","l":26,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.l1penalty_nonneg","k":"theorem","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":2,"d":"The L1 penalty is nonnegative.","s":"∀ {Param : Type u_2} [inst : Fintype Param] (β : Param → ℝ), 0 ≤ Causalean.ML.l1penalty β","l":46,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.lassoObjective","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"For a finite set of observations, a finite coefficient index set, a design matrix, an outcome vector, a penalty weight, and a coefficient vector, the lasso objective is the sum of squared residuals plus the penalty weight times the L1 penalty of the coefficient vector.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → (Param → ℝ) → ℝ","l":31,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.lasso_erm_excess_rate","k":"theorem","m":"Causalean.ML.Lasso.Rate","a":"ML/Lasso","t":1,"d":"Lasso ERM excess-risk rate over the L¹ ball. For linear predictors indexed by the L¹ ball, if the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the weight bound W is nonnegative, the feature map X is measurable, the constant t satisfies the calibration t·(Xinf·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical risk no larger than that of the comparator wstar, then the probability that the excess population risk of ŵ over wstar exceeds 4·(Xinf·W/√n)·√(2 log 2d) + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf W : ℝ}, 0 ≤ Xinf → 0 ≤ W → ∀ (X : Ω → LinftyBall Xinf), (Measurable fun ω => ↑(X ω)) → ∀ {t : ℝ}, t * (Xinf * W) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → L1Ball W) (wstar : L1Ball W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, ∑ j, (↑(ŵ ω)).ofLp j * (↑(X (ω k))).ofLp j ≤ (↑n)⁻¹ * ∑ k, ∑ j, (↑wstar).ofLp j * (↑(X (ω k))).ofLp j) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * (Xinf * W / √↑n * √(2 * Real.log (2 * ↑d))) + 2 * ε < ∫ (x : Ω), (fun ω' => ∑ j, (↑(ŵ ω)).ofLp j * (↑(X ω')).ofLp j) x ∂μ - ∫ (x : Ω), (fun ω' => ∑ j, (↑wstar).ofLp j * (↑(X ω')).ofLp j) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":73,"f":"Causalean/ML/Lasso/Rate.lean"},{"n":"Causalean.ML.lasso_erm_squaredLoss_excess_rate","k":"theorem","m":"Causalean.ML.Lasso.SquaredLoss","a":"ML/Lasso","t":1,"d":"Lasso ERM squared-loss excess-risk rate over the L¹ ball. For data pairing coordinatewise-bounded features with bounded responses, and predictors indexed by the coordinate L¹ ball of radius W, if the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the response bound Yb is nonnegative, the weight bound W is nonnegative, the feature coordinate map is measurable, the response coordinate map is measurable, the constant t satisfies the calibration t·((Xinf·W)² + 2·Yb·Xinf·W)² ≤ 1/2, the tolerance ε is nonnegative, and the estimator ŵ attains empirical squared loss no larger than that of the comparator wstar, then for the squared regression loss (∑ⱼ wⱼxⱼ - y)², the probability that the excess population risk of ŵ over wstar exceeds 4·((4(XinfW)² + 2·Yb·Xinf·W)/√n)·√(2 log 2d) + 2ε is at most exp(-ε²tn).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf Yb W : ℝ}, 0 ≤ Xinf → 0 ≤ Yb → 0 ≤ W → ∀ (X : Ω → Causalean.ML.LassoFeat d Xinf Yb), (Measurable fun ω => ↑(X ω).1) → (Measurable fun ω => ↑(X ω).2) → ∀ {t : ℝ}, t * ((Xinf * W) ^ 2 + 2 * Yb * (Xinf * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.LassoWeight d W) (wstar : Causalean.ML.LassoWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, (∑ j, (↑(ŵ ω)).ofLp j * (↑(X (ω k)).1).ofLp j - ↑(X (ω k)).2) ^ 2 ≤ (↑n)⁻¹ * ∑ k, (∑ j, (↑wstar).ofLp j * (↑(X (ω k)).1).ofLp j - ↑(X (ω k)).2) ^ 2) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 * ((4 * (Xinf * W) ^ 2 + 2 * Yb * Xinf * W) / √↑n * √(2 * Real.log (2 * ↑d))) + 2 * ε < ∫ (x : Ω), (fun ω' => (∑ j, (↑(ŵ ω)).ofLp j * (↑(X ω').1).ofLp j - ↑(X ω').2) ^ 2) x ∂μ - ∫ (x : Ω), (fun ω' => (∑ j, (↑wstar).ofLp j * (↑(X ω').1).ofLp j - ↑(X ω').2) ^ 2) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":46,"f":"Causalean/ML/Lasso/SquaredLoss.lean"},{"n":"Causalean.ML.layerMap","k":"def","m":"Causalean.ML.NeuralNet.FeedForward","a":"ML/NeuralNet","t":1,"d":"For a nonnegative network width, an activation function, a dense layer with that finite input and output width, and an input vector of that width, the one-layer network map applies the layer's affine transformation and then applies the activation function to each coordinate.","s":"{n : ℕ} → Causalean.ML.Activation → Causalean.ML.DenseLayer n n → (Fin n → ℝ) → Fin n → ℝ","l":21,"f":"Causalean/ML/NeuralNet/FeedForward.lean"},{"n":"Causalean.ML.linearPredict","k":"def","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"For a set of observations, a finite coefficient index set, a design matrix, and a coefficient vector, the linear prediction assigns to each observation its design-row weighted sum of coefficients.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Param] → Matrix Obs Param ℝ → (Param → ℝ) → Obs → ℝ","l":26,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.linear_predictor_sub_memLp_of_l4","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"A finite linear combination of features has finite L² norm under the covariate marginal whenever the feature vector has a finite fourth moment under the joint law.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → ∀ (β βstar : K → ℝ), MeasureTheory.MemLp (fun x => ∑ k, β k * φ.φ x k - ∑ k, βstar k * φ.φ x k) 2 (MeasureTheory.Measure.map Prod.fst P)","l":589,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.lipschitzAt0_clampedSq","k":"theorem","m":"Causalean.ML.Surrogate.ClampedSquare","a":"ML/Surrogate","t":1,"d":"For a nonnegative bound c, the clamped square clampedSq c is Lipschitz at 0 with constant 2c: it fixes 0 and is globally 2c-Lipschitz.","s":"∀ {c : ℝ}, 0 ≤ c → Causalean.Stat.Concentration.LipschitzAt0 (Causalean.ML.clampedSq c) (2 * c)","l":60,"f":"Causalean/ML/Surrogate/ClampedSquare.lean"},{"n":"Causalean.ML.logisticEmpRisk","k":"def","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"For a finite sample index set, a real inner-product feature space, a sample of feature vectors paired with binary outcomes, and a coefficient vector, the empirical logistic risk is the average, over the sample, of logistic score loss evaluated at the inner product of the coefficient vector and each feature vector.","s":"{ι : Type u_1} → {E : Type u_2} → [Fintype ι] → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (ι → E × Bool) → E → ℝ","l":49,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.logisticPredictor","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"For a covariate space, a finite feature index set, a feature map, and a coefficient vector, the logistic predictor maps each covariate value to the logistic transform of its linear feature score.","s":"{γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (K → ℝ) → γ → ℝ","l":57,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.logisticScoreLoss","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"For a binary outcome $y$ and a real-valued prediction score $t$, the logistic score loss is $\\log(1+\\exp t)-y_0t$, where $y_0$ is one when $y$ is true and zero when $y$ is false.","s":"Bool → ℝ → ℝ","l":39,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.logistic_exists_minimizer_on_compact","k":"theorem","m":"Causalean.ML.Binary.Logistic","a":"ML/Binary","t":1,"d":"For any labeled sample Z and a parameter set Θset that is nonempty and compact, the empirical logistic risk attains its minimum over Θset at some parameter βhat in Θset.","s":"∀ {ι : Type u_1} {E : Type u_2} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (Z : ι → E × Bool) {Θset : Set E}, Θset.Nonempty → IsCompact Θset → ∃ βhat ∈ Θset, IsMinOn (Causalean.ML.logisticEmpRisk Z) Θset βhat","l":129,"f":"Causalean/ML/Binary/Logistic.lean"},{"n":"Causalean.ML.margin_erm_surrogate_excess_rate","k":"theorem","m":"Causalean.ML.Margin.Rate","a":"ML/Margin","t":1,"d":"Margin surrogate ERM excess-risk rate. Fix nonnegative feature-, label-, and weight-ball radii Xb, Yb, W and a nonnegative Lipschitz constant L for a surrogate loss φ satisfying |φ(s) − φ(t)| ≤ L·|s − t|. For a measurable feature-label map X valued in the Xb-by-Yb ball, a concentration parameter t satisfying t·(L·Yb·Xb·W)² ≤ 1/2, and a nonnegative slack ε, if ŵ empirically minimizes the average surrogate loss over an i.i.d. sample of size n against any comparator wstar in the W-ball, then the probability that the excess population surrogate risk of ŵ over wstar exceeds 8·L·Yb·Xb·W/√n + 2ε is at most exp(-ε²·t·n).","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb Yb W L : ℝ}, 0 ≤ Xb → 0 ≤ Yb → 0 ≤ W → 0 ≤ L → ∀ (φ : ℝ → ℝ), (∀ (s t : ℝ), |φ s - φ t| ≤ L * |s - t|) → ∀ (X : Ω → Causalean.ML.KFeat d Xb Yb), Measurable X → ∀ {t : ℝ}, t * (L * (Yb * Xb * W)) ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ŵ : (Fin n → Ω) → Causalean.ML.KWeight d W) (wstar : Causalean.ML.KWeight d W), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, φ (↑(X (ω k)).2 * inner ℝ ↑(ŵ ω) ↑(X (ω k)).1) ≤ (↑n)⁻¹ * ∑ k, φ (↑(X (ω k)).2 * inner ℝ ↑wstar ↑(X (ω k)).1)) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 8 * L * (Yb * Xb * W) / √↑n + 2 * ε < ∫ (x : Ω), (fun ω' => φ (↑(X ω').2 * inner ℝ ↑(ŵ ω) ↑(X ω').1)) x ∂μ - ∫ (x : Ω), (fun ω' => φ (↑(X ω').2 * inner ℝ ↑wstar ↑(X ω').1)) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":28,"f":"Causalean/ML/Margin/Rate.lean"},{"n":"Causalean.ML.olsCoef","k":"def","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":1,"d":"For a finite set of observations, a finite coefficient index set whose equality can be decided, a design matrix, and an outcome vector, the closed-form ordinary least-squares coefficient vector is the product of the totalized inverse of the design matrix's cross-product matrix—equal to its ordinary inverse when that matrix is invertible—and the design matrix transposed times the outcome vector.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Obs → ℝ) → Param → ℝ","l":22,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsCoef_normalEq","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":1,"d":"For a design matrix X and response vector y, if XᵀX is invertible, i.e. its determinant is a unit, then the closed-form OLS coefficient (XᵀX)⁻¹Xᵀy solves the normal equations (XᵀX)β = Xᵀy.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), IsUnit (X.transpose * X).det → (X.transpose * X).mulVec (Causalean.ML.olsCoef X y) = X.transpose.mulVec y","l":115,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsCoef_unique","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":2,"d":"When XᵀX is invertible, every normal-equation solution equals the closed form OLS coefficient.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ), IsUnit (X.transpose * X).det → ∀ {β : Param → ℝ}, (X.transpose * X).mulVec β = X.transpose.mulVec y → β = Causalean.ML.olsCoef X y","l":124,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.olsObjective","k":"def","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"For a finite set of observations, a finite coefficient index set, a design matrix, an outcome vector, and a coefficient vector, the ordinary least-squares objective is the sum over observations of squared differences between the observed outcome and its linear prediction.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → (Param → ℝ) → ℝ","l":31,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.ols_is_squaredLoss_ERM_of_normalEq","k":"theorem","m":"Causalean.ML.Linear.Finite","a":"ML/Linear","t":1,"d":"For any coefficient vector β̂ satisfying the normal equations XᵀX β̂ = Xᵀy built from a finite design matrix X and outcome vector y, that vector minimizes the sum of squared residuals over every coefficient vector β.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) (βhat : Param → ℝ), (X.transpose * X).mulVec βhat = X.transpose.mulVec y → ∀ (β : Param → ℝ), Causalean.ML.olsObjective X y βhat ≤ Causalean.ML.olsObjective X y β","l":38,"f":"Causalean/ML/Linear/Finite.lean"},{"n":"Causalean.ML.ols_normalEq_of_minimizer","k":"theorem","m":"Causalean.ML.Linear.ClosedForm","a":"ML/Linear","t":2,"d":"A minimizer of the least-squares objective solves the normal equations.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) (βhat : Param → ℝ), (∀ (β : Param → ℝ), Causalean.ML.olsObjective X y βhat ≤ Causalean.ML.olsObjective X y β) → (X.transpose * X).mulVec βhat = X.transpose.mulVec y","l":32,"f":"Causalean/ML/Linear/ClosedForm.lean"},{"n":"Causalean.ML.populationGram","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Given an arbitrary covariate space, a finite set of feature coordinates, a feature map from covariates into those coordinates, and a probability or other measure on covariate--response pairs, the population feature Gram matrix has $(k,l)$ entry $\\int \\phi_k(x)\\phi_l(x)\\,dP(x,y)$. The definition is entrywise, so it does not require a matrix-valued integral.","s":"{γ : Type u_2} → {K : Type u_3} → [inst : MeasurableSpace γ] → [inst_1 : Fintype K] → Causalean.ML.FeatureMap γ K → MeasureTheory.Measure (γ × ℝ) → Matrix K K ℝ","l":87,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.populationGram_posSemidef","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The population feature Gram is positive semidefinite whenever the feature coordinates are measurable and the fourth moment of the feature norm is finite.","s":"∀ {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace γ] [inst_1 : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure P], (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (Causalean.ML.populationGram φ P).PosSemidef","l":99,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.populationRidgeObjective","k":"def","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"For a measurable covariate space, a finite feature index set, a joint covariate–response measure, a feature map, a penalty level, and a coefficient vector, the population ridge objective is the population squared prediction risk of the associated linear predictor plus the penalty level times the sum of squared coefficients.","s":"{X' : Type u_1} → {K : Type u_2} → [inst : MeasurableSpace X'] → [inst_1 : Fintype K] → MeasureTheory.Measure (X' × ℝ) → Causalean.ML.FeatureMap X' K → ℝ → (K → ℝ) → ℝ","l":28,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.populationRidge_minimizes","k":"theorem","m":"Causalean.ML.Ridge.Population","a":"ML/Ridge","t":1,"d":"With a nonnegative ridge penalty λ, for a probability measure P on features and outcome and a finite feature map φ, if the coefficient vector βstar satisfies the regularized population normal equations E[(Y − ⟪βstar,φ⟫)φₖ] = λβstarₖ for every feature k, the population squared-loss risks of the βstar- and β-predictors are both finite, and each feature is integrable against that residual, then the population ridge objective at βstar is at most its value at any other coefficient vector β.","s":"∀ {X' : Type u_1} {K : Type u_2} [inst : MeasurableSpace X'] [inst_1 : Fintype K] (P : MeasureTheory.Measure (X' × ℝ)) (φ : Causalean.ML.FeatureMap X' K) {lam : ℝ}, 0 ≤ lam → ∀ {βstar : K → ℝ}, Causalean.ML.IsPopulationRidge P φ lam βstar → ∀ (β : K → ℝ), (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, βstar k * φ.φ x k) → (Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P fun x => ∑ k, β k * φ.φ x k) → (∀ (k : K), MeasureTheory.Integrable (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) P) → Causalean.ML.populationRidgeObjective P φ lam βstar ≤ Causalean.ML.populationRidgeObjective P φ lam β","l":46,"f":"Causalean/ML/Ridge/Population.lean"},{"n":"Causalean.ML.populationRisk","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a measurable input space, a measurable label space, a pointwise loss, a joint input--label measure, and a prediction rule, the population risk is the integral under that joint measure of the loss comparing the rule's prediction at the input with the observed label.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → (X → Y) → ℝ","l":38,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.populationRiskP","k":"def","m":"Causalean.ML.Core.Risk","a":"ML/Core","t":1,"d":"For a parameter space, a measurable input space, a measurable label space, a parametric predictor, a pointwise loss, a joint input--label measure, and a parameter value, the parametric population risk is the population risk of the prediction rule selected by that parameter value.","s":"{Θ : Type u_1} → {X : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → Causalean.ML.Predictor Θ X Y → Causalean.ML.Loss Y → MeasureTheory.Measure (X × Y) → Θ → ℝ","l":45,"f":"Causalean/ML/Core/Risk.lean"},{"n":"Causalean.ML.populationRisk_eq_of_both_minimizers","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":2,"d":"Any two population-risk minimizers over the same class attain the same risk.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {H : Causalean.ML.HypothesisClass X Y} {loss : Causalean.ML.Loss Y} {P : MeasureTheory.Measure (X × Y)} {h₁ h₂ : X → Y}, Causalean.ML.IsPopulationRiskMinimizer H loss P h₁ → Causalean.ML.IsPopulationRiskMinimizer H loss P h₂ → Causalean.ML.populationRisk loss P h₁ = Causalean.ML.populationRisk loss P h₂","l":28,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.populationRisk_minimizer_eq_target","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":2,"d":"If a target is known to be the unique population-risk minimizer, every population-risk minimizer equals it.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {H : Causalean.ML.HypothesisClass X Y} {loss : Causalean.ML.Loss Y} {P : MeasureTheory.Measure (X × Y)} {target hstar : X → Y}, Causalean.ML.IsPopulationRiskMinimizer H loss P hstar → Causalean.ML.IsPopulationRiskMinimizer H loss P target → (∀ (h₁ h₂ : X → Y), Causalean.ML.IsPopulationRiskMinimizer H loss P h₁ → Causalean.ML.IsPopulationRiskMinimizer H loss P h₂ → h₁ = h₂) → hstar = target","l":36,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.populationTarget_pushforward","k":"theorem","m":"Causalean.ML.Core.Bridge","a":"ML/Core","t":1,"d":"Population-target pushforward. For a predictor M with loss loss and population law P, suppose every admissible parameter's prediction function is measurable, the parameter θhat is itself admissible, every admissible parameter attains finite population risk, and θhat minimizes the population risk over the admissible parameter set. Then the predictor M.predict θhat is a population-risk minimizer over the hypothesis class realized by M's image: a parametric population-risk minimizer pushes forward to an extensional minimizer over the realized function class.","s":"∀ {Θ : Type u_1} {X : Type u_2} {Y : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (M : Causalean.ML.Predictor Θ X Y) (loss : Causalean.ML.Loss Y) (P : MeasureTheory.Measure (X × Y)) (hmeas : ∀ θ ∈ M.paramSet, Measurable (M.predict θ)) {θhat : Θ}, θhat ∈ M.paramSet → (∀ θ ∈ M.paramSet, Causalean.ML.HasFinitePopulationRisk loss P (M.predict θ)) → IsMinOn (fun θ => Causalean.ML.populationRiskP M loss P θ) M.paramSet θhat → Causalean.ML.IsPopulationRiskMinimizer (Causalean.ML.imageClass M hmeas) loss P (M.predict θhat)","l":51,"f":"Causalean/ML/Core/Bridge.lean"},{"n":"Causalean.ML.properLoss_population_minimizer_recovers_eta","k":"theorem","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":1,"d":"Strictly proper population minimizers recover the regression function. For a strictly proper binary loss L on a measure space PX, with the true conditional probability η taking values in [0,1] and a candidate prediction rule q also taking values in [0,1], if q minimizes the population conditional risk over every [0,1]-valued prediction rule and the conditional risks of η and q are both integrable, then q equals η almost everywhere.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {L : ℝ → Bool → ℝ}, Causalean.ML.StrictProperBinaryLoss L → ∀ {PX : MeasureTheory.Measure X} (η : X → ℝ), (∀ (x : X), η x ∈ Set.Icc 0 1) → ∀ (q : X → ℝ), (∀ (x : X), q x ∈ Set.Icc 0 1) → IsMinOn (fun r => ∫ (x : X), η x * L (r x) true + (1 - η x) * L (r x) false ∂PX) {r | ∀ (x : X), r x ∈ Set.Icc 0 1} q → MeasureTheory.Integrable (fun x => η x * L (η x) true + (1 - η x) * L (η x) false) PX → MeasureTheory.Integrable (fun x => η x * L (q x) true + (1 - η x) * L (q x) false) PX → q =ᵐ[PX] η","l":62,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.properLoss_population_risk_le","k":"theorem","m":"Causalean.ML.Surrogate.GenericERM","a":"ML/Surrogate","t":2,"d":"Proper-loss integrated risk inequality. For a proper binary loss, the integral of the conditional risk obtained by predicting the true conditional probability is no larger than the corresponding integral for any pointwise [0,1]-valued comparator. The hypotheses keep both conditional-risk integrands integrable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {L : ℝ → Bool → ℝ}, Causalean.ML.ProperBinaryLoss L → ∀ {PX : MeasureTheory.Measure X} (η : X → ℝ), (∀ (x : X), η x ∈ Set.Icc 0 1) → ∀ (q : X → ℝ), (∀ (x : X), q x ∈ Set.Icc 0 1) → MeasureTheory.Integrable (fun x => η x * L (η x) true + (1 - η x) * L (η x) false) PX → MeasureTheory.Integrable (fun x => η x * L (q x) true + (1 - η x) * L (q x) false) PX → ∫ (x : X), η x * L (η x) true + (1 - η x) * L (η x) false ∂PX ≤ ∫ (x : X), η x * L (q x) true + (1 - η x) * L (q x) false ∂PX","l":46,"f":"Causalean/ML/Surrogate/GenericERM.lean"},{"n":"Causalean.ML.rademacherComplexity_l1_ball_le","k":"theorem","m":"Causalean.ML.Lasso.Rate","a":"ML/Lasso","t":1,"d":"Rademacher complexity of the L¹-ball linear class. If the dimension d is positive, the sample size n is positive, the coordinatewise feature bound Xinf is nonnegative, the weight bound W is nonnegative, and every feature coordinate is bounded in absolute value by Xinf, then the expected Rademacher complexity of the class of linear predictors with L¹ weight norm at most W is at most (Xinf·W/√n)·√(2 log 2d).","s":"∀ {d n : ℕ}, 0 < d → 0 < n → ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xinf W : ℝ}, 0 ≤ Xinf → 0 ≤ W → ∀ (X : Ω → EuclideanSpace ℝ (Fin d)), (∀ (ω : Ω) (j : Fin d), |(X ω).ofLp j| ≤ Xinf) → rademacherComplexity n (fun w a => ∑ j, (↑w).ofLp j * a.ofLp j) μ X ≤ Xinf * W / √↑n * √(2 * Real.log (2 * ↑d))","l":25,"f":"Causalean/ML/Lasso/Rate.lean"},{"n":"Causalean.ML.rademacherComplexity_l2_ball_le","k":"theorem","m":"Causalean.ML.Kernel.Rate","a":"ML/Kernel","t":1,"d":"Rademacher complexity of the L²-ball linear class. If the feature bound Xb is nonnegative, the weight bound W is nonnegative, and every feature vector has Euclidean norm at most Xb, then the expected Rademacher complexity of the class of linear predictors with weight norm at most W, on a sample of size n, is at most Xb·W/√n.","s":"∀ {d n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xb W : ℝ}, 0 ≤ Xb → 0 ≤ W → ∀ (X : Ω → EuclideanSpace ℝ (Fin d)), (∀ (ω : Ω), ‖X ω‖ ≤ Xb) → rademacherComplexity n (fun w a => inner ℝ (↑w) a) μ X ≤ Xb * W / √↑n","l":26,"f":"Causalean/ML/Kernel/Rate.lean"},{"n":"Causalean.ML.regLogisticCoef_isBigOp","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"The regularized-logistic coefficient error is O_p(n^{-1/2}): the strong-convexity basic inequality converts the centered-gradient bound into a coefficient-error bound.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (βhat : ℕ → Ω → K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (n : ℕ) (ω : Ω), Causalean.ML.regLogisticGrad φ S.Z lam n ω (βhat n ω) = 0) → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → Causalean.Stat.IsBigOp (fun n ω => ‖βhat n ω - βstar‖) (fun n => (√↑n)⁻¹) μ","l":375,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogisticGrad","k":"def","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"For an underlying sample-state space, a covariate space, a finite feature index set, a feature map, a sequence of observed covariate–response pairs indexed by sample state, a penalty level, a sample size, a sample state, and a coefficient vector, the empirical penalized-logistic gradient is the vector whose each coordinate equals the sample-average logistic score times that feature, plus twice the penalty level times the corresponding coefficient.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → (K → ℝ) → K → ℝ","l":35,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogisticGrad_coord_isBigOp","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"Each coordinate of the empirical penalized gradient at the population target is O_p(n^{-1/2}); the population score equations make the i.i.d. summands centered.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ} (βstar : K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → ∀ (k : K), Causalean.Stat.IsBigOp (fun n ω => Causalean.ML.regLogisticGrad φ S.Z lam n ω βstar k) (fun n => (√↑n)⁻¹) μ","l":315,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogistic_achievesL2Rate","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":1,"d":"Regularized-logistic root-n estimation rate. For a strictly positive regularization weight lam, suppose βstar solves the penalized population first-order condition for the logistic quasi-score under the feature map φ and law P, and that for every sample size n and outcome ω, the fitted coefficients βhat n ω solve the corresponding empirical penalized first-order condition on the i.i.d. sample S. Suppose further that every feature coordinate is measurable, that the fourth moment of the squared feature norm is integrable under P, and that each coordinate of the population logistic score at βstar is square-integrable under P. Then the fitted logistic predictor σ(⟨βhat n ω, φ⟩) achieves the L²-rate n^{-1/2} toward the population target predictor σ(⟨βstar, φ⟩), under P and the sampling law μ. Assembled from the coefficient rate, the 1/4-Lipschitz σ, and the shared linear-predictor L² bound.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (βhat : ℕ → Ω → K → ℝ), Causalean.ML.IsPopulationRegLogistic P φ lam βstar → (∀ (n : ℕ) (ω : Ω), Causalean.ML.regLogisticGrad φ S.Z lam n ω (βhat n ω) = 0) → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => ((∑ j, βstar j * φ.φ z.1 j).sigmoid - z.2) * φ.φ z.1 k) 2 P) → Causalean.ML.AchievesL2Rate (fun n ω => Causalean.ML.logisticPredictor φ (βhat n ω)) (Causalean.ML.logisticPredictor φ βstar) P (fun n => (√↑n)⁻¹) μ","l":430,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.regLogistic_basic_inequality","k":"theorem","m":"Causalean.ML.Binary.Rate","a":"ML/Binary","t":2,"d":"Strong-convexity basic inequality (deterministic). If β̂ solves the empirical penalized FOC ∇ₙ(β̂) = 0 and λ > 0, the coefficient error is controlled by the empirical gradient at the target: 2λ‖β̂ − β⋆‖ ≤ ∑ₖ |∇ₙ(β⋆)ₖ|.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (n : ℕ) (ω : Ω) {βhat : K → ℝ}, Causalean.ML.regLogisticGrad φ Z lam n ω βhat = 0 → 2 * lam * ‖βhat - βstar‖ ≤ ∑ k, |Causalean.ML.regLogisticGrad φ Z lam n ω βstar k|","l":152,"f":"Causalean/ML/Binary/Rate.lean"},{"n":"Causalean.ML.representer_theorem","k":"theorem","m":"Causalean.ML.Kernel.Ridge","a":"ML/Kernel","t":1,"d":"For a complete reproducing-kernel Hilbert space H, if (feval, representer) realizes H as an RKHS on X, i.e. evaluation at each point equals the inner product with its representer, the regularization level lam is strictly positive, and fhat minimizes the regularized empirical risk krrRisk over all of H for the sample (x, y), then fhat lies in the span of the sample representers representer x₁, …, representer xₙ, i.e. it is a finite linear combination of them.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {feval : H → X → ℝ} {representer : X → H}, Causalean.ML.IsRKHS X H feval representer → ∀ {n : ℕ} (x : Fin n → X) (y : Fin n → ℝ) {lam : ℝ}, 0 < lam → ∀ {fhat : H}, (∀ (g : H), Causalean.ML.krrRisk feval x y lam fhat ≤ Causalean.ML.krrRisk feval x y lam g) → ∃ α, fhat = ∑ i, α i • representer (x i)","l":34,"f":"Causalean/ML/Kernel/Ridge.lean"},{"n":"Causalean.ML.ridgeCoef","k":"def","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"For a finite set of observations, a finite coefficient index set whose equality can be decided, a design matrix, an outcome vector, and a ridge penalty weight, the closed-form ridge coefficient vector is the totalized inverse of the design matrix's cross-product matrix plus the penalty weight times the identity—equal to its ordinary inverse when that matrix is invertible—multiplied by the design matrix transposed times the outcome vector.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → [DecidableEq Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → Param → ℝ","l":23,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeCoef_normalEq","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"For a strictly positive ridge penalty λ, the closed-form ridge coefficient (XᵀX + λI)⁻¹Xᵀy, built from a finite design matrix X and outcome vector y, satisfies the ridge normal equations (XᵀX + λI)β̂ = Xᵀy.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 < lam → (X.transpose * X + lam • 1).mulVec (Causalean.ML.ridgeCoef X y lam) = X.transpose.mulVec y","l":47,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeCoef_unique","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":2,"d":"For λ > 0, every solution of the ridge normal equations is the closed-form ridge coefficient.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 < lam → ∀ {β : Param → ℝ}, (X.transpose * X + lam • 1).mulVec β = X.transpose.mulVec y → β = Causalean.ML.ridgeCoef X y lam","l":61,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeGram_posDef","k":"theorem","m":"Causalean.ML.Ridge.ClosedForm","a":"ML/Ridge","t":1,"d":"For a strictly positive ridge penalty λ, the ridge Gram matrix XᵀX + λI built from a finite design matrix X is positive definite.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) {lam : ℝ}, 0 < lam → (X.transpose * X + lam • 1).PosDef","l":35,"f":"Causalean/ML/Ridge/ClosedForm.lean"},{"n":"Causalean.ML.ridgeObjective","k":"def","m":"Causalean.ML.Ridge.Finite","a":"ML/Ridge","t":1,"d":"For a finite set of observations, a finite coefficient index set, a design matrix, an outcome vector, a ridge penalty weight, and a coefficient vector, the ridge objective is the sum of squared residuals plus the penalty weight times the sum of squared coefficients.","s":"{Obs : Type u_1} → {Param : Type u_2} → [Fintype Obs] → [Fintype Param] → Matrix Obs Param ℝ → (Obs → ℝ) → ℝ → (Param → ℝ) → ℝ","l":23,"f":"Causalean/ML/Ridge/Finite.lean"},{"n":"Causalean.ML.ridge_achievesL2Rate","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Ridge root-n estimation rate. For a strictly positive ridge penalty λ, a finite feature map φ, and an i.i.d. sample S from a distribution P on features and outcome, if the true coefficient vector βstar satisfies the regularized population ridge normal equations, each feature coordinate is measurable, the fourth moment of the squared feature norm is finite, and each per-coordinate score function is square-integrable, then the sample ridge predictor converges to the population ridge predictor at the root-n rate in the L²(P) sense.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] [inst_3 : DecidableEq K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → Causalean.ML.AchievesL2Rate (Causalean.ML.sampleRidgePredictor φ S.Z lam) (fun x => ∑ k, βstar k * φ.φ x k) P (fun n => (√↑n)⁻¹) μ","l":747,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.ridge_is_regularized_squaredLoss_ERM_of_normalEq","k":"theorem","m":"Causalean.ML.Ridge.Finite","a":"ML/Ridge","t":1,"d":"With a nonnegative ridge penalty λ, for any coefficient vector β̂ satisfying the ridge normal equations (XᵀX + λI)β̂ = Xᵀy built from a finite design matrix X and outcome vector y, that vector minimizes the ridge objective — the sum of squared residuals plus λ‖β‖² — over every coefficient vector β.","s":"∀ {Obs : Type u_1} {Param : Type u_2} [inst : Fintype Obs] [inst_1 : Fintype Param] [inst_2 : DecidableEq Param] (X : Matrix Obs Param ℝ) (y : Obs → ℝ) {lam : ℝ}, 0 ≤ lam → ∀ (βhat : Param → ℝ), (X.transpose * X + lam • 1).mulVec βhat = X.transpose.mulVec y → ∀ (β : Param → ℝ), Causalean.ML.ridgeObjective X y lam βhat ≤ Causalean.ML.ridgeObjective X y lam β","l":31,"f":"Causalean/ML/Ridge/Finite.lean"},{"n":"Causalean.ML.rkhs_ball_empiricalRademacher_le","k":"theorem","m":"Causalean.ML.Kernel.RKHSRademacher","a":"ML/Kernel","t":1,"d":"In a reproducing-kernel Hilbert space H, if (feval, representer) realizes H as an RKHS on X, κ is a nonnegative bound, r is a nonnegative radius, and every sampled kernel representer representer (xs k) has norm at most κ, then the empirical Rademacher complexity, on the sample xs, of the closed ball of radius r in H is at most κ·r/√n.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {feval : H → X → ℝ} {representer : X → H}, Causalean.ML.IsRKHS X H feval representer → ∀ {n : ℕ} {κ r : ℝ}, 0 ≤ κ → 0 ≤ r → ∀ (xs : Fin n → X), (∀ (k : Fin n), ‖representer (xs k)‖ ≤ κ) → empiricalRademacherComplexity n (fun f x => feval (↑f) x) xs ≤ κ * r / √↑n","l":465,"f":"Causalean/ML/Kernel/RKHSRademacher.lean"},{"n":"Causalean.ML.sampleRidgeCoef","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Given an arbitrary sample space, an arbitrary covariate space, a finite set of feature coordinates whose labels can be compared for equality, a feature map from covariates into those coordinates, a sequence of data samples indexed by sample-space outcomes, a real ridge-penalty level, a nonnegative sample size, and a sample-space outcome selecting one realized sequence, the sample ridge coefficient vector is $(\\widehat G_n+\\lambda I)^{-1}\\widehat C_n$. No nonsingularity condition is imposed on the regularized Gram matrix; its inverse is the total matrix-inverse operation used here.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → [DecidableEq K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → K → ℝ","l":63,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgeCoef_isBigOp","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The ridge coefficient error is O_p(n^{-1/2}): the regularized empirical-Gram inverse is O_p(1) (entrywise matrix LLN + inverse continuity at the PosDef limit) times the centered score mean O_p(n^{-1/2}).","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace γ] [inst_2 : Fintype K] [inst_3 : DecidableEq K] {μ : MeasureTheory.Measure Ω} (φ : Causalean.ML.FeatureMap γ K) (P : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω (γ × ℝ) μ P) [MeasureTheory.IsProbabilityMeasure μ] {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ), Causalean.ML.IsPopulationRidge P φ lam βstar → (∀ (k : K), Measurable fun x => φ.φ x k) → MeasureTheory.Integrable (fun z => (∑ k, φ.φ z.1 k ^ 2) ^ 2) P → (∀ (k : K), MeasureTheory.MemLp (fun z => (z.2 - ∑ j, βstar j * φ.φ z.1 j) * φ.φ z.1 k) 2 P) → (Causalean.ML.populationGram φ P + lam • 1).PosDef → Causalean.Stat.IsBigOp (fun n ω => ‖Causalean.ML.sampleRidgeCoef φ S.Z lam n ω - βstar‖) (fun n => (√↑n)⁻¹) μ","l":411,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgeCoef_sub_eq","k":"theorem","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":2,"d":"The ridge coefficient error factors through the regularized empirical Gram inverse and the centered empirical score Ŝₙ − λ β⋆ = (Ĉₙ − Ĝₙ β⋆) − λ β⋆.","s":"∀ {Ω : Type u_1} {γ : Type u_2} {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] (φ : Causalean.ML.FeatureMap γ K) (Z : ℕ → Ω → γ × ℝ) {lam : ℝ}, 0 < lam → ∀ (βstar : K → ℝ) (n : ℕ) (ω : Ω), Causalean.ML.sampleRidgeCoef φ Z lam n ω - βstar = (Causalean.ML.empiricalGram φ Z n ω + lam • 1)⁻¹.mulVec (Causalean.ML.empiricalCross φ Z n ω - (Causalean.ML.empiricalGram φ Z n ω).mulVec βstar - lam • βstar)","l":222,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sampleRidgePredictor","k":"def","m":"Causalean.ML.Ridge.Rate","a":"ML/Ridge","t":1,"d":"Given an arbitrary sample space, an arbitrary covariate space, a finite set of feature coordinates whose labels can be compared for equality, a feature map from covariates into those coordinates, a sequence of data samples indexed by sample-space outcomes, a real ridge-penalty level, a nonnegative sample size, and a sample-space outcome selecting one realized sequence, the sample ridge predictor maps each covariate value $x$ to $\\sum_k\\widehat\\beta_{n,k}\\phi_k(x)$, where $\\widehat\\beta_n$ is the corresponding sample ridge coefficient vector.","s":"{Ω : Type u_1} → {γ : Type u_2} → {K : Type u_3} → [inst : Fintype K] → [DecidableEq K] → Causalean.ML.FeatureMap γ K → (ℕ → Ω → γ × ℝ) → ℝ → ℕ → Ω → γ → ℝ","l":75,"f":"Causalean/ML/Ridge/Rate.lean"},{"n":"Causalean.ML.sigmoid","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"For a real-valued score $t$, the logistic sigmoid is $(1+\\exp(-t))^{-1}$.","s":"ℝ → ℝ","l":29,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.sigmoid_lt_one","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The logistic sigmoid is strictly below one at every score.","s":"∀ (t : ℝ), Causalean.ML.sigmoid t < 1","l":67,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.sigmoid_pos","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"The logistic sigmoid is strictly positive at every score.","s":"∀ (t : ℝ), 0 < Causalean.ML.sigmoid t","l":62,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.softThreshold","k":"def","m":"Causalean.ML.Lasso.Finite","a":"ML/Lasso","t":1,"d":"For a threshold level and a real-valued input, the soft-thresholded value is $\\max(z-\\lambda,0)-\\max(-z-\\lambda,0)$.","s":"ℝ → ℝ → ℝ","l":39,"f":"Causalean/ML/Lasso/Finite.lean"},{"n":"Causalean.ML.softThreshold_isMinOn","k":"theorem","m":"Causalean.ML.Lasso.Optimality","a":"ML/Lasso","t":1,"d":"For a real number a, if the regularization level lam is nonnegative, then the soft-thresholded value softThreshold lam a minimizes the one-dimensional penalized least-squares objective u ↦ (u − a)² + 2·lam·|u| over every real u.","s":"∀ {lam : ℝ}, 0 ≤ lam → ∀ (a u : ℝ), (Causalean.ML.softThreshold lam a - a) ^ 2 + 2 * lam * |Causalean.ML.softThreshold lam a| ≤ (u - a) ^ 2 + 2 * lam * |u|","l":20,"f":"Causalean/ML/Lasso/Optimality.lean"},{"n":"Causalean.ML.softplus","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"For a real-valued score $t$, the softplus transformation is $\\log(1+\\exp t)$.","s":"ℝ → ℝ","l":34,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.softplus_pos","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"Softplus is strictly positive at every score.","s":"∀ (t : ℝ), 0 < Causalean.ML.softplus t","l":76,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.square_loss_population_target_of_isL2Projection","k":"theorem","m":"Causalean.ML.Core.PopulationTarget","a":"ML/Core","t":1,"d":"For a joint covariate–response law P and a competitor function h, if the residual of m is orthogonal to every function of the covariate, i.e. m is an L² projection onto functions of the covariate, m is measurable, the competitor h is measurable, m attains finite squared population risk, h attains finite squared population risk, and the cross term (Y − m(X))·(m(X) − h(X)) is integrable, then the squared population risk of m is at most that of h: m minimizes squared population risk against any such competitor.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure (X × ℝ)} {m : X → ℝ}, Causalean.ML.IsL2Projection P m → ∀ (h : X → ℝ), Measurable m → Measurable h → Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P m → Causalean.ML.HasFinitePopulationRisk Causalean.ML.squaredLoss P h → MeasureTheory.Integrable (fun z => (z.2 - m z.1) * (m z.1 - h z.1)) P → Causalean.ML.populationRisk Causalean.ML.squaredLoss P m ≤ Causalean.ML.populationRisk Causalean.ML.squaredLoss P h","l":63,"f":"Causalean/ML/Core/PopulationTarget.lean"},{"n":"Causalean.ML.squaredLoss","k":"def","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":1,"d":"For a real-valued prediction $\\hat y$ and a real-valued observed outcome $y$, the squared loss is $(y-\\hat y)^2$.","s":"ℝ → ℝ → ℝ","l":19,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.squaredLoss_nonneg","k":"theorem","m":"Causalean.ML.Core.Losses","a":"ML/Core","t":2,"d":"Squared loss is always nonnegative.","s":"∀ (ŷ y : ℝ), 0 ≤ Causalean.ML.squaredLoss ŷ y","l":59,"f":"Causalean/ML/Core/Losses.lean"},{"n":"Causalean.ML.subgradientAt_zero_iff_isMinOn","k":"theorem","m":"Causalean.ML.Core.Convex","a":"ML/Core","t":2,"d":"Fermat's rule, subgradient form: 0 is a subgradient at x over s iff x minimizes f over s.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {f : E → ℝ} {s : Set E} {x : E}, x ∈ s → (Causalean.ML.SubgradientAt f s 0 x ↔ IsMinOn f s x)","l":127,"f":"Causalean/ML/Core/Convex.lean"},{"n":"Causalean.ML.summable_effectiveDimension_term","k":"theorem","m":"Causalean.ML.Kernel.EffectiveDimension","a":"ML/Kernel","t":2,"d":"For a trace-class operator (summable eigenvalues) the effective-dimension summands are summable, so effectiveDimension is a genuine real number.","s":"∀ {ι : Type u_1} {μ : ι → ℝ} {lam : ℝ}, (∀ (i : ι), 0 ≤ μ i) → 0 < lam → Summable μ → Summable fun i => μ i / (μ i + lam)","l":70,"f":"Causalean/ML/Kernel/EffectiveDimension.lean"},{"n":"Causalean.MarkovEquiv","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex population and two directed acyclic graphs on it, Markov equivalence holds exactly when, for every three finite vertex sets, the first and second sets are d-separated by the third in the first graph if and only if they are d-separated in the second graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":58,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.refl","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For any DAG G, G is Markov equivalent to itself.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V), Causalean.MarkovEquiv G G","l":72,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.MarkovEquiv G₂ G₁","l":76,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.MarkovEquiv.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.MarkovEquiv G₂ G₃ → Causalean.MarkovEquiv G₁ G₃","l":80,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineCodimensionIn","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a coordinate index set, a nonnegative integer $d$, a locus $Z$, and an ambient subset $X$ of the same complex affine space, the statement that $Z$ has affine codimension $d$ in $X$ means that every irreducible affine component of $Z$ is the initial member of a strictly increasing chain of $d+1$ irreducible affine-closed subsets whose final member is $X$, while at least one irreducible affine component of $Z$ is not the initial member of any strictly increasing chain of $d+2$ irreducible affine-closed subsets ending at $X$.","s":"{ι : Type u_1} → ℕ → Set (ι → ℂ) → Set (ι → ℂ) → Prop","l":32,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ChainDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a coordinate index set, a nonnegative integer $d$, and a subset $Z$ of the corresponding complex affine space, the statement that $Z$ has exact affine Zariski dimension $d$ means that there exists a strictly increasing chain of $d+1$ irreducible affine-closed subsets, all contained in $Z$, and no strictly increasing chain of $d+2$ irreducible affine-closed subsets, all contained in $Z$, exists.","s":"{ι : Type u_1} → ℕ → Set (ι → ℂ) → Prop","l":20,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ChainDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a coordinate index set and a subset $Z$ of the corresponding complex affine space, an irreducible affine-closed set is a set that equals its polynomial Zariski closure, is nonempty, and whenever it is the union of two affine-closed subsets, equals one of those two subsets.","s":"{ι : Type u_1} → Set (ι → ℂ) → Prop","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a coordinate index set, a candidate subset $C$ of the corresponding complex affine space, and a prescribed locus $Z$ in that space, an irreducible affine component of $Z$ is an irreducible affine-closed subset $C$ that is contained in $Z$ and maximal under inclusion among irreducible affine-closed subsets of $Z$: every such subset that contains $C$ equals $C$.","s":"{ι : Type u_1} → Set (ι → ℂ) → Set (ι → ℂ) → Prop","l":24,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a source-coordinate index set, a target-coordinate index set, and a map $f$ from the resulting source complex affine space to the resulting target complex affine space, the statement that $f$ is a polynomial map means that, for every target coordinate, there exists a multivariate complex polynomial in the source coordinates whose value at every source point equals that coordinate of $f$.","s":"{ι : Type u_1} → {κ : Type u_2} → ((ι → ℂ) → κ → ℂ) → Prop","l":20,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap.comp","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A composite of polynomial maps is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} {τ : Type u_3} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → τ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap (g ∘ f)","l":32,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap.eval_comp","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Substituting a polynomial map into a target polynomial yields a source polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ (Q : MvPolynomial κ ℂ), ∃ P, ∀ (x : ι → ℂ), (MvPolynomial.eval x) P = (MvPolynomial.eval (f x)) Q","l":56,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineLinearMap_isPolynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"An affine-linear map from a finite-coordinate source is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] (F : (ι → ℂ) →ₗ[ℂ] κ → ℂ) (c : κ → ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap fun x => c + F x","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineSpace_hasAffineZariskiDimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For any natural number d, complex affine d-space has affine Zariski dimension exactly d, in the sense of maximal chains of irreducible affine closed subsets.","s":"∀ (d : ℕ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d Set.univ","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineSubspace_hasAffineZariskiDimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"An affine translate of a d-dimensional linear subspace has exact irreducible-chain dimension d. For a complex linear subspace V of κ → ℂ and a base point x₀, if V has finite rank exactly d, then the affine translate {x | x - x₀ ∈ V} has irreducible-chain (Zariski) dimension exactly d.","s":"∀ {κ : Type u_1} [Finite κ] (V : Submodule ℂ (κ → ℂ)) (x₀ : κ → ℂ) (d : ℕ), Module.finrank ℂ ↥V = d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d {x | x - x₀ ∈ V}","l":59,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a coordinate index set and a subset $A$ of the corresponding complex affine space, its affine Zariski closure is the set of all points at which every complex multivariate polynomial that vanishes on every point of $A$ also vanishes.","s":"{ι : Type u_1} → Set (ι → ℂ) → Set (ι → ℂ)","l":23,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_eq_zeroLocus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A closed affine set is the zero locus of its vanishing ideal.","s":"∀ {ι : Type u_1} {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → Z = MvPolynomial.zeroLocus ℂ (MvPolynomial.vanishingIdeal ℂ Z)","l":50,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_extensive","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Every set is contained in its affine Zariski closure.","s":"∀ {ι : Type u_1} (A : Set (ι → ℂ)), A ⊆ Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A","l":31,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_idem","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Affine Zariski closure is idempotent.","s":"∀ {ι : Type u_1} (A : Set (ι → ℂ)), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A) = Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A","l":42,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_inter","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The intersection of two affine-closed sets is affine-closed.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (A ∩ B) = A ∩ B","l":56,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_mono","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Affine Zariski closure is monotone.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, A ⊆ B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A ⊆ Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B","l":36,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_nonvanishing_eq_univ","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a polynomial P in coordinates indexed by ι, if P is not the zero polynomial, then the set of points where P does not vanish is Zariski-dense in the whole affine space.","s":"∀ {ι : Type u_1} (P : MvPolynomial ι ℂ), P ≠ 0 → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | (MvPolynomial.eval x) P ≠ 0} = Set.univ","l":103,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_polynomial_image_of_dense","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The closure of the image of a dense set under a polynomial map equals the closure of the full range.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = Set.univ → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A) = Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f)","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_union","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The union of two affine-closed sets is affine-closed.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (A ∪ B) = A ∪ B","l":77,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure_zero_of_polynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.ZariskiClosure","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The zero set of one multivariate polynomial is affine-closed.","s":"∀ {ι : Type u_1} (P : MvPolynomial ι ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | (MvPolynomial.eval x) P = 0} = {x | (MvPolynomial.eval x) P = 0}","l":67,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/ZariskiClosure.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiDimension_iff_coordinateRingKrullDim","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a subset Z of ι → ℂ (with ι finite), if Z is Zariski-closed, then Z has irreducible-chain (Zariski) dimension d exactly when the Krull dimension of its coordinate ring — the polynomial ring on ι variables modulo Z's vanishing ideal — equals d.","s":"∀ {ι : Type u_1} [Finite ι] {d : ℕ} {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d Z ↔ ringKrullDim (MvPolynomial ι ℂ ⧸ MvPolynomial.vanishingIdeal ℂ Z) = ↑d)","l":56,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.algebraicIndependent_of_polynomialJacobianMinor_ne_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Over the complex numbers, a nonzero square Jacobian minor proves that the corresponding coordinate polynomials satisfy no nontrivial algebraic relation.","s":"∀ {ι : Type u_1} {κ : Type u_2} {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → AlgebraicIndependent ℂ fun a => f (rows a)","l":187,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebraAlgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a source-coordinate index set, a target-coordinate index set, and a family $f$ of complex multivariate polynomials indexed by target coordinates, the complex-algebra structure on the subalgebra generated by the coordinate polynomials of $f$ is the usual one inherited from the surrounding polynomial algebra.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → Algebra ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":23,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebra_trdeg_le_of_polynomial_factorization","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The algebraic dimension (transcendence degree) of the coordinate algebra generated by a polynomial map is at most k whenever each of its coordinates can be written as a polynomial expression in a fixed family of k polynomials. This is the factorization-based companion to a Jacobian-minor bound: it certifies an upper bound on dimension from an explicit polynomial reparametrization rather than from a derivative rank.","s":"∀ {ι κ τ : Type} [Finite τ] (f : κ → MvPolynomial ι ℂ) (g : τ → MvPolynomial ι ℂ) (h : κ → MvPolynomial τ ℂ), (∀ (k : κ), (MvPolynomial.bind₁ g) (h k) = f k) → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑(Nat.card τ)","l":208,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.coordinateSubalgebra_trdeg_le_of_surjection","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A surjective presentation by a polynomial algebra in a given number of generators certifies that the coordinate subalgebra has no larger transcendence degree.","s":"∀ {ι : Type u_1} {κ : Type u_2} {τ : Type u_3} [Finite τ] (f : κ → MvPolynomial ι ℂ) (present : MvPolynomial τ ℂ →ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)), Function.Surjective ⇑present → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑(Nat.card τ)","l":189,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.eq_C_of_forall_pderiv_eq_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial over a characteristic-zero field in finitely many variables whose partial derivatives all vanish equals its constant coefficient.","s":"∀ {σ : Type u_1} {K : Type u_2} [Finite σ] [inst : Field K] [CharZero K] (P : MvPolynomial σ K), (∀ (i : σ), (MvPolynomial.pderiv i) P = 0) → P = MvPolynomial.C (MvPolynomial.coeff 0 P)","l":137,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.eval_polynomialPullback","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For index types ι, κ, a polynomial parameterization f, a point x, and a polynomial P, substituting the coordinate polynomials f into P and evaluating at x agrees with evaluating P directly at the point obtained by evaluating f's coordinates at x.","s":"∀ {ι : Type u_1} {κ : Type u_2} (f : κ → MvPolynomial ι ℂ) (x : ι → ℂ) (P : MvPolynomial κ ℂ), (MvPolynomial.eval x) ((Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) P) = (MvPolynomial.eval (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateMap f x)) P","l":36,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.finiteCoordinateRange_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IrreducibleFiniteRange","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The set of complex coordinate vectors whose chosen coordinate belongs to a fixed finite set is closed in the affine Zariski topology.","s":"∀ {κ : Type u_1} (c : κ) (S : Finset ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | x c ∈ S} = {x | x c ∈ S}","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/IrreducibleFiniteRange.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.finiteTypeDomain_ringKrullDim_eq_trdeg","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A finitely generated integral algebra over a field has Krull dimension equal to its finite transcendence degree, the number of algebraically independent coordinates it contains.","s":"∀ (K : Type u_1) (A : Type u_2) [inst : Field K] [inst_1 : CommRing A] [inst_2 : Algebra K A] [IsDomain A] [Algebra.FiniteType K A] (d : ℕ), Algebra.trdeg K A = ↑d → ringKrullDim A = ↑d","l":31,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.hasAffineCodimensionIn_one_of_minimalPrime_span","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For subsets C, Z, X of ι → ℂ (with ι finite), suppose X is irreducible Zariski-closed, Z is a proper subset of X, C is an irreducible component of Z, and the vanishing ideal of C is a minimal prime over the vanishing ideal of X joined with the principal ideal generated by some polynomial P. Then Z has affine codimension exactly 1 in X.","s":"∀ {ι : Type u_1} [Finite ι] {C Z X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → Z ⊆ X → Z ≠ X → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent C Z → ∀ (P : MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineCodimensionIn 1 Z X","l":281,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.image_strictMono_on","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"If a function is injective on a set containing every member of a strictly increasing chain of sets, then the images of those sets form a strictly increasing chain.","s":"∀ {α : Type u_1} {β : Type u_2} {ι : Type u_3} [inst : Preorder ι] (f : α → β) {S : Set α}, Set.InjOn f S → ∀ {chain : ι → Set α}, StrictMono chain → (∀ (i : ι), chain i ⊆ S) → StrictMono fun i => f '' chain i","l":46,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducibleAffineClosed_iff_isPrime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"A nonempty polynomially closed complex set Z is irreducible exactly when the polynomial equations that vanish on it form a prime ideal.","s":"∀ {ι : Type u_1} [Finite ι] {Z : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure Z = Z → Z.Nonempty → (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z ↔ (MvPolynomial.vanishingIdeal ℂ Z).IsPrime)","l":51,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducibleAffineComponent_of_no_intermediate","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"An irreducible closed subset with no irreducible closed set strictly between it and an ambient irreducible variety is a component of every proper closed locus lying between the two.","s":"∀ {ι : Type u_1} {C Z X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → C ⊆ Z → Z ⊆ X → Z ≠ X → (¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X) → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineComponent C Z","l":197,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_affineClosure_polynomial_image_of_dense","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The closure of a polynomial image of a nonempty dense subset of finite complex affine space is irreducible.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = Set.univ → A.Nonempty → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A))","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_coordinate_constant_of_finite_range","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IrreducibleFiniteRange","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"On an irreducible affine-closed set Z, if a fixed coordinate c takes values only within a fixed finite set S at every point of Z, then that coordinate is in fact constant on Z, equal to some single value in S.","s":"∀ {κ : Type u_1} {Z : Set (κ → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z → ∀ (c : κ) (S : Finset ℂ), (∀ x ∈ Z, x c ∈ S) → ∃ z ∈ S, ∀ x ∈ Z, x c = z","l":41,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/IrreducibleFiniteRange.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_image_polynomial_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial embedding with a polynomial retraction sends irreducible affine-closed subsets to irreducible affine-closed images.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (f '' A)","l":74,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.irreducible_zeroLocus_of_prime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The zero locus of a prime ideal is irreducible affine-closed.","s":"∀ {ι : Type u_1} [Finite ι] (P : Ideal (MvPolynomial ι ℂ)) [P.IsPrime], Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (MvPolynomial.zeroLocus ℂ P)","l":123,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.isPolynomialMap_id","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The identity map is polynomial.","s":"∀ {ι : Type u_1}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap id","l":26,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.jacobianCoordinateSubalgebraAlgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex coordinate polynomials, the coordinate subalgebra inherits its complex-algebra structure, namely the structure induced by its inclusion in the polynomial ring in the input coordinates.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → Algebra ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.linearMap_isPolynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSubspaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A linear map from a finite-coordinate source is polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] (F : (ι → ℂ) →ₗ[ℂ] κ → ℂ), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap ⇑F","l":22,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSubspaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.map_quotient_strictMono","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"If one ideal is strictly contained in another and both contain a third ideal, their images in the quotient by that third ideal remain strictly ordered.","s":"∀ {R : Type u_1} [inst : CommRing R] (I A B : Ideal R), I ≤ A → A < B → Ideal.map (Ideal.Quotient.mk I) A < Ideal.map (Ideal.Quotient.mk I) B","l":46,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.no_intermediate_of_exact_affine_dimensions","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Exact dimensions differing by one exclude an intermediate irreducible closed set. The proof appends Y and X to a maximal-length chain inside C, contradicting the upper bound for X.","s":"∀ {ι : Type u_1} {d : ℕ} {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension (d + 1) X → ¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X","l":169,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.no_three_chain_of_minimalPrime_span","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A prime affine subvariety whose prime ideal is minimal over one additional equation in the coordinate ring of X admits no intermediate irreducible closed set. This is the endpoint-fixed form of Krull's principal ideal theorem needed by HasAffineCodimensionIn 1.","s":"∀ {ι : Type u_1} [Finite ι] {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → ∀ (P : MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes → ¬∃ chain, StrictMono chain ∧ (∀ (i : Fin 3), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (chain i)) ∧ chain 0 = C ∧ chain (Fin.last 2) = X","l":57,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.nonempty_zeroLocus_of_prime","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A prime ideal in a finite complex polynomial ring has a nonempty zero locus.","s":"∀ {ι : Type u_1} [Finite ι] (P : Ideal (MvPolynomial ι ℂ)) [P.IsPrime], (MvPolynomial.zeroLocus ℂ P).Nonempty","l":40,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.pderiv_polynomialPullback","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The partial derivative of a polynomial after polynomial substitution equals the sum of the substituted partial derivatives, each weighted by the corresponding derivative of the substituted polynomial.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype κ] (f : κ → MvPolynomial ι ℂ) (P : MvPolynomial κ ℂ) (i : ι), (MvPolynomial.pderiv i) ((Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) P) = ∑ j, (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f) ((MvPolynomial.pderiv j) P) * (MvPolynomial.pderiv i) (f j)","l":45,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex polynomials, one for each output coordinate, the polynomial coordinate map sends every complex-valued assignment to the input indices to the complex-valued assignment whose output-coordinate value is the corresponding polynomial evaluated at that assignment.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → (ι → ℂ) → κ → ℂ","l":16,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex coordinate polynomials, the coordinate subalgebra is the complex subalgebra of the polynomial ring in the input coordinates generated by that family.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → Subalgebra ℂ (MvPolynomial ι ℂ)","l":26,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex coordinate polynomials, the polynomial image closure is the smallest complex affine algebraic set containing every output vector obtained by evaluating those polynomials at a complex-valued input vector.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → Set (κ → ℂ)","l":31,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_coordinateRing","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex coordinate polynomials, the coordinate ring of the closure of the polynomial image is canonically isomorphic, as a complex algebra, to the subalgebra of the polynomial ring in the input coordinates generated by that family.","s":"{ι : Type u_1} → {κ : Type u_2} → (f : κ → MvPolynomial ι ℂ) → (MvPolynomial κ ℂ ⧸ MvPolynomial.vanishingIdeal ℂ (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)) ≃ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_jacobian","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"Given a finite polynomial parameterization f of κ-many coordinates by variables indexed by a finite set ι, together with d selected output coordinates rows and input variables cols, if the corresponding d × d Jacobian minor of f, formed from partial derivatives, is nonzero and the transcendence degree of the coordinate subalgebra generated by f is at most d, then the Zariski closure of the image of f has exact affine (irreducible-chain) dimension d.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) ≤ ↑d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":246,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_jacobian_and_surjection","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonzero Jacobian minor and a surjective presentation using the same number of generators identify the exact affine dimension of a polynomial image closure.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] {d : ℕ} (f : κ → MvPolynomial ι ℂ) (rows : Fin d → κ) (cols : Fin d → ι), Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor f rows cols ≠ 0 → ∀ (present : MvPolynomial (Fin d) ℂ →ₐ[ℂ] ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f)), Function.Surjective ⇑present → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":284,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_dimension_of_trdeg","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Transcendence","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a finite polynomial parameterization f of κ-many coordinates by variables indexed by a finite set ι, if the transcendence degree of the subalgebra of ℂ generated by the coordinate polynomials of f equals d, then the Zariski closure of the image of f has exact affine (irreducible-chain) dimension d.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] [Finite κ] (f : κ → MvPolynomial ι ℂ) (d : ℕ), Algebra.trdeg ℂ ↥(Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialCoordinateSubalgebra f) = ↑d → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f)","l":152,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Transcendence.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_isIrreducible","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.DenseImage","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a map f from the affine space ℂ^ι (with ι finite) to ℂ^κ that is coordinatewise polynomial, the Zariski closure of the range of f is irreducible as a polynomially closed set.","s":"∀ {ι : Type u_1} {κ : Type u_2} [Finite ι] {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f))","l":109,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/DenseImage.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure_vanishingIdeal","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CoordinateRing","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A target polynomial relation holds throughout a polynomial image closure exactly when substituting the parameterization's coordinate polynomials makes the relation identically zero.","s":"∀ {ι : Type u_1} {κ : Type u_2} (f : κ → MvPolynomial ι ℂ), MvPolynomial.vanishingIdeal ℂ (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialImageClosure f) = RingHom.ker (Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback f)","l":17,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CoordinateRing.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialJacobianMinor","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a commutative coefficient ring, two index sets, a nonnegative integer determining the minor size, a family of polynomials indexed by the output coordinates, a selection of that many output coordinates, and a selection of that many input coordinates, the polynomial Jacobian minor is the determinant of the square matrix whose entry in each selected row and column is the corresponding formal partial derivative, before evaluation at any point.","s":"{R : Type u_1} → {ι : Type u_2} → {κ : Type u_3} → [inst : CommRing R] → {d : ℕ} → (κ → MvPolynomial ι R) → (Fin d → κ) → (Fin d → ι) → MvPolynomial ι R","l":29,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialPullback","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Definitions","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For two index sets and a family of complex polynomials, one for each output coordinate, the polynomial pullback is the complex-algebra homomorphism that replaces every output-coordinate variable in a polynomial relation by its corresponding coordinate polynomial.","s":"{ι : Type u_1} → {κ : Type u_2} → (κ → MvPolynomial ι ℂ) → MvPolynomial κ ℂ →ₐ[ℂ] MvPolynomial ι ℂ","l":21,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Definitions.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialRetract_range_dimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialRetractDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For maps between the affine space ℂ^d and ℂ^κ, if f is coordinatewise polynomial, an accompanying map g back to ℂ^d is coordinatewise polynomial, and g is a left inverse of f, i.e. g (f x) = x for every x, then the range of f has exact affine (irreducible-chain) dimension d, matching the dimension of its source space.","s":"∀ {d : ℕ} {κ : Type u_1} {f : (Fin d → ℂ) → κ → ℂ} {g : (κ → ℂ) → Fin d → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.HasAffineZariskiDimension d (Set.range f)","l":114,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialRetractDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomialRing_dimension","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The Krull dimension of the ring of complex polynomials in d variables is d.","s":"∀ (d : ℕ), ringKrullDim (MvPolynomial (Fin d) ℂ) = ↑d","l":39,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_fixedPoints_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The fixed-point set of a polynomial endomorphism is affine-closed.","s":"∀ {ι : Type u_1} {f : (ι → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure {x | f x = x} = {x | f x = x}","l":94,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_image_closed_of_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A polynomial retract sends affine-closed source subsets to affine-closed images.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → ∀ {A : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f '' A) = f '' A","l":130,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_preimage_closed","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"The preimage of an affine-closed set under a polynomial map is affine-closed.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → ∀ {A : Set (κ → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (f ⁻¹' A) = f ⁻¹' A","l":75,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.polynomial_range_closed_of_retract","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.PolynomialMap","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For maps between the finite-dimensional complex affine spaces ℂ^ι and ℂ^κ, if f is coordinatewise polynomial, an accompanying map g back to ℂ^ι is coordinatewise polynomial, and g is a left inverse of f, i.e. g (f x) = x for every x, then the range of f is already Zariski-closed.","s":"∀ {ι : Type u_1} {κ : Type u_2} {f : (ι → ℂ) → κ → ℂ} {g : (κ → ℂ) → ι → ℂ}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsPolynomialMap g → Function.LeftInverse g f → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure (Set.range f) = Set.range f","l":110,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/PolynomialMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":1,"d":"For a finite index set, a subset of the associated complex affine space, and the hypothesis that this subset is irreducible and affine Zariski-closed, the associated prime-spectrum point is the prime ideal consisting of all complex polynomials that vanish on the subset.","s":"{ι : Type u_1} → [Finite ι] → (Z : Set (ι → ℂ)) → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Z → PrimeSpectrum (MvPolynomial ι ℂ)","l":24,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible_lt","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.AffineSpaceDimension","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"For two irreducible affine closed subsets of finite-dimensional complex affine space, strict inclusion reverses the strict order of their corresponding vanishing-ideal prime points.","s":"∀ {ι : Type u_1} [inst : Finite ι] {A B : Set (ι → ℂ)} (hA : Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed A) (hB : Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed B), A ⊂ B → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible B hB < Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.primeOfIrreducible A hA","l":31,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/AffineSpaceDimension.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.sum_X_mul_pderiv_eq_sum_degree","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"For a polynomial over a commutative semiring in finitely many variables, the sum of each variable times its partial derivative equals the sum of its monomials weighted by their total degrees.","s":"∀ {σ : Type u_1} {R : Type u_2} [inst : Fintype σ] [inst_1 : CommSemiring R] (P : MvPolynomial σ R), ∑ i, MvPolynomial.X i * (MvPolynomial.pderiv i) P = ∑ m ∈ P.support, Finsupp.degree m • (MvPolynomial.monomial m) (MvPolynomial.coeff m P)","l":112,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.totalDegree_pderiv_lt","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Jacobian","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonzero partial derivative of a polynomial has strictly smaller total degree than the original polynomial.","s":"∀ {σ : Type u_1} {R : Type u_2} [inst : CommSemiring R] {i : σ} {P : MvPolynomial σ R}, (MvPolynomial.pderiv i) P ≠ 0 → ((MvPolynomial.pderiv i) P).totalDegree < P.totalDegree","l":76,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Jacobian.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_mem_minimalPrimes_span_of_no_intermediate","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.CodimensionOne","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Geometric height one plus one equation through C identifies its prime ideal as a minimal prime over that equation in the coordinate ring of X. This is the Nullstellensatz converse used to manufacture the explicit minimal-prime certificates from the D0 dimension argument.","s":"∀ {ι : Type u_1} [Finite ι] {C X : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed C → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed X → C ⊆ X → ∀ P ∈ MvPolynomial.vanishingIdeal ℂ C, P ∉ MvPolynomial.vanishingIdeal ℂ X → (¬∃ Y, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.IsIrreducibleAffineClosed Y ∧ C ⊂ Y ∧ Y ⊂ X) → MvPolynomial.vanishingIdeal ℂ C ∈ (MvPolynomial.vanishingIdeal ℂ X ⊔ Ideal.span {P}).minimalPrimes","l":225,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/CodimensionOne.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_ne_top_of_nonempty","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"A nonempty affine set has a proper vanishing ideal.","s":"∀ {ι : Type u_1} {Z : Set (ι → ℂ)}, Z.Nonempty → MvPolynomial.vanishingIdeal ℂ Z ≠ ⊤","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.vanishingIdeal_strict_anti","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Strict inclusion of affine-closed sets strictly reverses their vanishing ideals.","s":"∀ {ι : Type u_1} {A B : Set (ι → ℂ)}, Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure A = A → Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.affineZariskiClosure B = B → A ⊂ B → MvPolynomial.vanishingIdeal ℂ B < MvPolynomial.vanishingIdeal ℂ A","l":136,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.zeroLocus_strict_anti","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.PolynomialImageDimension.Irreducibility","a":"Mathlib/AlgebraicGeometry/PolynomialImageDimension","t":2,"d":"Strict inclusion of prime ideals strictly reverses their complex zero loci.","s":"∀ {ι : Type u_1} [Finite ι] {P Q : Ideal (MvPolynomial ι ℂ)} [P.IsPrime] [Q.IsPrime], P < Q → MvPolynomial.zeroLocus ℂ Q ⊂ MvPolynomial.zeroLocus ℂ P","l":146,"f":"Causalean/Mathlib/AlgebraicGeometry/PolynomialImageDimension/Irreducibility.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","k":"structure","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A real rational map with an arbitrary index set of output coordinates stores one numerator and denominator multivariate polynomial for each output coordinate.","s":"Type u_5 → Type u_6 → Type (max u_5 u_6)","l":22,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.DefinedOn","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A rational map is defined on a domain when every coordinate denominator is nonzero at every point of that domain.","s":"{S : Type u_1} → {T : Type u_2} → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → Set (S → ℝ) → Prop","l":37,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.equalityPolynomial","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The cleared numerator for equality of one coordinate of two rational maps is the cross-product of their numerators and denominators.","s":"{S : Type u_1} → {T : Type u_2} → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → T → MvPolynomial S ℝ","l":55,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.eval","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Evaluating a rational map substitutes the source coordinates into each numerator and denominator and divides coordinatewise. Division is total: a coordinate whose denominator vanishes evaluates to zero, so this agrees with the rational map only where it is defined in the sense of DefinedOn.","s":"{S : Type u_1} → {T : Type u_2} → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → (S → ℝ) → T → ℝ","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.eval_eq_iff_equalityPolynomial_eq_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given two rational maps, a source assignment, an output coordinate, and nonzero first and second denominators, the rational values agree exactly when the cleared equality polynomial vanishes.","s":"∀ {S : Type u_1} {T : Type u_2} (f g : Causalean.Mathlib.AlgebraicGeometry.RationalMap S T) (x : S → ℝ) (t : T), (MvPolynomial.eval x) (f.den t) ≠ 0 → (MvPolynomial.eval x) (g.den t) ≠ 0 → (f.eval x t = g.eval x t ↔ (MvPolynomial.eval x) (f.equalityPolynomial g t) = 0)","l":60,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.eval_ofPolynomial","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"For a polynomial coordinate map and source assignment, the denominator-one rational representation has ordinary polynomial evaluation.","s":"∀ {S : Type u_1} {T : Type u_2} (p : T → MvPolynomial S ℝ) (x : S → ℝ), (Causalean.Mathlib.AlgebraicGeometry.RationalMap.ofPolynomial p).eval x = fun t => (MvPolynomial.eval x) (p t)","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.eval_renameSource","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"For a rational map, coordinate renaming, and renamed-source assignment, the renamed map evaluates as the original map on the pulled-back assignment.","s":"∀ {S : Type u_5} {U : Type u_6} {T : Type u_7} (f : Causalean.Mathlib.AlgebraicGeometry.RationalMap S T) (e : S → U) (x : U → ℝ), (f.renameSource e).eval x = f.eval (x ∘ e)","l":270,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.leftLift","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The left lift of a rational map makes it depend on the left coordinates of a disjoint-sum parameter space.","s":"{S : Type u_5} → {U : Type u_6} → {T : Type u_7} → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → Causalean.Mathlib.AlgebraicGeometry.RationalMap (S ⊕ U) T","l":277,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.ofPolynomial","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A polynomial coordinate map is viewed as a rational map with denominator one in every coordinate.","s":"{S : Type u_1} → {T : Type u_2} → (T → MvPolynomial S ℝ) → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T","l":42,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.renameSource","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Renaming the source variables of a rational map along a coordinate map renames every numerator and denominator polynomial.","s":"{S : Type u_5} → {U : Type u_6} → {T : Type u_7} → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T → (S → U) → Causalean.Mathlib.AlgebraicGeometry.RationalMap U T","l":264,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMap.rightLift","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The right lift of a rational map makes it depend on the right coordinates of a disjoint-sum parameter space.","s":"{S : Type u_5} → {U : Type u_6} → {T : Type u_7} → Causalean.Mathlib.AlgebraicGeometry.RationalMap U T → Causalean.Mathlib.AlgebraicGeometry.RationalMap (S ⊕ U) T","l":282,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A rational matrix map is a rational map whose output coordinates are row-column pairs.","s":"Type u_5 → Type u_6 → Type u_7 → Type (max u_5 u_7 u_6)","l":194,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap.evalMatrix","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Evaluating a rational matrix map and reshaping its pair-indexed output gives an ordinary matrix-valued function.","s":"{S : Type u_1} → {T : Type u_2} → {U : Type u_3} → Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap S T U → (S → ℝ) → Matrix T U ℝ","l":197,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar","k":"structure","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A real rational scalar coordinate is a numerator and denominator multivariate polynomial.","s":"Type u_2 → Type u_2","l":23,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.DefinedOn","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"For a rational scalar and domain, being defined on the domain means that its denominator never vanishes there.","s":"{S : Type u_1} → Causalean.Mathlib.AlgebraicGeometry.RationalScalar S → Set (S → ℝ) → Prop","l":37,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.IsFormallyNonconstant","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"For a rational scalar, formal nonconstancy means that at least one cleared partial-derivative numerator is a nonzero polynomial.","s":"{S : Type u_1} → Causalean.Mathlib.AlgebraicGeometry.RationalScalar S → Prop","l":48,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivativeNumerator","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"For a rational scalar and source coordinate, the cleared quotient-rule numerator is the polynomial numerator of that coordinate derivative.","s":"{S : Type u_1} → Causalean.Mathlib.AlgebraicGeometry.RationalScalar S → S → MvPolynomial S ℝ","l":42,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivativePolynomial","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"For a rational scalar, the full derivative certificate polynomial is the sum of squares of all cleared quotient-rule partial numerators.","s":"{S : Type u_1} → [Fintype S] → Causalean.Mathlib.AlgebraicGeometry.RationalScalar S → MvPolynomial S ℝ","l":53,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivativePolynomial_ne_zero_of_formallyNonconstant","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a rational scalar satisfying formal nonconstancy, the full derivative certificate polynomial is nonzero.","s":"∀ {S : Type u_1} [inst : Fintype S] (r : Causalean.Mathlib.AlgebraicGeometry.RationalScalar S), r.IsFormallyNonconstant → r.derivativePolynomial ≠ 0","l":219,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivativePolynomial_ne_zero_of_nonconstantOn_convex","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a rational scalar on a convex domain, a nowhere-vanishing denominator, two domain points, and distinct rational values, the full derivative certificate polynomial is nonzero.","s":"∀ {S : Type u_1} [inst : Fintype S] [DecidableEq S] (r : Causalean.Mathlib.AlgebraicGeometry.RationalScalar S) (D : Set (S → ℝ)), Convex ℝ D → r.DefinedOn D → ∀ (x y : S → ℝ), x ∈ D → y ∈ D → r.eval x ≠ r.eval y → r.derivativePolynomial ≠ 0","l":237,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivativePolynomial_ne_zero_of_witness","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a rational scalar, source assignment, coordinate, and a nonzero cleared derivative numerator there, the full derivative certificate polynomial is nonzero.","s":"∀ {S : Type u_1} [inst : Fintype S] (r : Causalean.Mathlib.AlgebraicGeometry.RationalScalar S) (x : S → ℝ) (i : S), (MvPolynomial.eval x) (r.derivativeNumerator i) ≠ 0 → r.derivativePolynomial ≠ 0","l":228,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.derivative_zero_locus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Given a rational scalar coordinate r on a domain where its denominator does not vanish, the points where its full Fréchet derivative vanishes are exactly one real polynomial zero locus within that domain.","s":"∀ {S : Type u_1} [inst : Fintype S] [DecidableEq S] (r : Causalean.Mathlib.AlgebraicGeometry.RationalScalar S) (D : Set (S → ℝ)), r.DefinedOn D → {x | x ∈ D ∧ fderiv ℝ r.eval x = 0} = D ∩ Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus r.derivativePolynomial","l":205,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.eval","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":1,"d":"For a rational scalar and source assignment, the evaluated scalar is its numerator evaluation divided by its denominator evaluation. Division is total: where the denominator vanishes the value is zero, so this is the rational function's value only on points where it is defined.","s":"{S : Type u_1} → Causalean.Mathlib.AlgebraicGeometry.RationalScalar S → (S → ℝ) → ℝ","l":30,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.RationalScalar.fderiv_eq_zero_iff_derivativePolynomial_eq_zero","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalDerivative","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a rational scalar, source assignment, and nonzero denominator at that assignment, the full Fréchet derivative vanishes exactly when its certificate polynomial vanishes.","s":"∀ {S : Type u_1} [inst : Fintype S] [DecidableEq S] (r : Causalean.Mathlib.AlgebraicGeometry.RationalScalar S) (x : S → ℝ), (MvPolynomial.eval x) r.den ≠ 0 → (fderiv ℝ r.eval x = 0 ↔ (MvPolynomial.eval x) r.derivativePolynomial = 0)","l":182,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalDerivative.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The conjunction polynomial of finitely many real polynomials is their sum of squares.","s":"{S : Type u_1} → {T : Type u_2} → [Fintype T] → (T → MvPolynomial S ℝ) → MvPolynomial S ℝ","l":75,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial_ne_zero_of_witness","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a finite polynomial family, source assignment, coordinate, and a nonzero value at that coordinate, the conjunction polynomial is nonzero.","s":"∀ {S : Type u_1} {T : Type u_2} [inst : Fintype T] (p : T → MvPolynomial S ℝ) (x : S → ℝ) (t : T), (MvPolynomial.eval x) (p t) ≠ 0 → Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial p ≠ 0","l":98,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.evalPolynomialMatrix","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Evaluating a matrix of multivariate polynomials substitutes the source point in every matrix entry.","s":"{S : Type u_1} → {T : Type u_2} → {U : Type u_3} → Matrix T U (MvPolynomial S ℝ) → (S → ℝ) → Matrix T U ℝ","l":203,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.eval_conjunctionPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"For a finite polynomial family and source assignment, the conjunction polynomial vanishes exactly when every constituent polynomial vanishes.","s":"∀ {S : Type u_1} {T : Type u_2} [inst : Fintype T] (p : T → MvPolynomial S ℝ) (x : S → ℝ), (MvPolynomial.eval x) (Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial p) = 0 ↔ ∀ (t : T), (MvPolynomial.eval x) (p t) = 0","l":80,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.eval_unionConjunctionPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"For a finite family of finite polynomial systems and source assignment, the union certificate vanishes exactly when one branch vanishes coordinatewise.","s":"∀ {S : Type u_1} {T : Type u_2} {B : Type u_4} [inst : Fintype B] [inst_1 : Fintype T] (p : B → T → MvPolynomial S ℝ) (x : S → ℝ), (MvPolynomial.eval x) (Causalean.Mathlib.AlgebraicGeometry.unionConjunctionPolynomial p) = 0 ↔ ∃ b, ∀ (t : T), (MvPolynomial.eval x) (p b t) = 0","l":116,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixInverseRationalMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The adjugate-over-determinant representation is the rational matrix map associated with the inverse of a square polynomial matrix.","s":"{S : Type u_1} → {T : Type u_2} → [Fintype T] → [DecidableEq T] → Matrix T T (MvPolynomial S ℝ) → Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap S T T","l":223,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixInverseRationalMap_definedOn","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a square polynomial matrix, domain, and a nowhere-vanishing evaluated determinant, the inverse representation is defined on the domain.","s":"∀ {S : Type u_1} {T : Type u_2} [inst : Fintype T] [inst_1 : DecidableEq T] (A : Matrix T T (MvPolynomial S ℝ)) (D : Set (S → ℝ)), (∀ x ∈ D, (MvPolynomial.eval x) A.det ≠ 0) → Causalean.Mathlib.AlgebraicGeometry.RationalMap.DefinedOn (Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixInverseRationalMap A) D","l":251,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixInverseRationalMap_eval","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a square polynomial matrix, source assignment, and nonzero evaluated determinant, the adjugate-over-determinant map evaluates to the matrix inverse.","s":"∀ {S : Type u_1} {T : Type u_2} [inst : Fintype T] [inst_1 : DecidableEq T] (A : Matrix T T (MvPolynomial S ℝ)) (x : S → ℝ), (MvPolynomial.eval x) A.det ≠ 0 → (Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixInverseRationalMap A).evalMatrix x = (Causalean.Mathlib.AlgebraicGeometry.evalPolynomialMatrix A x)⁻¹","l":230,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixRationalMap","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"A polynomial matrix is represented as a rational matrix map with denominator one entrywise.","s":"{S : Type u_1} → {T : Type u_2} → {U : Type u_3} → Matrix T U (MvPolynomial S ℝ) → Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap S T U","l":209,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixRationalMap_eval","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"For a polynomial matrix and source assignment, the rational-matrix representation agrees with entrywise polynomial evaluation.","s":"∀ {S : Type u_1} {T : Type u_2} {U : Type u_3} (A : Matrix T U (MvPolynomial S ℝ)) (x : S → ℝ), (Causalean.Mathlib.AlgebraicGeometry.polynomialMatrixRationalMap A).evalMatrix x = Causalean.Mathlib.AlgebraicGeometry.evalPolynomialMatrix A x","l":214,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.rationalMap_eq_locus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"Given two finite-coordinate rational maps f and g on a common domain where the denominators of the first map do not vanish and the denominators of the second map do not vanish, the whole-output equality condition is one real polynomial zero locus.","s":"∀ {S : Type u_1} {T : Type u_2} [inst : Fintype T] (f g : Causalean.Mathlib.AlgebraicGeometry.RationalMap S T) (D : Set (S → ℝ)), f.DefinedOn D → g.DefinedOn D → {x | x ∈ D ∧ f.eval x = g.eval x} = D ∩ Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial (f.equalityPolynomial g))","l":142,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.rationalMap_eq_union_locus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given finite families of rational maps on a common domain, with all first and second denominators nonvanishing, the union of their equality conditions is one polynomial zero locus.","s":"∀ {S : Type u_1} {T : Type u_2} {B : Type u_4} [inst : Fintype B] [inst_1 : Fintype T] (f g : B → Causalean.Mathlib.AlgebraicGeometry.RationalMap S T) (D : Set (S → ℝ)), (∀ (b : B), (f b).DefinedOn D) → (∀ (b : B), (g b).DefinedOn D) → {x | x ∈ D ∧ ∃ b, (f b).eval x = (g b).eval x} = D ∩ Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (Causalean.Mathlib.AlgebraicGeometry.unionConjunctionPolynomial fun b => (f b).equalityPolynomial (g b))","l":167,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.rationalMap_imageIntersection_locus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given finite-coordinate rational maps f and g on domains, with nonvanishing denominators for the first map and the second map, the pairs of parameters at which their images meet form one polynomial zero locus.","s":"∀ {S : Type u_5} {U : Type u_6} {T : Type u_7} [inst : Fintype T] (f : Causalean.Mathlib.AlgebraicGeometry.RationalMap S T) (g : Causalean.Mathlib.AlgebraicGeometry.RationalMap U T) (D : Set (S → ℝ)) (E : Set (U → ℝ)), f.DefinedOn D → g.DefinedOn E → {z | z ∈ Causalean.Mathlib.AlgebraicGeometry.sumProductDomain D E ∧ f.eval (z ∘ Sum.inl) = g.eval (z ∘ Sum.inr)} = Causalean.Mathlib.AlgebraicGeometry.sumProductDomain D E ∩ Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial (f.leftLift.equalityPolynomial g.rightLift))","l":294,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.rationalMatrixMap_imageIntersection_locus","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given rational matrix maps on domains, with nonvanishing denominators for both maps, the parameter pairs at which their matrix images meet form one polynomial zero locus.","s":"∀ {S : Type u_5} {U : Type u_6} {R : Type u_8} {C : Type u_9} [inst : Fintype R] [inst_1 : Fintype C] (f : Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap S R C) (g : Causalean.Mathlib.AlgebraicGeometry.RationalMatrixMap U R C) (D : Set (S → ℝ)) (E : Set (U → ℝ)), Causalean.Mathlib.AlgebraicGeometry.RationalMap.DefinedOn f D → Causalean.Mathlib.AlgebraicGeometry.RationalMap.DefinedOn g E → {z | z ∈ Causalean.Mathlib.AlgebraicGeometry.sumProductDomain D E ∧ f.evalMatrix (z ∘ Sum.inl) = g.evalMatrix (z ∘ Sum.inr)} = Causalean.Mathlib.AlgebraicGeometry.sumProductDomain D E ∩ Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (Causalean.Mathlib.AlgebraicGeometry.conjunctionPolynomial ((Causalean.Mathlib.AlgebraicGeometry.RationalMap.leftLift f).equalityPolynomial (Causalean.Mathlib.AlgebraicGeometry.RationalMap.rightLift g)))","l":322,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.sumProductDomain","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The sum-coordinate form of a product domain requires the left and right restrictions of one assignment to lie in their respective domains.","s":"{S : Type u_5} → {U : Type u_6} → Set (S → ℝ) → Set (U → ℝ) → Set (S ⊕ U → ℝ)","l":289,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.unionConjunctionPolynomial","k":"def","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":1,"d":"The union-of-conjunctions polynomial multiplies the sum-of-squares certificate for every branch.","s":"{S : Type u_1} → {T : Type u_2} → {B : Type u_4} → [Fintype B] → [Fintype T] → (B → T → MvPolynomial S ℝ) → MvPolynomial S ℝ","l":110,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.AlgebraicGeometry.unionConjunctionPolynomial_ne_zero_of_witnesses","k":"theorem","m":"Causalean.Mathlib.AlgebraicGeometry.RationalMap","a":"Mathlib/AlgebraicGeometry","t":2,"d":"Given a finite polynomial-system family, branchwise witness assignments, branchwise coordinates, and nonzero witness values, the union certificate is nonzero.","s":"∀ {S : Type u_1} {T : Type u_2} {B : Type u_4} [inst : Fintype B] [inst_1 : Fintype T] (p : B → T → MvPolynomial S ℝ) (witness : B → S → ℝ) (coordinate : B → T), (∀ (b : B), (MvPolynomial.eval (witness b)) (p b (coordinate b)) ≠ 0) → Causalean.Mathlib.AlgebraicGeometry.unionConjunctionPolynomial p ≠ 0","l":129,"f":"Causalean/Mathlib/AlgebraicGeometry/RationalMap.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A bounded window and an index determine whether the index is active in that window.","s":"{n : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n → ℕ → Prop","l":35,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ActiveAt","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A left endpoint, a right endpoint, and an index determine whether the index is active in the half-open interval.","s":"ℕ → ℕ → ℕ → Prop","l":31,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A deque is represented by its front-to-back list of natural-number indices. The type itself imposes no ordering or monotonicity; the monotone-deque invariant is carried separately by the validity predicates.","s":"Type","l":16,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a finite pass count have a family of monotone-window schedules, one for each pass.","s":"{α : Type u_1} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → ℕ → Type","l":18,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses.head_value_eq_windowMax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A fixed family of passes, a selected pass, a valid position in that pass, and a nonempty window there ensure that the scan head is an active argmax with the finite-window maximum value.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {passes : ℕ} (family : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses stream passes) (p : Fin passes) {k : ℕ} (hk : k < (family.schedule p).steps) (hne : ((family.schedule p).windows.get ⟨k, hk⟩).left < ((family.schedule p).windows.get ⟨k, hk⟩).right), ∃ step head tail, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream (family.schedule p))[k]? = some step ∧ step.window = (family.schedule p).windows.get ⟨k, hk⟩ ∧ step.after = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window head ∧ stream.value head = Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowMax stream ((family.schedule p).windows.get ⟨k, hk⟩) hne ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window i → stream.value i ≤ stream.value head","l":33,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses.memory_le_windowWidth","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A fixed family of passes, a selected pass, and a valid position in that pass ensure that the corresponding state stores no more than its window width.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {passes : ℕ} (family : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses stream passes) (p : Fin passes) {k : ℕ} (hk : k < (family.schedule p).steps), ∃ step, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream (family.schedule p))[k]? = some step ∧ List.length step.after ≤ ((family.schedule p).windows.get ⟨k, hk⟩).right - ((family.schedule p).windows.get ⟨k, hk⟩).left","l":68,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses.totalCost","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A fixed family of passes determines the sum of its individual scan costs.","s":"{α : Type u_1} → [LinearOrder α] → {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} → {passes : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses stream passes → ℕ","l":23,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses.totalCost_le","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A fixed family of passes ensures that its total cost is at most twice the pass count times stream length plus its total scheduled-window count.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {passes : ℕ} (family : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses stream passes), family.totalCost ≤ 2 * passes * stream.length + family.totalSteps","l":50,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses.totalSteps","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Passes","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A fixed family of passes determines its total number of scheduled windows.","s":"{α : Type u_1} → {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} → {passes : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.FixedPasses stream passes → ℕ","l":28,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Passes.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicatePasses","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key list, a step list, and a finite pass count determine a pass family with one predicate-window schedule per pass.","s":"{ι : Type u_1} → {κ : Type u_2} → List ι → List κ → ℕ → Type (max u_1 u_2)","l":451,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicatePasses.totalRawCost","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A fixed predicate-pass family and a raw score determine the sum of the raw costs of all passes.","s":"{ι : Type u_1} → {κ : Type u_2} → {α : Type u_3} → {keys : List ι} → {steps : List κ} → [LinearOrder α] → {passes : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicatePasses keys steps passes → (ι → α) → ℕ","l":456,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicatePasses.totalRawCost_le","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A fixed predicate-pass family, a raw score, and a fallback score ensure total raw cost is at most twice pass count times key count plus pass count times step count.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] {passes : ℕ} (family : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicatePasses keys steps passes) (score : ι → α) (fallback : α), family.totalRawCost score ≤ 2 * passes * keys.length + passes * steps.length","l":462,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A sparse raw-key list and an ordered step list determine a predicate window specification whose entry predicate, stay predicate, entry-prefix law, stay-suffix law, entry monotonicity across steps, and stay antitonicity across steps describe a monotone active window.","s":"{ι : Type u_1} → {κ : Type u_2} → List ι → List κ → Type (max u_1 u_2)","l":18,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.active_schedule_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a valid step position, and an in-bounds key position characterize activity in the compiled scheduled window by the raw predicates.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {k i : ℕ} (hk : k < steps.length) (hi : i < keys.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active (P.schedule.windows.get ⟨k, Eq.mpr (id (congrArg (LT.lt k) (List.length_map P.windowAt))) hk⟩) i ↔ P.entry (steps.get ⟨k, hk⟩) (keys.get ⟨i, hi⟩) ∧ P.stay (steps.get ⟨k, hk⟩) (keys.get ⟨i, hi⟩)","l":360,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.active_windowAt_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a step, and an in-bounds position satisfy the exact equivalence between position activity and the raw entry-and-stay predicates.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ) {i : ℕ} (hi : i < keys.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active (P.windowAt s) i ↔ P.entry s (keys.get ⟨i, hi⟩) ∧ P.stay s (keys.get ⟨i, hi⟩)","l":223,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.drop_leftEndpoint_eq_dropWhile","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule has the keys between its endpoints exactly equal to dropping the non-staying prefix from the entered prefix.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ), List.drop (P.leftEndpoint s) (List.take (P.rightEndpoint s) keys) = List.dropWhile (P.expiredTest s) (List.take (P.rightEndpoint s) keys)","l":172,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.entered","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a step determine the raw prefix that has entered by that step.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → List ι","l":51,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.entryTest","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a step, and a raw key determine the Boolean entry test used by the list operations. It decides the entry predicate classically, so it is a noncomputable specification rather than executable code.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → ι → Bool","l":37,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.expiredPrefix","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a step determine the expired prefix of entered keys, namely the initial segment that no longer satisfies the stay predicate.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → List ι","l":61,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.expiredTest","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a step, and a raw key determine the Boolean test that the key has expired, true exactly when the key does not stay at that step. It decides the stay predicate classically, so it is noncomputable.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → ι → Bool","l":44,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.leftEndpoint","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a step determine the left position endpoint, namely the length of the expired prefix among entered keys.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → ℕ","l":66,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.leftEndpoint_eq_add_newlyExpired","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, an earlier step, a later step, entry expansion, and stay shrinkage have later left endpoint equal to the earlier endpoint plus the exact takeWhile expiration-segment length.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s t : κ), (∀ x ∈ keys, P.entry s x → P.entry t x) → (∀ x ∈ keys, P.stay t x → P.stay s x) → P.leftEndpoint t = P.leftEndpoint s + (List.takeWhile (P.expiredTest t) (List.drop (P.leftEndpoint s) (List.take (P.rightEndpoint t) keys))).length","l":319,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.leftEndpoint_le_of_step_lt","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, two valid step positions, and their strict order ensure the earlier compiled left endpoint is no larger.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {i j : ℕ} (hi : i < steps.length) (hj : j < steps.length), i < j → P.leftEndpoint (steps.get ⟨i, hi⟩) ≤ P.leftEndpoint (steps.get ⟨j, hj⟩)","l":379,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.leftEndpoint_le_rightEndpoint","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule has every left endpoint bounded by its right endpoint.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ), P.leftEndpoint s ≤ P.rightEndpoint s","l":134,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.leftEndpoint_mono_of_predicates","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"Expanding entry and shrinking stay from one step to another make left endpoints nondecreasing.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {s t : κ}, (∀ x ∈ keys, P.entry s x → P.entry t x) → (∀ x ∈ keys, P.stay t x → P.stay s x) → P.leftEndpoint s ≤ P.leftEndpoint t","l":253,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.length_rawScan","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule and a raw score produce one raw trace entry per step.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α), (P.rawScan score).length = steps.length","l":151,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.newlyEntering_eq_drop_entered","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, an earlier step, a later step, and entry inclusion have newly entered raw keys exactly equal to the later entered prefix after dropping the earlier right endpoint.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s t : κ), (∀ x ∈ keys, P.entry s x → P.entry t x) → List.takeWhile (P.entryTest t) (List.drop (P.rightEndpoint s) keys) = List.drop (P.rightEndpoint s) (P.entered t)","l":274,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.newlyExpired_eq_takeWhile","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, an earlier step, a later step, entry expansion, and stay shrinkage have the keys crossing the left endpoint exactly equal to the takeWhile non-staying segment.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s t : κ), (∀ x ∈ keys, P.entry s x → P.entry t x) → (∀ x ∈ keys, P.stay t x → P.stay s x) → List.take (P.leftEndpoint t - P.leftEndpoint s) (List.drop (P.leftEndpoint s) (List.take (P.rightEndpoint t) keys)) = List.takeWhile (P.expiredTest t) (List.drop (P.leftEndpoint s) (List.take (P.rightEndpoint t) keys))","l":294,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rawScan","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a raw score determine the complete raw-key scan from all keys pending and an empty deque.","s":"{ι : Type u_1} → {κ : Type u_2} → {α : Type u_3} → {keys : List ι} → {steps : List κ} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → (ι → α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ)","l":145,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rawScanCost","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a raw score determine the raw scan cost, including one bookkeeping unit per step.","s":"{ι : Type u_1} → {κ : Type u_2} → {α : Type u_3} → {keys : List ι} → {steps : List κ} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → (ι → α) → ℕ","l":41,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rawScanFrom","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a raw score determine the raw trace obtained by folding over a supplied suffix of steps from a supplied state.","s":"{ι : Type u_1} → {κ : Type u_2} → {α : Type u_3} → {keys : List ι} → {steps : List κ} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → (ι → α) → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawState ι → List κ → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ)","l":135,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rawUpdate","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a raw score, a step, and a raw state determine the next recorded raw-key update. It advances entry by takeWhile, removes the non-staying prefixes by dropWhile, and pushes precisely the surviving new suffix.","s":"{ι : Type u_1} → {κ : Type u_2} → {α : Type u_3} → {keys : List ι} → {steps : List κ} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → (ι → α) → κ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawState ι → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ","l":111,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rightEndpoint","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a step determine the right position endpoint, namely the length of the entered raw-key prefix.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → ℕ","l":56,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rightEndpoint_eq_add_newlyEntering","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, an earlier step, a later step, and entry inclusion have later right endpoint equal to the earlier endpoint plus the exact takeWhile entering-segment length.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s t : κ), (∀ x ∈ keys, P.entry s x → P.entry t x) → P.rightEndpoint t = P.rightEndpoint s + (List.takeWhile (P.entryTest t) (List.drop (P.rightEndpoint s) keys)).length","l":284,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rightEndpoint_le_length","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule has every right endpoint bounded by the raw-key count.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ), P.rightEndpoint s ≤ keys.length","l":129,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rightEndpoint_le_of_step_lt","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, two valid step positions, and their strict order ensure the earlier compiled right endpoint is no larger.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {i j : ℕ} (hi : i < steps.length) (hj : j < steps.length), i < j → P.rightEndpoint (steps.get ⟨i, hi⟩) ≤ P.rightEndpoint (steps.get ⟨j, hj⟩)","l":371,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.rightEndpoint_mono_of_entry","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"Predicate inclusion from an earlier step to a later step makes right endpoints nondecreasing.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {s t : κ}, (∀ x ∈ keys, P.entry s x → P.entry t x) → P.rightEndpoint s ≤ P.rightEndpoint t","l":240,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.schedule","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule compiles to the monotone bounded-window schedule over key positions.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule keys.length","l":336,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.schedule_steps","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule compiles to one position window for each raw step.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps), P.schedule.steps = steps.length","l":347,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.schedule_window_get","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule and a valid step-list position compile the window at that position to the endpoints of the corresponding raw step.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) {k : ℕ} (hk : k < steps.length), P.schedule.windows.get ⟨k, Eq.mpr (id (congrArg (LT.lt k) (List.length_map P.windowAt))) hk⟩ = P.windowAt (steps.get ⟨k, hk⟩)","l":352,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.take_leftEndpoint_eq_expiredPrefix","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule has its left prefix exactly equal to the takeWhile segment of entered keys that fail the stay predicate.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ), List.take (P.leftEndpoint s) (List.take (P.rightEndpoint s) keys) = P.expiredPrefix s","l":163,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.take_rightEndpoint_eq_entered","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule has its right prefix exactly equal to the takeWhile entry segment.","s":"∀ {ι : Type u_1} {κ : Type u_2} {keys : List ι} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (s : κ), List.take (P.rightEndpoint s) keys = P.entered s","l":152,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.windowAt","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule and a step determine a bounded half-open position window, including when the raw-key list or the active window is empty.","s":"{ι : Type u_1} → {κ : Type u_2} → {keys : List ι} → {steps : List κ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps → κ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window keys.length","l":144,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule.windowAt_empty","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateSchedule","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule over an empty raw-key list has both endpoints equal to zero at every step.","s":"∀ {ι : Type u_1} {κ : Type u_2} {steps : List κ} (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule [] steps) (s : κ), (P.windowAt s).left = 0 ∧ (P.windowAt s).right = 0","l":388,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateSchedule.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PushBatch","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A batch insertion records its final deque, every pushed index, and every index removed from the back while processing the batch.","s":"Type","l":47,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawDequeTrace","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key type determines a deque-update trace with the deque before updating, the deque after expiration, the final deque, the pushed keys, the front-popped keys, and the back-popped keys.","s":"Type u_4 → Type u_4","l":86,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawPushBatch","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key type determines a batch record with the final key deque, the pushed keys, and the keys removed from the back.","s":"Type u_4 → Type u_4","l":47,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawState","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key type determines a predicate-scan state with the unentered suffix and the current raw-key deque.","s":"Type u_4 → Type u_4","l":80,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key type and a step type determine a predicate-scan step record that extends a deque update with the current step, the pending suffix before updating, the pending suffix afterward, and the complete entered prefix before expiration filtering.","s":"Type u_4 → Type u_5 → Type (max u_4 u_5)","l":97,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace.dequeTrace","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate step trace determines its exact generic-shaped deque trace.","s":"{ι : Type u_1} → {κ : Type u_2} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawDequeTrace ι","l":107,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A finite schedule for a stream of given length contains bounded windows with nondecreasing left and right endpoints.","s":"ℕ → Type","l":79,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule.left_le_of_get_lt","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A finite schedule, two valid schedule positions, and their strict order ensure that the earlier left endpoint is no larger.","s":"∀ {n : ℕ} (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule n) {i j : ℕ} (hi : i < schedule.steps) (hj : j < schedule.steps), i < j → (schedule.windows.get ⟨i, hi⟩).left ≤ (schedule.windows.get ⟨j, hj⟩).left","l":89,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule.right_le_of_get_lt","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A finite schedule, two valid schedule positions, and their strict order ensure that the earlier right endpoint is no larger.","s":"∀ {n : ℕ} (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule n) {i j : ℕ} (hi : i < schedule.steps) (hj : j < schedule.steps), i < j → (schedule.windows.get ⟨i, hi⟩).right ≤ (schedule.windows.get ⟨j, hj⟩).right","l":96,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule.steps","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A finite schedule determines its number of scheduled windows.","s":"{n : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule n → ℕ","l":86,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream has a step record containing its before and after deques and exact push, front-pop, and back-pop event lists.","s":"{α : Type u_1} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → Type","l":15,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A finite ordered stream of values in a type records its length and value at each natural-number index.","s":"Type u_1 → Type u_1","l":17,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceBackPopped","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a scan trace determine the aggregate back-pop log.","s":"{α : Type u_1} → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream) → List ℕ","l":23,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceFrontPopped","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a scan trace determine the aggregate front-pop log.","s":"{α : Type u_1} → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream) → List ℕ","l":19,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TracePushed","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a scan trace determine the aggregate push log.","s":"{α : Type u_1} → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream) → List ℕ","l":15,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a bounded window, and a deque determine whether the deque is valid for that window.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → Prop","l":76,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid.head_argmax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid bounded-window deque and a nonempty window ensure that its head is an active argmax.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream window q → window.left < window.right → ∃ head tail, q = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active window head ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active window i → stream.value i ≤ stream.value head","l":62,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid.head_value_eq_windowMax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A valid bounded-window deque and a nonempty window ensure that its head is an active argmax whose value equals the finite-window maximum.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream window q → ∀ (hne : window.left < window.right), ∃ head tail, q = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active window head ∧ stream.value head = Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowMax stream window hne ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active window i → stream.value i ≤ stream.value head","l":74,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid.length_le_windowWidth","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid bounded-window deque has stored length at most its window width.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream window q → List.length q ≤ window.right - window.left","l":319,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, two raw endpoints, and a deque satisfy the validity invariant when every retained index is active for the endpoints and within the stream, retained indices strictly increase from front to back, their stream values strictly decrease from front to back, and every active in-stream index that is not retained is dominated by a later retained index whose value is at least as large.","s":"{α : Type u_1} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → ℕ → ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → Prop","l":64,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt.head_argmax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid raw-window deque, a nonempty raw interval, and a right endpoint within the stream ensure that the deque head is active and maximizes the stream value over that interval.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left right : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left right q → left < right → right ≤ stream.length → ∃ head tail, q = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ActiveAt left right head ∧ head < stream.length ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ActiveAt left right i → i < stream.length → stream.value i ≤ stream.value head","l":35,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt.nodup","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid raw-window deque has no duplicate indices.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left right : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left right q → List.Nodup q","l":172,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt.to_max_right","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid raw-window deque remains valid when its right endpoint is enlarged to at least its left endpoint.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left right : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left right q → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left (max right left) q","l":130,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window","k":"structure","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A contiguous half-open window in a stream of given length has a left endpoint no larger than its right endpoint, which does not exceed the stream length.","s":"ℕ → Type","l":23,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.card_indexFinset","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A bounded window has as many active indices as its width.","s":"∀ {n : ℕ} (w : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n), w.indexFinset.card = w.right - w.left","l":57,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.indexFinset","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A bounded window determines its finite set of active indices.","s":"{n : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n → Finset ℕ","l":43,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.indexFinset_nonempty_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A bounded window has an active index exactly when its left endpoint is strictly below its right endpoint.","s":"∀ {n : ℕ} (w : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n), w.indexFinset.Nonempty ↔ w.left < w.right","l":73,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.indices","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A bounded window determines its increasing list of active indices.","s":"{n : ℕ} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n → List ℕ","l":39,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.indices_eq_nil_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A bounded window has an empty index list exactly when its two endpoints coincide.","s":"∀ {n : ℕ} (w : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n), w.indices = [] ↔ w.left = w.right","l":62,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.mem_indexFinset_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A bounded window has the same active indices in its finite-set enumeration and in its active-window condition.","s":"∀ {n i : ℕ} (w : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n), i ∈ w.indexFinset ↔ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active w i","l":52,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window.mem_indices_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Basic","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A bounded window has the same active indices in its list enumeration and in its active-window condition.","s":"∀ {n i : ℕ} (w : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window n), i ∈ w.indices ↔ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active w i","l":46,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Basic.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.dequeOperations","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a scan trace determine the total number of deque mutations.","s":"{α : Type u_1} → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream) → ℕ","l":27,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.dequeOperations_le_two_mul_length","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that total deque mutations are at most twice the stream length.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.dequeOperations stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule) ≤ 2 * stream.length","l":301,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expireFront","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A left endpoint and a deque determine the deque after front expiration.","s":"ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","l":24,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expireFront_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid old raw-window deque and a nondecreasing new left endpoint ensure that front expiration preserves the full raw-window invariant, including an empty intermediate interval.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {oldLeft oldRight newLeft : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream oldLeft oldRight q → oldLeft ≤ newLeft → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream newLeft oldRight (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expireFront newLeft q)","l":91,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expiredFront","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A left endpoint and a deque determine the recorded front pops.","s":"ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → List ℕ","l":28,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expiredFront_append_expireFront","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A left endpoint and a deque have a reported front-pop prefix followed by the retained deque exactly equal to the original deque.","s":"∀ (left : ℕ) (q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expiredFront left q ++ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.expireFront left q = q","l":90,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.initialDeque","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"The initial deque is empty before any stream index enters.","s":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","l":21,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.initialDeque_validAt","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream has an empty initial deque satisfying the zero-width invariant.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream 0 0 Causalean.Mathlib.Algorithms.MonotoneWindowDeque.initialDeque","l":81,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key list, a raw-key score, and an out-of-bounds fallback score determine the total position stream used by the generic deque. The fallback is never observed at a valid position.","s":"{ι : Type u_1} → {α : Type u_3} → List ι → (ι → α) → α → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α","l":20,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.length_scan","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that the scan has exactly one trace step for each scheduled window.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule).length = schedule.steps","l":28,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.mapStepTrace","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key list maps one generic position-level step to the exact raw-key deque trace by safe lookup in every list-valued field.","s":"{ι : Type u_1} → {α : Type u_3} → List ι → {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawDequeTrace ι","l":166,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.mem_prunedBack_dominated","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, all old indices being earlier than the new index, and an index recorded as a back pop ensure that the popped index is earlier and no larger in value than the new index.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque} {i j : ℕ}, (∀ k ∈ q, k < i) → j ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.prunedBack stream q i → j < i ∧ stream.value j ≤ stream.value i","l":161,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.mem_push_iff","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a deque, a new index, and a queried index have the stated membership characterization after one push.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque) (i k : ℕ), k ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push stream q i ↔ k ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack stream q i ∨ k = i","l":121,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.peakStored","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a scan trace determine the peak stored deque size.","s":"{α : Type u_1} → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream) → ℕ","l":38,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.peakStored_le_length","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that peak deque storage is at most the stream length.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.peakStored stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule) ≤ stream.length","l":343,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.positionsToKeys","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw-key list and a list of natural positions determine the raw keys obtained by safe lookup of every in-bounds position.","s":"{ι : Type u_1} → List ι → List ℕ → List ι","l":27,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a deque, and a new index determine the deque after rightmost-stable back pruning, which removes ties in favor of the newer index.","s":"{α : Type u_1} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","l":32,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack_append_prunedBack","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a deque, and a new index have a retained prefix and reported back-pop suffix exactly equal to the original deque.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque) (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack stream q i ++ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.prunedBack stream q i = q","l":96,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack_index_ordered","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a new index, and strictly ordered old deque indices ensure that back pruning preserves strict index order.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque} {i : ℕ}, List.Pairwise (fun x1 x2 => x1 < x2) q → List.Pairwise (fun x1 x2 => x1 < x2) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack stream q i)","l":108,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack_prefix","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a deque, and a new index ensure that back pruning retains a prefix of the old deque.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque) (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack stream q i <+: q","l":102,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack_value_decreasing","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a new index, and strictly decreasing old deque values ensure that back pruning preserves strict value decrease.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque} {i : ℕ}, List.Pairwise (fun j k => stream.value k < stream.value j) q → List.Pairwise (fun j k => stream.value k < stream.value j) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pruneBack stream q i)","l":114,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.prunedBack","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a deque, and a new index determine the recorded back pops.","s":"{α : Type u_1} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → ℕ → List ℕ","l":37,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a deque, and a new index determine the deque after one rightmost-stable insertion.","s":"{α : Type u_1} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","l":42,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pushAll","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream determines the recorded batch insertion from an initial deque and a list of entering indices.","s":"{α : Type u_1} → [LinearOrder α] → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → List ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PushBatch","l":54,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pushAll_interval_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid deque after front expiration, a nondecreasing right endpoint, a new endpoint within the stream, and a valid new raw interval ensure that inserting newly entered active indices restores the invariant.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left oldRight newRight : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left oldRight q → oldRight ≤ newRight → newRight ≤ stream.length → left ≤ newRight → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left newRight (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pushAll stream q (List.range' (max oldRight left) (newRight - max oldRight left))).state","l":224,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pushAll_range_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid raw-window deque, a left endpoint no larger than the range start, and a range ending within the stream ensure that pushing the whole consecutive range preserves the invariant.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left start count : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left start q → left ≤ start → start + count ≤ stream.length → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left (start + count) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.pushAll stream q (List.range' start count)).state","l":205,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push_index_ordered","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, strictly ordered old deque indices, and all old indices being earlier than the new one ensure that one push preserves strict index order.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque} {i : ℕ}, List.Pairwise (fun x1 x2 => x1 < x2) q → (∀ j ∈ q, j < i) → List.Pairwise (fun x1 x2 => x1 < x2) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push stream q i)","l":127,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push_succ_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A valid raw-window deque, a left endpoint no larger than the current right endpoint, and room for one more stream index ensure that pushing the right-boundary index preserves the invariant for the enlarged window.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {left right : ℕ} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left right q → left ≤ right → right < stream.length → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.ValidAt stream left (right + 1) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push stream q right)","l":152,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push_value_decreasing","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and strictly decreasing old deque values ensure that one rightmost-stable push leaves strictly decreasing values.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque} {i : ℕ}, List.Pairwise (fun j k => stream.value k < stream.value j) q → List.Pairwise (fun j k => stream.value k < stream.value j) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.push stream q i)","l":139,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Deque.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawDequeOperations","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate-scan trace determines its exact number of deque mutations.","s":"{ι : Type u_1} → {κ : Type u_2} → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ) → ℕ","l":36,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPeakStored","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate-scan trace determines the largest raw deque length reached.","s":"{ι : Type u_1} → {κ : Type u_2} → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ) → ℕ","l":47,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPeakStored_eq_peakStored","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, and a fallback score have raw peak storage exactly equal to generic compiled position-scan peak storage.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPeakStored (P.rawScan score) = Causalean.Mathlib.Algorithms.MonotoneWindowDeque.peakStored (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream keys score fallback) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream keys score fallback) P.schedule)","l":256,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPeakStored_le_length","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, and a fallback score ensure raw peak deque storage is at most the number of sparse keys.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPeakStored (P.rawScan score) ≤ keys.length","l":296,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPruneBack","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw score, a raw deque, and a new key determine the deque prefix after deleting its suffix of scores no larger than the new score.","s":"{ι : Type u_1} → {α : Type u_3} → [LinearOrder α] → (ι → α) → List ι → ι → List ι","l":32,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPrunedBack","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw score, a raw deque, and a new key determine the keys removed from the back by the deterministic rightmost-stable tie policy.","s":"{ι : Type u_1} → {α : Type u_3} → [LinearOrder α] → (ι → α) → List ι → ι → List ι","l":37,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPrunedBack_score_le","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A key removed by rightmost-stable back pruning has score no larger than the newly pushed key, so equal-score ties deterministically favor the newer occurrence.","s":"∀ {ι : Type u_1} {α : Type u_3} [inst : LinearOrder α] (score : ι → α) (q : List ι) (x y : ι), y ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPrunedBack score q x → score y ≤ score x","l":73,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPush","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw score, a raw deque, and a new key determine one rightmost-stable monotone push; equal old scores are removed in favor of the new key.","s":"{ι : Type u_1} → {α : Type u_3} → [LinearOrder α] → (ι → α) → List ι → ι → List ι","l":42,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPushAll","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw score, together with an initial raw deque and a list of input keys, determines the recorded batch of rightmost-stable pushes, given by leaving an empty input list unchanged and pushing the first key before recursing on the remainder.","s":"{ι : Type u_1} → {α : Type u_3} → [LinearOrder α] → (ι → α) → List ι → List ι → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawPushBatch ι","l":54,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPush_getLast?_eq","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"One raw rightmost-stable push has the newly pushed key as its final element, including when all old entries tie with it.","s":"∀ {ι : Type u_1} {α : Type u_3} [inst : LinearOrder α] (score : ι → α) (q : List ι) (x : ι), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawPush score q x).getLast? = some x","l":67,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawScanCost_eq_scanCost","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, and a fallback score have raw operation cost exactly equal to the generic compiled position-scan cost.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α), P.rawScanCost score = Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scanCost (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream keys score fallback) P.schedule","l":216,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawScanCost_le","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, and a fallback score ensure raw scan cost is at most twice the number of sparse keys plus the number of steps.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α), P.rawScanCost score ≤ 2 * keys.length + steps.length","l":287,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawScan_head_argmax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a raw score, a fallback score, a valid step position, and a nonempty raw predicate window ensure the raw deque head is an active score maximizer, with equal maxima resolved at the greatest key position.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α) {k : ℕ} (hk : k < steps.length), (∃ i, P.entry (steps.get ⟨k, hk⟩) (keys.get i) ∧ P.stay (steps.get ⟨k, hk⟩) (keys.get i)) → ∃ trace head tail headPos, (P.rawScan score)[k]? = some trace ∧ trace.step = steps.get ⟨k, hk⟩ ∧ trace.after = head :: tail ∧ head = keys.get headPos ∧ P.entry (steps.get ⟨k, hk⟩) head ∧ P.stay (steps.get ⟨k, hk⟩) head ∧ (∀ (i : Fin keys.length), P.entry (steps.get ⟨k, hk⟩) (keys.get i) → P.stay (steps.get ⟨k, hk⟩) (keys.get i) → score (keys.get i) ≤ score head) ∧ ∀ (i : Fin keys.length), P.entry (steps.get ⟨k, hk⟩) (keys.get i) → P.stay (steps.get ⟨k, hk⟩) (keys.get i) → score (keys.get i) = score head → ↑i ≤ ↑headPos","l":343,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawScan_memory_le_windowWidth","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, a fallback score, and a valid step position ensure the raw deque stored there is no wider than its compiled window.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α) {k : ℕ} (hk : k < steps.length), ∃ trace, (P.rawScan score)[k]? = some trace ∧ trace.after.length ≤ P.rightEndpoint (steps.get ⟨k, hk⟩) - P.leftEndpoint (steps.get ⟨k, hk⟩)","l":305,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawTraceBackPopped","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate-scan trace determines the aggregate list of back-popped raw keys.","s":"{ι : Type u_1} → {κ : Type u_2} → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ) → List ι","l":31,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawTraceFrontPopped","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate-scan trace determines the aggregate list of front-popped raw keys.","s":"{ι : Type u_1} → {κ : Type u_2} → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ) → List ι","l":26,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.rawTracePushed","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateAccounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A raw predicate-scan trace determines the aggregate list of pushed raw keys.","s":"{ι : Type u_1} → {κ : Type u_2} → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace ι κ) → List ι","l":22,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateAccounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a monotone window schedule determine the complete scan trace from the empty initial state.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream)","l":23,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scanCost","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a monotone window schedule determine the scan cost, including one bookkeeping unit per window.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length → ℕ","l":33,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scanCost_le","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a monotone window schedule ensure that scan cost is at most twice stream length plus the number of scheduled windows.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scanCost stream schedule ≤ 2 * stream.length + schedule.steps","l":311,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scanFrom","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream determines the scan trace produced from a prior right endpoint, deque, and list of windows.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length) → List (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream)","l":14,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_back_pop_count_le_one","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and an index ensure that the index is back-popped at most once in the aggregate scan log.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) (i : ℕ), List.count i (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceBackPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)) ≤ 1","l":220,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_front_pop_count_le_one","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and an index ensure that the index is front-popped at most once in the aggregate scan log.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) (i : ℕ), List.count i (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceFrontPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)) ≤ 1","l":211,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_head_argmax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, a valid schedule position, and a nonempty window at that position ensure that the scan head is active and maximizes the stream value in that window.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) {k : ℕ} (hk : k < schedule.steps), (schedule.windows.get ⟨k, hk⟩).left < (schedule.windows.get ⟨k, hk⟩).right → ∃ step head tail, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)[k]? = some step ∧ step.window = schedule.windows.get ⟨k, hk⟩ ∧ step.after = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window head ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window i → stream.value i ≤ stream.value head","l":95,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_head_value_eq_windowMax","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a monotone window schedule, a valid schedule position, and a nonempty window at that position ensure that the scan head is an active argmax whose value equals the finite-window maximum.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) {k : ℕ} (hk : k < schedule.steps) (hne : (schedule.windows.get ⟨k, hk⟩).left < (schedule.windows.get ⟨k, hk⟩).right), ∃ step head tail, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)[k]? = some step ∧ step.window = schedule.windows.get ⟨k, hk⟩ ∧ step.after = head :: tail ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window head ∧ stream.value head = Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowMax stream (schedule.windows.get ⟨k, hk⟩) hne ∧ ∀ (i : ℕ), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Active step.window i → stream.value i ≤ stream.value head","l":113,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_memory_le_windowWidth","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and a valid schedule position ensure that the corresponding scan state stores no more indices than its window width.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) {k : ℕ} (hk : k < schedule.steps), ∃ step, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)[k]? = some step ∧ List.length step.after ≤ (schedule.windows.get ⟨k, hk⟩).right - (schedule.windows.get ⟨k, hk⟩).left","l":330,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_pop_logs_disjoint","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that no index appears in both front-pop and back-pop logs.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceFrontPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)).Disjoint (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceBackPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule))","l":229,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_popped_was_pushed","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and an index recorded as a front or back pop ensure that the index was previously pushed.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) {i : ℕ}, i ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceFrontPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule) ∨ i ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceBackPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule) → i ∈ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TracePushed stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)","l":245,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_predicateSchedule_after_eq","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A predicate schedule, a raw score, a valid step position, and a fallback score have the generic deque after that step mapped through key lookup equal to the raw predicate-scan deque.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α) {k : ℕ}, k < steps.length → ∃ generic raw, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream keys score fallback) P.schedule)[k]? = some generic ∧ (P.rawScan score)[k]? = some raw ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.positionsToKeys keys generic.after = raw.after","l":735,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_predicateSchedule_state_eq","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateRawScan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A predicate schedule, a raw-key score, and a fallback score have the generic position scan mapped through raw-key lookup exactly equal to the corresponding raw-key predicate scan, including every deque state and push/pop log.","s":"∀ {ι : Type u_1} {κ : Type u_2} {α : Type u_3} {keys : List ι} {steps : List κ} [inst : LinearOrder α] (P : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.PredicateWindowSchedule keys steps) (score : ι → α) (fallback : α), List.map (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.mapStepTrace keys) (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.keyedStream keys score fallback) P.schedule) = List.map Causalean.Mathlib.Algorithms.MonotoneWindowDeque.RawStepTrace.dequeTrace (P.rawScan score)","l":717,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/PredicateRawScan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_push_count_le_one","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and an index ensure that the index is pushed at most once in the aggregate scan log.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) (i : ℕ), List.count i (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TracePushed stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)) ≤ 1","l":200,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_total_pops_le_pushes","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that total front and back pops are no more numerous than total pushes.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceFrontPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)).length + (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TraceBackPopped stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)).length ≤ (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TracePushed stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)).length","l":261,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_total_pushes_le_length","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Accounting","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and a monotone window schedule ensure that total pushes are at most the stream length.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.TracePushed stream (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)).length ≤ stream.length","l":283,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Accounting.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan_valid_at","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Scan","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a monotone window schedule, and a valid schedule position ensure that the corresponding scan step has a valid deque for its window.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (schedule : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Schedule stream.length) {k : ℕ} (hk : k < schedule.steps), ∃ step, (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.scan stream schedule)[k]? = some step ∧ step.window = schedule.windows.get ⟨k, hk⟩ ∧ Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream step.window step.after","l":72,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Scan.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, the prior right endpoint, a new bounded window, and the prior deque determine the recorded monotone-deque update, expiring the front before inserting newly entered active indices.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → ℕ → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.StepTrace stream","l":27,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update_initial_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream and its first bounded window ensure that updating from the empty initial interval produces a valid deque.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length), Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream window (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update stream 0 window []).after","l":257,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update_preserves","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A valid old-window deque, a nondecreasing left endpoint, and a nondecreasing right endpoint ensure that one update produces a valid new-window deque.","s":"∀ {α : Type u_1} [inst : LinearOrder α] {stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α} {oldWindow newWindow : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length} {q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque}, Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream oldWindow q → oldWindow.left ≤ newWindow.left → oldWindow.right ≤ newWindow.right → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Valid stream newWindow (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update stream oldWindow.right newWindow q).after","l":241,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update_pushed","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Update","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a prior right endpoint, a new window, and a prior deque have an update push log equal to the newly entered active interval.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (oldRight : ℕ) (window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length) (q : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Deque), (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.update stream oldRight window q).pushed = List.range' (max oldRight window.left) (window.right - max oldRight window.left)","l":82,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Update.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowMax","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream, a bounded window, and strictly ordered window endpoints determine the window maximum.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → (window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length) → window.left < window.right → α","l":29,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowValues","k":"def","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":1,"d":"A stream and a bounded window determine the finite set of values observed in that window.","s":"{α : Type u_1} → [LinearOrder α] → (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) → Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length → Finset α","l":16,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowValues_nonempty","k":"theorem","m":"Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Correctness","a":"Mathlib/Algorithms/MonotoneWindowDeque","t":2,"d":"A stream, a bounded window, and strictly ordered window endpoints ensure that the window's finite value set is nonempty.","s":"∀ {α : Type u_1} [inst : LinearOrder α] (stream : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Stream α) (window : Causalean.Mathlib.Algorithms.MonotoneWindowDeque.Window stream.length), window.left < window.right → (Causalean.Mathlib.Algorithms.MonotoneWindowDeque.windowValues stream window).Nonempty","l":21,"f":"Causalean/Mathlib/Algorithms/MonotoneWindowDeque/Correctness.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsExtremalDecomposition","k":"structure","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"An extremal certificate given by two finite measures on the real line, labelled positive and negative. Each has mass 1/2, is symmetric and supported on [-1,1]; their degree-K moments agree, while their absolute first moments differ by E_K.","s":"ℕ → Type","l":417,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsExtremalDecomposition.toMomentMatchedPriors","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a nonnegative moment-degree limit and an absolute-value extremal decomposition at that limit, the associated pair of moment-matched priors has first prior equal to twice the decomposition's negative measure and second prior equal to twice its positive measure.","s":"{K : ℕ} → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsExtremalDecomposition K → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K","l":597,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors","k":"structure","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"Two symmetric probability measures on [-1,1] whose moments match through degree K and whose absolute first moments have the oriented gap 2 E_K.","s":"ℕ → Type","l":580,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.circleDouble","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"The circle-doubling map is the continuous map from the additive circle of period π to itself that sends each point to twice that point.","s":"C(AddCircle Real.pi, AddCircle Real.pi)","l":575,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.continuous_fejerKernel","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a Fejér order, the normalized Fejér kernel is continuous.","s":"∀ (n : ℕ), Continuous (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerKernel n)","l":37,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a positive integer order, the continuous-linear cusp functional maps a continuous complex-valued function on the circle of period π to its value at zero minus its de la Vallée--Poussin mean of that order.","s":"(n : ℕ) → 0 < n → C(AddCircle Real.pi, ℂ) →L[ℂ] ℂ","l":340,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a nonnegative frequency at most that order, the cusp functional annihilates the Fourier character.","s":"∀ (n k : ℕ) (hn : 0 < n), k ≤ n → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier ↑k) = 0","l":347,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_nat_high","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a nonnegative frequency at least twice that order, the cusp functional has multiplier one.","s":"∀ (n k : ℕ) (hn : 0 < n), 2 * n ≤ k → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier ↑k) = 1","l":437,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_nat_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a nonnegative frequency, the real part of the cusp multiplier is nonnegative.","s":"∀ (n k : ℕ) (hn : 0 < n), 0 ≤ ((Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier ↑k)).re","l":363,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_neg_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a negative frequency magnitude at most that order, the cusp functional annihilates the Fourier character.","s":"∀ (n k : ℕ) (hn : 0 < n), k ≤ n → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier (-↑k)) = 0","l":354,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_neg_nat_high","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a negative frequency magnitude at least twice that order, the cusp functional has multiplier one.","s":"∀ (n k : ℕ) (hn : 0 < n), 2 * n ≤ k → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier (-↑k)) = 1","l":450,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_fourier_neg_nat_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a negative frequency magnitude, the real part of the cusp multiplier is nonnegative.","s":"∀ (n k : ℕ) (hn : 0 < n), 0 ≤ ((Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) (fourier (-↑k))).re","l":400,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctional_norm_apply_le","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a continuous input, the cusp functional is bounded by four times the uniform norm.","s":"∀ (n : ℕ) (hn : 0 < n) (f : C(AddCircle Real.pi, ℂ)), ‖(Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) f‖ ≤ 4 * ‖f‖","l":463,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.exists_fourierPoly_sinDouble","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a polynomial of degree at most a positive order, there is a corresponding Fourier polynomial that agrees after sine composition and is annihilated by the cusp functional.","s":"∀ (p : Polynomial ℝ) (n : ℕ) (hn : 0 < n), p.natDegree ≤ n → ∃ q, (∀ (t : ℝ), q ↑t = ↑(Polynomial.eval (Real.sin (2 * t)) p)) ∧ (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.cuspFunctionalCLM n hn) q = 0","l":522,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerKernel","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a nonnegative integer order and a point on the circle of period π, the normalized Fejér kernel is the squared complex modulus of the one-sided Fourier sum at that point, divided by the order; at order zero, this quotient is defined to be zero.","s":"ℕ → AddCircle Real.pi → ℝ","l":33,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a nonnegative integer order and a continuous complex-valued function on the circle of period π, the Fejér mean is the Haar integral of the function multiplied by the normalized Fejér kernel of that order.","s":"ℕ → C(AddCircle Real.pi, ℂ) → ℂ","l":44,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMeanCLM","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a positive integer order, the continuous-linear Fejér mean functional maps each continuous complex-valued function on the circle of period π to its Fejér mean of that order.","s":"(n : ℕ) → 0 < n → C(AddCircle Real.pi, ℂ) →L[ℂ] ℂ","l":233,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMeanCLM_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive Fejér order and a continuous input, the packaged linear map equals the integral definition.","s":"∀ (n : ℕ) (hn : 0 < n) (f : C(AddCircle Real.pi, ℂ)), (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMeanCLM n hn) f = Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean n f","l":268,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean_fourier_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive Fejér order and a nonnegative frequency, the Fejér mean has the stated triangular multiplier.","s":"∀ (n k : ℕ), 0 < n → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean n (fourier ↑k) = ↑(if k < n then ↑(n - k) / ↑n else 0)","l":88,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean_fourier_neg_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive Fejér order and a negative frequency magnitude, the Fejér mean has the same triangular multiplier.","s":"∀ (n k : ℕ), 0 < n → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean n (fourier (-↑k)) = ↑(if k < n then ↑(n - k) / ↑n else 0)","l":154,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive Fejér order, a continuous input, and a uniform norm bound, the Fejér mean obeys that bound.","s":"∀ (n : ℕ), 0 < n → ∀ (f : C(AddCircle Real.pi, ℂ)) (E : ℝ), (∀ (t : AddCircle Real.pi), ‖f t‖ ≤ E) → ‖Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerMean n f‖ ≤ E","l":200,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejer_integrand_expand","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a Fejér order, Fourier frequency, and circle point, the kernel-weighted character has the stated finite Fourier expansion.","s":"∀ (n k : ℕ) (t : AddCircle Real.pi), ↑(Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerKernel n t) * (fourier ↑k) t = (↑n)⁻¹ * ∑ r ∈ Finset.range n, ∑ s ∈ Finset.range n, (fourier (↑r - ↑s + ↑k)) t","l":63,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fourier_comp_circleDouble","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a Fourier frequency, composition with circle doubling doubles that frequency.","s":"∀ (j : ℤ), (fourier j).comp Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.circleDouble = fourier (2 * j)","l":579,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fourier_pair_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For two real coefficients, a frequency, and a circle coordinate, the conjugate Fourier pair equals the corresponding real sine--cosine mode.","s":"∀ (A B : ℝ) (k : ℕ) (t : ℝ), (↑A - ↑B * Complex.I) / 2 * (fourier ↑k) ↑t + (↑A + ↑B * Complex.I) / 2 * (fourier (-↑k)) ↑t = ↑(A * Real.cos (2 * ↑k * t) + B * Real.sin (2 * ↑k * t))","l":494,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.instFactLtRealOfNatPi_causalean","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"","s":"Fact (0 < Real.pi)","l":27,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.integral_fejerKernel","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive Fejér order, the normalized kernel integrates to one.","s":"∀ (n : ℕ), 0 < n → ∫ (t : AddCircle Real.pi), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.fejerKernel n t ∂AddCircle.haarAddCircle = 1","l":176,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.integral_fourier","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a Fourier frequency, its Haar integral is one at frequency zero and zero otherwise.","s":"∀ (k : ℤ), ∫ (x : AddCircle Real.pi), (fourier k) x ∂AddCircle.haarAddCircle = if k = 0 then 1 else 0","l":48,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.oneSidedFourierSum","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a nonnegative integer order, the one-sided Fourier sum is the continuous complex-valued function on the circle obtained by summing the Fourier characters with integer frequencies from zero through one less than that order.","s":"ℕ → C(AddCircle Real.pi, ℂ)","l":29,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.valleePoussinMeanCLM","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a positive integer order, the continuous-linear de la Vallée--Poussin mean maps a continuous complex-valued function on the circle of period π to twice its Fejér mean of order twice the given order, minus its Fejér mean of the given order.","s":"(n : ℕ) → 0 < n → C(AddCircle Real.pi, ℂ) →L[ℂ] ℂ","l":275,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.valleePoussinMean_fourier_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a frequency at most that order, the de la Vallée--Poussin mean preserves the nonnegative Fourier character.","s":"∀ (n k : ℕ) (hn : 0 < n), k ≤ n → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.valleePoussinMeanCLM n hn) (fourier ↑k) = 1","l":280,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.valleePoussinMean_fourier_neg_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Fejer","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive order and a frequency magnitude at most that order, the de la Vallée--Poussin mean preserves the negative Fourier character.","s":"∀ (n k : ℕ) (hn : 0 < n), k ≤ n → (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.FejerCertificate.valleePoussinMeanCLM n hn) (fourier (-↑k)) = 1","l":310,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Fejer.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSupportedOnUnitInterval","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a measure on the real line, support on the unit interval means that the complement of the closed interval from −1 to 1 has measure zero.","s":"MeasureTheory.Measure ℝ → Prop","l":34,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSymmetric","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a Borel measure on the real line, symmetry about zero means that reflecting every point through zero leaves the measure unchanged.","s":"MeasureTheory.Measure ℝ → Prop","l":30,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a nonnegative integer degree bound, the best uniform absolute-value approximation error is the infimum of the uniform errors of all real polynomials whose degree is at most that bound.","s":"ℕ → ℝ","l":40,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_antitone","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For two degree limits with the first no larger than the second, allowing the larger degree cannot increase the best error.","s":"∀ {K L : ℕ}, K ≤ L → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs L ≤ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":101,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_eq_sInf","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a polynomial degree limit, the best absolute-value approximation error is the infimum over all admissible polynomial errors.","s":"∀ (K : ℕ), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K = sInf {e | ∃ p, p.natDegree ≤ K ∧ e = Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs p}","l":78,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_lower","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Rate","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive polynomial degree, the cusp of absolute value forces the stated inverse-degree lower bound.","s":"∀ (K : ℕ), 0 < K → 1 / 100 / ↑K ≤ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":624,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Rate.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a polynomial degree limit, the best approximation error cannot be negative.","s":"∀ (K : ℕ), 0 ≤ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":88,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_order","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Rate","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"There are universal positive constants that sandwich the best absolute-value approximation error between constant multiples of the reciprocal degree for every positive degree.","s":"∃ c C, 0 < c ∧ c ≤ C ∧ ∀ (K : ℕ), 0 < K → c / ↑K ≤ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K ∧ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K ≤ C / ↑K","l":685,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Rate.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_two_mul_add_one","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a nonnegative integer, allowing the odd degree 2m+1 gives the same best error as degree 2m.","s":"∀ (m : ℕ), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs (2 * m + 1) = Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs (2 * m)","l":323,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs_upper","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Rate","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a positive polynomial degree, the best absolute-value approximation error is at most one divided by that degree.","s":"∀ (K : ℕ), 0 < K → Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K ≤ 1 / ↑K","l":402,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Rate.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.exists_absExtremalDecomposition","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a polynomial degree limit, a normalized extremal signed-measure decomposition exists.","s":"∀ (K : ℕ), Nonempty (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsExtremalDecomposition K)","l":443,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.exists_bestPolynomialAbs","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a polynomial degree limit, some admissible polynomial attains the best absolute-value approximation error.","s":"∀ (K : ℕ), ∃ p, p.natDegree ≤ K ∧ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs p = Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":164,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.exists_bestPolynomialAbs_interval","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a polynomial degree limit, some admissible polynomial bounds every absolute-value residual by the best error on the unit interval.","s":"∀ (K : ℕ), ∃ p, p.natDegree ≤ K ∧ ∀ x ∈ Set.Icc (-1) 1, ||x| - Polynomial.eval x p| ≤ Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":237,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.exists_symmetric_momentMatched_absGap","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a positive even degree, a symmetric moment-matched probability-prior pair with the exact absolute-moment gap exists. The positivity and evenness assumptions are not used by the proof; the same conclusion holds for every degree via exists_absExtremalDecomposition.","s":"∀ {K : ℕ}, 0 < K → Even K → Nonempty (Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K)","l":639,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorOne_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its second prior is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), MeasureTheory.IsProbabilityMeasure P.ν₁","l":657,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorOne_mass","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its second prior has total mass one.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), P.ν₁ Set.univ = 1","l":671,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorOne_supported","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its second prior is supported on the unit interval.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSupportedOnUnitInterval P.ν₁","l":685,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorOne_symmetric","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its second prior is symmetric about zero.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSymmetric P.ν₁","l":699,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorZero_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its first prior is a probability measure.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), MeasureTheory.IsProbabilityMeasure P.ν₀","l":650,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorZero_mass","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its first prior has total mass one.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), P.ν₀ Set.univ = 1","l":664,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorZero_supported","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its first prior is supported on the unit interval.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSupportedOnUnitInterval P.ν₀","l":678,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.priorZero_symmetric","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, its first prior is symmetric about zero.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.IsSymmetric P.ν₀","l":692,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.prior_absMoment_gap","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair, the oriented difference in absolute first moments is twice the best approximation error.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K), ∫ (x : ℝ), |x| ∂P.ν₁ - ∫ (x : ℝ), |x| ∂P.ν₀ = 2 * Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.bestUniformApproxErrorAbs K","l":714,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.prior_moments_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Duality","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a packaged prior pair and a moment order no larger than the degree limit, the two priors have equal moments of that order.","s":"∀ {K : ℕ} (P : Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.AbsMomentMatchedPriors K) {j : ℕ}, j ≤ K → ∫ (x : ℝ), x ^ j ∂P.ν₀ = ∫ (x : ℝ), x ^ j ∂P.ν₁","l":706,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"For a real polynomial, its uniform absolute-value approximation error is the supremum, over every real number in the closed interval from $-1$ through $1$, of the absolute difference between the polynomial's value and that number's absolute value.","s":"Polynomial ℝ → ℝ","l":33,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs_eq_sSup","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a real polynomial, its uniform absolute-value approximation error is the supremum of its pointwise residual on the unit interval.","s":"∀ (p : Polynomial ℝ), Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs p = sSup ((fun x => ||x| - Polynomial.eval x p|) '' Set.Icc (-1) 1)","l":48,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs_le_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":2,"d":"For a real polynomial and a proposed error bound, the bound holds exactly when it bounds every residual on the unit interval.","s":"∀ {p : Polynomial ℝ} {e : ℝ}, Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.uniformApproxErrorAbs p ≤ e ↔ ∀ x ∈ Set.Icc (-1) 1, ||x| - Polynomial.eval x p| ≤ e","l":57,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.unitInterval","k":"def","m":"Causalean.Mathlib.Analysis.AbsoluteValueMomentPriorDuality.Basic","a":"Mathlib/Analysis/AbsoluteValueMomentPriorDuality","t":1,"d":"The unit interval is the closed set of real numbers from $-1$ through $1$, inclusive.","s":"Set ℝ","l":30,"f":"Causalean/Mathlib/Analysis/AbsoluteValueMomentPriorDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.argumentPrinciple_circle","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"Argument principle for a circle. For a positive radius R, if the function f is complex-analytic on a neighborhood of the closed disk of radius R centered at c and f is nonzero on the boundary circle, then the normalized logarithmic-derivative integral of f around that circle equals the number of zeros of f strictly inside the disk, counted with analytic multiplicity.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = ↑(Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R)","l":105,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.eventuallyEq_logDeriv_add_order_div_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Near a finite-order zero of an analytic complex function, its logarithmic derivative is the zero multiplicity divided by displacement from the zero plus the logarithmic derivative of an analytic factor that does not vanish there.","s":"∀ {f : ℂ → ℂ} {a : ℂ}, AnalyticAt ℂ f a → analyticOrderAt f a ≠ ⊤ → ∃ g, AnalyticAt ℂ g a ∧ g a ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhdsWithin a {a}ᶜ, logDeriv f z = ↑(analyticOrderNatAt f a) / (z - a) + logDeriv g z","l":54,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.finiteSupport_orderWithinBall","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Under disk analyticity and boundary nonvanishing, the function assigning each interior zero its analytic multiplicity and all other points zero has finite support.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → Function.HasFiniteSupport fun z => if z ∈ Metric.ball c R then analyticOrderNatAt f z else 0","l":58,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.finite_interiorZeros","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A complex function analytic on a neighborhood of a closed disk and nonzero on its boundary has only finitely many zeros strictly inside that disk.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.interiorZeros f c R).Finite","l":28,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.interiorZeros","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For a complex-valued function, a complex center, and a real radius, the interior-zero set consists exactly of the complex numbers that are zeros of the function and lie strictly inside the open disk with that center and radius.","s":"(ℂ → ℂ) → ℂ → ℝ → Set ℂ","l":47,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For a complex-valued function, a complex center, and a real radius, the normalized logarithmic-derivative circle integral is the integral of the function's logarithmic derivative around the positively oriented circle with that center and radius, divided by $2\\pi i$.","s":"(ℂ → ℂ) → ℂ → ℝ → ℂ","l":30,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_centeredMonomial","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For a positive radius R such that the point a lies strictly inside the open disk of radius R centered at c, the normalized logarithmic-derivative integral around that circle of the monomial z ↦ (z - a)^n equals n.","s":"∀ {c a : ℂ} {R : ℝ} {n : ℕ}, 0 < R → a ∈ Metric.ball c R → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral (fun z => (z - a) ^ n) c R = ↑n","l":83,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_eq_of_rouche","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Under Rouché's strict boundary inequality, two analytic complex functions have equal normalized logarithmic-derivative circle integrals.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, ‖g z - f z‖ < ‖f z‖) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral g c R","l":253,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_eq_of_straightLineHomotopy","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Two analytic complex functions have equal normalized logarithmic-derivative circle integrals when every function on their straight-line interpolation is nonzero on the boundary circle.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ t ∈ Set.Icc 0 1, ∀ z ∈ Metric.sphere c R, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy f g t z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral g c R","l":35,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_exists_nat","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"The normalized logarithmic-derivative integral of a boundary-zero-free analytic function is a nonnegative whole number, viewed as a complex number.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → ∃ n, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R = ↑n","l":281,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral_re_pos_of_exists_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A boundary-zero-free analytic function with an interior zero has normalized logarithmic-derivative integral with strictly positive real part.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (∃ z ∈ Metric.ball c R, f z = 0) → 0 < (Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.normalizedLogDerivCircleIntegral f c R).re","l":324,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.rouche_circle","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"Rouché's theorem for a circle. For a positive radius R, if both f and g are complex-analytic on a neighborhood of the closed disk of radius R centered at c and on the boundary circle the discrepancy ‖g z - f z‖ is everywhere strictly smaller than ‖f z‖, then f and g have the same multiplicity-weighted number of zeros strictly inside the disk.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, ‖g z - f z‖ < ‖f z‖) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount g c R","l":222,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For two complex-valued functions, a real interpolation time, and a complex argument, the straight-line homotopy is the value at that argument of the affine combination giving weight $1-t$ to the first function and weight $t$ to the second.","s":"(ℂ → ℂ) → (ℂ → ℂ) → ℝ → ℂ → ℂ","l":28,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount","k":"def","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Basic","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":1,"d":"For a complex-valued function, a complex center, and a real radius, the zero-multiplicity count is the sum of the analytic multiplicities of all zeros of the function lying strictly inside the open disk with that center and radius, with the standard totalized finite sum used when the support is not finite.","s":"(ℂ → ℂ) → ℂ → ℝ → ℕ","l":39,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount_eq_of_straightLineHomotopy","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.Homotopy","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"Two analytic complex functions have the same multiplicity-weighted number of interior zeros when every function on their straight-line interpolation is nonzero on the boundary circle.","s":"∀ {f g : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → AnalyticOnNhd ℂ g (Metric.closedBall c R) → (∀ t ∈ Set.Icc 0 1, ∀ z ∈ Metric.sphere c R, Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.straightLineHomotopy f g t z ≠ 0) → Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R = Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount g c R","l":200,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/Homotopy.lean"},{"n":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount_pos_of_exists_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.ArgumentPrinciple","a":"Mathlib/Analysis/ArgumentPrincipleCircle","t":2,"d":"A boundary-zero-free analytic function that vanishes somewhere strictly inside the disk has a strictly positive multiplicity-weighted interior zero count.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {R : ℝ}, 0 < R → AnalyticOnNhd ℂ f (Metric.closedBall c R) → (∀ z ∈ Metric.sphere c R, f z ≠ 0) → (∃ z ∈ Metric.ball c R, f z = 0) → 0 < Causalean.Mathlib.Analysis.ArgumentPrincipleCircle.zeroMultiplicityCount f c R","l":289,"f":"Causalean/Mathlib/Analysis/ArgumentPrincipleCircle/ArgumentPrinciple.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE","k":"def","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a nonnegative integer degree bound and a real-valued function of a real argument, the assertion that the function is a real trigonometric polynomial of degree at most the bound means that there exist two real coefficient sequences such that, for every real argument $t$, the function equals $\\sum_{k=0}^{n}\\{a_k\\cos(kt)+b_k\\sin(kt)\\}$.","s":"ℕ → (ℝ → ℝ) → Prop","l":45,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.add","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The sum of two real trigonometric polynomials of degree at most n is again a real trigonometric polynomial of degree at most n.","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => f t + g t","l":69,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.card_simple_add_double_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"Multiplicity-refined zero-count lemma for real trigonometric polynomials (the sharp input to Szegő's inequality; not in Mathlib). Suppose f is a real trigonometric polynomial of degree at most n that is not identically zero, S is a finite set of zeros of f lying in the half-open period [c, c + 2π), and there is a further point t₀ in that same period and not belonging to S that is itself a zero of f at which the derivative of f also vanishes, i.e. t₀ is a zero of order at least 2. Then the cardinality of S, plus 2 for the double zero at t₀, is at most 2 n.","s":"∀ {n : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∃ t, f t ≠ 0) → ∀ {c : ℝ} {S : Finset ℝ}, ↑S ⊆ Set.Ico c (c + 2 * Real.pi) → (∀ t ∈ S, f t = 0) → ∀ {t₀ : ℝ}, t₀ ∈ Set.Ico c (c + 2 * Real.pi) → t₀ ∉ S → f t₀ = 0 → HasDerivAt f 0 t₀ → S.card + 2 ≤ 2 * n","l":296,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.card_zeros_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"Zero-count lemma for real trigonometric polynomials (the load-bearing input to Szegő's inequality; not in Mathlib). If f is a real trigonometric polynomial of degree at most n that is not identically zero, then, for a finite set S of zeros of f contained in a half-open period [c, c + 2π), the cardinality of S is at most 2 n.","s":"∀ {n : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∃ t, f t ≠ 0) → ∀ {c : ℝ} {S : Finset ℝ}, ↑S ⊆ Set.Ico c (c + 2 * Real.pi) → (∀ t ∈ S, f t = 0) → S.card ≤ 2 * n","l":245,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.congr","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A function pointwise equal to a real trigonometric polynomial of degree at most n is itself a real trigonometric polynomial of degree at most n.","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → (∀ (t : ℝ), g t = f t) → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g","l":100,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.const_mul","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"Multiplying a real trigonometric polynomial of degree at most n by a scalar preserves the degree bound.","s":"∀ {n : ℕ} {f : ℝ → ℝ} (c : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => c * f t","l":85,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.mono","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A degree-≤n trig polynomial is also a degree-≤m trig polynomial when n ≤ m (pad the higher coefficients with zeros).","s":"∀ {n m : ℕ} {f : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → n ≤ m → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE m f","l":62,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE.sub","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The difference of two degree-≤n trig polynomials is a degree-≤n trig polynomial (subtract the coefficient sequences termwise).","s":"∀ {n : ℕ} {f g : ℝ → ℝ}, Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n f → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n g → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => f t - g t","l":92,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.cosComp_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a real polynomial R, if its degree is at most β, then the even trigonometric transform t ↦ R(cos t) is a real trigonometric polynomial of degree at most β.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE β fun t => Polynomial.eval (Real.cos t) R","l":121,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.cos_nat_mul_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The function t ↦ cos(n t) is a real trigonometric polynomial of degree at most n.","s":"∀ (n : ℕ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun t => Real.cos (↑n * t)","l":107,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.deriv_cosComp","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Basic","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For any real polynomial R and any point t, the derivative of s ↦ R(cos s) at t equals R'(cos t) · (-sin t).","s":"∀ (R : Polynomial ℝ) (t : ℝ), deriv (fun s => Polynomial.eval (Real.cos s) R) t = Polynomial.eval (Real.cos t) (Polynomial.derivative R) * -Real.sin t","l":48,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.exp_mul_I_injOn_Ico","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"On any half-open real interval spanning one full period, the complex exponential evaluated at an imaginary real input is injective.","s":"∀ {c s t : ℝ}, s ∈ Set.Ico c (c + 2 * Real.pi) → t ∈ Set.Ico c (c + 2 * Real.pi) → Complex.exp (↑s * Complex.I) = Complex.exp (↑t * Complex.I) → s = t","l":216,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.hasDerivAt_cosComp","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Basic","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For any real polynomial R and any point t, the map s ↦ R(cos s) has derivative R'(cos t) · (-sin t) at t.","s":"∀ (R : Polynomial ℝ) (t : ℝ), HasDerivAt (fun s => Polynomial.eval (Real.cos s) R) (Polynomial.eval (Real.cos t) (Polynomial.derivative R) * -Real.sin t) t","l":35,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.isTrigPolyLE_const","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.TrigPoly","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"Constant functions are trigonometric polynomials of degree ≤ n (they use only the k = 0 term).","s":"∀ (n : ℕ) (c : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE n fun x => c","l":55,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/TrigPoly.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp","k":"def","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a nonnegative integer frequency, a prescribed value, a prescribed derivative, a real base point, and a real argument, the Szegő comparison interpolant is $Q₀\\cos(β(t-t₀))+(Q₁/β)\\sin(β(t-t₀))$.","s":"ℕ → ℝ → ℝ → ℝ → ℝ → ℝ","l":40,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_abs_le","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and values Q₀, Q₁, base point t₀, and evaluation point t, the Szegő interpolant's value is bounded in absolute value by its amplitude A = √(Q₀² + (Q₁/β)²).","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ t : ℝ), |Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t| ≤ √(Q₀ ^ 2 + (Q₁ / ↑β) ^ 2)","l":142,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_amplitude","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant is an elementary cosine wave of amplitude A = √(Q₀² + (Q₁/β)²): there is a phase φ with S(t) = A · cos(β(t − t₀) − φ) for every t.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), ∃ φ, ∀ (t : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t = √(Q₀ ^ 2 + (Q₁ / ↑β) ^ 2) * Real.cos (↑β * (t - t₀) - φ)","l":131,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_amplitude_core","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"A cosine-sine wave is an elementary wave of amplitude A = √(Q₀² + q²): for every real frequency ω, there is a phase φ with Q₀ cos(ω(t − t₀)) + q sin(ω(t − t₀)) = A cos(ω(t − t₀) − φ) for all t.","s":"∀ (ω Q₀ q t₀ : ℝ), ∃ φ, ∀ (t : ℝ), Q₀ * Real.cos (ω * (t - t₀)) + q * Real.sin (ω * (t - t₀)) = √(Q₀ ^ 2 + q ^ 2) * Real.cos (ω * (t - t₀) - φ)","l":97,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_continuous","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":2,"d":"The Szegő interpolant is continuous as a function of its argument.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Continuous (Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀)","l":188,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"The Szegő comparison interpolant S built at base point t₀ from a prescribed value Q₀ and derivative Q₁ reproduces the prescribed derivative there: provided the degree parameter β is at least 1, the derivative of S at t₀ equals Q₁.","s":"∀ (β : ℕ), 1 ≤ β → ∀ (Q₀ Q₁ t₀ : ℝ), HasDerivAt (fun t => Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t) Q₁ t₀","l":59,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant is a real trigonometric polynomial of degree at most β.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE β (Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀)","l":44,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp_self","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Interp","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"For a degree parameter β and [values Q₀, Q₁ and base point t₀] (hyp:Q₀,Q₁,t₀), the Szegő interpolant reproduces its prescribed value at the base point: S(t₀) = Q₀.","s":"∀ (β : ℕ) (Q₀ Q₁ t₀ : ℝ), Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szegoInterp β Q₀ Q₁ t₀ t₀ = Q₀","l":52,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Interp.lean"},{"n":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.szego_deriv_sq_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.BernsteinSzegoTrig.Szego","a":"Mathlib/Analysis/BernsteinSzegoTrig","t":1,"d":"The sharp Bernstein/Szegő differential inequality (the deliverable; not available in Mathlib). For a real polynomial R, if its degree is at most β and the even trigonometric transform t ↦ R(cos t) is bounded in absolute value by M at every t, then at every t the squared derivative of that transform plus β² times its squared value is at most β² · M², equivalently |d/dt R(cos t)| ≤ β · √(M² − R(cos t)²).","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ (M : ℝ), (∀ (t : ℝ), |Polynomial.eval (Real.cos t) R| ≤ M) → ∀ (t : ℝ), deriv (fun s => Polynomial.eval (Real.cos s) R) t ^ 2 + ↑β ^ 2 * Polynomial.eval (Real.cos t) R ^ 2 ≤ ↑β ^ 2 * M ^ 2","l":244,"f":"Causalean/Mathlib/Analysis/BernsteinSzegoTrig/Szego.lean"},{"n":"Causalean.Mathlib.Analysis.C1CloseOn","k":"def","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"Two real functions are uniformly C¹-close on a set when both values and first within-derivatives differ by less than the same radius everywhere on the set.","s":"Set ℝ → ℝ → (ℝ → ℝ) → (ℝ → ℝ) → Prop","l":32,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.C2CloseOn","k":"def","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"Two real functions are uniformly C²-close when they are uniformly C¹-close and their second within-derivatives differ by less than the same radius.","s":"Set ℝ → ℝ → (ℝ → ℝ) → (ℝ → ℝ) → Prop","l":38,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedIntervalArithmetic","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.API","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"This interface packages sound rational interval arithmetic and total certified-real refinement behind a paper-independent interface.","s":"Type","l":14,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/API.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A certified real name represents a real number by a sequence of rational interval enclosures indexed by precision that is nested — each successive enclosure a subinterval of the one before — and always contains the represented value, together with a rule (an arbitrary function, not required to be computable) selecting, for any requested positive rational error, a precision level whose enclosure is no wider than that error.","s":"Type","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two certified real numbers, their certified sum denotes the sum of their named real numbers, adds their rational interval enclosures at each common precision, and selects a precision sufficient to make each input enclosure no wider than half the requested tolerance.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":106,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.approx_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Nestedness extends from successive precisions to every pair of ordered precisions.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) {m n : ℕ}, m ≤ n → (x.approx n).Subinterval (x.approx m)","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.exists_refinement","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every positive rational tolerance admits an explicitly returned enclosing interval of at most that width.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : ℚ), 0 < ε → ∃ n, (x.approx n).Contains x.value ∧ (x.approx n).width ≤ ε","l":66,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a certified real number, its certified negation denotes the negative of the real number named by the certificate, negates every rational interval enclosure, and retains the same precision-selection rule.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":89,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.ofRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number, its certified real name denotes that rational number and uses its degenerate one-point rational interval at every precision.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":72,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a certified real number and a requested positive rational tolerance, its refinement is the rational interval returned at the precision selected by that certificate's modulus for the requested tolerance.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":48,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Refinement always encloses the real value named by the certificate.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (x.refine ε).Contains x.value","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.refine_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a certified real number x and a requested positive rational tolerance ε, refining x to the precision selected by ε's modulus yields an enclosure whose width is at most ε.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (x.refine ε).width ≤ ↑ε","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two certified real numbers, their certified difference is their certified sum after negating the second certificate; it therefore denotes the first named real number minus the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":135,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.sub_value","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Certified subtraction denotes the difference of the two named real values.","s":"∀ (x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal), (x.sub y).value = x.value - y.value","l":140,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal.value_eq_of_common_approximations","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Two certified names whose every approximation is shared denote the same real value.","s":"∀ (x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal), (∀ (n : ℕ), x.approx n = y.approx n) → x.value = y.value","l":145,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a complex-valued function, a complex center, and a real radius, the circle contour integral is the integral from $0$ to $1$ of the parameterized contour integrand. No integrability is required: for an integrand that is not interval integrable the integral is zero by convention, so this is the contour integral only when the parameterized integrand is integrable.","s":"(ℂ → ℂ) → ℂ → ℝ → ℂ","l":92,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral_enclosed","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Applying the complex mesh theorem to the parameterized circle integrand encloses the contour integral.","s":"∀ {f : ℂ → ℂ} {c : ℂ} {r : ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r s - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r t‖ ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand f c r (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleContourIntegral f c r)","l":293,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleIntegrand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a complex-valued function, a complex center, a real radius, and a real parameter, the parameterized contour integrand is the value of the function at the corresponding circle point multiplied by the circle tangent there.","s":"(ℂ → ℂ) → ℂ → ℝ → ℝ → ℂ","l":88,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleMap","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a complex center, a real radius, and a real parameter, the circle point is $c+r\\exp(2\\pi u i)$.","s":"ℂ → ℝ → ℝ → ℂ","l":80,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.circleTangent","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a real radius and a real parameter, the circle tangent is $2\\pi i r\\exp(2\\pi u i)$.","s":"ℝ → ℝ → ℂ","l":84,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.continuousOn_of_lipschitz_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A function satisfying a finite Lipschitz bound on the unit interval is continuous there.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] {g : ℝ → E} {C : ℝ}, 0 ≤ C → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ C * |s - t|) → ContinuousOn g (Set.Icc 0 1)","l":250,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational node intervals, a rational Lipschitz constant, the condition that the constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the infimum enclosure has lower endpoint equal to the minimum node lower endpoint minus $L/n$ and upper endpoint equal to the minimum node upper endpoint.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":274,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified infimum enclosure on the unit mesh. Fix a nonnegative Lipschitz constant and a positive number of mesh nodes, and suppose the real function is Lipschitz on [0, 1] with that constant and each rational node interval contains the function's value at the corresponding mesh point. Then the infimum enclosure built from those node intervals contains the true infimum of the function over [0, 1].","s":"∀ {f : ℝ → ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, |f s - f t| ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (f (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure nodes L hL n hn).Contains (sInf (f '' Set.Icc 0 1))","l":288,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of complex rational node rectangles, a rational Lipschitz constant, the condition that this constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the integral enclosure widens the trapezoidal enclosure by $L/(2n)$ in both coordinates.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":28,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified trapezoidal enclosure of an integral. Fix a nonnegative Lipschitz constant and a positive node count, and suppose the complex-valued integrand is Lipschitz on [0, 1] with that constant and each complex rational rectangle contains the integrand's value at the corresponding mesh point. Then the trapezoidal enclosure built from those rectangles contains the true integral of the integrand over [0, 1].","s":"∀ {g : ℝ → ℂ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).Contains (∫ (u : ℝ) in 0..1, g u)","l":236,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Each coordinate width of the certified integral is bounded by the uniform node width plus one mesh-scale Lipschitz allowance.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).re.width ≤ w ∧ (nodes k).im.width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).re.width ≤ w + L / ↑n ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure nodes L hL n hn).im.width ≤ w + L / ↑n","l":261,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.maxHiUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational intervals, the maximum upper endpoint up to a terminal node index is computed recursively: at index zero it is the upper endpoint of the initial interval, and at each successor index it is the larger of the preceding result and the new upper endpoint.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":117,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.maxLoUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational intervals, the maximum lower endpoint up to a terminal node index is computed recursively: at index zero it is the lower endpoint of the initial interval, and at each successor index it is the larger of the preceding result and the new lower endpoint.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":110,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a mesh size and a node index, the mesh point is the real number $k/n$. It is a mesh point for positive mesh size; for $n=0$ every point is zero by the division convention.","s":"ℕ → ℕ → ℝ","l":77,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A node of a nonempty uniform mesh lies in the unit parameter interval.","s":"∀ {n k : ℕ}, 0 < n → k ≤ n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k ∈ Set.Icc 0 1","l":225,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.minHiUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational intervals, the minimum upper endpoint up to a terminal node index is computed recursively: at index zero it is the upper endpoint of the initial interval, and at each successor index it is the smaller of the preceding result and the new upper endpoint.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":103,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.minLoUpTo","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational intervals, the minimum lower endpoint up to a terminal node index is computed recursively: at index zero it is the lower endpoint of the initial interval, and at each successor index it is the smaller of the preceding result and the new lower endpoint.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → ℚ","l":96,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.norm_integral_sub_trapezoid_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A Lipschitz complex function on the unit parameter interval differs from its deterministic trapezoidal rule by at most half a Lipschitz mesh unit.","s":"∀ {g : ℝ → ℂ} {L : ℚ}, 0 ≤ L → ∀ {n : ℕ}, 0 < n → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖g s - g t‖ ≤ ↑L * |s - t|) → ‖(∫ (u : ℝ) in 0..1, g u) - 1 / ↑(2 * ↑n) * ∑ k ∈ Finset.range n, (g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k) + g (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n (k + 1)))‖ ≤ ↑L / (2 * ↑n)","l":75,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of rational node intervals, a rational Lipschitz constant, the condition that the constant is nonnegative, a mesh size, and the condition that the mesh size is positive, the supremum enclosure has lower endpoint equal to the maximum node lower endpoint and upper endpoint equal to the maximum node upper endpoint plus $L/n$.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → (L : ℚ) → 0 ≤ L → (n : ℕ) → 0 < n → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":281,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational node enclosures and a Lipschitz bound enclose the supremum of a real function on the unit mesh interval.","s":"∀ {f : ℝ → ℝ} {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {L : ℚ} (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, |f s - f t| ≤ ↑L * |s - t|) → (∀ k ≤ n, (nodes k).Contains (f (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint n k))) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure nodes L hL n hn).Contains (sSup (f '' Set.Icc 0 1))","l":333,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.trapezoidEnclosure","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a sequence of complex rational rectangles and a mesh size, the trapezoidal enclosure is the unscaled trapezoidal rectangle sum multiplied by $1/(2n)$. It is meaningful for positive mesh size, as required by the integral enclosure built from it; at mesh size zero the scale factor is zero by the division convention.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":24,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.trapezoidSum","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Quadrature","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a sequence of complex rational rectangles, the unscaled trapezoidal rectangle sum at a nonnegative index is computed recursively: at zero it is the zero rectangle, and at each successor it adds the two rectangles at the new cell's endpoints to the preceding sum.","s":"(ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Quadrature.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.width_infEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The infimum enclosure width is at most one mesh error plus the uniform node-enclosure width.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure nodes L hL n hn).width ≤ w + L / ↑n","l":370,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.width_supEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The supremum enclosure width is at most one mesh error plus the uniform node-enclosure width.","s":"∀ {nodes : ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {w L : ℚ}, 0 ≤ w → ∀ (hL : 0 ≤ L) {n : ℕ} (hn : 0 < n), (∀ k ≤ n, (nodes k).width ≤ w) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure nodes L hL n hn).width ≤ w + L / ↑n","l":383,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified complex name represents a complex number by a sequence of rational rectangle enclosures indexed by fuel that is nested — each successive enclosure a subrectangle of the one before — and always contains the represented value, together with a computable rule selecting, for any requested positive rational error, a fuel level whose enclosure meets that coordinate-width tolerance.","s":"Type","l":13,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For two certified complex names, their certified sum denotes the sum of their exact complex values and uses the coordinatewise sum of their equal-fuel rectangles.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":69,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.norm","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex name, its certified modulus name denotes the absolute value of its exact complex value and uses the recursive modulus enclosures with the stated precision rule.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":304,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex name, the modulus approximation at each nonnegative fuel level is computed recursively: at zero it is the initial rectangle's modulus enclosure, and at each successor it is the intersection of the preceding approximation with the new modulus enclosure.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":101,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive modulus approximations contain the norm and are adjacent-fuel nested.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (fuel : ℕ), (z.normApprox fuel).Contains ‖z.value‖ ∧ (z.normApprox (fuel + 1)).Subinterval (z.normApprox fuel)","l":117,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.normPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex name and a requested positive rational tolerance, the modulus precision is the maximum of a fuel level that refines the input rectangle to a tolerance determined by its initial magnitude bound and an explicit Newton-iteration fuel bound.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.norm_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex input and a requested positive rational tolerance ε, evaluating the certified-modulus approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (z.normApprox (z.normPrecision ε)).width ≤ ↑ε","l":161,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.ofRatPair","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational real coordinate and a rational imaginary coordinate, the certified complex name of their complex point has that point as its exact value and the corresponding singleton rectangle at every fuel level.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":55,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.refine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex name and a requested positive rational tolerance, the refined rational rectangle is the rectangle selected by that name's tolerance-to-fuel rule.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex.refine_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Names","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Effective refinement preserves containment and meets its requested width.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (z.refine ε).Contains z.value ∧ (z.refine ε).width ≤ ↑ε","l":50,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Names.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified complex map packages an exact complex function with an executable interval-evaluation program over rational input rectangles and a fuel level whose primitive-operation count is recorded, proved sound — every executable result contains the exact value at every enclosed input. Increasing fuel refines the enclosure to a subrectangle of the previous one, and refining the input rectangle cannot enlarge the output enclosure. The map carries [a nonnegative Lipschitz bound on how input width amplifies into output width](hyp:amplification, amplification_nonneg) and a nonincreasing sequence of algorithmic-error bounds in the fuel, such that the evaluated width never exceeds the algorithmic error plus the amplified input width, together with a computable fuel rule meeting any requested positive algorithmic-error target.","s":"Type","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For two certified complex maps, their certified pointwise sum denotes the sum of their exact values, evaluates by adding their output rectangles at a common fuel level, and adds their operation counts, width amplifications, and algorithmic errors.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":105,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.algorithmError_le_of_modulus","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Fuel sufficiency bound. For a certified complex-map evaluation algorithm and a requested positive rational error tolerance, if the number of iterations supplied is at least the algorithm's certified error modulus at that tolerance, then the algorithm's declared error allowance at that number of iterations, the additive term in its certified output-width bound, is at most the requested tolerance.","s":"∀ {f : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap} {ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} {fuel : ℕ}, f.errorModulus ε ≤ fuel → f.algorithmError fuel ≤ ↑ε","l":335,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.constant","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational real coordinate and a rational imaginary coordinate, the constant certified complex map denotes the corresponding complex constant and returns its singleton rational rectangle for every input rectangle and fuel level.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":81,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.identity","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The identity certified complex map denotes every complex input itself, returns the input rectangle at every fuel level, has no primitive operations or algorithmic error, and has unit input-width amplification.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":62,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.mulWithBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given two certified complex maps, rational magnitude bounds for their executable output rectangles, the condition that both bounds are nonnegative, the condition that every output rectangle of the first map has maximum coordinate magnitude at most its bound, and the analogous condition for the second map, the certified pointwise product denotes the product of their exact values and uses the product-width propagation formula for its error and width amplification.","s":"(f g : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap) → (Bf Bg : ℚ) → 0 ≤ Bf → 0 ≤ Bg → (∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (f.eval I fuel).maxAbs ≤ Bf) → (∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (g.eval I fuel).maxAbs ≤ Bg) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":213,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Program","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For two certified complex maps, their certified pointwise difference denotes the difference of their exact values, evaluates by subtracting their output rectangles at a common fuel level, and adds their operation counts, width amplifications, and algorithmic errors.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplexMap","l":158,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Program.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A certified program schedule packages an exact schedule for evaluating a given contour program to a stated separation bound, together with the guarantees that its operation count matches the program's structural operation count, a common positive error target selected canonically from the tolerance, value bounds, and separation, an input precision equal to the uniform circle precision for that target, and that its fuel dominates the circle-exponential fuel requirement, the numerator's algorithmic-error modulus, and the denominator's algorithmic-error modulus, while the compositional arithmetic propagation bound fits within the schedule's node budget.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Type","l":342,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A finite rational contour program consists of a certified interval extension of the numerator function and of the denominator function, together with a rational circle radius that is strictly positive.","s":"Type","l":22,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.canonicalNodeTarget","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, uniform bounds on its numerator and denominator values, a positive rational separation bound, and a positive rational tolerance, the canonical node target is the smaller of one and the tolerance divided by three times the node-error scale.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.canonicalScheduled","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a contour program, uniform bounds on its numerator and denominator values, a positive rational separation bound, a positive rational tolerance, a rational magnitude bound, and the condition that this magnitude bound is nonnegative, the canonical certified schedule derives the operation count, input precision, fuel, mesh, and error budgets from these quantities.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) → (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → (magnitude : ℚ) → 0 ≤ magnitude → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation","l":401,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational real numerator coordinate, a rational imaginary numerator coordinate, a rational circle radius, and the condition that the radius is strictly positive, the constant-over-unit contour program has the indicated constant numerator, unit denominator, and radius.","s":"ℚ → ℚ → (radius : ℚ) → 0 < radius → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","l":39,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnitBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational real numerator coordinate, a rational imaginary numerator coordinate, a rational circle radius, and the condition that the radius is strictly positive, the value bounds for the constant-over-unit contour program bound its numerator coordinates by the larger absolute numerator coordinate and its denominator coordinates by one.","s":"(numeratorRe numeratorIm radius : ℚ) → (hradius : 0 < radius) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit numeratorRe numeratorIm radius hradius)","l":306,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnitCertificate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational real numerator coordinate, a rational imaginary numerator coordinate, a rational circle radius, the condition that the radius is strictly positive, and a schedule, the constant-over-unit denominator certificate certifies separation one and a squared-modulus lower bound of one at every scheduled endpoint.","s":"(numeratorRe numeratorIm radius : ℚ) → (hradius : 0 < radius) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.constantUnit numeratorRe numeratorIm radius hradius) schedule","l":452,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.evaluate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a schedule, and a certificate that its denominator is separated from zero at the scheduled endpoints, the evaluated contour rectangle is the deterministic integral enclosure of its integrand-node rectangles, widened symmetrically by half the schedule's quadrature budget.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":290,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.integrandNodeFin","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a schedule, a certificate that its denominator is separated from zero at the scheduled endpoints, and an endpoint index among the mesh plus one endpoints, the finite-index integrand node rectangle is the guarded quotient of the evaluated numerator and denominator rectangles multiplied by the node rectangle.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → Fin (schedule.mesh + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":471,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.integrandNodes","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a schedule, a certificate that its denominator is separated from zero at the scheduled endpoints, and a nonnegative index, the total integrand-node rectangle is the guarded evaluated quotient times the node rectangle when the index is at most the mesh size, and otherwise is the terminal finite-index node rectangle.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":480,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.mapWidthBounds","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program and a rational common error target, the pair of map-width bounds consists of the target plus the target multiplied by the numerator map's width amplification, and the analogous quantity for the denominator map. These are width bounds for a nonnegative target, which is how the evaluation uses them (with a positive rational target); a negative target gives negative numbers.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℚ → ℚ × ℚ","l":81,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeBox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a schedule, and a nonnegative node index, the node rectangle is the scheduled rational rectangle for the circle point at that index.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":47,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodePropagationBound","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, uniform bounds on its numerator and denominator values, a rational separation bound, and a rational common error target, the node-error propagation bound is obtained by forming the two map-width bounds, adding the numerator bound to its width bound, adding the denominator bound to its width bound, forming a bound for the division numerator, forming its width bound, forming the denominator squared-modulus width bound, forming the quotient width bound, forming the quotient magnitude bound, and combining quotient and circle-point errors. The quotient terms divide by the separation, so this is a valid propagation bound only for positive separation, as assumed by the soundness results; at separation zero those terms are zero by the division convention.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → ℚ → ℚ → ℚ","l":86,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeScale","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, uniform bounds on its numerator and denominator values, and a positive rational separation bound, the node-error scale is one plus the absolute value of the node-error propagation bound evaluated at target one.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℚ","l":102,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeTrace","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, its value bounds, a positive separation target, a certified schedule, and a nonnegative node index, the node trace lists one scheduled trace event for each structural operation of the program.","s":"(program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) → {bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program} → {separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation → ℕ → List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","l":374,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.nodeTrace_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a target error separation, a certified schedule for that program, and a stage index k, the executable node trace at stage k has exactly one event per structural operation of the program, and every event in it records the full schedule together with that schedule's operation count.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) {bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program} {separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat} (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (k : ℕ), (program.nodeTrace scheduled k).length = program.operationCount ∧ ∀ event ∈ program.nodeTrace scheduled k, event.schedule = scheduled.schedule ∧ event.schedule.operationCount = program.operationCount","l":382,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.normalizedContourIntegral","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, the normalized contour integral is its circle contour integral for the exact numerator-to-denominator quotient, divided by $2\\pi i$. It is the contour integral of the quotient when the denominator does not vanish on the circle (as guaranteed by the separation hypotheses of the evaluation theorems); otherwise the quotient and the integral are Lean's totalized values, which can be zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℂ","l":499,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.normalizedIntegrand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, a schedule, and a real circle parameter, the normalized integrand is the quotient of the exact numerator and denominator at the corresponding circle point, multiplied by that point.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℝ → ℂ","l":492,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram.operationCount","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a contour program, the structural operation count of one contour node is the sum of the operation counts of its numerator and denominator maps plus two operations for guarded division and multiplication by the circle point.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → ℕ","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Uniform semantic magnitude data for one contour program: a nonnegative bound that covers every real and imaginary coordinate of the exact numerator function everywhere on the circle, and a nonnegative bound that covers the denominator function likewise.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Type","l":52,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A denominator certificate for a contour program under a fixed schedule provides a common squared-modulus separation bound that is strictly positive, together with the guarantees that every mesh endpoint's denominator interval evaluation is executable and bounded away from zero and that the same separation bound lies below every such endpoint's lower squared-modulus bound.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → Type","l":325,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourProgram.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A contour schedule packages a requested rational error tolerance, an operation count, an input precision, a mesh size that is nonempty, a fuel budget for transcendental evaluation, and a nonnegative magnitude-amplification bound, together with three nonnegative error budgets — [for node evaluation](hyp:nodeBudget, nodeBudget_nonneg), for the uniform mesh, and for the final quadrature widening — such that the three budgets together fit within the tolerance and the mesh budget absorbs the magnitude-amplified discretization error.","s":"Type","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule.canonical","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a positive rational tolerance, a natural-number operation count, a rational magnitude bound, and evidence that this bound is nonnegative, the canonical contour schedule allocates one third of the tolerance to each error budget and selects its input precision, mesh size, and fuel from the stated rational data.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ → (magnitude : ℚ) → 0 ≤ magnitude → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule","l":59,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"A trace event records the shared schedule, the index of the evaluated endpoint, and the ordinal of the primitive operation performed. Retaining the schedule object itself means execution and semantic audit cannot silently disagree about fuel, mesh, or precision.","s":"Type","l":123,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.abs_sub_intervalMid_le_radius","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every real point enclosed by a rational interval is at most half its width away from the interval's rational midpoint.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → |x - ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I)| ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius I)","l":484,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the arctangent remainder scale is $|q|^{2f+3}/(2f+3)$.","s":"ℚ → ℕ → ℚ","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the arctangent Taylor partial sum is the alternating rational series through degree $2f+1$.","s":"ℚ → ℕ → ℚ","l":30,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.atanRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the raw arctangent interval has endpoints equal to the Taylor partial sum minus and plus the remainder scale, respectively.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":41,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational complex input rectangle and a natural-number fuel level, the complex-exponential rectangle at that level is the raw complex-exponential rectangle at level zero, and at every successor level is the intersection of the preceding rectangle and the new raw complex-exponential rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":26,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpInnerTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a natural-number stage index, the inner complex-exponential tolerance is the stage tolerance divided by sixteen times the complex-exponential magnitude scale.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpInputTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a natural-number stage index, the complex-exponential input tolerance is the inner tolerance divided by four times the complex-exponential magnitude scale.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":134,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpMagnitude","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number, its complex-exponential magnitude scale is two plus the absolute upper endpoint of the level-zero scalar-exponential interval evaluated at the maximum absolute real-coordinate endpoint of its level-zero enclosure.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℚ","l":118,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number, the certified complex-exponential name has exact value the complex exponential of the input value, uses the recursively refined complex-exponential rectangles as approximations, and uses the stated precision rule for requested positive rational widths.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex","l":562,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex number and a natural-number stage index, the complex-exponential name approximation at that level is the stage-zero rectangle, and at every successor level is the intersection of the preceding name approximation and the new stage rectangle.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":164,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive complex-exponential name approximations remain sound and adjacent stages are nested by their defining finite intersection.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z n).Contains (Complex.exp z.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z n)","l":173,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a positive rational target width, the complex-exponential precision is the denominator of that target.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":536,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex input rectangle and a natural-number fuel level, the raw complex-exponential rectangle multiplies the real-exponential interval of the real coordinate by the cosine and sine intervals of the imaginary coordinate to form its real and imaginary coordinates.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a natural-number stage index, the complex-exponential stage rectangle evaluates the complex exponential on that stage's input rectangle using that stage's fuel.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":160,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageFuel","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a natural-number stage index, the complex-exponential stage fuel is the stage input rectangle, the inner tolerance divided by eight, the resulting trigonometric Taylor-fuel bound, and the larger of that bound and the two scalar-exponential precision bounds.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → ℕ","l":145,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageInput","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified complex number and a natural-number stage index, the complex-exponential stage input rectangle is the input's approximation at the precision required by that stage's input tolerance.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":141,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a natural-number stage index, the complex-exponential stage tolerance is the positive rational number $1/(n+1)$.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":123,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every scheduled stage has width at most its reciprocal stage tolerance; the proof combines input refinement, scalar Taylor convergence, magnitude amplification, and explicit rectangle-product propagation.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStage z n).width ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpStageTolerance n)","l":268,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent complex exponential outputs for one fixed input rectangle are nested by coordinatewise finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel)","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Compositional complex exponential evaluation encloses the exact complex exponential.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel).Contains (Complex.exp z)","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex exponential width is bounded explicitly by the widths and endpoint magnitudes of its certified exponential and trigonometric factors.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp I fuel).width ≤ 2 * (max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I.re fuel).maxAbs 0 * max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I.im fuel).width (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I.im fuel).width + max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I.im fuel).maxAbs (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I.im fuel).maxAbs * max (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I.re fuel).width 0)","l":67,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ComplexExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified complex input and a requested positive rational tolerance ε, evaluating the certified complex-exponential approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (z : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedComplex) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpNameApprox z (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExpPrecision z ε)).width ≤ ↑ε","l":539,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ComplexExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the cosine remainder scale is $|q|^{2f+2}/(2f+2)!$.","s":"ℚ → ℕ → ℚ","l":229,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational input interval and a natural-number fuel level, the cosine image interval at that level is the raw cosine image interval at level zero, and at every successor level is the intersection of the preceding cosine image interval and the new raw cosine image interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":280,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational input interval and a natural-number fuel level, the raw cosine image interval expands the midpoint cosine interval by the input interval's radius.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":268,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent cosine outputs are nested by recursive finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel)","l":541,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Cosine interval evaluation encloses cosine throughout the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel).Contains (Real.cos x)","l":524,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Cosine output width is the shrinking input diameter plus its explicit Taylor error.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosInterval I fuel).width ≤ I.width + 2 * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosError (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel","l":570,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified real number, the certified cosine name has exact value the cosine of the input value, uses the recursively refined cosine intervals as approximations, and uses the trigonometric precision rule for requested positive rational widths.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":257,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified real number and a natural-number fuel level, the cosine name approximation at that level is the cosine interval of the level-zero input enclosure, and at every successor level is the intersection of the preceding cosine approximation and the cosine interval of the corresponding input enclosure.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive cosine-name approximations remain sound and adjacent-fuel nested.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x fuel).Contains (Real.cos x.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x fuel)","l":53,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosName_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Refinement of the input name plus Taylor fuel makes cosine output width effective.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosNameApprox x (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision x ε)).width ≤ ↑ε","l":195,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the cosine Taylor partial sum is the alternating rational cosine series through degree $2f$.","s":"ℚ → ℕ → ℚ","l":224,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the raw cosine interval has endpoints equal to the cosine Taylor partial sum minus and plus the cosine remainder scale, respectively.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":242,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Raw rational cosine bounds contain the exact cosine at every rational center.","s":"∀ (q : ℚ) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.cosRaw q fuel).Contains (Real.cos ↑q)","l":470,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational input interval and a natural-number fuel level, the exponential image interval at that level is the raw exponential image interval at level zero, and at every successor level is the intersection of the preceding exponential image interval and the new raw exponential image interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":33,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational input interval and a natural-number fuel level, the raw exponential image interval is the input midpoint, the scalar exponential interval at that midpoint, the upper endpoint of the scalar exponential interval at the input's maximum absolute endpoint magnitude, and the midpoint exponential interval expanded by that upper bound times the input radius.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent midpoint exponential outputs are nested by finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel)","l":96,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The magnitude-dependent exponential interval contains the exponential of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel).Contains (Real.exp x)","l":88,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.IntervalExp","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational interval I and a Taylor fuel/iteration count, the width of the certified interval exponential is bounded by the scalar Taylor truncation error at the interval's midpoint plus twice the interval's radius amplified by a certified bound on the exponential's magnitude over the interval.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.expInterval I fuel).width ≤ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel).width + 2 * (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar I.maxAbs fuel).hi * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius I","l":106,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/IntervalExp.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational interval, its rational midpoint is the arithmetic mean of its lower and upper endpoints.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":251,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalRadius","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational interval, its rational radius is half its width.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":256,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a natural-number fuel level, the π interval at that level is the raw π interval at level zero, and at every successor level is the intersection of the preceding π interval and the new raw π interval.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":58,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent π fuel values are nested by construction through finite intersection.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval fuel)","l":134,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every recursively intersected π rectangle contains π.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval fuel).Contains Real.pi","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a requested positive rational tolerance ε, evaluating the certified π-enclosure at the precision level selected for that tolerance yields an enclosure whose width is no larger than ε.","s":"∀ (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piInterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piPrecision ε)).width ≤ ↑ε","l":163,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"The certified real name for π has exact value π, uses the recursively refined π intervals as approximations, and uses the specified π precision to achieve each requested positive rational width.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":206,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a positive rational target width, the π precision is eight times one plus the denominator of that target.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":63,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a natural-number fuel level, the raw π interval is four times the difference between four times the arctangent interval at $1/5$ and the arctangent interval at $1/239$.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":50,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every raw Machin rectangle contains the mathematical constant π.","s":"∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.piRaw fuel).Contains Real.pi","l":112,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.power_div_factorial_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the Taylor fuel dominates a rational magnitude and denominator scale, the corresponding rational power-over-factorial remainder is explicitly small.","s":"∀ (q : ℚ) (A d N r : ℕ), |q| ≤ ↑A → 1 ≤ A → 1 ≤ d → 1 ≤ r → 32 * d * A ^ 2 ≤ N → |q| ^ (2 * N + r) / ↑(2 * N + r).factorial ≤ 1 / ↑(4 * d)","l":98,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinError","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the sine remainder scale is $|q|^{2f+3}/(2f+3)!$.","s":"ℚ → ℕ → ℚ","l":220,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational input interval and a natural-number fuel level, the sine image interval at that level is the raw sine image interval at level zero, and at every successor level is the intersection of the preceding sine image interval and the new raw sine image interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":275,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinIntervalRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational input interval and a natural-number fuel level, the raw sine image interval expands the midpoint sine interval by the input interval's radius.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":261,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Adjacent sine outputs are nested by recursive finite intersection.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel)","l":531,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Sine interval evaluation encloses sine throughout the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel).Contains (Real.sin x)","l":517,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Sine output width is the shrinking input diameter plus its explicit Taylor error.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinInterval I fuel).width ≤ I.width + 2 * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinError (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.intervalMid I) fuel","l":551,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified real number, the certified sine name has exact value the sine of the input value, uses the recursively refined sine intervals as approximations, and uses the trigonometric precision rule for requested positive rational widths.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":248,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified real number and a natural-number fuel level, the sine name approximation at that level is the sine interval of the level-zero input enclosure, and at every successor level is the intersection of the preceding sine approximation and the sine interval of the corresponding input enclosure.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Recursive sine-name approximations remain sound and adjacent-fuel nested.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x fuel).Contains (Real.sin x.value) ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x (fuel + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x fuel)","l":33,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinName_width_at_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a certified real input and a requested positive rational tolerance ε, evaluating the certified sine name-approximation at the precision level selected for that tolerance yields an output enclosure whose width is no larger than ε.","s":"∀ (x : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinNameApprox x (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision x ε)).width ≤ ↑ε","l":139,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the sine Taylor partial sum is the alternating rational sine series through degree $2f+1$.","s":"ℚ → ℕ → ℚ","l":215,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational argument and a natural-number fuel level, the raw sine interval has endpoints equal to the sine Taylor partial sum minus and plus the sine remainder scale, respectively.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":233,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Raw rational sine bounds contain the exact sine at every rational center.","s":"∀ (q : ℚ) (fuel : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.sinRaw q fuel).Contains (Real.sin ↑q)","l":456,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Transcendental.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.trigNamePrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TrigNames","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a certified real number and a positive rational target width, the trigonometric name precision is computed by taking half the target width as an internal input tolerance and returning the larger of the input precision the certified real requires for that tolerance and an explicit Taylor-fuel bound based on the target denominator and the initial enclosure's magnitude.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":27,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/TrigNames.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.certified_contour_evaluation","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Certified finite-precision evaluation of a contour integral. Fix a contour-integral program, value bounds for it, a target separation, a certified mesh schedule built to meet that target, and a certificate that the denominator stays bounded away from zero, whose guaranteed separation covers the target one; assume also that the program's denominator never vanishes anywhere on the parametrized circle and the normalized integrand is Lipschitz in the circle parameter with the schedule's constant. Then the rational rectangle obtained by evaluating the program both contains the true normalized contour integral and has real-part width at most the schedule's requested tolerance.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand scheduled.schedule s - program.normalizedIntegrand scheduled.schedule t‖ ≤ ↑scheduled.schedule.magnitude * |s - t|) → (program.evaluate scheduled.schedule certificate).Contains program.normalizedContourIntegral ∧ (program.evaluate scheduled.schedule certificate).re.width ≤ ↑scheduled.schedule.tolerance","l":358,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.certified_contour_evaluation_inverseMax","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Specializing the generic contour certificate to reciprocal-max tolerance gives the width required by canonical finite-rational statistics.","s":"∀ (n : ℕ) (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation), scheduled.schedule.tolerance = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.inverseMaxTolerance n → ∀ (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand scheduled.schedule s - program.normalizedIntegrand scheduled.schedule t‖ ≤ ↑scheduled.schedule.magnitude * |s - t|) → (program.evaluate scheduled.schedule certificate).re.width ≤ 1 / max (↑n) 1","l":392,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a contour schedule and a natural-number endpoint index, the rational rectangle enclosing the corresponding pure-imaginary circle angle has real coordinate zero and imaginary coordinate $2πk/m$, where $m$ is the schedule's mesh size.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":138,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The angle rectangle contains the exact pure-imaginary angle at every endpoint from zero through the terminal mesh endpoint.","s":"∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle schedule k).Contains (↑2 * ↑Real.pi * (↑↑k / ↑schedule.mesh) * Complex.I)","l":389,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleExpFuel","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational radius, a natural-number mesh size, and a positive rational target width, the circle exponential fuel is the larger of the zero-input exponential precision and an explicit trigonometric Taylor-fuel bound computed from the internal circle tolerance.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":162,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInnerTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational radius and a positive rational target width, the internal circle tolerance is the smaller of $\\mathrm{target}/(256(|r|+1))$ and $1/1024$.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":148,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInputPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational radius and a positive rational target width, the circle input precision is the π-approximation precision selected for their internal circle tolerance.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":158,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational radius, a contour schedule, and a natural-number endpoint index, the scheduled circle-node rectangle is the rational-interval evaluation of $r\\exp(2πik/m)$ using that schedule's angle enclosure and fuel.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":144,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a contour schedule, a mesh-endpoint index, and a primitive-operation index, the circle-node trace event records exactly those three quantities.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.TraceEvent","l":134,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_endpoint","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The terminal endpoint k = mesh, which is explicitly used by the trapezoidal program, is certified and denotes the same point as endpoint zero.","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule schedule.mesh).Contains ↑radius ∧ Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode radius schedule schedule.mesh = ↑radius","l":417,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_schedule_correspondence","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Reading operation count, input precision, fuel, and mesh from the trace and from execution gives definitionally the same data because both retain the shared schedule object.","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (k operation : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.operationCount = schedule.operationCount ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.inputPrecision = schedule.inputPrecision ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.fuel = schedule.fuel ∧ (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule.mesh = schedule.mesh ∧ Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNodeEvent schedule k operation).schedule k = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k","l":456,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational radius, a quadrature schedule, and a mesh index k, if k does not exceed the schedule's mesh size, then the computed rational circle-node rectangle at index k contains the exact complex circle point radius · exp(2πi · k / mesh).","s":"∀ (radius : ℚ) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).Contains (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode radius schedule k)","l":407,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Circle-node width propagation exposes radius scaling and the scheduled complex-exponential approximation width.","s":"∀ (radius : ℚ), 0 ≤ radius → ∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (k : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).width ≤ radius * (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Transcendental.complexExp (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleAngle schedule k) schedule.fuel).width","l":433,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode_width_at_selected_precision","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The explicit circle input precision and exponential fuel make every actually evaluated endpoint rectangle no wider than the requested target.","s":"∀ (radius : ℚ), 0 ≤ radius → ∀ (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (target : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), schedule.inputPrecision = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleInputPrecision radius target → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleExpFuel radius schedule.mesh target ≤ schedule.fuel → ∀ {k : ℕ}, k ≤ schedule.mesh → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.circleNode radius schedule k).width ≤ ↑target","l":220,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.denominator_ne_zero_at_nodes","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Denominator separation at all certified nodes implies nonvanishing of the exact denominator at all mesh endpoints.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule) {k : ℕ}, k ≤ schedule.mesh → program.denominator.value (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode program.radius schedule k) ≠ 0","l":299,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.evaluate_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Under a Lipschitz bound and denominator nonvanishing on the full circle, the returned rational rectangle contains the normalized exact contour integral.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule), (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → (∀ s ∈ Set.Icc 0 1, ∀ t ∈ Set.Icc 0 1, ‖program.normalizedIntegrand schedule s - program.normalizedIntegrand schedule t‖ ≤ ↑schedule.magnitude * |s - t|) → (program.evaluate schedule certificate).Contains program.normalizedContourIntegral","l":317,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.evaluate_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Uniform scheduled node widths propagate through mesh and final widening, so both result coordinates fit within the three split budgets.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → (program.evaluate scheduled.schedule certificate).width ≤ ↑scheduled.schedule.tolerance","l":333,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.exactCircleNode","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CircleSchedule","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational radius, a contour schedule, and a natural-number endpoint index, the exact circle node is the complex number $r\\exp(2πik/m)$, where $m$ is the schedule's mesh size.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule → ℕ → ℂ","l":384,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/CircleSchedule.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.integrandNodes_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Every certified finite endpoint rectangle contains the exact normalized integrand.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program schedule) {k : ℕ}, k ≤ schedule.mesh → (program.integrandNodes schedule certificate k).Contains (program.normalizedIntegrand schedule (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint schedule.mesh k))","l":53,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.integrandNodes_width_le_propagation","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The scheduled node-width bound follows compositionally from circle input precision, both map error moduli, their amplification fields, rational magnitude bounds, guarded-division separation, and primitive width lemmas.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (bounds : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourValueBounds program) (separation : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat) (scheduled : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.CertifiedProgramSchedule program bounds separation) (certificate : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.DenominatorCertificate program scheduled.schedule), ↑separation ≤ certificate.separation → ∀ {k : ℕ}, k ≤ scheduled.schedule.mesh → (program.integrandNodes scheduled.schedule certificate k).width ≤ program.nodePropagationBound bounds ↑separation ↑scheduled.target","l":69,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.inverseMaxTolerance","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a nonnegative integer, the reciprocal-maximum tolerance is the positive rational number $1/\\max(n,1)$.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","l":388,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.midpoint_error_le_half_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Containment in a rational real interval bounds its midpoint error by half the interval width, turning the reciprocal-max width into a statistic error bound.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → |↑((I.lo + I.hi) / 2) - x| ≤ ↑I.width / 2","l":414,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.normalizedContourIntegral_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourEvaluation","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The canceled unit-parameter integral equals the normalized exact circle contour integral whenever the denominator is nonzero along the circle.","s":"∀ (program : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.ContourProgram) (schedule : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Schedule), (∀ u ∈ Set.Icc 0 1, program.denominator.value (↑program.radius * Complex.exp (↑2 * ↑Real.pi * ↑u * Complex.I)) ≠ 0) → program.normalizedContourIntegral = ∫ (u : ℝ) in 0..1, program.normalizedIntegrand schedule u","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/ContourEvaluation.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational rectangle in the complex plane consists of a rational interval enclosing the real coordinate and a rational interval enclosing the imaginary coordinate.","s":"Type","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.Contains","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given a rational rectangle and a complex number, rectangle membership means that both the real and imaginary coordinates of the number lie in their respective rational intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℂ → Prop","l":30,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.Subinterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Given two rational rectangles in the complex plane, the refinement relation holds exactly when each coordinate interval of the first is contained in the corresponding coordinate interval of the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Prop","l":34,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational rectangles in the complex plane, their sum rectangle is obtained by adding their real-coordinate intervals and their imaginary-coordinate intervals separately.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":38,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.add_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Complex rectangle addition encloses sums of enclosed complex numbers.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.add J).Contains (z + w)","l":53,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.conj","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex rectangle, its conjugate rectangle retains its real coordinate interval and negates its imaginary coordinate interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":34,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a numerator rectangle, a denominator rectangle, and evidence that the denominator's squared-modulus interval excludes zero, the guarded quotient rectangle is the product of the numerator rectangle and the denominator's conjugate, with both resulting coordinate intervals divided by the denominator's squared-modulus interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → (J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval) → J.normSq.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_maxAbs","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the squared-modulus denominator has positive lower bound δ, guarded division has maximum coordinate magnitude at most the numerator magnitude divided by δ.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} (hJ : J.normSq.AwayFromZero) {δ : ℚ}, 0 < δ → δ ≤ J.normSq.lo → (I.div J hJ).maxAbs ≤ (I.mul J.conj).maxAbs / δ","l":370,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Guarded division of complex rational rectangles. For two complex rational rectangles enclosing complex values z and w respectively, provided the denominator rectangle's squared-modulus interval is bounded away from zero, the rectangle obtained by dividing the first rectangle by the second under that guard contains the quotient z / w .","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ} (hJ : J.normSq.AwayFromZero), I.Contains z → J.Contains w → (I.div J hJ).Contains (z / w)","l":99,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.div_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"If the squared-modulus denominator has certified lower bound δ, guarded division propagates widths with the usual inverse-square factor.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} (hJ : J.normSq.AwayFromZero) {δ : ℚ}, 0 < δ → δ ≤ J.normSq.lo → (I.div J hJ).width ≤ (I.mul J.conj).maxAbs * J.normSq.width / δ ^ 2 + (I.mul J.conj).width / δ","l":409,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.expand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational rectangle in the complex plane, a rational error allowance, and the condition that this allowance is nonnegative, the expanded rectangle widens each coordinate interval by that allowance.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → (e : ℚ) → 0 ≤ e → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":46,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.expand_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Coordinatewise widening preserves containment.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} {e : ℚ} (he : 0 ≤ e), I.Contains z → (I.expand e he).Contains z","l":65,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.ext","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval}, x.re = y.re → x.im = y.im → x = y","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.ext_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval}, x = y ↔ x.re = y.re ∧ x.im = y.im","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex rectangle, its maximum coordinate endpoint magnitude is the larger of the maximum absolute endpoint magnitudes of its real and imaginary coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℚ","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs_conj","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Conjugation preserves the maximum coordinate endpoint magnitude.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.conj.maxAbs = I.maxAbs","l":125,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.maxAbs_le_of_contains_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Semantic coordinate bounds and rectangle width give an executable maximum endpoint-magnitude bound.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} {C w : ℚ}, I.Contains z → max |z.re| |z.im| ≤ ↑C → I.width ≤ w → I.maxAbs ≤ C + w","l":56,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given two rational complex rectangles, their product rectangle is obtained from the Cartesian complex-product formula using outward rational interval operations on the coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":29,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_maxAbs","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle multiplication has twice the product of operand coordinate magnitude bounds as an executable magnitude bound.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.mul J).maxAbs ≤ 2 * I.maxAbs * J.maxAbs","l":145,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle multiplication encloses products of enclosed values.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.mul J).Contains (z * w)","l":81,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.mul_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Each coordinate width after complex multiplication is bounded explicitly by the operand widths and endpoint magnitudes.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.mul J).width ≤ 2 * (I.maxAbs * J.width + J.maxAbs * I.width)","l":306,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex rectangle, its negative rectangle negates both coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":23,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.neg_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle negation encloses negated complex values.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → I.neg.Contains (-z)","l":70,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a complex rational rectangle and a natural-number square-root fuel level, applying the rational square-root bounds at that fuel level to the rectangle's squared-modulus interval produces an executable enclosure of the complex norm.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":472,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The executable modulus interval contains the norm of every enclosed complex value.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → ∀ (fuel : ℕ), (I.normInterval fuel).Contains ‖z‖","l":478,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex rectangle, its squared-modulus interval is the sum of the sign-aware squares of its real and imaginary coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":37,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_lo_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"For a rational complex rectangle, the lower endpoint of its squared-modulus interval is nonnegative.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), 0 ≤ I.normSq.lo","l":463,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The squared-modulus interval contains the squared complex norm.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → I.normSq.Contains (‖z‖ ^ 2)","l":92,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.normSq_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The squared-modulus interval width is controlled by rectangle magnitude and rectangle width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.normSq.width ≤ 4 * I.maxAbs * I.width","l":353,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational real coordinate and a rational imaginary coordinate, the complex point rectangle is the zero-width rectangle containing $x+iy$.","s":"ℚ → ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The rational point rectangle contains the corresponding complex number.","s":"∀ (x y : ℚ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.point x y).Contains (↑↑x + ↑↑y * Complex.I)","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_maxAbs_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Refining an interval cannot increase its maximum endpoint magnitude.","s":"∀ {K L : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, K.Subinterval L → K.maxAbs ≤ L.maxAbs","l":299,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_sq_hi_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The upper endpoint of a squared interval is bounded by the square of its maximum endpoint magnitude.","s":"∀ (K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), K.sq.hi ≤ K.maxAbs ^ 2","l":284,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.rat_sq_width_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Squaring an interval has width at most twice its maximum endpoint magnitude times its width.","s":"∀ (K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), K.sq.width ≤ 2 * K.maxAbs * K.width","l":278,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational scalar and a rational rectangle in the complex plane, the scalar-multiple rectangle is obtained by multiplying each coordinate interval by that scalar.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":42,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational scalar multiplication encloses the corresponding complex scalar multiple.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ} (q : ℚ), I.Contains z → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.smulRat q I).Contains (↑q * z)","l":58,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a first rational complex rectangle and a second rational complex rectangle, their difference rectangle adds the first rectangle to the coordinatewise negative of the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":26,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.sub_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle subtraction encloses differences of enclosed values.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z w : ℂ}, I.Contains z → J.Contains w → (I.sub J).Contains (z - w)","l":76,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.tighten","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given two rational complex rectangles, their coordinatewise tightening is formed by tightening the real coordinate intervals and tightening the imaginary coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":45,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.tighten_sound_left","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Tightening two rectangles containing a common value preserves that value and produces a subrectangle of the first argument.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval} {z : ℂ}, I.Contains z → J.Contains z → (I.tighten J).Contains z ∧ (I.tighten J).Subinterval I","l":129,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"Given a rational complex rectangle, its width is the larger of the widths of its real and imaginary coordinate intervals.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval → ℚ","l":13,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Complex rectangle addition has exactly the maximum of the two summed coordinate widths.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), (I.add J).width = max (I.re.width + J.re.width) (I.im.width + J.im.width)","l":139,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.width_conj","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Rectangles","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Conjugation preserves rectangle width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval), I.conj.width = I.width","l":121,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Rectangles.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval.zero","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Mesh","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The zero rectangle is the rational rectangle whose real and imaginary coordinate intervals are both the singleton interval containing zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":50,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a nonnegative integer, a family of $n+1$ rational candidate intervals, and a strictly positive rational width tolerance, the finite-refinement interval is the candidate at the least index whose width is at most the tolerance, or the zeroth candidate when no candidate meets that condition.","s":"{n : ℕ} → (Fin (n + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":86,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"If a candidate meets the requested width, finite refinement returns the least such candidate and meets the width.","s":"∀ {n : ℕ} {candidates : Fin (n + 1) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (∃ i, (candidates i).width ≤ ↑ε) → (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine candidates ε).width ≤ ↑ε","l":91,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.instMeasurableSpaceComplexRatInterval","k":"instance","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The measurable structure on complex rational rectangles is the discrete $\\sigma$-algebra, so every collection of complex rational rectangles is measurable.","s":"MeasurableSpace Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":40,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.instMeasurableSpaceRatInterval","k":"instance","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The measurable structure on rational intervals is the discrete $\\sigma$-algebra, so every collection of rational intervals is measurable.","s":"MeasurableSpace Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":35,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a nonnegative integer and a real-valued score for each of the $n+1$ indices from zero through $n, the least-score index is obtained by forming the finite set of pairs consisting of each score and its index, ordered first by score and then by index, certifying that this finite set is nonempty, and returning the index in its least pair.","s":"{n : ℕ} → (Fin (n + 1) → ℝ) → Fin (n + 1)","l":112,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex_minimal","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The selected least-score index has score no larger than every competing score.","s":"∀ {n : ℕ} (score : Fin (n + 1) → ℝ) (i : Fin (n + 1)), score (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score) ≤ score i","l":122,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex_tie","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Equal minimal scores are resolved in favor of the smaller original index.","s":"∀ {n : ℕ} (score : Fin (n + 1) → ℝ) {i : Fin (n + 1)}, score i = score (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex score ≤ i","l":147,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a nonnegative integer and a Boolean-valued family indexed by the $n+1$ indices from zero through $n, the least-successful-index rule returns the smallest index whose Boolean value is true when such an index exists, and index zero when none exists.","s":"{n : ℕ} → (Fin (n + 1) → Bool) → Fin (n + 1)","l":45,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_accepts","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"When at least one Boolean test succeeds, the returned index succeeds.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool}, (∃ i, accept i = true) → accept (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept) = true","l":51,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"If no Boolean test succeeds, the total search returns index zero.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool}, (∀ (i : Fin (n + 1)), accept i = false) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept = 0","l":75,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue_le","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every successful index is no smaller than the successful index returned by the search.","s":"∀ {n : ℕ} {accept : Fin (n + 1) → Bool} {i : Fin (n + 1)}, accept i = true → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue accept ≤ i","l":64,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_finiteRefine","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Finite refinement is measurable as a function of a finite vector of rational interval candidates.","s":"∀ {n : ℕ} (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), Measurable fun candidates => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.finiteRefine candidates ε","l":104,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_infEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational infimum enclosure is measurable as a function of its finite rational node vector.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.infEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":238,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_integralEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational complex integral enclosure is measurable as a function of its finite node rectangles.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.integralEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":258,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_leastScoreIndex","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a sample-size index n, the function that selects, from a vector of n+1 real scores, the smallest index attaining the least score is measurable with respect to the Borel σ-algebra on the score vector space and the discrete σ-algebra on the finite index set.","s":"∀ {n : ℕ}, Measurable Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastScoreIndex","l":174,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_leastTrue","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A finite least-success search is measurable when every individual success event is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} (accept : Fin (n + 1) → Ω → Bool), (∀ (i : Fin (n + 1)), Measurable (accept i)) → Measurable fun ω => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.leastTrue fun i => accept i ω","l":226,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch.measurable_supEnclosure","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.FiniteSearch","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The rational supremum enclosure is measurable as a function of its finite rational node vector.","s":"∀ {n : ℕ} (hn : 0 < n) (L : ℚ) (hL : 0 ≤ L), Measurable fun nodes => Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.supEnclosure (fun k => if hk : k < n + 1 then nodes ⟨k, hk⟩ else nodes 0) L hL n hn","l":248,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/FiniteSearch.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A positive rational number is a rational number that is strictly greater than zero.","s":"Type","l":15,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/CertifiedReal.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","k":"structure","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"A rational closed interval consists of a rational lower endpoint and a rational upper endpoint, with the lower endpoint no greater than the upper endpoint.","s":"Type","l":18,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.AwayFromZero","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, the property of being separated from zero holds precisely when the interval lies strictly below zero or strictly above zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":75,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Contains","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval and a real number, the containment relation holds precisely when the real number is no smaller than the interval's lower endpoint and no larger than its upper endpoint, after the rational endpoints are viewed as real numbers.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℝ → Prop","l":32,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Contains.mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A real point contained in a subinterval is contained in the enclosing interval.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Subinterval J → I.Contains x → J.Contains x","l":126,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Subinterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, the subinterval relation holds precisely when the second interval's lower endpoint is no greater than the first interval's lower endpoint and the first interval's upper endpoint is no greater than the second interval's upper endpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":42,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, their interval sum has lower endpoint equal to the sum of the lower endpoints, upper endpoint equal to the sum of the upper endpoints, and valid endpoint order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Addition is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.add J).Subinterval (I'.add J')","l":306,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.add_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval addition encloses the sum of any enclosed real operands.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.add J).Contains (x + y)","l":142,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.contains_point_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Containment in a point interval is equality with that rational point.","s":"∀ (q : ℚ) (x : ℝ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point q).Contains x ↔ x = ↑q","l":107,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a numerator rational interval and a denominator rational interval whose denominator interval is separated from zero, their interval quotient is the interval product of the numerator interval and the interval reciprocal of the denominator interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → (J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → J.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":85,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Division is inclusion-isotone when both denominator intervals avoid zero.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → ∀ (hJ0 : J.AwayFromZero) (hJ0' : J'.AwayFromZero), (I.div J hJ0).Subinterval (I'.div J' hJ0')","l":383,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.div_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval division encloses every quotient whose denominator interval avoids zero.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ} (hJ : J.AwayFromZero), I.Contains x → J.Contains y → (I.div J hJ).Contains (x / y)","l":265,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.expand","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, a rational widening amount that is nonnegative, the expanded interval has lower endpoint equal to the original lower endpoint minus the widening amount, upper endpoint equal to the original upper endpoint plus the widening amount, and valid endpoint order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → (e : ℚ) → 0 ≤ e → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":99,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ext","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, x.lo = y.lo → x.hi = y.hi → x = y","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ext_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"","s":"∀ {x y : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, x = y ↔ x.lo = y.lo ∧ x.hi = y.hi","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.hull","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, their rational interval hull has the smaller lower endpoint, the larger upper endpoint, and these endpoints in weakly increasing order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval that is separated from zero, its interval reciprocal has lower endpoint equal to the reciprocal of the original upper endpoint, upper endpoint equal to the reciprocal of the original lower endpoint, and valid endpoint order.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → I.AwayFromZero → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":78,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Reciprocal preserves inclusion when both intervals are certified away from zero.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → ∀ (hI : I.AwayFromZero) (hJ : J.AwayFromZero), (I.inv hI).Subinterval (J.inv hJ)","l":367,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.inv_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval reciprocal encloses the reciprocal of every enclosed real operand.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} (hI : I.AwayFromZero), I.Contains x → (I.inv hI).Contains x⁻¹","l":248,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, its maximum endpoint magnitude is the larger of the absolute values of its two rational endpoints.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":39,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Addition increases endpoint magnitude by at most the sum of operand bounds.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.add J).maxAbs ≤ I.maxAbs + J.maxAbs","l":240,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_le_of_contains_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Containment of one real value together with interval width controls both endpoint magnitudes.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} {C w : ℚ}, I.Contains x → |x| ≤ ↑C → I.width ≤ w → I.maxAbs ≤ C + w","l":209,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_mul","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Interval multiplication has the product of operand endpoint magnitudes as an executable magnitude bound.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.mul J).maxAbs ≤ I.maxAbs * J.maxAbs","l":256,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Negation preserves the maximum endpoint magnitude.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.neg.maxAbs = I.maxAbs","l":236,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.maxAbs_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Subtraction increases endpoint magnitude by at most the sum of operand bounds.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.sub J).maxAbs ≤ I.maxAbs + J.maxAbs","l":251,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, their interval product has lower endpoint equal to the least of the four products of one endpoint from each interval, upper endpoint equal to the greatest of those four products, and valid endpoint order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":65,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Multiplication is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.mul J).Subinterval (I'.mul J')","l":323,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.mul_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For rational intervals I and J, if x is a real number contained in I and y is a real number contained in J, then the interval-multiplication enclosure mul I J contains the real product x * y.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.mul J).Contains (x * y)","l":161,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.ne_zero_of_contains","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A real number contained in an interval separated from zero is nonzero.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.AwayFromZero → I.Contains x → x ≠ 0","l":239,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, its interval negation has lower endpoint equal to the negative of the original upper endpoint, upper endpoint equal to the negative of the original lower endpoint, and valid endpoint order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":58,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Negation preserves interval inclusion.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.neg.Subinterval J.neg","l":312,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.neg_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval negation encloses the negation of every enclosed real operand.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → I.neg.Contains (-x)","l":149,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, its natural-power interval sequence assigns the point interval at one to exponent zero and the interval product of the preceding power and the original interval to each positive exponent.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":88,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Natural interval powers preserve interval inclusion.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → ∀ (n : ℕ), (I.npow n).Subinterval (J.npow n)","l":360,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.npow_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Repeated interval multiplication encloses every natural power of an enclosed real number.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (n : ℕ), (I.npow n).Contains (x ^ n)","l":270,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number, the point interval at that number is the rational interval whose lower endpoint is that number, upper endpoint is that number, and endpoints are ordered because they are equal.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":46,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The point interval soundly encloses its rational value viewed as a real.","s":"∀ (q : ℚ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.point q).Contains ↑q","l":138,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational interval, its squared interval is the interval from the square of its upper endpoint to the square of its lower endpoint when it lies strictly below zero, the interval from the square of its lower endpoint to the square of its upper endpoint when it lies strictly above zero, and the interval from zero to the larger squared endpoint when it contains zero.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":16,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Squaring is inclusion-isotone for rational intervals.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.sq.Subinterval J.sq","l":48,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sq_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The sign-aware rational square enclosure contains the square of every real number contained in the input interval.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → I.sq.Contains (x ^ 2)","l":25,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational interval whose lower endpoint is nonnegative and a Newton iteration count, the square-root interval has lower endpoint equal to the Newton lower bound of the original lower endpoint, upper endpoint equal to the Newton upper bound of the original upper endpoint, and valid endpoint order.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → 0 ≤ I.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":180,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The executable square-root interval encloses the square root of every nonnegative real contained in its input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (hI : 0 ≤ I.lo) {x : ℝ}, I.Contains x → ∀ (fuel : ℕ), (I.sqrtInterval hI fuel).Contains √x","l":190,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtInterval_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"Square-root interval width is exactly the endpoint Newton gap, exposing both input-diameter and iteration-error contributions.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hI : 0 ≤ I.lo) (fuel : ℕ), (I.sqrtInterval hI fuel).width = Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper I.hi fuel - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower I.lo fuel","l":202,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational number and an iteration count, the Newton lower square-root bound is the number divided by its Newton upper iterate at that count.","s":"ℚ → ℕ → ℚ","l":83,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a rational number, the Newton upper-iterate sequence assigns the absolute value of the number plus one at iteration zero and one half of the sum of the preceding iterate and the number divided by that iterate at each later iteration.","s":"ℚ → ℕ → ℚ","l":76,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_gap_rate","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The Newton upper-minus-lower gap is bounded by an explicit reciprocal fuel rate.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (n : ℕ), Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n ≤ (q + 1) / (↑n + 1)","l":138,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_iterates_converge","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":2,"d":"The gap between the Newton lower and upper square-root bounds converges effectively to zero.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), ∃ n, Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n - Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n ≤ ↑ε","l":168,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrt_iterates_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Complex.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex","t":1,"d":"For a nonnegative rational number q and any fuel count n, the Newton-iterate rational lower bound sqrtLower q n and upper bound sqrtUpper q n both enclose the real square root of q: sqrtLower q n ≤ √q ≤ sqrtUpper q n.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (n : ℕ), ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtLower q n) ≤ √↑q ∧ √↑q ≤ ↑(Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sqrtUpper q n)","l":103,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Complex/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, their interval difference is the interval sum of the first interval and the interval negation of the second.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":62,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subtraction is inclusion-isotone in both interval arguments.","s":"∀ {I I' J J' : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval I' → J.Subinterval J' → (I.sub J).Subinterval (I'.sub J')","l":317,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.sub_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Rational interval subtraction encloses the difference of any enclosed real operands.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → (I.sub J).Contains (x - y)","l":156,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.subinterval_refl","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subinterval is reflexive.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.Subinterval I","l":116,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.subinterval_trans","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subinterval is transitive.","s":"∀ {I J K : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → J.Subinterval K → I.Subinterval K","l":120,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For two rational intervals, their conditional tightening is their exact intersection when its lower endpoint does not exceed its upper endpoint, and the first interval otherwise.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":93,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening two intervals that share an enclosed real point returns their intersection and continues to enclose that point.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Contains x","l":277,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.tighten_subinterval_left","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening by another sound interval produces a subinterval of the first interval.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Subinterval I","l":292,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval, its width is the upper rational endpoint minus the lower rational endpoint.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℚ","l":36,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_add","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Addition makes widths add exactly.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.add J).width = I.width + J.width","l":390,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_expand","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Widening an interval by a nonnegative amount on each side increases its width by twice that amount.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (e : ℚ) (he : 0 ≤ e), (I.expand e he).width = I.width + 2 * e","l":404,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_mono","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Inclusion of rational intervals cannot increase their width.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, I.Subinterval J → I.width ≤ J.width","l":133,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Negation preserves interval width exactly.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), I.neg.width = I.width","l":395,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every rational interval has nonnegative width.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), 0 ≤ I.width","l":103,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.width_sub","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Basic","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Subtraction makes widths add exactly.","s":"∀ (I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), (I.sub J).width = I.width + J.width","l":400,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.abs_expReduced_le_one","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Range reduction puts the exponential Taylor argument in the closed unit interval.","s":"∀ (q : ℚ), |Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReduced q| ≤ 1","l":58,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.abs_logCoordinate_lt_one","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"A positive rational logarithm argument has atanh coordinate of absolute value strictly below one.","s":"∀ (q : ℚ), 0 < q → |Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logCoordinate q| < 1","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval and a precision index, the exponential interval evaluation has lower endpoint equal to the lower endpoint of the scalar exponential enclosure at the input interval's lower endpoint, upper endpoint equal to the upper endpoint of the scalar exponential enclosure at the input interval's upper endpoint, and valid endpoint order.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":31,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Exponential interval evaluation is nested in its precision argument.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I n)","l":112,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Exponential interval evaluation encloses the exponential of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expInterval I n).Contains (Real.exp x)","l":71,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number, the certified-real representation of its exponential has value equal to the real exponential of that rational number, approximating intervals given by the scalar exponential enclosure sequence, nested approximations, containment of the exact value, the exponential precision index as its modulus, and the corresponding target-width guarantee.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":478,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a nonnegative polynomial degree, the exponential Taylor partial sum is the sum of the terms $q^k/k!$ for every integer $k$ from zero through $n$.","s":"ℚ → ℕ → ℚ","l":27,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational exponential argument and a strictly positive rational target width, the exponential precision index is the product of one plus the target-width denominator and two copies of one plus the exponential scaling factor.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational exponential argument and a nonnegative polynomial degree, the raw exponential enclosure is the natural-power interval obtained by raising the reduced exponential enclosure to the exponential scaling factor.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":45,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The raw rational exponential interval encloses the real exponential of the rational input.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRaw q n).Contains (Real.exp ↑q)","l":80,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReduced","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational exponential argument, the reduced exponential argument is the argument divided by its exponential scaling factor.","s":"ℚ → ℚ","l":24,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expReducedRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a nonnegative polynomial degree, the reduced exponential enclosure has lower endpoint equal to the Taylor partial sum minus the remainder radius, upper endpoint equal to the Taylor partial sum plus the remainder radius, and valid endpoint order.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":36,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRemainder","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a nonnegative polynomial degree, the exponential remainder radius is $|q|^{n+1}(n+2)/((n+1)!(n+1))$.","s":"ℚ → ℕ → ℚ","l":31,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational exponential argument, the scalar exponential enclosure sequence assigns the raw exponential enclosure at degree zero to index zero and the conditional tightening of the preceding enclosure with the next raw enclosure to each positive index.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":49,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Scalar exponential enclosures are nested as precision increases.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q n)","l":121,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every tightened scalar exponential interval encloses the real exponential.","s":"∀ (q : ℚ) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q n).Contains (Real.exp ↑q)","l":113,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational input q and a requested positive rational tolerance ε, evaluating the certified scalar exponential at the precision level selected for that tolerance yields an enclosure whose width is no larger than ε.","s":"∀ (q : ℚ) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScalar q (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expPrecision q ε)).width ≤ ↑ε","l":332,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expScale","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational exponential argument, the exponential scaling factor is the larger of one and the absolute value of the argument's numerator.","s":"ℚ → ℕ","l":21,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.inv_den_le_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The reciprocal of a positive rational's denominator is no larger than the rational itself.","s":"∀ (u : ℚ), 0 < u → 1 / ↑u.den ≤ u","l":315,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logCoordinate","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number, the logarithm coordinate is $(q-1)/(q+1)$. It is the atanh coordinate of the logarithm for positive $q$, the regime used by the logarithm enclosures; at $q=-1$ the denominator vanishes and the value is zero by the division convention.","s":"ℚ → ℚ","l":19,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational interval whose lower endpoint is strictly positive and a precision index, the logarithm interval evaluation has lower endpoint equal to the lower endpoint of the scalar logarithm enclosure at the original lower endpoint, upper endpoint equal to the upper endpoint of the scalar logarithm enclosure at the original upper endpoint, and valid endpoint order.","s":"(I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → 0 < I.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":48,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Positive logarithm interval evaluation is nested in its precision argument.","s":"∀ (I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hI : 0 < I.lo) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI n)","l":120,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Positive logarithm interval evaluation encloses the logarithm of every enclosed real input.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ} (hI : 0 < I.lo), I.Contains x → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logInterval I hI n).Contains (Real.log x)","l":78,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logName","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number that is strictly positive, the certified-real representation of its logarithm has value equal to the real logarithm of that rational number, approximating intervals given by the scalar logarithm enclosure sequence, nested approximations, containment of the exact value, the logarithm precision index as its modulus, and the corresponding target-width guarantee.","s":"(q : ℚ) → 0 < q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedReal","l":289,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPartial","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a nonnegative polynomial degree, the logarithm Taylor partial sum is twice the sum, for every integer $k$ from zero through $n$, of the $(2k+1)$st power of the logarithm coordinate divided by $2k+1$. It approximates the logarithm of $q$ only for positive $q$; no positivity is imposed by the definition itself.","s":"ℚ → ℕ → ℚ","l":22,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPrecision","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a strictly positive rational target width, the logarithm precision index is the product of one plus the target-width denominator and two copies of one plus the sum of the absolute numerator and denominator of the rational number.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat → ℕ","l":123,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number that is strictly positive and a nonnegative polynomial degree, the raw logarithm enclosure has lower endpoint equal to the logarithm partial sum minus its remainder bound, upper endpoint equal to the logarithm partial sum plus its remainder bound, and valid endpoint order.","s":"(q : ℚ) → 0 < q → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":105,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"The raw rational logarithm interval encloses the real logarithm of its positive rational input.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRaw q hq n).Contains (Real.log ↑q)","l":170,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logRemainder","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a rational number and a nonnegative polynomial degree, the logarithm remainder bound is $2|z|^{2n+3}/((2n+3)(1-|z|^2))$, where $z=(q-1)/(q+1)$. It bounds the series tail only for positive $q$, where $|z|<1$; at $q=0$ (where $|z|=1$) and at $q=-1$ the formula divides by zero and gives zero by the division convention.","s":"ℚ → ℕ → ℚ","l":27,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a strictly positive rational number, the scalar logarithm enclosure sequence assigns the raw logarithm enclosure at degree zero to index zero and the conditional tightening of the preceding enclosure with the next raw enclosure to each positive index.","s":"(q : ℚ) → 0 < q → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":118,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Scalar logarithm enclosures are nested as precision increases.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq n)","l":214,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Every tightened scalar logarithm interval encloses the real logarithm.","s":"∀ (q : ℚ) (hq : 0 < q) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq n).Contains (Real.log ↑q)","l":206,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar_width","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Logarithm","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Precision-driven logarithm enclosure width. For a positive rational number and a requested positive rational tolerance, evaluating the scalar logarithm enclosure at the precision level determined from those two inputs produces an interval no wider than the requested tolerance.","s":"∀ (q : ℚ) (hq : 0 < q) (ε : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.PosRat), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logScalar q hq (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.logPrecision q ε)).width ≤ ↑ε","l":228,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Logarithm.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a base rational interval and an exponent rational interval whose base interval has a strictly positive lower endpoint, the real-power interval sequence assigns the raw real-power interval at precision zero to index zero and the conditional tightening of the preceding interval with the next raw real-power interval to each positive index.","s":"(base : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → 0 < base.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":58,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval_nested","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Real-power interval evaluation is nested in its precision argument.","s":"∀ (base exponent : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) (hbase : 0 < base.lo) (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase (n + 1)).Subinterval (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase n)","l":129,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval_sound","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"Certified real-power interval evaluation. For a base rational interval whose lower endpoint is strictly positive that encloses a real base value, and an exponent rational interval enclosing a real exponent value, the real-power interval evaluation of the base and exponent, at any precision level, contains the true value of the base raised to that exponent.","s":"∀ {base exponent : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ} (hbase : 0 < base.lo), base.Contains x → exponent.Contains y → ∀ (n : ℕ), (Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowInterval base exponent hbase n).Contains (x ^ y)","l":100,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.rpowRaw","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Operations","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"For a base rational interval and an exponent rational interval whose base interval has a strictly positive lower endpoint, and a precision index, the raw real-power interval is the exponential interval of the product of the exponent interval and the logarithm interval of the base.","s":"(base : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → 0 < base.lo → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":54,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Operations.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.tighten_subinterval_right","k":"theorem","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Exponential","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":2,"d":"Tightening two enclosures of the same real value is contained in either input enclosure.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → J.Contains x → (I.tighten J).Subinterval J","l":127,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/Exponential.lean"},{"n":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.certifiedIntervalArithmetic","k":"def","m":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.API","a":"Mathlib/Analysis/CertifiedContourIntervalArithmetic","t":1,"d":"The concrete certified interval-arithmetic package provides outwardly sound rational interval operations for addition, subtraction, multiplication, division away from zero, exponentiation, logarithms and real powers on their positive domains, together with certified-real refinement of any requested positive rational precision.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CertifiedIntervalArithmetic","l":63,"f":"Causalean/Mathlib/Analysis/CertifiedContourIntervalArithmetic/API.lean"},{"n":"Causalean.Mathlib.Analysis.ContDiffAt.hasLocallyBoundedFDerivAt","k":"theorem","m":"Causalean.Mathlib.Analysis.LocallyBoundedDerivative","a":"Mathlib/Analysis","t":1,"d":"Given a function between real normed vector spaces, a point in its domain, and continuous differentiability at that point, the function has a locally bounded Fréchet derivative there.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {f : E → F} {x : E}, ContDiffAt ℝ 1 f x → Causalean.Mathlib.Analysis.HasLocallyBoundedFDerivAt f x","l":19,"f":"Causalean/Mathlib/Analysis/LocallyBoundedDerivative.lean"},{"n":"Causalean.Mathlib.Analysis.ContinuousLinearMap.abs_singularValues_add_sub_singularValues_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SingularValueWeyl","a":"Mathlib/Analysis","t":2,"d":"Given a baseline continuous linear map, an additive continuous perturbation, and a singular-value index, the absolute change in the indexed singular value is at most the perturbation's operator norm.","s":"∀ {V : Type u_1} {W : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V] [inst_3 : NormedAddCommGroup W] [inst_4 : InnerProductSpace ℝ W] [inst_5 : FiniteDimensional ℝ W] (A D : V →L[ℝ] W) (j : ℕ), |(↑(A + D)).singularValues j - (↑A).singularValues j| ≤ ‖D‖","l":225,"f":"Causalean/Mathlib/Analysis/SingularValueWeyl.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.abs_czTrig_le_czSup","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Every trigonometric value is bounded by the sup-norm: |czTrig R t| ≤ czSup R for all t ∈ [0, π].","s":"∀ (R : Polynomial ℝ) {t : ℝ}, t ∈ Set.Icc 0 Real.pi → |Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t| ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":88,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshLHS_le_czSup","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Sup transfer. The [-1, 1] sup-norm of R equals the sup-norm of its trigonometric transform on [0, π]: since t ↦ -cos t maps [0, π] *onto* [-1, 1], sup_{x∈[-1,1]} |R x| ≤ czSup R (in fact with equality).","s":"∀ (R : Polynomial ℝ), sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":122,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"For a real polynomial and a nonnegative mesh order, the mesh maximum is the largest absolute value of the polynomial over all Chebyshev–Lobatto nodes with indices from zero through that order.","s":"Polynomial ℝ → ℕ → ℝ","l":48,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial leaves unchanged its mesh maximum at the Chebyshev–Lobatto nodes.","s":"∀ (R : Polynomial ℝ) (k : ℕ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax (-R) k = Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":184,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The mesh maximum of the absolute values of a real polynomial at the Chebyshev–Lobatto nodes is nonnegative.","s":"∀ (R : Polynomial ℝ) (k : ℕ), 0 ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":160,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"For a nonnegative mesh order and a nonnegative node index, the Chebyshev–Lobatto node is $-\\cos(\\pi j/k)$. When the mesh order is positive, the indices from zero through that order give its endpoints $-1$ and $1$ as well as its intermediate nodes.","s":"ℕ → ℕ → ℝ","l":40,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"For a real polynomial, the cosine-transform sup norm is the supremum of the absolute value of its cosine-parametrized transform over $[0,\\pi]$. Equivalently, it is the supremum of the polynomial's absolute value over $[-1,1]$.","s":"Polynomial ℝ → ℝ","l":54,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_attained","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The sup-norm czSup R is attained at some point of [0, π]: there is t₀ ∈ [0, π] with |czTrig R t₀| = czSup R. This is the extreme-value theorem applied to the continuous map |czTrig R| on the compact interval [0, π].","s":"∀ (R : Polynomial ℝ), ∃ t₀ ∈ Set.Icc 0 Real.pi, |Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t₀| = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":69,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial leaves unchanged the supremum of the absolute value of its trigonometric transform over the cosine interval.","s":"∀ (R : Polynomial ℝ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup (-R) = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":172,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"czSup R is nonnegative (it is a supremum of absolute values).","s":"∀ (R : Polynomial ℝ), 0 ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R","l":82,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig","k":"def","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"For a real polynomial and a real argument, the cosine-parametrized polynomial transform is the polynomial evaluated at $-\\cos(t)$. As the argument ranges over $[0,\\pi]$, its evaluation point ranges over all of $[-1,1]$.","s":"Polynomial ℝ → ℝ → ℝ","l":46,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_arccos_lipschitz","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Arccos-Lipschitz reformulation of Szegő's inequality. For a real polynomial of degree at most β with a strictly positive trigonometric sup-norm on [0, π], and for any two points in [0, π], the arccosine of the polynomial's normalized trigonometric transform is β-Lipschitz between those two points: the difference of the two arccosine values is bounded by β times the distance between the points.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → 0 < Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ∀ {s t : ℝ}, s ∈ Set.Icc 0 Real.pi → t ∈ Set.Icc 0 Real.pi → |Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t / Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R) - Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s / Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R)| ≤ ↑β * |t - s|","l":276,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_arccos_lipschitz_regularized","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"For a real polynomial of degree at most β, the arccosine of its cosine-polynomial transform normalized by its supremum plus a positive constant is Lipschitz on [0, π] with constant β.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ {δ : ℝ}, 0 < δ → ∀ {s t : ℝ}, s ∈ Set.Icc 0 Real.pi → t ∈ Set.Icc 0 Real.pi → |Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t / (Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R + δ)) - Real.arccos (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s / (Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R + δ))| ≤ ↑β * |t - s|","l":160,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_at_meshParam","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The trigonometric transform evaluated at the mesh parameter t = π j / k recovers the node value: czTrig R (π j / k) = R(czNode k j).","s":"∀ (R : Polynomial ℝ) (k j : ℕ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R (Real.pi * ↑j / ↑k) = Polynomial.eval (Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode k j) R","l":56,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_continuous","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"czTrig R is continuous (composition of the polynomial evaluation with t ↦ -cos t).","s":"∀ (R : Polynomial ℝ), Continuous (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R)","l":62,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_maximizer_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Maximizer node bound (the packaged output consumed by the mesh file). For a real polynomial of degree at most β with a strictly positive trigonometric sup-norm, suppose a point t₀ in [0, π] is a maximizer where the trigonometric transform attains the sup-norm, another point s also lies in [0, π], and the two points are close enough that β times their distance is at most π/2. Then the trigonometric transform at s is bounded below by the sup-norm times the cosine of β times the distance between the two points.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → 0 < Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ∀ {t₀ s : ℝ}, t₀ ∈ Set.Icc 0 Real.pi → s ∈ Set.Icc 0 Real.pi → Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t₀ = Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R → ↑β * |t₀ - s| ≤ Real.pi / 2 → Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R * Real.cos (↑β * |t₀ - s|) ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R s","l":332,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Negating a real polynomial negates its trigonometric transform at every point.","s":"∀ (R : Polynomial ℝ) (t : ℝ), Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig (-R) t = -Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t","l":167,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig_szego_deriv","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Bernstein","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Szegő's differential inequality (the crux; not available in Mathlib). For a real polynomial of degree at most β and any real point t, the derivative of the polynomial's trigonometric transform at that point is bounded in absolute value by β times the square root of the sup-norm squared minus the transform's value at that point squared.","s":"∀ (R : Polynomial ℝ) (β : ℕ), R.natDegree ≤ β → ∀ (t : ℝ), |deriv (Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R) t| ≤ ↑β * √(Causalean.Mathlib.Analysis.EhlichZellerMesh.czSup R ^ 2 - Causalean.Mathlib.Analysis.EhlichZellerMesh.czTrig R t ^ 2)","l":97,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Bernstein.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.ehlichZeller_mesh_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Ehlich–Zeller mesh (norming) inequality. For a real polynomial of degree at most β and a mesh order k strictly exceeding β, with Chebyshev–Lobatto nodes x_j = -cos(π j / k) for j = 0, …, k, the polynomial's supremum absolute value on [-1, 1] is bounded by the secant of π β / (2k) times the maximum of its absolute values at the mesh nodes.","s":"∀ (R : Polynomial ℝ) (β k : ℕ), R.natDegree ≤ β → β < k → sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ 1 / Real.cos (Real.pi * ↑β / (2 * ↑k)) * Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":272,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.eval_node_le_czMeshMax","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"The mesh maximum lower-bounds every node value it ranges over.","s":"∀ (R : Polynomial ℝ) (k : ℕ) {j : ℕ}, j ∈ Finset.range (k + 1) → |Polynomial.eval (Causalean.Mathlib.Analysis.EhlichZellerMesh.czNode k j) R| ≤ Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":138,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.exists_meshParam_close","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"Node-selection lemma. For k ≥ 1 and any t₀ ∈ [0, π] there is a mesh index j ≤ k whose parameter π j / k is within π / (2k) of t₀: |t₀ − π j / k| ≤ π / (2k). (The k mesh parameters partition [0, π] into subintervals of length π / k, so every point is within a half-step of a node.)","s":"∀ (k : ℕ), 0 < k → ∀ {t₀ : ℝ}, t₀ ∈ Set.Icc 0 Real.pi → ∃ j ∈ Finset.range (k + 1), |t₀ - Real.pi * ↑j / ↑k| ≤ Real.pi / (2 * ↑k)","l":62,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.meshParam_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":2,"d":"For a positive mesh order, every valid Chebyshev–Lobatto mesh parameter lies in the interval from zero to π.","s":"∀ (k : ℕ), 0 < k → ∀ {j : ℕ}, j ∈ Finset.range (k + 1) → Real.pi * ↑j / ↑k ∈ Set.Icc 0 Real.pi","l":190,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.EhlichZellerMesh.oversampled_norming","k":"theorem","m":"Causalean.Mathlib.Analysis.EhlichZellerMesh.Mesh","a":"Mathlib/Analysis/EhlichZellerMesh","t":1,"d":"Oversampled Chebyshev–Lobatto norming (constant depending only on the oversampling ratio). For an oversampling ratio strictly greater than one, a polynomial degree bound of at least one, a real polynomial whose degree does not exceed that bound, and a mesh order at least the oversampling ratio times the degree bound, the polynomial's supremum absolute value on [-1, 1] is bounded by the secant of π / (2c) times the maximum of its absolute values at the mesh nodes — a norming constant depending only on the oversampling ratio, uniformly in the degree bound.","s":"∀ (R : Polynomial ℝ) (β k : ℕ) (c : ℝ), 1 < c → 1 ≤ β → R.natDegree ≤ β → c * ↑β ≤ ↑k → sSup ((fun x => |Polynomial.eval x R|) '' Set.Icc (-1) 1) ≤ 1 / Real.cos (Real.pi / (2 * c)) * Causalean.Mathlib.Analysis.EhlichZellerMesh.czMeshMax R k","l":309,"f":"Causalean/Mathlib/Analysis/EhlichZellerMesh/Mesh.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative number of sampled coordinates minus one, a collection of real sampling nodes indexed from zero through that number, and a nonnegative degree bound, the node-evaluation linear map sends a coefficient vector to the values at all sampling nodes of its polynomial of degree at most that bound.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → (β : ℕ) → (Fin (β + 1) → ℝ) →ₗ[ℝ] Fin (k + 1) → ℝ","l":67,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Unfolding lemma for Ev (definitional).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (b : Fin (β + 1) → ℝ) (j : Fin (k + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j = ∑ i, b i * p j ^ ↑i","l":85,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_injective","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Node evaluation is injective. For k + 1 pairwise distinct real nodes and a degree bound β at most k, the linear map sending a degree-≤ β coefficient vector to its values at the nodes is injective.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → Function.Injective ⇑(Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β)","l":129,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev_single","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Node evaluation of the ℓ-th coordinate vector recovers the monomial column (p j)^ℓ.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (ℓ : Fin (β + 1)) (j : Fin (k + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) (Pi.single ℓ 1) j = p j ^ ↑ℓ","l":174,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative number of nodes minus one, a collection of real nodes indexed from zero through that number, and a nonnegative degree bound, the moment system of admissible weight vectors consists of precisely those real weight vectors whose weighted power sum at every nonnegative degree no greater than the bound is zero at degree zero and one at every positive degree.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set (Fin (k + 1) → ℝ)","l":51,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative degree bound and a real coefficient vector indexed from zero through that bound, the associated coefficient polynomial is the sum of each coefficient times the corresponding monomial. Its degree is at most the stated bound.","s":"{β : ℕ} → (Fin (β + 1) → ℝ) → Polynomial ℝ","l":43,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_contrast","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The contrast functional computes the endpoint contrast of coeffPoly b.","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), Polynomial.eval 1 (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) - Polynomial.eval 0 (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b","l":118,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_eval","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Evaluating coeffPoly b at t gives the polynomial value ∑ i, b i * t^i.","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ) (t : ℝ), Polynomial.eval t (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = ∑ i, b i * t ^ ↑i","l":61,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_eval_node","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Node values of coeffPoly b coincide with Ev p β b.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} (b : Fin (β + 1) → ℝ) (j : Fin (k + 1)), Polynomial.eval (p j) (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b) = (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j","l":90,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly_natDegree_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"coeffPoly b has degree at most β (each monomial X^i has i ≤ β).","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.coeffPoly b).natDegree ≤ β","l":51,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative degree bound, the endpoint-contrast linear functional maps a coefficient vector to the value at one minus the value at zero of its associated polynomial.","s":"(β : ℕ) → (Fin (β + 1) → ℝ) →ₗ[ℝ] ℝ","l":95,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Unfolding lemma for contrastL (definitional).","s":"∀ {β : ℕ} (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b = ∑ i, b i * if ↑i = 0 then 0 else 1","l":113,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_le_dual_mul_ninf","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Boundedness estimate (the Hahn–Banach hypothesis). For k + 1 pairwise distinct real nodes, a degree bound β at most k, and any coefficient vector b, the endpoint-contrast functional evaluated at b is bounded above by the dual supremum times the sup-norm of b's node values.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf ((Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b)","l":262,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_le_dual_mul_ninf_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Hahn-Banach domination estimate on the node-value subspace, using feasibility instead of node-evaluation injectivity: contrastL β b ≤ sSup (dualValSet p β) * ninf (Ev p β b).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf ((Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b)","l":83,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL_single","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The contrast of the ℓ-th coordinate vector is 1^ℓ - 0^ℓ.","s":"∀ {β : ℕ} (ℓ : Fin (β + 1)), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) (Pi.single ℓ 1) = if ↑ℓ = 0 then 0 else 1","l":186,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative number of nodes minus one, a collection of real nodes indexed from zero through that number, and a nonnegative degree bound, the set of achievable dual values consists of the absolute differences between a real polynomial's values at one and zero, for every polynomial of degree at most the bound whose absolute value at every node is at most one.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set ℝ","l":71,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_bddAbove","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The dual set is bounded above (by any primal value; the primal set is nonempty for distinct nodes and β ≤ k).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty → BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":124,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_bddAbove_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Feasibility of the moment system bounds every dual value by any feasible primal norm. This is the weak-duality estimate: if w ∈ MomentSol p β, then every node-bounded degree-≤ β polynomial has endpoint contrast at most ∑ j, |w j|, so the dual value set is bounded above.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":47,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The dual set is nonempty: the zero polynomial contributes the value 0 (degree 0 ≤ β, trivially node-bounded, contrast 0).","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β).Nonempty","l":82,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_le_primal","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Weak duality. If a real number is the ℓ¹ norm of some feasible weight vector and another real number is the endpoint contrast attained by some node-bounded degree-≤ β polynomial, then the second number is no larger than the first: every dual value is bounded by every primal value.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} {s t : ℝ}, s ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β → t ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β → t ≤ s","l":97,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Dual value is nonnegative. 0 ∈ dualValSet p β (the zero polynomial) and the set is bounded above, so its supremum is ≥ 0.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, BddAbove (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) → 0 ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":250,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dual_nonneg_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"With a feasible primal system, the dual supremum is nonnegative because 0 is dual-feasible.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → 0 ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":57,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.exists_moment_le_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.StrongDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality (existence of an optimal weight). For k + 1 pairwise distinct real nodes and a degree bound β at most k, there exists a feasible weight vector — one solving the moment system — whose ℓ¹ norm does not exceed the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → ∃ w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β, ∑ j, |w j| ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":21,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/StrongDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.exists_moment_le_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality from moment-system feasibility. If the moment system for the given nodes and degree bound is feasible, i.e. it has at least one solution, then there exists a feasible weight vector whose ℓ¹ norm is at most the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → ∃ w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β, ∑ j, |w j| ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":152,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_eq_sup_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Duality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Finite-dimensional ℓ¹/ℓ∞ duality. For k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k, the least ℓ¹ norm of a weight vector that reproduces the endpoint contrast r ↦ r.eval 1 - r.eval 0 of every degree-≤ β polynomial through its node values equals the largest such contrast attained by a degree-≤ β polynomial bounded by 1 at every node.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":22,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_eq_sup_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Finite-dimensional ℓ¹/ℓ∞ duality under feasibility alone. If the moment system for the given nodes and degree bound has at least one solution, then the least ℓ¹ norm among feasible weight vectors equals the largest endpoint contrast attained by a degree-bounded polynomial bounded by 1 at every node.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":303,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.l1_repr_sq_eq_sup_dual_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Duality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Squared form. Under the same hypotheses as l1_repr_eq_sup_dual — k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k — the square of the min-norm-representation value equals the square of the dual sup.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ^ 2 = sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ^ 2","l":36,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Duality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.le_ninf","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Each coordinate is bounded by the sup norm: |x j| ≤ ninf x.","s":"∀ {k : ℕ} (x : Fin (k + 1) → ℝ) (j : Fin (k + 1)), |x j| ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":204,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.momentSol_contrast_eq_sum_eval","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"A feasible moment vector represents the coefficient contrast on node-value vectors: for every coefficient vector b, contrastL β b equals the pairing of w with the node-evaluation vector Ev p β b.","s":"∀ {k β : ℕ} {p w : Fin (k + 1) → ℝ}, w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β → ∀ (b : Fin (β + 1) → ℝ), (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.contrastL β) b = ∑ j, w j * (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Ev p β) b j","l":69,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Feasibility of the moment system (Vandermonde). For k + 1 pairwise distinct real interpolation nodes and a degree bound β at most k, the moment system — the set of weight vectors that reproduce the endpoint contrast of every degree-≤ β polynomial through the node values — has a solution.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty","l":104,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative number of coordinates minus one and a real vector indexed from zero through that number, the sup norm is the largest absolute coordinate value.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℝ","l":197,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_add_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Subadditivity (triangle inequality) of the sup norm.","s":"∀ {k : ℕ} (x y : Fin (k + 1) → ℝ), Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf (x + y) ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x + Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf y","l":234,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The sup norm is nonnegative.","s":"∀ {k : ℕ} (x : Fin (k + 1) → ℝ), 0 ≤ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":208,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_sign","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The sup norm of a ±1 sign vector is 1.","s":"∀ {k : ℕ} (s : Fin (k + 1) → ℝ), (∀ (j : Fin (k + 1)), |s j| = 1) → Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf s = 1","l":244,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf_smul","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.HahnBanachSetup","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Positive homogeneity (with absolute value) of the sup norm.","s":"∀ {k : ℕ} (c : ℝ) (x : Fin (k + 1) → ℝ), Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf (c • x) = |c| * Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.ninf x","l":212,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/HahnBanachSetup.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet","k":"def","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"For a nonnegative number of nodes minus one, a collection of real nodes indexed from zero through that number, and a nonnegative degree bound, the set of achievable primal norms consists of the sums of absolute weights over all weight vectors that satisfy the corresponding moment system.","s":"{k : ℕ} → (Fin (k + 1) → ℝ) → ℕ → Set ℝ","l":62,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_bddBelow","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Every element of the primal set is ≥ 0 (a sum of absolute values), so the set is bounded below by 0.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, BddBelow (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":92,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"The primal set is nonempty whenever the moment system is solvable.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty","l":149,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonempty_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"If the moment system is feasible, the primal norm set is nonempty.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty","l":40,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.Basic","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"0 is a lower bound for the primal set.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ} {s : ℝ}, s ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β → 0 ≤ s","l":99,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.repr_identity","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Representation identity. If a weight vector solves the moment system for the given nodes and degree bound and a real polynomial has degree at most β, then the polynomial's endpoint contrast, r.eval 1 - r.eval 0, is reproduced exactly by the weighted sum of its values at the nodes.","s":"∀ {k β : ℕ} {p w : Fin (k + 1) → ℝ}, w ∈ Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β → ∀ {r : Polynomial ℝ}, r.natDegree ≤ β → Polynomial.eval 1 r - Polynomial.eval 0 r = ∑ j, w j * Polynomial.eval (p j) r","l":29,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sInf_primal_le_sSup_dual","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.StrongDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Strong duality (inequality form). For k + 1 pairwise distinct real nodes and a degree bound β at most k, the least ℓ¹ norm among feasible weight vectors is at most the dual supremum.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, Function.Injective p → β ≤ k → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":29,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/StrongDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sInf_primal_le_sSup_dual_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Inequality form of strong duality under moment-system feasibility.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β) ≤ sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β)","l":285,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sSup_dual_le_sInf_primal","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.WeakDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":1,"d":"Weak duality (inequality form). If the primal set of achievable ℓ¹ norms is nonempty, then the dual supremum is at most the primal infimum: every dual value is bounded by every primal value, hence so is the supremum of dual values by the infimum of primal values.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β).Nonempty → sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ≤ sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":131,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/WeakDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.sSup_dual_le_sInf_primal_of_momentSol_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.NonemptyDuality","a":"Mathlib/Analysis/FiniteDimL1LinfDuality","t":2,"d":"Weak duality as an sSup ≤ sInf inequality under moment-system feasibility.","s":"∀ {k β : ℕ} {p : Fin (k + 1) → ℝ}, (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.MomentSol p β).Nonempty → sSup (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.dualValSet p β) ≤ sInf (Causalean.Mathlib.Analysis.FiniteDimL1LinfDuality.primalNormSet p β)","l":294,"f":"Causalean/Mathlib/Analysis/FiniteDimL1LinfDuality/NonemptyDuality.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.AlternationDualCertificate","k":"structure","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Alternation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"An alternation dual certificate consists of a best degree-L approximant, L+2 ordered extrema of its residual, and the normalized signed Lagrange weights on those nodes, with the exact oriented target evaluation. the stated inputs establish the described certificate.","s":"(ℝ → ℝ) → ℝ → ℝ → ℕ → Type","l":23,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Alternation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.AlternationDualCertificate.toFiniteMomentDual","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Alternation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Forgetting the best approximant and orientation from an alternation certificate yields the paper-independent finite moment-dual package. the stated inputs establish the defined object.","s":"{f : ℝ → ℝ} → {r s : ℝ} → {L : ℕ} → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.AlternationDualCertificate f r s L → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.FiniteMomentDual f r s L","l":135,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Alternation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.EquioscillationWitness","k":"structure","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Equioscillation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"An equioscillation witness records a best degree-L polynomial and L+2 ordered interval points where its residual has the common optimal magnitude with alternating signs. the stated inputs establish the described certificate.","s":"(ℝ → ℝ) → ℝ → ℝ → ℕ → Type","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Equioscillation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.FiniteMomentDual","k":"structure","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Alternation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"A finite moment dual packages L+2 ordered interval nodes and normalized signed weights that annihilate moments through degree L and separate the target by exactly its best uniform approximation error. the stated inputs establish the described certificate.","s":"(ℝ → ℝ) → ℝ → ℝ → ℕ → Type","l":122,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Alternation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.IsAlternatingExtrema","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ExtremalSignPolynomial","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"A residual has L+2 alternating uniform extrema when it attains its supremum magnitude with alternating signs at that many strictly ordered points of the interval. the stated inputs establish the defined object.","s":"(ℝ → ℝ) → ℝ → ℝ → (L : ℕ) → (Fin (L + 2) → ℝ) → ℝ → Prop","l":241,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ExtremalSignPolynomial.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.abs_eval_derivative_le_chebyshev_at_root","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevZeroDerivativeComparison","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a degree-at-most-L real polynomial has absolute value at most one at all L+1 extrema of T_L, then at every real zero of T_L its derivative is no larger in absolute value than the derivative of T_L. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (Q : Polynomial ℝ), Q.natDegree ≤ L → (∀ i ≤ L, |Polynomial.eval (Polynomial.Chebyshev.node L i) Q| ≤ 1) → ∀ {x : ℝ}, Polynomial.eval x (Polynomial.Chebyshev.T ℝ ↑L) = 0 → |Polynomial.eval x (Polynomial.derivative Q)| ≤ |Polynomial.eval x (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|","l":190,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevZeroDerivativeComparison.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.abs_eval_derivative_le_chebyshev_endpoint_of_root_control","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.HalfPlaneDerivativeComparison","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Suppose a degree-at-most-L real polynomial has derivative bounded by T_L' at every real zero of T_L. At a point x ∈ [-1,1], if the complex modulus of T_L on the vertical line through x is dominated by the vertical line through 1, then the polynomial's derivative at x is bounded by |T_L'(1)|. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (Q : Polynomial ℝ), Q.natDegree ≤ L → (∀ (z : ℝ), Polynomial.eval z (Polynomial.Chebyshev.T ℝ ↑L) = 0 → |Polynomial.eval z (Polynomial.derivative Q)| ≤ |Polynomial.eval z (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|) → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → (∀ (y : ℝ), ‖Polynomial.eval (↑x + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖ ≤ ‖Polynomial.eval (1 + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖) → |Polynomial.eval x (Polynomial.derivative Q)| ≤ |Polynomial.eval 1 (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/HalfPlaneDerivativeComparison.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.abs_eval_derivative_le_sq_of_chebyshev_root_control","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevRootDerivativePropagation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a degree-at-most-L real polynomial has derivative no larger than T_L' at every zero of T_L, then its derivative is bounded by L² throughout [-1,1]. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (Q : Polynomial ℝ), Q.natDegree ≤ L → (∀ (z : ℝ), Polynomial.eval z (Polynomial.Chebyshev.T ℝ ↑L) = 0 → |Polynomial.eval z (Polynomial.derivative Q)| ≤ |Polynomial.eval z (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|) → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → |Polynomial.eval x (Polynomial.derivative Q)| ≤ ↑L ^ 2","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevRootDerivativePropagation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.PairingRearrangement","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"adjacentPairProduct t xs multiplies a * b + t over consecutive pairs a, b of xs. A final unpaired entry contributes no factor. the stated inputs establish the defined object.","s":"ℝ → List ℝ → ℝ","l":21,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/PairingRearrangement.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct_chebyshevPairedChordList","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordFactorization","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Pairing the two chord squares belonging to each Chebyshev root converts a*b + 4y² into four times the squared distance from cos θ + iy to that root. the stated inputs establish the stated conclusion.","s":"∀ (L : ℕ) (θ y : ℝ), Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct (4 * y ^ 2) (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList L θ) = ∏ k ∈ Finset.range L, 4 * ‖↑(Real.cos θ - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖ ^ 2","l":26,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordFactorization.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct_le_of_perm_sorted","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.PairingRearrangement","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Let xs be a decreasing list of nonnegative real numbers of even length. For every permutation ys of xs, the consecutive-pair product of ys is at most the consecutive-pair product of xs itself. the stated inputs establish the stated conclusion.","s":"∀ {t : ℝ}, 0 ≤ t → ∀ {xs ys : List ℝ}, Even xs.length → List.Pairwise (fun a b => b ≤ a) xs → (∀ a ∈ xs, 0 ≤ a) → ys.Perm xs → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct t ys ≤ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct t xs","l":234,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/PairingRearrangement.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Basic","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The best degree-L uniform polynomial-approximation error is the infimum of the uniform errors of all real polynomials of degree at most L. the stated inputs establish the defined object.","s":"(ℝ → ℝ) → ℝ → ℝ → ℕ → ℝ","l":35,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.BestApproximation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"Every degree-at-most-L polynomial has uniform error at least the best degree-L error. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ {L : ℕ} {Q : Polynomial ℝ}, Q.natDegree ≤ L → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError f r s L ≤ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.uniformApproxError f r s Q","l":152,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/BestApproximation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.BestApproximation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For a continuous target on a nondegenerate interval, the best bounded-degree uniform approximation error is nonnegative. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ (L : ℕ), 0 ≤ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError f r s L","l":165,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/BestApproximation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevDerivativeWeight","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevDerivativeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The Chebyshev differentiation weight at x and node index i is the derivative at x of the corresponding Lagrange cardinal polynomial. the stated inputs establish the defined object.","s":"ℕ → ℝ → ℕ → ℝ","l":24,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevDerivativeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordDefinitions","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The 2L chord squares associated with an abscissa angle θ, listed in the two-element pairs belonging to the L Chebyshev root angles. The degree and angle determine the defined object.","s":"ℕ → ℝ → List ℝ","l":32,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordDefinitions.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList_add_int_mul_pi_div_perm","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordPermutation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Shifting an angle by any integer multiple of π/L permutes the paired list of the 2L Chebyshev chord squares. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (θ : ℝ) (m : ℤ), (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList L (θ + ↑m * (Real.pi / ↑L))).Perm (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList L θ)","l":182,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordPermutation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevRootAngle","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordDefinitions","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The angle (2k+1)π/(2L) of the k-th degree-L Chebyshev root. the stated inputs establish the defined object.","s":"ℕ → ℕ → ℝ","l":18,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordDefinitions.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordDefinitions","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The k-th cosine root used in the degree-L Chebyshev product, indexed by 0 ≤ k < L. the stated inputs establish the defined object.","s":"ℕ → ℕ → ℝ","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordDefinitions.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.continuousOn_rationalTarget","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Rational","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"If the pole -a is outside a closed interval, the rational target x ↦ x/(x+a) is continuous on that interval. the stated inputs establish the stated conclusion.","s":"∀ {a r s : ℝ}, -a ∉ Set.Icc r s → ContinuousOn (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.rationalTarget a) (Set.Icc r s)","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Rational.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.cosineChordSq","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordDefinitions","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The squared chord length from 1 to the point of the unit circle with argument θ. The stated angle determines the defined object.","s":"ℝ → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordDefinitions.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.crossingPairProduct_le_adjacentPairProduct","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.PairingRearrangement","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For four decreasing real numbers, replacing the crossing pairs (a,c) and (b,d) by the adjacent pairs (a,b) and (c,d) cannot decrease the product after adding the same nonnegative constant to each pair. the stated inputs establish the stated conclusion.","s":"∀ {a b c d t : ℝ}, 0 ≤ t → b ≤ a → c ≤ b → d ≤ c → (a * c + t) * (b * d + t) ≤ (a * b + t) * (c * d + t)","l":27,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/PairingRearrangement.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.derivative_eval_pullbackToUnitInterval","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Affine","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"The derivative of the affine pullback is the original derivative evaluated at the corresponding point, multiplied by the interval half-length. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) (r s t : ℝ), Polynomial.eval t (Polynomial.derivative (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.pullbackToUnitInterval Q r s)) = (s - r) / 2 * Polynomial.eval ((s - r) / 2 * t + (r + s) / 2) (Polynomial.derivative Q)","l":49,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.duffinSchaeffer_derivative_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.DuffinSchaeffer","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a real polynomial of degree at most L, with L > 0, is bounded by C at every degree-L Chebyshev extremum, then its derivative is bounded by L² C throughout [-1,1]. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) {L : ℕ}, 0 < L → Q.natDegree ≤ L → ∀ {C : ℝ}, 0 ≤ C → (∀ i ≤ L, |Polynomial.eval (Polynomial.Chebyshev.node L i) Q| ≤ C) → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → |Polynomial.eval x (Polynomial.derivative Q)| ≤ ↑L ^ 2 * C","l":20,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/DuffinSchaeffer.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.eval_derivative_eq_sum_chebyshevDerivativeWeight","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevDerivativeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"The derivative of every degree-at-most-L polynomial is the weighted sum of its values at the L+1 Chebyshev extrema, with the Chebyshev differentiation weights. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) {L : ℕ}, Q.natDegree ≤ L → ∀ (x : ℝ), Polynomial.eval x (Polynomial.derivative Q) = ∑ i ∈ Finset.range (L + 1), Polynomial.eval (Polynomial.Chebyshev.node L i) Q * Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevDerivativeWeight L x i","l":29,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevDerivativeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.eval_pullbackToUnitInterval","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Affine","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"Evaluating the affine pullback at t equals evaluating the original polynomial at the corresponding point of [r,s]. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) (r s t : ℝ), Polynomial.eval t (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.pullbackToUnitInterval Q r s) = Polynomial.eval ((s - r) / 2 * t + (r + s) / 2) Q","l":28,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_alternationDualCertificate","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Alternation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Every continuous real target on a nondegenerate compact interval admits a finite Chebyshev alternation dual certificate for degree-L approximation. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ (L : ℕ), Nonempty (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.AlternationDualCertificate f r s L)","l":50,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Alternation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_bestPolynomial","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.BestApproximation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"A continuous real target on a nondegenerate compact interval has a real polynomial of degree at most L attaining the best uniform error. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ (L : ℕ), ∃ Q, Q.natDegree ≤ L ∧ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.uniformApproxError f r s Q = Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.bestUniformApproxError f r s L","l":74,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/BestApproximation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_chebyshevRootReflection","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevRootReflection","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a degree-at-most-L real polynomial has derivative bounded by T_L' at every real zero of T_L, then at any real base point there is a reflected degree-at-most-L complex polynomial with the same vertical boundary modulus as the translated T_L and with derivative at zero dominating the original derivative at the base point. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (Q : Polynomial ℝ), Q.natDegree ≤ L → (∀ (z : ℝ), Polynomial.eval z (Polynomial.Chebyshev.T ℝ ↑L) = 0 → |Polynomial.eval z (Polynomial.derivative Q)| ≤ |Polynomial.eval z (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|) → ∀ (x : ℝ), ∃ R, R.natDegree ≤ L ∧ (∀ (y : ℝ), ‖Polynomial.eval (↑y * Complex.I) R‖ = ‖Polynomial.eval (↑x + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖) ∧ |Polynomial.eval x (Polynomial.derivative Q)| ≤ ‖Polynomial.eval 0 (Polynomial.derivative R)‖","l":174,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevRootReflection.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_equioscillationWitness","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Equioscillation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Every continuous real target on a nondegenerate compact interval has a best degree-L polynomial whose residual equioscillates at L+2 strictly ordered interval points. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ (L : ℕ), Nonempty (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.EquioscillationWitness f r s L)","l":40,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Equioscillation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_finiteMomentDual","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Alternation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"Every continuous real target on a nondegenerate compact interval admits L+2 ordered, normalized finite weights annihilating moments through degree L and attaining the best approximation error in absolute value. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn f (Set.Icc r s) → ∀ (L : ℕ), Nonempty (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.FiniteMomentDual f r s L)","l":158,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Alternation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_int_abs_sub_mul_pi_div_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.CenteredRemainder","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Every real angle differs from an integer multiple of π/L by at most half of that positive step. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (θ : ℝ), ∃ m, |θ - ↑m * (Real.pi / ↑L)| ≤ Real.pi / (2 * ↑L)","l":17,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/CenteredRemainder.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_rationalFiniteMomentDual","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Rational","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"On a nondegenerate interval avoiding -a, there are L+2 ordered, normalized finite weights that match moments through degree L and whose rational-target separation is the exact best degree-L uniform error. the stated inputs establish the stated conclusion.","s":"∀ {a r s : ℝ}, r < s → -a ∉ Set.Icc r s → ∀ (L : ℕ), Nonempty (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.FiniteMomentDual (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.rationalTarget a) r s L)","l":35,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Rational.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_rootProduct_dominating_abscissa_past_chebyshevZeros","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordGeometry","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Given a positive degree, a point of the unit interval, and a real height, some abscissa in that interval, weakly to the right of all degree-L Chebyshev roots, has at least as large a vertical root-distance product.","s":"∀ {L : ℕ}, 0 < L → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → ∀ (y : ℝ), ∃ x₀ ∈ Set.Icc (-1) 1, (∀ k ∈ Finset.range L, Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k ≤ x₀) ∧ ∏ k ∈ Finset.range L, ‖↑(x - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖ ≤ ∏ k ∈ Finset.range L, ‖↑(x₀ - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖","l":24,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordGeometry.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_signPolynomial_of_no_alternatingExtrema","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ExtremalSignPolynomial","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a continuous residual on a nondegenerate interval has positive supremum norm but no L+2 alternating extrema, some polynomial of degree at most L has the residual's strict sign at every norm-attaining point. the stated inputs establish the stated conclusion.","s":"∀ {g : ℝ → ℝ} {r s : ℝ}, r < s → ContinuousOn g (Set.Icc r s) → ∀ (L : ℕ), 0 < Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s → (¬∃ nodes orientation, Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.IsAlternatingExtrema g r s L nodes orientation) → ∃ Q, Q.natDegree ≤ L ∧ ∀ x ∈ Set.Icc r s, |g x| = Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s → 0 < g x * Polynomial.eval x Q","l":253,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ExtremalSignPolynomial.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_sorted_chebyshevChordList_with_adjacentPairProduct_eq_of_centered","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevChordOrdering","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"At a centered angle |φ| ≤ π/(2L), decreasingly sorting all 2L chord squares can be done without changing the product obtained from the root-angle pairing, for any common additive term. The degree, its positivity, the angle, and the additive term establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ {φ t : ℝ}, |φ| ≤ Real.pi / (2 * ↑L) → ∃ zs, zs.Perm (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList L φ) ∧ List.Pairwise (fun a b => b ≤ a) zs ∧ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct t zs = Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.adjacentPairProduct t (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevPairedChordList L φ)","l":36,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevChordOrdering.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.exists_strict_uniformImprovement","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ExtremalPerturbation","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If a polynomial has the strict sign of a positive-norm continuous residual at every norm-attaining point of a compact interval, then subtracting a sufficiently small positive multiple of that polynomial strictly decreases the residual's uniform norm. the stated inputs establish the stated conclusion.","s":"∀ {g : ℝ → ℝ} {r s : ℝ}, r ≤ s → ContinuousOn g (Set.Icc r s) → ∀ {Q : Polynomial ℝ}, 0 < Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s → (∀ x ∈ Set.Icc r s, |g x| = Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s → 0 < g x * Polynomial.eval x Q) → ∃ t, 0 < t ∧ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => g x - t * Polynomial.eval x Q) r s < Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ExtremalPerturbation.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Basic","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The compact-interval supremum norm of a real function is the supremum of its absolute values on the indicated closed interval. the stated inputs establish the defined object.","s":"(ℝ → ℝ) → ℝ → ℝ → ℝ","l":24,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm_le_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Basic","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"For a continuous function on a nonempty closed interval, its interval supremum norm is at most C exactly when every pointwise absolute value is at most C. the stated inputs establish the stated conclusion.","s":"∀ {g : ℝ → ℝ} {r s C : ℝ}, ContinuousOn g (Set.Icc r s) → r ≤ s → (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm g r s ≤ C ↔ ∀ x ∈ Set.Icc r s, |g x| ≤ C)","l":42,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm_pullbackToUnitInterval","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Affine","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"On a nondegenerate interval, affine pullback preserves the compact supremum norm exactly. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) {r s : ℝ}, r < s → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun t => Polynomial.eval t (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.pullbackToUnitInterval Q r s)) (-1) 1 = Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => Polynomial.eval x Q) r s","l":58,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.markov_derivative_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Markov","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"If r < s and a real polynomial has degree at most L, then its derivative supremum norm on [r,s] is at most 2 L²/(s-r) times the polynomial's supremum norm on that interval. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) {r s : ℝ}, r < s → ∀ (L : ℕ), Q.natDegree ≤ L → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => Polynomial.eval x (Polynomial.derivative Q)) r s ≤ 2 * ↑L ^ 2 / (s - r) * Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => Polynomial.eval x Q) r s","l":20,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Markov.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.markov_derivative_unitInterval","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.MarkovUnit","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"A real polynomial of degree at most L has derivative supremum norm on [-1,1] at most L² times its own supremum norm there. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) (L : ℕ), Q.natDegree ≤ L → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => Polynomial.eval x (Polynomial.derivative Q)) (-1) 1 ≤ ↑L ^ 2 * Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.intervalSupNorm (fun x => Polynomial.eval x Q) (-1) 1","l":24,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/MarkovUnit.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.natDegree_pullbackToUnitInterval_le","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Affine","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"Affine pullback does not increase the natural degree of a real polynomial. the stated inputs establish the stated conclusion.","s":"∀ (Q : Polynomial ℝ) (r s : ℝ), (Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.pullbackToUnitInterval Q r s).natDegree ≤ Q.natDegree","l":36,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.nestedPairProduct_le_adjacentPairProduct","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.PairingRearrangement","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For four decreasing real numbers, replacing the outer/inner pairs (a,d) and (b,c) by the adjacent pairs (a,b) and (c,d) cannot decrease the product after adding the same nonnegative constant to each pair. the stated inputs establish the stated conclusion.","s":"∀ {a b c d t : ℝ}, 0 ≤ t → b ≤ a → c ≤ b → d ≤ c → (a * d + t) * (b * c + t) ≤ (a * b + t) * (c * d + t)","l":43,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/PairingRearrangement.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.norm_eval_chebyshev_eq_leadingCoeff_mul_rootProduct","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevRootProduct","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"For positive degree, the complex norm of a Chebyshev evaluation is its leading-coefficient norm times the product of the distances to the explicit cosine roots. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (x y : ℝ), ‖Polynomial.eval (↑x + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖ = ‖(Polynomial.Chebyshev.T ℂ ↑L).leadingCoeff‖ * ∏ k ∈ Finset.range L, ‖↑(x - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖","l":52,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevRootProduct.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.norm_eval_chebyshev_le_endpoint_vertical","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevVerticalModulus","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Above any point x of [-1,1], the complex modulus of the degree-L Chebyshev polynomial is no larger than its modulus at the point with the same imaginary part above the endpoint 1. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ} {x : ℝ}, x ∈ Set.Icc (-1) 1 → ∀ (y : ℝ), ‖Polynomial.eval (↑x + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖ ≤ ‖Polynomial.eval (1 + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖","l":20,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevVerticalModulus.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.norm_eval_derivative_zero_le_chebyshev_endpoint_of_vertical","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.HalfPlaneBoundaryDerivative","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"A degree-at-most-L complex polynomial dominated on the imaginary axis by z ↦ T_L(1+z) has derivative at zero no larger than |T_L'(1)|. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ (R : Polynomial ℂ), R.natDegree ≤ L → (∀ (y : ℝ), ‖Polynomial.eval (↑y * Complex.I) R‖ ≤ ‖Polynomial.eval (1 + ↑y * Complex.I) (Polynomial.Chebyshev.T ℂ ↑L)‖) → ‖Polynomial.eval 0 (Polynomial.derivative R)‖ ≤ |Polynomial.eval 1 (Polynomial.derivative (Polynomial.Chebyshev.T ℝ ↑L))|","l":197,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/HalfPlaneBoundaryDerivative.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.LagrangeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The normalized signed Lagrange weight at a node is its barycentric nodal weight divided by the sum of the absolute barycentric weights. the stated inputs establish the defined object.","s":"{n : ℕ} → (Fin n → ℝ) → Fin n → ℝ","l":22,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/LagrangeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight_alternates","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.LagrangeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For L+2 strictly increasing real nodes, the normalized Lagrange weight at index i has sign (-1)^(L+1-i). the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ} {nodes : Fin (L + 2) → ℝ}, StrictMono nodes → ∀ (i : Fin (L + 2)), Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight nodes i = (-1) ^ (L + 1 - ↑i) * |Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight nodes i|","l":48,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/LagrangeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.prod_norm_sub_chebyshevZero_le_endpoint","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevRootProduct","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"The product of distances from a point above [-1,1] to the degree-L Chebyshev roots is no larger than the corresponding product above the endpoint 1 at the same height. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → ∀ (y : ℝ), ∏ k ∈ Finset.range L, ‖↑(x - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖ ≤ ∏ k ∈ Finset.range L, ‖↑(1 - Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevZero L k) + ↑y * Complex.I‖","l":34,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevRootProduct.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.pullbackToUnitInterval","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Affine","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"Pull a polynomial on [r,s] back along the affine map sending [-1,1] onto [r,s]. the stated inputs establish the defined object.","s":"Polynomial ℝ → ℝ → ℝ → Polynomial ℝ","l":21,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.rationalTarget","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Rational","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The rational target with parameter a is the function x ↦ x/(x+a). the stated inputs establish the defined object.","s":"ℝ → ℝ → ℝ","l":19,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Rational.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.sum_abs_chebyshevDerivativeWeight_le_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.ChebyshevDerivativeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"On [-1,1], the total absolute mass of the degree-L Chebyshev differentiation weights is at most L² when L is positive. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ}, 0 < L → ∀ {x : ℝ}, x ∈ Set.Icc (-1) 1 → ∑ i ∈ Finset.range (L + 1), |Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.chebyshevDerivativeWeight L x i| ≤ ↑L ^ 2","l":55,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/ChebyshevDerivativeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.sum_abs_normalizedLagrangeWeight_eq_one","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.LagrangeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For L+2 distinct real nodes, the normalized signed Lagrange weights have total absolute mass one. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ} {nodes : Fin (L + 2) → ℝ}, Function.Injective nodes → ∑ i, |Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight nodes i| = 1","l":29,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/LagrangeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.sum_normalizedLagrangeWeight_mul_eval_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.LagrangeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"For L+2 distinct real nodes, the normalized signed Lagrange weights annihilate the values of every real polynomial of degree at most L. the stated inputs establish the stated conclusion.","s":"∀ {L : ℕ} {nodes : Fin (L + 2) → ℝ}, Function.Injective nodes → ∀ {Q : Polynomial ℝ}, Q.natDegree ≤ L → ∑ i, Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight nodes i * Polynomial.eval (nodes i) Q = 0","l":143,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/LagrangeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.sum_normalizedLagrangeWeight_mul_pow_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.LagrangeWeights","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"For L+2 distinct real nodes, the normalized signed Lagrange weights annihilate the monomial x^j whenever j ≤ L. the stated inputs establish the stated conclusion.","s":"∀ {L j : ℕ} {nodes : Fin (L + 2) → ℝ}, Function.Injective nodes → j ≤ L → ∑ i, Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.normalizedLagrangeWeight nodes i * nodes i ^ j = 0","l":118,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/LagrangeWeights.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.uniformApproxError","k":"def","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Basic","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":1,"d":"The uniform error of a polynomial against a real target on a closed interval is the compact-interval supremum norm of their residual. the stated inputs establish the defined object.","s":"(ℝ → ℝ) → ℝ → ℝ → Polynomial ℝ → ℝ","l":29,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.uniformApproxError_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.Basic","a":"Mathlib/Analysis/FinitePolynomialAlternationDuality","t":2,"d":"On a nonempty interval, the uniform approximation error is nonnegative. the stated inputs establish the stated conclusion.","s":"∀ {f : ℝ → ℝ} {r s : ℝ}, r ≤ s → ∀ (Q : Polynomial ℝ), 0 ≤ Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.uniformApproxError f r s Q","l":60,"f":"Causalean/Mathlib/Analysis/FinitePolynomialAlternationDuality/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.HasLocallyBoundedFDerivAt","k":"def","m":"Causalean.Mathlib.Analysis.LocallyBoundedDerivative","a":"Mathlib/Analysis","t":1,"d":"Given a function between real normed vector spaces and a point in its domain, the property of having a locally bounded Fréchet derivative means that one neighborhood of the point has a uniform nonnegative bound on derivative norms.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → [inst : NormedAddCommGroup F] → [NormedSpace ℝ F] → (E → F) → E → Prop","l":14,"f":"Causalean/Mathlib/Analysis/LocallyBoundedDerivative.lean"},{"n":"Causalean.Mathlib.Analysis.IsOrthogonalProjectorOntoTop","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix and a candidate matrix, the property of being the orthogonal projector onto the upper-root eigenspace means that the candidate is symmetric, idempotent, and fixes exactly the upper-root eigenvectors.","s":"Matrix (Fin 2) (Fin 2) ℝ → Matrix (Fin 2) (Fin 2) ℝ → Prop","l":25,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.IsTrigPolyLE","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A frequency limit and a real-valued function determine the assertion that the function is a real trigonometric polynomial within that limit. It is an abbreviation of Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE, re-exposed in this namespace.","s":"ℕ → (ℝ → ℝ) → Prop","l":16,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affineJackson_approx_four","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"For integer order K that is strictly positive, a rectangle center, positive coordinate radii, a function, a nonnegative Lipschitz constant, continuity on the centered rectangle, and the stated Lipschitz bound in normalized coordinates, there is a degree-controlled polynomial that approximates the function throughout the rectangle within one hundred twenty-eight times the Lipschitz constant divided by K.","s":"∀ {K : ℕ}, 0 < K → ∀ (c r : Fin 4 → ℝ), (∀ (i : Fin 4), 0 < r i) → ∀ (f : (Fin 4 → ℝ) → ℝ) (L : ℝ), 0 ≤ L → ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r) → (∀ y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r, ∀ z ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r, |f y - f z| ≤ L * ∑ i, |(y i - z i) / r i|) → ∃ p, (∀ m ∈ p.support, ∀ (i : Fin 4), m i ≤ 2 * (K - 1)) ∧ p.totalDegree ≤ 8 * (K - 1) ∧ ∀ y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r, |(MvPolynomial.eval y) p - f y| ≤ 128 * L / ↑K","l":473,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affineJackson_coeffBound_four","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, a rectangle center, positive coordinate radii, a function, a nonnegative uniform bound, continuity on the centered rectangle, and the corresponding uniform bound throughout the rectangle, the affine Jackson construction has a four-variable polynomial representation with the stated evaluation, support, degree, and exponential coefficient one-norm bounds.","s":"∀ {K : ℕ}, 0 < K → ∀ (c r : Fin 4 → ℝ), (∀ (i : Fin 4), 0 < r i) → ∀ (f : (Fin 4 → ℝ) → ℝ) (B : ℝ), 0 ≤ B → ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r) → (∀ y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r, |f y| ≤ B) → ∃ p q, (∀ (y : Fin 4 → ℝ), (MvPolynomial.eval y) p = (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r y)) q) ∧ (∀ (x : Fin 4 → ℝ), (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint x)) q = Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K (fun z => f (Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint c r z)) x) ∧ (∀ m ∈ p.support, ∀ (i : Fin 4), m i ≤ 2 * (K - 1)) ∧ p.totalDegree ≤ 8 * (K - 1) ∧ Causalean.Mathlib.Analysis.JacksonApproximation.mvCoeffL1 p ≤ 2 ^ (40 * K + 20) * B * ∏ i, max 1 ((1 + |c i|) / r i) ^ (2 * (K - 1))","l":542,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affineJackson_exists_mvPolynomial_four","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, a rectangle center, positive coordinate radii, and a function that is continuous on the centered rectangle, the affine Jackson construction has a four-variable polynomial representation with the stated evaluation, support, and degree bounds.","s":"∀ {K : ℕ}, 0 < K → ∀ (c r : Fin 4 → ℝ), (∀ (i : Fin 4), 0 < r i) → ∀ (f : (Fin 4 → ℝ) → ℝ), ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r) → ∃ p q, (∀ (y : Fin 4 → ℝ), (MvPolynomial.eval y) p = (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r y)) q) ∧ (∀ (x : Fin 4 → ℝ), (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint x)) q = Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K (fun z => f (Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint c r z)) x) ∧ (∀ m ∈ p.support, ∀ (i : Fin 4), m i ≤ 2 * (K - 1)) ∧ p.totalDegree ≤ 8 * (K - 1)","l":435,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension, a rectangle center, coordinate radii, and normalized coordinates determine the corresponding point under the center-plus-radius affine map.","s":"{d : ℕ} → (Fin d → ℝ) → (Fin d → ℝ) → (Fin d → ℝ) → Fin d → ℝ","l":15,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint_mem_centeredRectangle","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, a center, coordinate radii, and normalized coordinates, if every radius is nonnegative and the normalized point lies in the normalized cube, then its affine image lies in the centered rectangle.","s":"∀ {d : ℕ} (c r z : Fin d → ℝ), (∀ (i : Fin d), 0 < r i) → z ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d → Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint c r z ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r","l":61,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint_normalizedPoint","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, a center, coordinate radii, and a point, if every radius is positive, then applying the affine map to the normalized point recovers the original point.","s":"∀ {d : ℕ} (c r y : Fin d → ℝ), (∀ (i : Fin d), 0 < r i) → Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint c r (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r y) = y","l":50,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension, a center, and coordinate radii determine the closed centered rectangle consisting of points whose coordinatewise distance from the center does not exceed the corresponding radius. No sign condition is imposed on the radii; a negative radius makes the set empty, and the approximation results assume positive radii.","s":"{d : ℕ} → (Fin d → ℝ) → (Fin d → ℝ) → Set (Fin d → ℝ)","l":31,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.chebyshev_coeffL1_le","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a nonnegative integer index, the coefficient one-norm of the corresponding first-kind Chebyshev polynomial is at most three to that index.","s":"∀ (n : ℕ), Causalean.Mathlib.Analysis.JacksonApproximation.polyCoeffL1 (Polynomial.Chebyshev.T ℝ ↑n) ≤ 3 ^ n","l":160,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.continuous_jackson","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the normalized Jackson kernel is continuous on the real line.","s":"∀ (K : ℕ), Continuous (Causalean.Mathlib.Analysis.JacksonApproximation.jackson K)","l":621,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.continuous_jraw","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the raw Jackson kernel is continuous on the real line.","s":"∀ (K : ℕ), Continuous (Causalean.Mathlib.Analysis.JacksonApproximation.jraw K)","l":483,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.cosLift","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension, a real function on normalized coordinates, and a point in period coordinates determine the cosine lift of the function by coordinatewise cosine precomposition.","s":"{d : ℕ} → ((Fin d → ℝ) → ℝ) → (Fin d → ℝ) → ℝ","l":52,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension and a point in period coordinates determine the coordinatewise cosine point.","s":"{d : ℕ} → (Fin d → ℝ) → Fin d → ℝ","l":39,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint_mem_normalizedCube","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension and a point in period coordinates, the coordinatewise cosine point lies in the normalized cube.","s":"∀ {d : ℕ} (t : Fin d → ℝ), Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint t ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d","l":44,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.cosine_sum_exists_polynomial","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"Given a frequency limit and real cosine coefficients, the finite cosine sum has an ordinary polynomial representation of bounded degree whose coefficient one-norm is bounded by the stated weighted sum.","s":"∀ (n : ℕ) (a : ℕ → ℝ), ∃ p, p.natDegree ≤ n ∧ (∀ (t : ℝ), Polynomial.eval (Real.cos t) p = ∑ k ∈ Finset.range (n + 1), a k * Real.cos (↑k * t)) ∧ Causalean.Mathlib.Analysis.JacksonApproximation.polyCoeffL1 p ≤ ∑ k ∈ Finset.range (n + 1), |a k| * 3 ^ k","l":212,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.even_trigPoly_exists_polynomial","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"Given a frequency limit and a real function, if the function is a trigonometric polynomial within that limit and is symmetric about zero, then it is an ordinary polynomial in the cosine coordinate with degree at most that limit.","s":"∀ {n : ℕ} {q : ℝ → ℝ}, Causalean.Mathlib.Analysis.JacksonApproximation.IsTrigPolyLE n q → Function.Even q → ∃ p, p.natDegree ≤ n ∧ ∀ (t : ℝ), Polynomial.eval (Real.cos t) p = q t","l":97,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.integrableOn_jackson","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the normalized Jackson kernel is integrable over the standard period.","s":"∀ (K : ℕ), MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.JacksonApproximation.jackson K) (Set.Icc (-Real.pi) Real.pi) MeasureTheory.volume","l":635,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.integrableOn_jraw","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the raw Jackson kernel is integrable over the standard period.","s":"∀ (K : ℕ), MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.JacksonApproximation.jraw K) (Set.Icc (-Real.pi) Real.pi) MeasureTheory.volume","l":505,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.integrableOn_tensorJackson","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K and a finite dimension, the tensor Jackson kernel is integrable over the standard period box.","s":"∀ (K d : ℕ), MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d) (Causalean.Mathlib.Analysis.JacksonApproximation.periodBox d) MeasureTheory.volume","l":86,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.integral_periodBox_prod","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"Given a finite dimension, one integrand for each coordinate, and integrability of every coordinate integrand over the standard period, the integral of their product over the period box equals the product of their one-dimensional integrals.","s":"∀ {d : ℕ} (g : Fin d → ℝ → ℝ), (∀ (i : Fin d), MeasureTheory.IntegrableOn (g i) (Set.Icc (-Real.pi) Real.pi) MeasureTheory.volume) → ∫ (u : Fin d → ℝ) in Causalean.Mathlib.Analysis.JacksonApproximation.periodBox d, ∏ i, g i (u i) = ∏ i, ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, g i t","l":99,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"An integer order and a real argument determine the normalized order-four Jackson kernel, equal to the raw kernel divided by its mass over the standard period. It is normalized for positive order; at order zero the raw kernel and its mass vanish and the value is zero by the division convention.","s":"ℕ → ℝ → ℝ","l":32,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_even","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the normalized Jackson kernel is symmetric about zero.","s":"∀ (K : ℕ), Function.Even (Causalean.Mathlib.Analysis.JacksonApproximation.jackson K)","l":648,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_first_moment","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the normalized absolute first moment is at most thirty-two divided by K.","s":"∀ (K : ℕ), 0 < K → ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, |t| * Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t ≤ 32 / ↑K","l":841,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_integral_eq_one","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the normalized Jackson kernel has unit mass over the standard period.","s":"∀ (K : ℕ), 0 < K → ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t = 1","l":654,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the normalized Jackson kernel is a real trigonometric polynomial with frequency at most twice the predecessor of K.","s":"∀ (K : ℕ), 0 < K → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE (2 * (K - 1)) (Causalean.Mathlib.Analysis.JacksonApproximation.jackson K)","l":966,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"At integer order K that is strictly positive and real argument t, the normalized Jackson kernel is nonnegative.","s":"∀ (K : ℕ), 0 < K → ∀ (t : ℝ), 0 ≤ Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t","l":642,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jackson_second_moment","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"For integer order K that is strictly positive, the normalized second moment is at most sixty-four divided by K squared.","s":"∀ (K : ℕ), 0 < K → ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, t ^ 2 * Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t ≤ 64 / ↑K ^ 2","l":894,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A nonnegative integer order and a real argument determine the raw order-four Jackson kernel, obtained from the fourth power of the Dirichlet sine quotient with its removable singularities filled continuously.","s":"ℕ → ℝ → ℝ","l":18,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"An integer order determines the raw Jackson mass, the integral of the raw kernel over the standard period from minus π to π.","s":"ℕ → ℝ","l":26,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the raw Jackson mass equals two π thirds times K times two K squared plus one.","s":"∀ (K : ℕ), 0 < K → Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass K = 2 * Real.pi / 3 * ↑K * (2 * ↑K ^ 2 + 1)","l":537,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass_lower","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the raw Jackson mass is at least four π thirds times K cubed.","s":"∀ (K : ℕ), 0 < K → 4 * Real.pi / 3 * ↑K ^ 3 ≤ Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass K","l":587,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the raw Jackson mass is strictly positive.","s":"∀ (K : ℕ), 0 < K → 0 < Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass K","l":612,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass_upper","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the raw Jackson mass is at most two π times K cubed.","s":"∀ (K : ℕ), 0 < K → Causalean.Mathlib.Analysis.JacksonApproximation.jrawMass K ≤ 2 * Real.pi * ↑K ^ 3","l":597,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw_eq_of_sin_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"At order K and argument t, if the denominator sine vanishes, then the raw kernel equals K to the fourth power.","s":"∀ (K : ℕ) {t : ℝ}, Real.sin (t / 2) = 0 → Causalean.Mathlib.Analysis.JacksonApproximation.jraw K t = ↑K ^ 4","l":46,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw_eq_of_sin_ne_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"At order K and argument t, if the denominator sine is nonzero, then the raw kernel equals the displayed fourth power of the sine quotient.","s":"∀ (K : ℕ) {t : ℝ}, Real.sin (t / 2) ≠ 0 → Causalean.Mathlib.Analysis.JacksonApproximation.jraw K t = (Real.sin (↑K * t / 2) / Real.sin (t / 2)) ^ 4","l":39,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw_even","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the raw Jackson kernel is symmetric about zero.","s":"∀ (K : ℕ), Function.Even (Causalean.Mathlib.Analysis.JacksonApproximation.jraw K)","l":518,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw_isTrigPolyLE","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K that is strictly positive, the raw Jackson kernel is a real trigonometric polynomial with frequency at most twice the predecessor of K.","s":"∀ (K : ℕ), 0 < K → Causalean.Mathlib.Analysis.BernsteinSzegoTrig.IsTrigPolyLE (2 * (K - 1)) (Causalean.Mathlib.Analysis.JacksonApproximation.jraw K)","l":905,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.jraw_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"At integer order K and real argument t, the raw Jackson kernel is nonnegative.","s":"∀ (K : ℕ) (t : ℝ), 0 ≤ Causalean.Mathlib.Analysis.JacksonApproximation.jraw K t","l":511,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.measurable_jackson","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the normalized Jackson kernel is Borel measurable.","s":"∀ (K : ℕ), Measurable (Causalean.Mathlib.Analysis.JacksonApproximation.jackson K)","l":629,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.measurable_jraw","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Kernel","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K, the raw Jackson kernel is Borel measurable.","s":"∀ (K : ℕ), Measurable (Causalean.Mathlib.Analysis.JacksonApproximation.jraw K)","l":499,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Kernel.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.measurable_tensorJackson","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For integer order K and a finite dimension, the tensor Jackson kernel is measurable.","s":"∀ (K d : ℕ), Measurable (Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d)","l":78,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.mvCoeffL1","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.CoefficientEnvelopeFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension and a real multivariate polynomial determine its coefficient one-norm, the sum of the absolute values of all monomial coefficients.","s":"{d : ℕ} → MvPolynomial (Fin d) ℝ → ℝ","l":16,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/CoefficientEnvelopeFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.mvPolynomial_affine_substitution_four","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"Given a four-variable polynomial, a center, coordinate radii that are strictly positive, a coordinate-degree limit, and the corresponding support bound, affine substitution produces a four-variable polynomial with the stated evaluation, support, total-degree, and coefficient one-norm bounds.","s":"∀ (q : MvPolynomial (Fin 4) ℝ) (c r : Fin 4 → ℝ) (n : ℕ), (∀ (i : Fin 4), 0 < r i) → (∀ m ∈ q.support, ∀ (i : Fin 4), m i ≤ n) → ∃ p, (∀ (y : Fin 4 → ℝ), (MvPolynomial.eval y) p = (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r y)) q) ∧ (∀ m ∈ p.support, ∀ (i : Fin 4), m i ≤ n) ∧ p.totalDegree ≤ q.totalDegree ∧ Causalean.Mathlib.Analysis.JacksonApproximation.mvCoeffL1 p ≤ Causalean.Mathlib.Analysis.JacksonApproximation.mvCoeffL1 q * ∏ i, max 1 ((1 + |c i|) / r i) ^ n","l":240,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension determines the normalized cube, whose coordinates all lie between minus one and one.","s":"(d : ℕ) → Set (Fin d → ℝ)","l":24,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension, a rectangle center, coordinate radii, and a point in the rectangle's ambient space determine its normalized coordinates by subtracting the center and dividing coordinatewise by the radii. This is a normalization for positive radii, as assumed by the approximation results; a zero radius gives coordinate zero by the division convention.","s":"{d : ℕ} → (Fin d → ℝ) → (Fin d → ℝ) → (Fin d → ℝ) → Fin d → ℝ","l":22,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint_affinePoint","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, a center, coordinate radii, and normalized coordinates, if every radius is positive, then normalizing the affine image recovers the original normalized coordinates.","s":"∀ {d : ℕ} (c r z : Fin d → ℝ), (∀ (i : Fin d), 0 < r i) → Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r (Causalean.Mathlib.Analysis.JacksonApproximation.affinePoint c r z) = z","l":39,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint_mem_normalizedCube","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.AffineFour","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, a center, coordinate radii, and a point, if every radius is positive and the point lies in the centered rectangle, then its normalized coordinates lie in the normalized cube.","s":"∀ {d : ℕ} (c r y : Fin d → ℝ), (∀ (i : Fin d), 0 < r i) → y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.centeredRectangle c r → Causalean.Mathlib.Analysis.JacksonApproximation.normalizedPoint c r y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d","l":73,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/AffineFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.periodBox","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension determines the standard period box, whose coordinates all lie between minus π and π.","s":"(d : ℕ) → Set (Fin d → ℝ)","l":18,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.polyCoeffL1","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A real polynomial determines its coefficient one-norm, the sum of the absolute values of its nonzero coefficients.","s":"Polynomial ℝ → ℝ","l":24,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.polynomial_cos_is_even_trigPoly","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TrigExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"Given a frequency limit and a real polynomial whose degree is at most that limit, composition with cosine is an even trigonometric polynomial within the same frequency limit.","s":"∀ {n : ℕ} (p : Polynomial ℝ), p.natDegree ≤ n → (Causalean.Mathlib.Analysis.JacksonApproximation.IsTrigPolyLE n fun t => Polynomial.eval (Real.cos t) p) ∧ Function.Even fun t => Polynomial.eval (Real.cos t) p","l":148,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TrigExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"A finite dimension, an integer kernel order, a real function on normalized coordinates, and a point in period coordinates determine the tensor Jackson convolution, the kernel-weighted integral of the translated cosine lift over the period box. It is a kernel-weighted average for positive order when that integrand is integrable; the integral is Lean's Bochner integral, which is zero for a non-integrable integrand, and at order zero in positive dimension the kernel vanishes and so does the convolution (in dimension zero the kernel is the empty product, equal to one).","s":"{d : ℕ} → ℕ → ((Fin d → ℝ) → ℝ) → (Fin d → ℝ) → ℝ","l":59,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution_approx_lipschitz","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TensorExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"For a finite dimension, integer order K that is strictly positive, a function, a nonnegative Lipschitz constant, continuity on the normalized cube, the stated local coordinatewise Lipschitz bound, and a point in period coordinates, tensor Jackson convolution approximates the cosine lift there within thirty-two times dimension times the Lipschitz constant divided by K.","s":"∀ {d K : ℕ}, 0 < K → ∀ (f : (Fin d → ℝ) → ℝ) (L : ℝ), 0 ≤ L → ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d) → (∀ x ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d, ∀ y ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d, |f x - f y| ≤ L * ∑ i, |x i - y i|) → ∀ (x : Fin d → ℝ), |Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K f x - f (Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint x)| ≤ 32 * ↑d * L / ↑K","l":624,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TensorExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution_const","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, a real constant, and a point in period coordinates, tensor convolution preserves the constant function.","s":"∀ {d K : ℕ}, 0 < K → ∀ (c : ℝ) (x : Fin d → ℝ), Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K (fun x => c) x = c","l":215,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution_exists_mvPolynomial","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.TensorExtraction","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, and a function that is continuous on the normalized cube, its tensor Jackson convolution is represented after the cosine change of coordinates by a multivariate polynomial with the stated coordinate and total-degree bounds.","s":"∀ {d K : ℕ}, 0 < K → ∀ (f : (Fin d → ℝ) → ℝ), ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d) → ∃ p, (∀ (x : Fin d → ℝ), (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint x)) p = Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K f x) ∧ (∀ m ∈ p.support, ∀ (i : Fin d), m i ≤ 2 * (K - 1)) ∧ p.totalDegree ≤ d * (2 * (K - 1))","l":606,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/TensorExtraction.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution_exists_mvPolynomial_four_coeffBound","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.CoefficientEnvelopeFour","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"For integer order K that is strictly positive, a function on four normalized coordinates that is continuous on the normalized cube, a nonnegative uniform bound, and the corresponding bound on the function throughout the cube, its tensor convolution has a four-variable polynomial representation with the stated support, degree, and exponential coefficient one-norm bounds.","s":"∀ {K : ℕ}, 0 < K → ∀ (f : (Fin 4 → ℝ) → ℝ) (B : ℝ), 0 ≤ B → ContinuousOn f (Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube 4) → (∀ z ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube 4, |f z| ≤ B) → ∃ p, (∀ (x : Fin 4 → ℝ), (MvPolynomial.eval (Causalean.Mathlib.Analysis.JacksonApproximation.cosPoint x)) p = Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K f x) ∧ (∀ m ∈ p.support, ∀ (i : Fin 4), m i ≤ 2 * (K - 1)) ∧ p.totalDegree ≤ 8 * (K - 1) ∧ Causalean.Mathlib.Analysis.JacksonApproximation.mvCoeffL1 p ≤ 2 ^ (40 * K + 20) * B","l":356,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/CoefficientEnvelopeFour.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, a function that is nonnegative throughout the normalized cube, and a selected point in period coordinates, the tensor convolution is nonnegative at that point.","s":"∀ {d K : ℕ}, 0 < K → ∀ {f : (Fin d → ℝ) → ℝ}, (∀ z ∈ Causalean.Mathlib.Analysis.JacksonApproximation.normalizedCube d, 0 ≤ f z) → ∀ (x : Fin d → ℝ), 0 ≤ Causalean.Mathlib.Analysis.JacksonApproximation.tensorConvolution K f x","l":203,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson","k":"def","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"An integer kernel order, a finite dimension, and a point in period coordinates determine the tensor Jackson kernel, the product of the one-dimensional kernels across coordinates. For positive order it is a normalized kernel in every dimension. At order zero it inherits the degeneracy of the one-dimensional kernel in positive dimension (it vanishes), while in dimension zero it is the empty product, equal to one with unit mass, for every order.","s":"ℕ → (d : ℕ) → (Fin d → ℝ) → ℝ","l":30,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson_first_moment_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, and a selected coordinate, the tensor kernel's absolute first moment in that coordinate equals the one-dimensional first moment.","s":"∀ {d K : ℕ}, 0 < K → ∀ (i : Fin d), ∫ (u : Fin d → ℝ) in Causalean.Mathlib.Analysis.JacksonApproximation.periodBox d, |u i| * Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d u = ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, |t| * Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t","l":123,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson_integral_eq_one","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":1,"d":"For a finite dimension and integer order K that is strictly positive, the tensor Jackson kernel has unit mass over the period box.","s":"∀ {d K : ℕ}, 0 < K → ∫ (u : Fin d → ℝ) in Causalean.Mathlib.Analysis.JacksonApproximation.periodBox d, Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d u = 1","l":114,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, and a point in period coordinates, the tensor Jackson kernel is nonnegative.","s":"∀ {d K : ℕ}, 0 < K → ∀ (u : Fin d → ℝ), 0 ≤ Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d u","l":71,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson_second_moment_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.JacksonApproximation.Tensor","a":"Mathlib/Analysis/JacksonApproximation","t":2,"d":"For a finite dimension, integer order K that is strictly positive, and a selected coordinate, the tensor kernel's second moment in that coordinate equals the one-dimensional second moment.","s":"∀ {d K : ℕ}, 0 < K → ∀ (i : Fin d), ∫ (u : Fin d → ℝ) in Causalean.Mathlib.Analysis.JacksonApproximation.periodBox d, u i ^ 2 * Causalean.Mathlib.Analysis.JacksonApproximation.tensorJackson K d u = ∫ (t : ℝ) in Set.Icc (-Real.pi) Real.pi, t ^ 2 * Causalean.Mathlib.Analysis.JacksonApproximation.jackson K t","l":163,"f":"Causalean/Mathlib/Analysis/JacksonApproximation/Tensor.lean"},{"n":"Causalean.Mathlib.Analysis.Matrix.singularValues_mul_mul_transpose_lower_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given finite real left and right matrices, an injective finite real square core, and full-column-rank assumptions for the two rectangular matrices, the last signal singular value of their transpose product is bounded below by the product of the three least signal singular values.","s":"∀ {m : Type u_4} {n : Type u_5} {k : Type u_6} [inst : Fintype m] [inst_1 : Fintype n] [inst_2 : Fintype k] [inst_3 : DecidableEq n] [inst_4 : DecidableEq k] [Nonempty k] (A : Matrix m k ℝ) (D : Matrix k k ℝ) (B : Matrix n k ℝ), Function.Injective ⇑(Matrix.toEuclideanLin A) → Function.Injective ⇑(Matrix.toEuclideanLin D) → Function.Injective ⇑(Matrix.toEuclideanLin B) → (Matrix.toEuclideanLin A).singularValues (Fintype.card k - 1) * (Matrix.toEuclideanLin D).singularValues (Fintype.card k - 1) * (Matrix.toEuclideanLin B).singularValues (Fintype.card k - 1) ≤ (Matrix.toEuclideanLin (A * D * B.transpose)).singularValues (Fintype.card k - 1)","l":175,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator","k":"def","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Definitions","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"Given two real-valued functions, a parameter, and an input point, the affine denominator is given by their displayed linear interpolation at that parameter.","s":"{α : Type u_1} → (α → ℝ) → (α → ℝ) → ℝ → α → ℝ","l":13,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Definitions.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator_reciprocal_eq_tsum","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Affine","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"If the affine denominator at the expansion center is nonzero and the normalized affine perturbation has absolute value below one, then the reciprocal affine denominator equals its centered geometric power series.","s":"∀ {α : Type u_1} {a b : α → ℝ} {t t₀ : ℝ} {x : α}, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x ≠ 0 → |(t - t₀) * (b x - a x) / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x| < 1 → (Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x)⁻¹ = ∑' (n : ℕ), (-(b x - a x) / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x) ^ n / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x * (t - t₀) ^ n","l":177,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator_slope_div_le_inv_radius","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Affine","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"If the parameter-ball radius and separation margin are positive and the affine denominator is uniformly separated from zero throughout that ball and the integration set, then the affine slope divided by its central denominator is bounded by the reciprocal radius.","s":"∀ {α : Type u_1} {K : Set α} {a b : α → ℝ} {t₀ ε r : ℝ}, 0 < r → 0 < ε → (∀ (t : ℝ), |t - t₀| < r → ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|) → ∀ x ∈ K, |(b x - a x) / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x| ≤ r⁻¹","l":68,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator_uniformly_nonzero_near","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Affine","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"If the separation margin is positive, the slope bound is nonnegative, the affine denominator is separated from zero at the reference parameter, and its slope is uniformly bounded on the integration set, then it remains separated from zero by half that margin on an explicit parameter neighborhood.","s":"∀ {α : Type u_1} {K : Set α} {a b : α → ℝ} {t₀ ε L : ℝ}, 0 < ε → 0 ≤ L → (∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x|) → (∀ x ∈ K, |b x - a x| ≤ L) → ∃ r > 0, ∀ (t : ℝ), |t - t₀| < r → ∀ x ∈ K, ε / 2 ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|","l":15,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator_uniformly_nonzero_near_of_slope_div_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Affine","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"If the separation margin is positive, the relative-slope bound is nonnegative, the central affine denominator is uniformly separated from zero, and the slope-to-denominator ratio is uniformly bounded, then the denominator remains separated by half the margin on an explicit parameter neighborhood.","s":"∀ {α : Type u_1} {K : Set α} {a b : α → ℝ} {t₀ ε Q : ℝ}, 0 < ε → 0 ≤ Q → (∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x|) → (∀ x ∈ K, |(b x - a x) / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x| ≤ Q) → ∃ r > 0, ∀ (t : ℝ), |t - t₀| < r → ∀ x ∈ K, ε / 2 ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|","l":117,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator_uniformly_nonzero_on_open_near","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Affine","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"If the parameter set is open, the separation margin is positive, the affine denominator is uniformly separated from zero throughout that set and the integration set, and the reference parameter lies in the open set, then some positive ball around it stays in the parameter set and retains the same uniform separation.","s":"∀ {α : Type u_1} {K : Set α} {a b : α → ℝ} {O : Set ℝ} {ε : ℝ}, IsOpen O → 0 < ε → (∀ t ∈ O, ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|) → ∀ {t₀ : ℝ}, t₀ ∈ O → ∃ r > 0, Metric.ball t₀ r ⊆ O ∧ ∀ t ∈ Metric.ball t₀ r, ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|","l":56,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Affine.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticAt_integral_of_powerSeries_domination","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.PowerSeries","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"Given a finite measure, an integrand family, its coefficient functions, coefficient envelopes, an expansion center and radius, a positive radius, measurable coefficients, nonnegative envelopes, almost-everywhere coefficient domination, a summable radius-weighted envelope, and an almost-everywhere pointwise power-series expansion throughout that radius, integrating the family produces a real-analytic function at the expansion center.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : ℝ → α → ℝ) (c : ℕ → α → ℝ) (M : ℕ → ℝ) (t₀ r : ℝ), 0 < r → (∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (c n) μ) → (∀ (n : ℕ), 0 ≤ M n) → (∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖c n x‖ ≤ M n) → (Summable fun n => M n * r ^ n) → (∀ (t : ℝ), |t - t₀| < r → ∀ᵐ (x : α) ∂μ, f t x = ∑' (n : ℕ), c n x * (t - t₀) ^ n) → AnalyticAt ℝ (fun t => ∫ (x : α), f t x ∂μ) t₀","l":57,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/PowerSeries.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticAt_setIntegral_polynomial_div_affine","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Main","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"Given a measure, a measurable integration set of finite measure, a polynomial degree, coefficient functions, and affine endpoint functions, an expansion center, positive separation margin, nonnegative slope bound, and coefficient bounds, positivity and nonnegativity of those bounds, measurability of the coefficients and endpoints, uniform coefficient and slope bounds on the integration set, and uniform separation of the central denominator from zero, the polynomial-over-affine set integral is real analytic at the expansion center.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {K : Set α}, MeasurableSet K → μ K ≠ ⊤ → ∀ (N : ℕ) (c : Fin (N + 1) → α → ℝ) (a b : α → ℝ) (t₀ ε L : ℝ) (C : Fin (N + 1) → ℝ), 0 < ε → 0 ≤ L → (∀ (i : Fin (N + 1)), Measurable (c i)) → Measurable a → Measurable b → (∀ (i : Fin (N + 1)), ∀ x ∈ K, |c i x| ≤ C i) → (∀ x ∈ K, |b x - a x| ≤ L) → (∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x|) → AnalyticAt ℝ (fun t => ∫ (x : α) in K, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator N c t x / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x ∂μ) t₀","l":17,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Main.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticAt_setIntegral_polynomial_div_affine_of_slope_div_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Main","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"Given a measure, a measurable integration set of finite measure, a polynomial degree, coefficient functions, and affine endpoint functions, an expansion center, positive separation margin, nonnegative relative-slope bound, and coefficient bounds, positivity and nonnegativity of those bounds, measurability of the coefficients and endpoints, a uniform coefficient bound, a uniform relative affine-slope bound, and uniform central denominator separation, the polynomial-over-affine set integral is real analytic at the expansion center.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {K : Set α}, MeasurableSet K → μ K ≠ ⊤ → ∀ (N : ℕ) (c : Fin (N + 1) → α → ℝ) (a b : α → ℝ) (t₀ ε Q : ℝ) (C : Fin (N + 1) → ℝ), 0 < ε → 0 ≤ Q → (∀ (i : Fin (N + 1)), Measurable (c i)) → Measurable a → Measurable b → (∀ (i : Fin (N + 1)), ∀ x ∈ K, |c i x| ≤ C i) → (∀ x ∈ K, |(b x - a x) / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x| ≤ Q) → (∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t₀ x|) → AnalyticAt ℝ (fun t => ∫ (x : α) in K, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator N c t x / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x ∂μ) t₀","l":190,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Main.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticAt_setIntegral_polynomial_div_affine_of_uniform_nonzero_near","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Main","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"Given a measure, a measurable integration set of finite measure, a polynomial degree, coefficient functions, and affine endpoint functions, a center, positive separation margin and radius, and coefficient bounds, positive numerical bounds, measurability of the coefficients and endpoints, a uniform coefficient bound, and uniform denominator separation throughout the parameter ball, the polynomial-over-affine set integral is real analytic at the ball center.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {K : Set α}, MeasurableSet K → μ K ≠ ⊤ → ∀ (N : ℕ) (c : Fin (N + 1) → α → ℝ) (a b : α → ℝ) (t₀ ε r : ℝ) (C : Fin (N + 1) → ℝ), 0 < ε → 0 < r → (∀ (i : Fin (N + 1)), Measurable (c i)) → Measurable a → Measurable b → (∀ (i : Fin (N + 1)), ∀ x ∈ K, |c i x| ≤ C i) → (∀ (t : ℝ), |t - t₀| < r → ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|) → AnalyticAt ℝ (fun t => ∫ (x : α) in K, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator N c t x / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x ∂μ) t₀","l":359,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Main.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticOnNhd_integral_one_add_tx_div_two_add_tx","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Examples","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"The scalar integral of one plus the parameter times the integration variable, divided by two plus that product, is real analytic on the open interval from minus one to one.","s":"AnalyticOnNhd ℝ (fun t => ∫ (x : ℝ) in Set.Icc 0 1, (1 + t * x) / (2 + t * x)) (Set.Ioo (-1) 1)","l":14,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Examples.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticOnNhd_setIntegral_polynomial_div_affine","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Main","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"Given a measure, a measurable integration set of finite measure, a polynomial degree, coefficient functions, and affine endpoint functions, an open parameter set, positive separation margin, nonnegative slope bound, and coefficient bounds, openness and valid numerical bounds, measurability of the coefficients and endpoints, uniform coefficient and slope bounds on the integration set, and uniform denominator separation over the parameter and integration sets, the polynomial-over-affine set integral is real analytic throughout the open parameter set.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {K : Set α}, MeasurableSet K → μ K ≠ ⊤ → ∀ (N : ℕ) (c : Fin (N + 1) → α → ℝ) (a b : α → ℝ) (O : Set ℝ) (ε L : ℝ) (C : Fin (N + 1) → ℝ), IsOpen O → 0 < ε → 0 ≤ L → (∀ (i : Fin (N + 1)), Measurable (c i)) → Measurable a → Measurable b → (∀ (i : Fin (N + 1)), ∀ x ∈ K, |c i x| ≤ C i) → (∀ x ∈ K, |b x - a x| ≤ L) → (∀ t ∈ O, ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|) → AnalyticOnNhd ℝ (fun t => ∫ (x : α) in K, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator N c t x / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x ∂μ) O","l":172,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Main.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.analyticOnNhd_setIntegral_polynomial_div_affine_of_uniform_nonzero","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Main","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"Given a measure, a measurable integration set of finite measure, a polynomial degree, coefficient functions, and affine endpoint functions, an open parameter set, a positive separation margin, and coefficient bounds, openness and positivity, measurability of the coefficients and endpoints, a uniform coefficient bound, and uniform denominator separation throughout the parameter and integration sets, the polynomial-over-affine set integral is real analytic throughout the open parameter set.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {K : Set α}, MeasurableSet K → μ K ≠ ⊤ → ∀ (N : ℕ) (c : Fin (N + 1) → α → ℝ) (a b : α → ℝ) (O : Set ℝ) (ε : ℝ) (C : Fin (N + 1) → ℝ), IsOpen O → 0 < ε → (∀ (i : Fin (N + 1)), Measurable (c i)) → Measurable a → Measurable b → (∀ (i : Fin (N + 1)), ∀ x ∈ K, |c i x| ≤ C i) → (∀ t ∈ O, ∀ x ∈ K, ε ≤ |Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x|) → AnalyticOnNhd ℝ (fun t => ∫ (x : α) in K, Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator N c t x / Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.affineDenominator a b t x ∂μ) O","l":381,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Main.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.integral_eq_tsum_of_powerSeries_domination","k":"theorem","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.PowerSeries","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":2,"d":"Given a finite measure, an integrand family, its coefficient functions, coefficient envelopes, an expansion center, radius, and evaluation parameter, a positive radius and an evaluation inside it, measurable coefficients, nonnegative envelopes, almost-everywhere coefficient domination, a summable radius-weighted envelope, and an almost-everywhere pointwise power-series expansion, the integral equals the series of coefficient integrals at the evaluation parameter.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : ℝ → α → ℝ) (c : ℕ → α → ℝ) (M : ℕ → ℝ) (t₀ r t : ℝ), 0 < r → |t - t₀| < r → (∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (c n) μ) → (∀ (n : ℕ), 0 ≤ M n) → (∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖c n x‖ ≤ M n) → (Summable fun n => M n * r ^ n) → (∀ (s : ℝ), |s - t₀| < r → ∀ᵐ (x : α) ∂μ, f s x = ∑' (n : ℕ), c n x * (s - t₀) ^ n) → ∫ (x : α), f t x ∂μ = ∑' (n : ℕ), (∫ (x : α), c n x ∂μ) * (t - t₀) ^ n","l":16,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/PowerSeries.lean"},{"n":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.polynomialNumerator","k":"def","m":"Causalean.Mathlib.Analysis.ParametricRationalIntegralAnalyticity.Definitions","a":"Mathlib/Analysis/ParametricRationalIntegralAnalyticity","t":1,"d":"Given a degree bound, coefficient functions, a parameter, and an input point, the polynomial numerator is given by the displayed finite polynomial evaluated at that parameter and point.","s":"{α : Type u_1} → (N : ℕ) → (Fin (N + 1) → α → ℝ) → ℝ → α → ℝ","l":18,"f":"Causalean/Mathlib/Analysis/ParametricRationalIntegralAnalyticity/Definitions.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"A real matrix whose n columns are vectors with p coordinates. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → Type","l":20,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The ordered multi-index space used for a degree-d tensor-power lift. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → Type","l":26,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"A finite matrix has orthonormal columns when its transpose times itself is the identity. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → [Fintype κ] → [DecidableEq κ] → Matrix ι κ ℝ → Prop","l":15,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"A real scalar family, over an arbitrary index type, has pairwise gap at least gap. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → (ι → ℝ) → ℝ → Prop","l":19,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Two finite matrices have the same column space when their associated Euclidean linear maps have equal ranges. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → [Fintype κ] → [DecidableEq κ] → Matrix ι κ ℝ → Matrix ι κ ℝ → Prop","l":20,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"An order-r real tensor on a p-dimensional coordinate space, represented as an array. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → Type","l":23,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"A finite real vector with p coordinates. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → Type","l":17,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.abs_trace_le_card_mul_operatorNorm","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Projectors","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The trace of a real square matrix is at most the dimension times its Euclidean operator norm in absolute value. The stated conclusion follows.","s":"∀ {n : ℕ} (A : Matrix (Fin n) (Fin n) ℝ), |A.trace| ≤ ↑n * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm A","l":18,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Projectors.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.blockIndex","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Concatenate two degree-d indices and one degree-q index into an order-d+d+q index. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p d q : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d → Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d → (Fin q → Fin p) → Fin (d + d + q) → Fin p","l":67,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedDenominator_leastSingularValue","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Given a factor matrix, its coefficients, a denominator probe, orthonormal lifted coordinates, positive degree and margins, a lifted singular-value margin, coefficient lower bounds, and probe-loading lower bounds, the compressed denominator contraction has least singular value at least the assembled margin. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d q : ℕ} [NeZero n] (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (lam : Fin n → ℝ) (u : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ) {sigma kappa : ℝ}, 0 < q → 0 < sigma → 0 < kappa → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace U (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → (∀ (j : Fin n), kappa ≤ |lam j|) → (∀ (j : Fin n), sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j)) → kappa * sigma ^ (q + 2) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue ((U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C * Matrix.diagonal fun j => lam j * Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j) ^ q) * (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C).transpose)","l":293,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedLift_condition_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"A degree lift with unit columns and least singular value at least sigma has compressed diagonalizer condition number at most sqrt n / sigma. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ} [NeZero n] (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ) {sigma : ℝ}, 0 < d → 0 < sigma → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace U (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C)⁻¹ ≤ √↑n / sigma","l":272,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedLift_inverse_operatorNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The inverse of the square coordinate matrix of a well-conditioned degree lift has operator norm at most the reciprocal of the lifted least-singular-value margin. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ} [NeZero n] (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ) {sigma : ℝ}, 0 < sigma → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace U (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C)⁻¹ ≤ sigma⁻¹","l":260,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedLift_leastSingularValue","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"If a lifted factor has least singular value at least sigma, then its coordinates in any orthonormal basis of its column space have the same lower singular-value bound. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ} [NeZero n] (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ) {sigma : ℝ}, 0 < sigma → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace U (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C)","l":222,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedLift_operatorNorm_le_sqrt","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The square coordinate matrix of a degree lift with unit columns has operator norm at most the square root of the number of factor columns. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ} (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ), 0 < d → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) ≤ √↑n","l":237,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.compressedNumerator_operatorNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Every compressed numerator contraction has operator norm at most n * Lambda when the factor columns and the contraction probe are unit and the coefficients are bounded by Lambda. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d q : ℕ} (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (lam : Fin n → ℝ) (u w : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) (U : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ) {Lambda : ℝ}, 0 < d → 0 < q → 0 ≤ Lambda → Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm u = 1 → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm w = 1 → (∀ (j : Fin n), |lam j| ≤ Lambda) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast (Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C lam) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilProbes u w) * U) ≤ ↑n * Lambda","l":379,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Contract the final q modes of an order-d+d+q tensor against a family of q probes. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p d q : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor p (d + d + q) → (Fin q → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) → Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) ℝ","l":72,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast_decompositionTensor","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Contracting a weighted symmetric rank-one decomposition against q-1 copies of u and one copy of w gives the lifted factor matrix times the corresponding diagonal loadings times its transpose. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d q : ℕ}, 0 < q → ∀ (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (lam : Fin n → ℝ) (u w : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p), Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast (Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C lam) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilProbes u w) = (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C * Matrix.diagonal fun j => lam j * Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j) ^ (q - 1) * Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot w (Matrix.col C j)) * (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C).transpose","l":192,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast_frobeniusNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Contracting a finite tensor against probes of norm at most one cannot increase its Frobenius norm. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p d q : ℕ} (T : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor p (d + d + q)) (probes : Fin q → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p), (∀ (a : Fin q), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (probes a) ≤ 1) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast T probes) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm T","l":128,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast_operatorNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The operator norm of a contracted tensor is bounded by the original tensor's Frobenius norm when every contraction probe has norm at most one. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p d q : ℕ} (T : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor p (d + d + q)) (probes : Fin q → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p), (∀ (a : Fin q), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (probes a) ≤ 1) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractLast T probes) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm T","l":181,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractedMargin","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The contracted denominator singular-value margin assembled from coefficient, probe, and lifted-factor margins. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℝ → ℝ → ℝ","l":15,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The coordinate projector selected by coordinate j of a diagonalizer S: S times the j-th standard diagonal unit matrix times the inverse of S. It is the rank-one spectral projector when S is invertible; for singular S the inverse is zero by convention and the value is the zero matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{n : ℕ} → Matrix (Fin n) (Fin n) ℝ → Fin n → Matrix (Fin n) (Fin n) ℝ","l":30,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector_operatorNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Every coordinate projector of a diagonalizer has Euclidean operator norm at most the declared condition-number envelope. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} (S : Matrix (Fin n) (Fin n) ℝ) (j : Fin n) {chi : ℝ}, IsUnit S.det → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S⁻¹ ≤ chi → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S j) ≤ chi","l":38,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector_perturbation_of_matched_eigenvalue","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Projectors","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Matched simple eigenvalues of two nearby diagonalizable matrices have close coordinate projectors, and the perturbed projector remains controlled by twice the reference condition envelope. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (S S' : Matrix (Fin n) (Fin n) ℝ) (values values' : Fin n → ℝ) (j j' : Fin n) {sigma chi delta : ℝ}, IsUnit S.det → IsUnit S'.det → 0 < sigma → 0 < chi → 0 ≤ delta → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap values sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap values' sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S⁻¹ ≤ chi → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S' values' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S values) ≤ delta → |values' j' - values j| ≤ chi * delta → 6 * chi * delta < sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S' j' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S j) ≤ 6 * chi ^ 2 * delta / sigma ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S j) ≤ chi ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S' j') ≤ 2 * chi","l":437,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Projectors.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The symmetric tensor represented by weighted rank-one tensor powers of the columns. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p n : ℕ} → (r : ℕ) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n → (Fin n → ℝ) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor p r","l":57,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The conjugation of the diagonal matrix of the prescribed real entries by S, that is S times the diagonal times the inverse of S. It is diagonalized by S with those entries when S is invertible; for singular S the inverse is zero by convention and the value is the zero matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{n : ℕ} → Matrix (Fin n) (Fin n) ℝ → (Fin n → ℝ) → Matrix (Fin n) (Fin n) ℝ","l":23,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix_eigenvalue_localization","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Every prescribed eigenvalue of a nearby explicitly diagonalized matrix lies within chi * delta of some eigenvalue of the reference diagonalization. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (S S' : Matrix (Fin n) (Fin n) ℝ) (values values' : Fin n → ℝ) {chi delta : ℝ}, IsUnit S.det → IsUnit S'.det → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S⁻¹ ≤ chi → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S' values' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S values) ≤ delta → ∀ (j' : Fin n), ∃ j, |values' j' - values j| ≤ chi * delta","l":85,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The Euclidean dot product of two finite real vectors. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → ℝ","l":50,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.exists_permutation_factorMatrix_frobeniusNorm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Quantitative symmetric-tensor-pencil local inverse. Given two finite factor matrices, their coefficient vectors, two probes, the gap and coefficient margins, positive ambient dimensions and degrees, an admissible singular-value margin, positive and compatible coefficient bounds, unit probes, unit factor columns, two-sided coefficient bounds, positive denominator loadings, well-conditioned lifted directions, separated pencil ratios, and a tensor perturbation below the explicit local radius, one column permutation makes the factor-matrix Frobenius error at most the stated Lipschitz factor times the tensor Frobenius error. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d q : ℕ} (C C' : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (lam lam' : Fin n → ℝ) (u v : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) (sigma kappa Lambda : ℝ), 0 < p → 0 < n → 0 < d → 0 < q → 0 < sigma ∧ sigma ≤ 1 → 0 < kappa → kappa ≤ Lambda → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm u = 1 → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm v = 1 → ((∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) ∧ ∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C' j) = 1) → ((∀ (j : Fin n), kappa ≤ |lam j| ∧ |lam j| ≤ Lambda) ∧ ∀ (j : Fin n), kappa ≤ |lam' j| ∧ |lam' j| ≤ Lambda) → ((∀ (j : Fin n), sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j)) ∧ ∀ (j : Fin n), sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C' j)) → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) ∧ sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C') → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap (fun j => Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot v (Matrix.col C j) / Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j)) sigma ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap (fun j => Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot v (Matrix.col C' j) / Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C' j)) sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C' lam' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C lam) < Causalean.Mathlib.Analysis.SymmetricTensorPencil.localInverseRadius n q sigma kappa Lambda → ∃ pi, Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.permuteColumns C' pi - C) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.factorRecoveryConstant p n q sigma kappa Lambda * Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C' lam' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.decompositionTensor (d + d + q) C lam)","l":165,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.exists_permutation_matching_of_localization","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.SpectralMatching","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"One-sided localization between two equally sized finite scalar families becomes a permutation matching when the source family is separated by more than twice the localization radius. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} (values values' : Fin n → ℝ) {gap radius : ℝ}, 0 ≤ radius → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap values' gap → (∀ (j' : Fin n), ∃ j, |values' j' - values j| ≤ radius) → 2 * radius < gap → ∃ pi, ∀ (j : Fin n), |values' (pi j) - values j| ≤ radius","l":198,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/SpectralMatching.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.exists_permutation_projector_matching","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Projectors","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Given two diagonalizers, their simple spectra, invertibility, positive separation, conditioning, and error margins, separation of both spectra, a diagonalizer condition bound, a pencil perturbation bound, and the required small-error condition, one permutation matches the simple spectral projectors with explicit eigenvalue and operator-norm bounds. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (S S' : Matrix (Fin n) (Fin n) ℝ) (values values' : Fin n → ℝ) {sigma chi delta : ℝ}, IsUnit S.det → IsUnit S'.det → 0 < sigma → 0 < chi → 0 ≤ delta → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap values sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.PairwiseGap values' sigma → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S * Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm S⁻¹ ≤ chi → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S' values' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S values) ≤ delta → 6 * chi * delta < sigma → ∃ pi, ∀ (j : Fin n), |values' (pi j) - values j| < sigma / 3 ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S' (pi j) - Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S j) ≤ 6 * chi ^ 2 * delta / sigma ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S j) ≤ chi ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S' (pi j)) ≤ 2 * chi","l":678,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Projectors.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.factorRecoveryConstant","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The final factor-matrix Frobenius Lipschitz factor for the quantitative tensor-pencil inverse. It is a fixed multiple of the trace-recovery factor and inherits its unguarded conventions: it is a valid Lipschitz factor only in the positive-margin regime of the recovery theorems. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → ℕ → ℝ → ℝ → ℝ → ℝ","l":46,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The Euclidean/Frobenius norm of a finite real array. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ℝ","l":29,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm_liftedColumn_eq_one","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Unit columns have unit degree-d tensor lifts for every positive lifting degree. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ}, 0 < d → ∀ (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n), (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) → ∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm ((Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C).col j) = 1","l":263,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm_sub_le_sqrt_mul","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"A uniform coordinatewise error bound gives a Euclidean error bound larger by at most the square root of the number of coordinates. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p : ℕ} (x y : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) {b : ℝ}, 0 ≤ b → (∀ (i : Fin p), |x i - y i| ≤ b) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (x - y) ≤ √↑p * b","l":314,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.injective_of_pos_le_leastColumnSingularValue","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"A positive lower bound for the last column singular value implies full column rank. Under the listed assumptions, the stated conclusion follows.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype ι] [inst_1 : Fintype κ] [inst_2 : DecidableEq κ] [Nonempty κ] (A : Matrix ι κ ℝ) {sigma : ℝ}, 0 < sigma → sigma ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue A → Function.Injective ⇑(Matrix.toEuclideanLin A)","l":157,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.inverse_operatorNorm_le_reciprocal","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"A positive lower bound on the least singular value of a real square matrix makes its determinant a unit and bounds the operator norm of its inverse by the reciprocal margin. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (A : Matrix (Fin n) (Fin n) ℝ) {eta : ℝ}, 0 < eta → eta ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue A → IsUnit A.det ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm A⁻¹ ≤ eta⁻¹","l":27,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.inverse_perturbation_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"If a square matrix with least singular value at least eta is perturbed by at most e < eta / 2, the perturbed matrix stays invertible and its inverse norm is at most 2 / eta. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (A A' : Matrix (Fin n) (Fin n) ℝ) {eta e : ℝ}, 0 < eta → 0 ≤ e → eta ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue A → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (A' - A) ≤ e → e < eta / 2 → IsUnit A'.det ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm A'⁻¹ ≤ 2 / eta ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (A'⁻¹ - A⁻¹) ≤ 2 * e / eta ^ 2","l":70,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The last domain-indexed singular value of a finite real matrix, i.e. the singular value at index one less than the number of columns. For a matrix with no columns the index is zero by natural subtraction; the results using it assume a nonempty column index. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → [Fintype κ] → [DecidableEq κ] → Matrix ι κ ℝ → ℝ","l":43,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue_transpose_mul_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Compressing a full-column-rank matrix in an orthonormal basis of its column space preserves its least column singular value. Under the listed assumptions, the stated conclusion follows.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype ι] [inst_1 : Fintype κ] [DecidableEq ι] [inst_3 : DecidableEq κ] [Nonempty κ] (U V : Matrix ι κ ℝ), Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.SameColumnSpace U V → Function.Injective ⇑(Matrix.toEuclideanLin V) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue (U.transpose * V) = Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue V","l":87,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedConditionEnvelope","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The condition-number envelope for a lifted matrix with n unit columns and least singular value at least sigma, namely the square root of n divided by sigma. The formula is unguarded: it is a condition-number envelope only for positive sigma, and a zero margin gives zero by the division-by-zero convention. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℝ → ℝ","l":20,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The matrix whose columns are the ordered degree-d tensor powers of the factor columns. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p n : ℕ} → (d : ℕ) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n → Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Fin n) ℝ","l":62,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.localInverseRadius","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The admissible tensor perturbation radius for the quantitative tensor-pencil inverse. The formula is unguarded: it is an admissible radius only in the positive-margin regime assumed by the inverse theorem, and degenerate margins give zero by the division-by-zero convention. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → ℝ → ℝ → ℝ → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.localInverse_constants_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"For positive dimensions and admissible positive margins, the contracted margin, lifted condition envelope, pencil constant, local radius, trace constant, and factor constant are all strictly positive. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n q : ℕ}, 0 < p → 0 < n → 0 < q → ∀ {sigma kappa Lambda : ℝ}, 0 < sigma → 0 < kappa → kappa ≤ Lambda → 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractedMargin q sigma kappa ∧ 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedConditionEnvelope n sigma ∧ 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilPerturbationConstant n (Causalean.Mathlib.Analysis.SymmetricTensorPencil.contractedMargin q sigma kappa) Lambda ∧ 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.localInverseRadius n q sigma kappa Lambda ∧ 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceRecoveryConstant n q sigma kappa Lambda ∧ 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.factorRecoveryConstant p n q sigma kappa Lambda","l":122,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The Frobenius norm of a finite real matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → [Fintype κ] → Matrix ι κ ℝ → ℝ","l":33,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm_le_sqrt_mul_of_columns","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"If corresponding columns have Euclidean error at most b, their matrix Frobenius error is at most sqrt n * b. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n : ℕ} (A B : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) {b : ℝ}, 0 ≤ b → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col A j - Matrix.col B j) ≤ b) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm (A - B) ≤ √↑n * b","l":355,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm_liftedDirections","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The Frobenius norm of a lifted factor matrix with unit original columns is the square root of the number of columns. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n d : ℕ}, 0 < d → ∀ (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n), (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Matrix.col C j) = 1) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.liftedDirections d C) = √↑n","l":288,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeFinite","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Normalize a finite real array by its Euclidean norm, with the conventional zero value when the input is zero. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ι → ℝ","l":17,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Normalize a finite coordinate vector by its Euclidean norm, with the conventional zero value when the input is zero. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p","l":22,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec_inv_smul_eq","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Positively rescaling a unit vector and then normalizing returns the original vector. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p : ℕ} (c : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) (a : ℝ), Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm c = 1 → 0 < a → Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec (a⁻¹ • c) = c","l":343,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec_sub_normalizeVec_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Normalization is quantitatively stable away from zero: the distance between normalized vectors is at most twice their original distance divided by the smaller input norm. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p : ℕ} (x y : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p), 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm x → 0 < Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm y → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec x - Causalean.Mathlib.Analysis.SymmetricTensorPencil.normalizeVec y) ≤ 2 * Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (x - y) / min (Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm x) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm y)","l":333,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilPerturbationConstant","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The explicit Lipschitz coefficient for a right pencil whose denominator has lower singular value eta and whose numerator norm is at most n * Lambda. The formula is unguarded: it is a Lipschitz coefficient only for positive eta, and eta = 0 gives zero by the division-by-zero convention. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℝ → ℝ → ℝ","l":20,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilProbes","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The probe family containing q-1 copies of u followed by one copy of w, for positive q; for q = 0 it is the empty family. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p q : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Fin q → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p","l":78,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.permuteColumns","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Permute the columns of a finite factor matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p n : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n → Equiv.Perm (Fin n) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n","l":385,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.rankOneTensor","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The order-r rank-one tensor power of a vector. With its explicit inputs, the defined object is given by the displayed formula.","s":"(r : ℕ) → {p : ℕ} → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Tensor p r","l":53,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.recoverRankOneLift","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Apply a candidate projector to a degree-d rank-one lift and normalize the resulting lifted direction. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p d : ℕ} → Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) ℝ → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d → ℝ","l":129,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.recoverRankOneLift_error_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"A matched nearby projector recovers the normalized rank-one lifted direction with error at most 2 * delta / (1 - delta). Under the listed assumptions, the stated conclusion follows.","s":"∀ {p d : ℕ}, 0 < d → ∀ (P P' : Matrix (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.LiftIndex p d) ℝ) (c : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) {delta : ℝ}, Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm c = 1 → ((P.mulVec fun I => ∏ k, c (I k)) = fun I => ∏ k, c (I k)) → 0 ≤ delta → delta < 1 → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (P' - P) ≤ delta → Causalean.Mathlib.Analysis.SymmetricTensorPencil.finiteFrobeniusNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.recoverRankOneLift P' c - fun I => ∏ k, c (I k)) ≤ 2 * delta / (1 - delta)","l":135,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The right generalized-eigenvalue pencil formed from a numerator and denominator matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{n : ℕ} → Matrix (Fin n) (Fin n) ℝ → Matrix (Fin n) (Fin n) ℝ → Matrix (Fin n) (Fin n) ℝ","l":15,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil_eq_diagonalization","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"Exact contracted factorizations produce a simultaneous diagonalization of every right pencil, with diagonal entries equal to probe-loading ratios. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} (S : Matrix (Fin n) (Fin n) ℝ) (lam den num : Fin n → ℝ), IsUnit S.det → (∀ (j : Fin n), lam j * den j ≠ 0) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil ((S * Matrix.diagonal fun j => lam j * num j) * S.transpose) ((S * Matrix.diagonal fun j => lam j * den j) * S.transpose) = (S * Matrix.diagonal fun j => num j / den j) * S⁻¹","l":194,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil_perturbation_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Pencil","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Given reference and perturbed numerator and denominator contractions, a positive denominator margin, nonnegative numerator and error bounds, a denominator singular-value bound, a numerator norm bound, numerator and denominator perturbation bounds, and a small-error condition, both denominator matrices are invertible and the two right generalized-eigenvalue pencils differ by the explicit Lipschitz bound. Under the listed assumptions, the stated conclusion follows.","s":"∀ {n : ℕ} [NeZero n] (Aw Au Aw' Au' : Matrix (Fin n) (Fin n) ℝ) {eta Lambda e : ℝ}, 0 < eta → 0 ≤ Lambda → 0 ≤ e → eta ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.leastColumnSingularValue Au → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm Aw ≤ ↑n * Lambda → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Aw' - Aw) ≤ e → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Au' - Au) ≤ e → e < eta / 2 → IsUnit Au.det ∧ IsUnit Au'.det ∧ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil Aw' Au' - Causalean.Mathlib.Analysis.SymmetricTensorPencil.rightPencil Aw Au) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.pencilPerturbationConstant n eta Lambda * e","l":148,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Pencil.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The Euclidean operator norm of a finite real square matrix. With its explicit inputs, the defined object is given by the displayed formula.","s":"{ι : Type u_1} → [Fintype ι] → [DecidableEq ι] → Matrix ι ι ℝ → ℝ","l":38,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm_compress_le","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Conditioning","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"An orthonormal compression cannot increase the Euclidean operator norm of a square matrix. Under the listed assumptions, the stated conclusion follows.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype ι] [inst_1 : Fintype κ] [inst_2 : DecidableEq ι] [inst_3 : DecidableEq κ] (U : Matrix ι κ ℝ) (A : Matrix ι ι ℝ), Causalean.Mathlib.Analysis.SymmetricTensorPencil.OrthonormalColumns U → Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (U.transpose * A * U) ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm A","l":133,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Conditioning.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm_le_matrixFrobenius","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Basic","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"The Euclidean operator norm of a finite real square matrix is at most its entrywise Frobenius norm. The stated conclusion follows.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (A : Matrix ι ι ℝ), Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm A ≤ Causalean.Mathlib.Analysis.SymmetricTensorPencil.matrixFrobeniusNorm A","l":104,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.standardBasis","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The standard coordinate vector in a finite real coordinate space. With its explicit inputs, the defined object is given by the displayed formula.","s":"(p : ℕ) → Fin p → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p","l":13,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Trace coordinates evaluate each coordinate pencil against the j-th matrix of a supplied matrix family, by the trace of their product. No condition is imposed on the family; in the recovery argument it is a family of rank-one spectral projectors. With its explicit inputs, the defined object is given by the displayed formula.","s":"{p n : ℕ} → (Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Matrix (Fin n) (Fin n) ℝ) → (Fin n → Matrix (Fin n) (Fin n) ℝ) → Fin n → Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p","l":203,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates_diagonalization","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":2,"d":"For an exact simultaneous diagonalization, trace coordinates recover a factor column divided by its positive denominator-probe loading. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n : ℕ} (C : Causalean.Mathlib.Analysis.SymmetricTensorPencil.FactorMatrix p n) (u : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p) (S : Matrix (Fin n) (Fin n) ℝ), IsUnit S.det → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j) ≠ 0) → Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates (fun w => Causalean.Mathlib.Analysis.SymmetricTensorPencil.diagonalizableMatrix S fun j => Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot w (Matrix.col C j) / Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j)) (Causalean.Mathlib.Analysis.SymmetricTensorPencil.coordinateProjector S) = fun j => (Causalean.Mathlib.Analysis.SymmetricTensorPencil.dot u (Matrix.col C j))⁻¹ • Matrix.col C j","l":236,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates_perturbation_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Recovery","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"Given reference and perturbed coordinate pencils, reference and perturbed matrix families (arbitrary matrices; projectors in the intended use), coordinate-pencil error bounds, projector error bounds, perturbed-projector norm bounds, and reference-pencil norm bounds, every trace coordinate obeys the explicit product error bound. Under the listed assumptions, the stated conclusion follows.","s":"∀ {p n : ℕ} (G G' : Causalean.Mathlib.Analysis.SymmetricTensorPencil.Vec p → Matrix (Fin n) (Fin n) ℝ) (P P' : Fin n → Matrix (Fin n) (Fin n) ℝ) {pencilError projectorError projectorNorm pencilNorm : ℝ}, (∀ (i : Fin p), Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (G' (Causalean.Mathlib.Analysis.SymmetricTensorPencil.standardBasis p i) - G (Causalean.Mathlib.Analysis.SymmetricTensorPencil.standardBasis p i)) ≤ pencilError) → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (P' j - P j) ≤ projectorError) → (∀ (j : Fin n), Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (P' j) ≤ projectorNorm) → (∀ (i : Fin p), Causalean.Mathlib.Analysis.SymmetricTensorPencil.squareOperatorNorm (G (Causalean.Mathlib.Analysis.SymmetricTensorPencil.standardBasis p i)) ≤ pencilNorm) → ∀ (i : Fin p) (j : Fin n), |Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates G' P' j i - Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceCoordinates G P j i| ≤ ↑n * (pencilError * projectorNorm + pencilNorm * projectorError)","l":255,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Recovery.lean"},{"n":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.traceRecoveryConstant","k":"def","m":"Causalean.Mathlib.Analysis.SymmetricTensorPencil.Main","a":"Mathlib/Analysis/SymmetricTensorPencil","t":1,"d":"The coordinatewise trace-recovery Lipschitz factor for the quantitative tensor-pencil inverse. The formula is unguarded: it is a valid Lipschitz factor only in the positive-margin regime assumed by the recovery theorems, and degenerate margins give zero by the division-by-zero convention. With its explicit inputs, the defined object is given by the displayed formula.","s":"ℕ → ℕ → ℝ → ℝ → ℝ → ℝ","l":36,"f":"Causalean/Mathlib/Analysis/SymmetricTensorPencil/Main.lean"},{"n":"Causalean.Mathlib.Analysis.UniformC1On","k":"def","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"Uniform C¹ convergence on a real set means uniform convergence of both function values and their first within-derivatives.","s":"Set ℝ → (ℕ → ℝ → ℝ) → (ℝ → ℝ) → Prop","l":26,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.UniformlyOn","k":"def","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"A sequence of real-valued functions converges uniformly on K when every positive error tolerance eventually controls all points of K.","s":"{α : Type u_1} → Set α → (ℕ → α → ℝ) → (α → ℝ) → Prop","l":21,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.WaldInput","k":"def","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":1,"d":"Given a finite outcome-coordinate type, an input to the rank-one Wald functional consists of a real two-row matrix, an outcome direction, and a two-dimensional target direction.","s":"Type u_2 → Type u_2","l":17,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide","k":"inductive","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"The type of radial sides adjacent to the unit circle has an inside alternative, representing its interior, and an outside alternative, representing its strict exterior.","s":"Type","l":31,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide.measurableSet_region","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"For a chosen radial side, its radial region is Lebesgue measurable.","s":"∀ (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide), MeasurableSet side.region","l":64,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide.region","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"The region associated with a radial side is the closed interior of the unit circle for the inside side, and the strict exterior of the unit circle for the outside side.","s":"Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide → Set Causalean.Mathlib.Analysis.WeightedCircularTube.Plane","l":37,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.CoversUnitCircle","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a finite set of planar sites and a real radius, the property of covering the unit circle means that every point of the unit circle belongs to an open ball of radius $r$ centred at one of the sites.","s":"Finset Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ → Prop","l":29,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.IsMaximalSeparated","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a finite set of planar sites and a real separation scale, the property of being a maximal separated set on the unit circle means that the sites lie on the unit circle, are separated by at least that scale, and are maximal among all sets with both properties.","s":"Finset Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ → Prop","l":25,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.IsSeparatedAtLeast","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a real separation scale and a set of planar points, the property of being separated at least at that scale means that every two distinct points in the set are at distance at least the separation scale.","s":"ℝ → Set Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → Prop","l":21,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.Plane","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"The Euclidean plane is represented by the complex numbers, equipped with their standard two-dimensional Euclidean metric and Lebesgue measure.","s":"Type","l":20,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.annularNormalizer","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a real exponent and a real tube width, the weighted annular normalizer is the two-dimensional Lebesgue integral of the radial power weight with that exponent over the annular tube of that width. The integral is the real-valued Bochner integral, which is zero when the weight is not integrable there; the results below use it for exponents above two, where it is integrable.","s":"ℝ → ℝ → ℝ","l":320,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.annularNormalizer_finite_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two and the tube width is positive and below one, the radial power weight is integrable on that annular tube and its weighted normalizer is strictly positive.","s":"∀ {κ δ : ℝ}, 2 < κ → 0 < δ → δ < 1 → MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ) (Causalean.Mathlib.Analysis.WeightedCircularTube.annularTube δ) MeasureTheory.volume ∧ 0 < Causalean.Mathlib.Analysis.WeightedCircularTube.annularNormalizer κ δ","l":324,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.annularTube","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a real tube width, the annular tube is the set of planar points whose absolute radial distance from the unit circle is strictly less than $\\delta$.","s":"ℝ → Set Causalean.Mathlib.Analysis.WeightedCircularTube.Plane","l":296,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.continuous_powerWeight","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the radial power weight is continuous on the entire Euclidean plane, including the unit circle.","s":"∀ {κ : ℝ}, 2 < κ → Continuous (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ)","l":78,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.dist_mul_one_of_mem_unitCircle","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"If a complex multiplier lies on the unit circle, then multiplication of a planar point by it preserves its distance from the reference point one.","s":"∀ {u : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane}, u ∈ Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle → ∀ (z : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane), dist (u * z) u = dist z 1","l":120,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.instDecidableEqCircleSide","k":"instance","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"","s":"DecidableEq Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide","l":35,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.integrableOn_powerWeight_annularTube","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two and the tube width is below one, the radial power weight is integrable over that annular tube.","s":"∀ {κ δ : ℝ}, 2 < κ → δ < 1 → MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ) (Causalean.Mathlib.Analysis.WeightedCircularTube.annularTube δ) MeasureTheory.volume","l":303,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.integrableOn_powerWeight_sideBall","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the radial power weight is integrable over the one-sided ball determined by the chosen side, center, and radius.","s":"∀ {κ : ℝ}, 2 < κ → ∀ (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane) (h : ℝ), MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ) (Causalean.Mathlib.Analysis.WeightedCircularTube.sideBall side x h) MeasureTheory.volume","l":101,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.isCompact_unitCircle","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"The unit circle is compact in the Euclidean plane.","s":"IsCompact Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle","l":55,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.measurableSet_annularTube","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"For a tube width, the corresponding annular tube is Lebesgue measurable.","s":"∀ (δ : ℝ), MeasurableSet (Causalean.Mathlib.Analysis.WeightedCircularTube.annularTube δ)","l":299,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.measurableSet_sideBall","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"For a chosen side, a planar center, and a radius, the corresponding one-sided ball is Lebesgue measurable.","s":"∀ (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane) (h : ℝ), MeasurableSet (Causalean.Mathlib.Analysis.WeightedCircularTube.sideBall side x h)","l":72,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.measurable_powerWeight","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the radial power weight is Lebesgue measurable.","s":"∀ {κ : ℝ}, 2 < κ → Measurable (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ)","l":84,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a real exponent parameter and a planar point, the radial power weight is the absolute radial offset raised to the power $κ-2$. On the unit circle itself the offset is zero, and zero to a negative power is zero by convention, so for exponents below two the weight is zero there rather than infinite; this affects only a Lebesgue-null set.","s":"ℝ → Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ","l":43,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the radial power weight at a planar point vanishes exactly when that point is on the unit circle.","s":"∀ {κ : ℝ}, 2 < κ → ∀ (z : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane), Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ z = 0 ↔ z ∈ Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle","l":94,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"For an exponent parameter and a planar point, the radial power weight is nonnegative.","s":"∀ (κ : ℝ) (z : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane), 0 ≤ Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ z","l":89,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.radialOffset","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a planar point, its signed radial offset is its Euclidean distance from the origin minus one; it is negative inside the unit circle and positive outside it.","s":"Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.radialOffset_mul_of_mem_unitCircle","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"If a complex multiplier lies on the unit circle, then multiplying any planar point by it preserves signed radial offset.","s":"∀ {u : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane}, u ∈ Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle → ∀ (z : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane), Causalean.Mathlib.Analysis.WeightedCircularTube.radialOffset (u * z) = Causalean.Mathlib.Analysis.WeightedCircularTube.radialOffset z","l":113,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.reference_powerWeighted_sideBall_bounds","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the allowed exponent ceiling exceeds two and the maximum scale is positive and below one, there are positive constants, uniform over both radial sides, all exponents between two and that ceiling, and all positive scales up to that maximum, which bound the reference one-sided weighted mass above and below by constant multiples of the scale to the exponent.","s":"∀ {κMax h0 : ℝ}, 2 < κMax → 0 < h0 → h0 < 1 → ∃ c C, 0 < c ∧ c ≤ C ∧ ∀ (κ : ℝ) (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (h : ℝ), 2 < κ → κ ≤ κMax → 0 < h → h ≤ h0 → c * h ^ κ ≤ Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side 1 h ∧ Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side 1 h ≤ C * h ^ κ","l":177,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.sideBall","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a chosen radial side, a planar center, and a real radius, the one-sided ball is the open Euclidean ball with that center and radius, intersected with the region on the chosen side of the unit circle.","s":"Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide → Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ → Set Causalean.Mathlib.Analysis.WeightedCircularTube.Plane","l":49,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"For a real exponent, a choice of the inside or outside of the unit circle, a planar centre, and a real radius, the weighted side-ball mass is the two-dimensional Lebesgue integral of the radial power weight with that exponent over the corresponding one-sided open ball. The integral is the real-valued Bochner integral, which is zero when the weight is not integrable there; the results below use it for exponents above two, where it is integrable.","s":"ℝ → Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide → Causalean.Mathlib.Analysis.WeightedCircularTube.Plane → ℝ → ℝ","l":20,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass_eq_reference","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"For a chosen radial side and a center on the unit circle, rotating that center to the reference point one leaves its weighted side-ball mass unchanged.","s":"∀ {κ h : ℝ} (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) {x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane}, x ∈ Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle → Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side x h = Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side 1 h","l":143,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass_integrable","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the radial power weight is integrable over the one-sided ball determined by the chosen side, center, and radius and hence its weighted mass is finite.","s":"∀ {κ : ℝ}, 2 < κ → ∀ (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane) (h : ℝ), MeasureTheory.IntegrableOn (Causalean.Mathlib.Analysis.WeightedCircularTube.powerWeight κ) (Causalean.Mathlib.Analysis.WeightedCircularTube.sideBall side x h) MeasureTheory.volume","l":127,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the exponent κ exceeds two, the weighted mass of the one-sided ball determined by the chosen side, center, and radius is nonnegative.","s":"∀ {κ : ℝ}, 2 < κ → ∀ (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane) (h : ℝ), 0 ≤ Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side x h","l":135,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle","k":"def","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"The unit circle is the set of planar points whose Euclidean distance from the origin is one.","s":"Set Causalean.Mathlib.Analysis.WeightedCircularTube.Plane","l":24,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_exists_finite_maximalSeparated","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"At every positive separation radius, there exists a finite inclusion-maximal separated subset of the unit circle.","s":"∀ {r : ℝ}, 0 < r → ∃ sites, Causalean.Mathlib.Analysis.WeightedCircularTube.IsMaximalSeparated sites r","l":117,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_exists_maximalSeparated_with_card_bounds","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"There are positive universal constants with an ordered small-scale range such that at every positive scale in that range one can choose a finite site set maximal at separation three times that scale, whose open three-scale balls cover the unit circle and whose size is comparable to the reciprocal scale.","s":"∃ cpack Cpack hstar, 0 < cpack ∧ cpack ≤ Cpack ∧ 0 < hstar ∧ ∀ (h : ℝ), 0 < h → h ≤ hstar → ∃ sites, Causalean.Mathlib.Analysis.WeightedCircularTube.IsMaximalSeparated sites (3 * h) ∧ Causalean.Mathlib.Analysis.WeightedCircularTube.CoversUnitCircle sites (3 * h) ∧ cpack / h ≤ ↑sites.card ∧ ↑sites.card ≤ Cpack / h","l":298,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_maximalSeparated_card_bounds","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"There are positive universal constants with an ordered small-scale range such that every finite site set maximal at separation three times a positive scale in that range has open three-scale balls covering the unit circle and a number of sites comparable to the reciprocal scale.","s":"∃ cpack Cpack hstar, 0 < cpack ∧ cpack ≤ Cpack ∧ 0 < hstar ∧ ∀ (h : ℝ) (sites : Finset Causalean.Mathlib.Analysis.WeightedCircularTube.Plane), 0 < h → h ≤ hstar → Causalean.Mathlib.Analysis.WeightedCircularTube.IsMaximalSeparated sites (3 * h) → Causalean.Mathlib.Analysis.WeightedCircularTube.CoversUnitCircle sites (3 * h) ∧ cpack / h ≤ ↑sites.card ∧ ↑sites.card ≤ Cpack / h","l":278,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_maximalSeparated_covers","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Packing","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"When the separation radius is positive and a finite site set is inclusion-maximal at that non-strict separation radius, its open balls of that radius cover the unit circle.","s":"∀ {sites : Finset Causalean.Mathlib.Analysis.WeightedCircularTube.Plane} {r : ℝ}, 0 < r → Causalean.Mathlib.Analysis.WeightedCircularTube.IsMaximalSeparated sites r → Causalean.Mathlib.Analysis.WeightedCircularTube.CoversUnitCircle sites r","l":94,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Packing.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.Basic","a":"Mathlib/Analysis/WeightedCircularTube","t":2,"d":"The unit circle contains at least one planar point.","s":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle.Nonempty","l":60,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/Basic.lean"},{"n":"Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle_powerWeighted_sideBall_bounds","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCircularTube.SideMass","a":"Mathlib/Analysis/WeightedCircularTube","t":1,"d":"When the allowed exponent ceiling exceeds two and the maximum scale is positive and below one, there are positive constants, uniform over every unit-circle center, either radial side, all exponents strictly greater than two and at most that ceiling, and all positive scales up to that maximum, which bound one-sided weighted mass above and below by constant multiples of the scale to the exponent.","s":"∀ {κMax h0 : ℝ}, 2 < κMax → 0 < h0 → h0 < 1 → ∃ c C, 0 < c ∧ c ≤ C ∧ ∀ (κ : ℝ) (x : Causalean.Mathlib.Analysis.WeightedCircularTube.Plane) (side : Causalean.Mathlib.Analysis.WeightedCircularTube.CircleSide) (h : ℝ), 2 < κ → κ ≤ κMax → x ∈ Causalean.Mathlib.Analysis.WeightedCircularTube.unitCircle → 0 < h → h ≤ h0 → c * h ^ κ ≤ Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side x h ∧ Causalean.Mathlib.Analysis.WeightedCircularTube.sideBallMass κ side x h ≤ C * h ^ κ","l":277,"f":"Causalean/Mathlib/Analysis/WeightedCircularTube/SideMass.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_neg_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a symmetric band around zero produces a value whose magnitude is at most the half-width of the band.","s":"∀ {B : ℝ}, 0 ≤ B → ∀ (x : ℝ), |Causalean.Mathlib.Analysis.clipIcc (-B) B x| ≤ B","l":113,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_sub_clipIcc_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping two numbers to the same interval never increases the distance between them.","s":"∀ (a b x y : ℝ), |Causalean.Mathlib.Analysis.clipIcc a b x - Causalean.Mathlib.Analysis.clipIcc a b y| ≤ |x - y|","l":64,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_clipIcc_sub_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"Clipping toward an in-interval target does not increase distance. If a real number lies in the closed interval from a to b, then for any real x, clipping x into that interval moves it no farther from the target than x itself was.","s":"∀ {a b t : ℝ}, t ∈ Set.Icc a b → ∀ (x : ℝ), |Causalean.Mathlib.Analysis.clipIcc a b x - t| ≤ |x - t|","l":79,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_min_max_sub_min_max_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Capping from above and then flooring from below is a contraction as well: two numbers put through that order of operations are never pushed further apart. Unlike clipIcc_eq_min_max, this needs no assumption on the two thresholds, which is what makes it usable for a propensity band [q, 1 - q] whose definition carries no q ≤ 1/2 hypothesis.","s":"∀ (a b x y : ℝ), |min b (max a x) - min b (max a y)| ≤ |x - y|","l":69,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.abs_singularValues_add_sub_singularValues_le_opNorm","k":"theorem","m":"Causalean.Mathlib.Analysis.SingularValueWeyl","a":"Mathlib/Analysis","t":1,"d":"Given a baseline linear map, an additive perturbation, and a singular-value index, the absolute change in the indexed singular value is at most the perturbation's operator norm.","s":"∀ {V : Type u_1} {W : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V] [inst_3 : NormedAddCommGroup W] [inst_4 : InnerProductSpace ℝ W] [inst_5 : FiniteDimensional ℝ W] (A D : V →ₗ[ℝ] W) (j : ℕ), |(A + D).singularValues j - A.singularValues j| ≤ ‖LinearMap.toContinuousLinearMap D‖","l":206,"f":"Causalean/Mathlib/Analysis/SingularValueWeyl.lean"},{"n":"Causalean.Mathlib.Analysis.abs_weighted_inner_le","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":1,"d":"Weighted Cauchy-Schwarz inequality on a finite support, square-root form. With nonnegative weights, the absolute value of the weighted inner product of two real-valued functions is at most the product of the square roots of their weighted sums of squares.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f g : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → |∑ i ∈ s, w i * (f i * g i)| ≤ √(∑ i ∈ s, w i * f i ^ 2) * √(∑ i ∈ s, w i * g i ^ 2)","l":67,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.abs_weighted_mean_le_sqrt_weighted_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":2,"d":"Weighted mean is dominated by the weighted root-mean-square. For weights that are nonnegative and sum to at most one, the absolute value of the weighted average of a real-valued function is at most the square root of the weighted average of its square.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → ∀ (c : ℝ), ∑ i ∈ s, w i ≤ c → |∑ i ∈ s, w i * f i| ≤ √c * √(∑ i ∈ s, w i * f i ^ 2)","l":82,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.affine_of_continuous_of_second_diff_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.FrechetFunctionalEquation","a":"Mathlib/Analysis","t":1,"d":"Fréchet functional equation, degree one. If f is a continuous function from the reals into a normed real vector space and its second forward difference vanishes: f(x+2s) + f(x) = 2·f(x+s) for all real x and s, then f is affine — there exist vectors a and b such that f(x) = a + x·b for every real x.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E}, Continuous f → (∀ (x s : ℝ), f (x + s + s) + f x = f (x + s) + f (x + s)) → ∃ a b, ∀ (x : ℝ), f x = a + x • b","l":27,"f":"Causalean/Mathlib/Analysis/FrechetFunctionalEquation.lean"},{"n":"Causalean.Mathlib.Analysis.algebraicRankOnePseudoInverse","k":"def","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real matrix with two rows, the choice-free algebraic rank-one pseudoinverse transposes the algebraically projected matrix and divides by the upper Gram root.","s":"{E : Type u_1} → [Fintype E] → Matrix (Fin 2) E ℝ → Matrix E (Fin 2) ℝ","l":69,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.algebraicRankOnePseudoInverse_isMoorePenrose","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real two-row matrix with a strict positive upper left-Gram root, the algebraic rank-one pseudoinverse satisfies all four Moore--Penrose equations for the algebraic rank-one truncation.","s":"∀ {E : Type u_1} [inst : Fintype E] {H : Matrix (Fin 2) E ℝ}, Causalean.Mathlib.Analysis.leftGram H ∈ Causalean.Mathlib.Analysis.strictGapSet → 0 < Causalean.Mathlib.Analysis.lambda₁ (Causalean.Mathlib.Analysis.leftGram H) → have A := Causalean.Mathlib.Analysis.algebraicRankOneTruncation H; have Aplus := Causalean.Mathlib.Analysis.algebraicRankOnePseudoInverse H; A * Aplus * A = A ∧ Aplus * A * Aplus = Aplus ∧ (A * Aplus).IsHermitian ∧ (Aplus * A).IsHermitian","l":93,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.algebraicRankOneTruncation","k":"def","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real matrix with two rows, the choice-free algebraic rank-one truncation applies the algebraic upper projector of its left Gram matrix. It has rank one under a strict gap between the two roots of the left Gram matrix; without a gap the projector's division by the zero gap makes it the zero matrix.","s":"{E : Type u_1} → [Fintype E] → Matrix (Fin 2) E ℝ → Matrix (Fin 2) E ℝ","l":64,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.algebraicWaldFunctional","k":"def","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":1,"d":"Given a rank-one Wald input, the choice-free algebraic Wald functional applies the algebraic rank-one pseudoinverse to the target direction and pairs the result with the outcome direction.","s":"{E : Type u_1} → [Fintype E] → Causalean.Mathlib.Analysis.WaldInput E → ℝ","l":25,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.algebraicWaldFunctional_hasLocallyBoundedFDerivAt","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":1,"d":"Given a rank-one Wald input in the regularity region, the algebraic Wald functional has a bounded Fréchet derivative on some neighborhood of that input.","s":"∀ {E : Type u_1} [inst : Fintype E] {x : Causalean.Mathlib.Analysis.WaldInput E}, x ∈ Causalean.Mathlib.Analysis.waldRegularSet → Causalean.Mathlib.Analysis.HasLocallyBoundedFDerivAt Causalean.Mathlib.Analysis.algebraicWaldFunctional x","l":72,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.bernD","k":"def","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":1,"d":"For a real number, the negative Bernoulli entropy is $t\\log t + (1-t)\\log(1-t)$.","s":"ℝ → ℝ","l":61,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.bernoulliKL_eq_bregman","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":2,"d":"Bernoulli Kullback–Leibler divergence equals the Bregman remainder of the negative entropy function at the second probability.","s":"∀ {p q : ℝ}, p ≠ 0 → 1 - p ≠ 0 → q ≠ 0 → 1 - q ≠ 0 → p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q)) = Causalean.Mathlib.Analysis.bernD p - Causalean.Mathlib.Analysis.bernD q - (Real.log q - Real.log (1 - q)) * (p - q)","l":182,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.bernoulli_kl_le_four_sq_sub_of_mem_quarter_band","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":1,"d":"For probabilities p and q both restricted to the band [1/4, 3/4], the Bernoulli Kullback–Leibler divergence p · log(p / q) + (1 - p) · log((1 - p) / (1 - q)) is bounded above by 4 · (p - q) ^ 2.","s":"∀ {p q : ℝ}, 1 / 4 ≤ p → p ≤ 3 / 4 → 1 / 4 ≤ q → q ≤ 3 / 4 → p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q)) ≤ 4 * (p - q) ^ 2","l":192,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc","k":"def","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"For two real endpoints and a real input, the clipped value is the larger of the first endpoint and the smaller of the second endpoint and the input. When the first endpoint does not exceed the second, it is the input pushed into their closed interval.","s":"ℝ → ℝ → ℝ → ℝ","l":31,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_eq_min_max","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping from below first and then from above gives the same answer as clipping from above first and then from below. This reconciles the two spellings of the same operation that occur in practice, max a (min b x) and min b (max a x).","s":"∀ {a b : ℝ}, a ≤ b → ∀ (x : ℝ), Causalean.Mathlib.Analysis.clipIcc a b x = min b (max a x)","l":53,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping leaves untouched any number that already lies inside the interval.","s":"∀ {a b x : ℝ}, x ∈ Set.Icc a b → Causalean.Mathlib.Analysis.clipIcc a b x = x","l":49,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"A clipped value always lies in the interval it was clipped to (provided the interval is nonempty, i.e. its left endpoint does not exceed its right endpoint).","s":"∀ {a b : ℝ}, a ≤ b → ∀ (x : ℝ), Causalean.Mathlib.Analysis.clipIcc a b x ∈ Set.Icc a b","l":44,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_neg_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a symmetric band around zero leaves untouched any number whose magnitude is already within the half-width of the band.","s":"∀ {x B : ℝ}, |x| ≤ B → Causalean.Mathlib.Analysis.clipIcc (-B) B x = x","l":118,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.clipIcc_sub_sq_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":1,"d":"Squared version: clipping toward an in-interval target does not increase squared distance. If a real number lies in the closed interval from a to b, then for any real x, the squared distance from the clipped value of x to the target is at most the squared distance from x itself to the target.","s":"∀ {a b t : ℝ}, t ∈ Set.Icc a b → ∀ (x : ℝ), (Causalean.Mathlib.Analysis.clipIcc a b x - t) ^ 2 ≤ (x - t) ^ 2","l":89,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.closedHalfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Replacing the open diameter of an upper half-disc by the closed diameter does not affect the Cartesian cosine integral.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 ≤ z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * (z.1 / Causalean.Mathlib.Analysis.planarRadius z) = 0","l":223,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.coe_projIcc_eq_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping into an interval is exactly Mathlib's projection onto that interval, read as a plain real number instead of as an element of the interval. This is the bridge that makes every fact about the projection available to a development that never leaves the reals.","s":"∀ {a b : ℝ} (hab : a ≤ b) (x : ℝ), ↑(Set.projIcc a b hab x) = Causalean.Mathlib.Analysis.clipIcc a b x","l":38,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_algebraicRankOnePseudoInverse","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":2,"d":"Given a real two-row matrix whose left Gram matrix has a strict positive upper root, the algebraic rank-one pseudoinverse varies continuously differentiably near that matrix.","s":"∀ {E : Type u_1} [inst : Fintype E] {H : Matrix (Fin 2) E ℝ}, Causalean.Mathlib.Analysis.leftGram H ∈ Causalean.Mathlib.Analysis.strictGapSet → 0 < Causalean.Mathlib.Analysis.lambda₁ (Causalean.Mathlib.Analysis.leftGram H) → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.algebraicRankOnePseudoInverse H","l":166,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_algebraicRankOneTruncation","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":2,"d":"Given a real two-row matrix whose left Gram matrix has distinct roots, the algebraic rank-one truncation varies continuously differentiably near that matrix.","s":"∀ {E : Type u_1} [inst : Fintype E] {H : Matrix (Fin 2) E ℝ}, Causalean.Mathlib.Analysis.leftGram H ∈ Causalean.Mathlib.Analysis.strictGapSet → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.algebraicRankOneTruncation H","l":153,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_algebraicWaldFunctional","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":2,"d":"Given a rank-one Wald input in the regularity region, the algebraic Wald functional varies continuously differentiably near that input.","s":"∀ {E : Type u_1} [inst : Fintype E] {x : Causalean.Mathlib.Analysis.WaldInput E}, x ∈ Causalean.Mathlib.Analysis.waldRegularSet → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.algebraicWaldFunctional x","l":45,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_lambda₁","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix with a strict gap between its explicit roots, the upper root varies continuously differentiably near that matrix.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G ∈ Causalean.Mathlib.Analysis.strictGapSet → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.lambda₁ G","l":132,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix with a strict gap between its explicit roots, the lower root varies continuously differentiably near that matrix.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G ∈ Causalean.Mathlib.Analysis.strictGapSet → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.lambda₂ G","l":145,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffAt_topProjector","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix with distinct explicit roots, its algebraic upper projector varies continuously differentiably near that matrix.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G ∈ Causalean.Mathlib.Analysis.strictGapSet → ContDiffAt ℝ 1 Causalean.Mathlib.Analysis.topProjector G","l":188,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffOn_algebraicWaldFunctional","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":2,"d":"The algebraic Wald functional is continuously differentiable throughout its regularity region.","s":"∀ {E : Type u_1} [inst : Fintype E], ContDiffOn ℝ 1 Causalean.Mathlib.Analysis.algebraicWaldFunctional Causalean.Mathlib.Analysis.waldRegularSet","l":66,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffOn_lambda₁","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"The upper explicit root is continuously differentiable throughout the strict-gap region.","s":"ContDiffOn ℝ 1 Causalean.Mathlib.Analysis.lambda₁ Causalean.Mathlib.Analysis.strictGapSet","l":168,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffOn_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"The lower explicit root is continuously differentiable throughout the strict-gap region.","s":"ContDiffOn ℝ 1 Causalean.Mathlib.Analysis.lambda₂ Causalean.Mathlib.Analysis.strictGapSet","l":173,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.contDiffOn_topProjector","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"The algebraic upper projector is continuously differentiable throughout the strict-gap region.","s":"ContDiffOn ℝ 1 Causalean.Mathlib.Analysis.topProjector Causalean.Mathlib.Analysis.strictGapSet","l":204,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.contDiff_one_fderiv_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"x ↦ fderiv ℝ f x d is Cⁿ when f is Cⁿ⁺¹. It is the composition of the derivative map with the continuous linear evaluation L ↦ L d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {n : ℕ} {f : E → ℝ}, ContDiff ℝ (↑n + 1) f → ∀ (d : E), ContDiff ℝ ↑n fun x => (fderiv ℝ f x) d","l":63,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a fixed interval is a continuous operation.","s":"∀ (a b : ℝ), Continuous (Causalean.Mathlib.Analysis.clipIcc a b)","l":102,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_convexProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set varies continuously with the point being projected.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), Continuous (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":108,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_monomial","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A multivariate real monomial varies continuously with its coordinate vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (e : ι → ℕ), Continuous (Causalean.Mathlib.Analysis.monomial e)","l":56,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_monomialCombination","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A finite linear combination of multivariate real monomials is continuous.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) (z : Fin p → ℝ), Continuous fun u => ∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u","l":63,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.continuous_secondDirDeriv","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"The curvature modulus is continuous in the base point. Hence bounded on any compact set, which supplies the BddAbove hypothesis of le_ciSup.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (d : E), Continuous (Causalean.Mathlib.Analysis.secondDirDeriv f d)","l":89,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.convexOn_deriv2_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":1,"d":"A convex function of one real variable has nonnegative second derivative at interior points of its domain. If g is convex on the set S, differentiable at every point of S, x lies in the interior of S, and the derivative of g is itself differentiable at x, then the second derivative of g at x is nonnegative.","s":"∀ {S : Set ℝ} {g : ℝ → ℝ} {x : ℝ}, ConvexOn ℝ S g → (∀ y ∈ S, DifferentiableAt ℝ g y) → x ∈ interior S → DifferentiableAt ℝ (deriv g) x → 0 ≤ deriv (deriv g) x","l":97,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"For a real Hilbert space and a target subset that is nonempty, closed, and convex, the metric projection assigns each point its nearest point in that target subset.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [CompleteSpace E] → (K : Set E) → K.Nonempty → IsClosed K → Convex ℝ K → E → E","l":35,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every point in a nonempty closed convex target set is unchanged by metric projection onto that set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), ∀ x ∈ K, Causalean.Mathlib.Analysis.convexProj K hne hc hconv x = x","l":75,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_le_of_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Projecting a point onto a nonempty closed convex set cannot increase its distance from any point already in that set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x y : E), y ∈ K → ‖Causalean.Mathlib.Analysis.convexProj K hne hc hconv x - y‖ ≤ ‖x - y‖","l":61,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_lipschitz","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set is nonexpansive: projected distances never exceed the original distances.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), LipschitzWith 1 (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":83,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The metric projection of every point belongs to its nonempty closed convex target set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x : E), Causalean.Mathlib.Analysis.convexProj K hne hc hconv x ∈ K","l":48,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_spec","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"A point's metric projection onto a nonempty closed convex set belongs to that set and attains the smallest possible distance from the point among all points in the set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x : E), Causalean.Mathlib.Analysis.convexProj K hne hc hconv x ∈ K ∧ ‖x - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x‖ = ⨅ y, ‖x - ↑y‖","l":40,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.convexProj_variational","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The residual from a point to its metric projection has nonpositive inner product with every feasible direction based at the projection.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K) (x y : E), y ∈ K → inner ℝ (x - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x) (y - Causalean.Mathlib.Analysis.convexProj K hne hc hconv x) ≤ 0","l":53,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.derivWithin_logRatio","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"On a nondegenerate compact interval, the within-derivative of a positive log ratio is the difference of the two logarithmic derivatives.","s":"∀ {a b : ℝ}, a < b → ∀ {q p : ℝ → ℝ}, DifferentiableOn ℝ q (Set.Icc a b) → DifferentiableOn ℝ p (Set.Icc a b) → (∀ x ∈ Set.Icc a b, 0 < q x) → (∀ x ∈ Set.Icc a b, 0 < p x) → ∀ {x : ℝ}, x ∈ Set.Icc a b → derivWithin (Causalean.Mathlib.Analysis.logRatio q p) (Set.Icc a b) x = derivWithin q (Set.Icc a b) x / q x - derivWithin p (Set.Icc a b) x / p x","l":151,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.deriv_deriv_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"Second directional derivative along a line. The second derivative of the line restriction at parameter t is the second directional derivative of f at the moving base point q + t • d. Note this is an identity of deriv (not derivWithin): it holds at every t because the line restriction is differentiable on all of ℝ.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (q d : E) (t : ℝ), deriv (deriv fun s => f (q + s • d)) t = Causalean.Mathlib.Analysis.secondDirDeriv f d (q + t • d)","l":76,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.deriv_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"First directional derivative along a line.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} (q d : E) (t : ℝ), DifferentiableAt ℝ f (q + t • d) → deriv (fun s => f (q + s • d)) t = (fderiv ℝ f (q + t • d)) d","l":56,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep","k":"def","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":1,"d":"For a real curvature bound, a real slope margin, and a real available interval length, the quadratic-model step is the available interval length when the curvature bound is zero and otherwise the smaller of that length and the slope margin divided by the curvature bound.","s":"ℝ → ℝ → ℝ → ℝ","l":24,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_mem_Icc","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"The clipped quadratic-model step is feasible whenever the interval length, slope margin, and curvature bound are nonnegative.","s":"∀ {M c T : ℝ}, 0 ≤ M → 0 ≤ c → 0 ≤ T → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T","l":42,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_of_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With zero curvature bound, the clipped quadratic-model step is the endpoint.","s":"∀ (c T : ℝ), Causalean.Mathlib.Analysis.descentStep 0 c T = T","l":31,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.descentStep_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With a positive curvature bound, the clipped quadratic-model step is the smaller of the available interval length and the unconstrained quadratic-model optimizer.","s":"∀ {M : ℝ}, 0 < M → ∀ (c T : ℝ), Causalean.Mathlib.Analysis.descentStep M c T = min T (c / M)","l":36,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.det_sub_lambda_mul_one_eq_zero","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"Given a real symmetric two-by-two matrix and one of the two root labels, subtracting that labelled explicit root from the diagonal has zero determinant.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → ∀ (i : Fin 2), (G - (if i = 0 then Causalean.Mathlib.Analysis.lambda₁ G else Causalean.Mathlib.Analysis.lambda₂ G) • 1).det = 0","l":74,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.differentiable_fderiv_apply","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"x ↦ fderiv ℝ f x d is differentiable when f is C².","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ}, ContDiff ℝ 2 f → ∀ (d : E), Differentiable ℝ fun x => (fderiv ℝ f x) d","l":71,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.eigenvalue_eq_lambda₁_or_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":1,"d":"Given a real symmetric two-by-two matrix, a real number, a nonzero two-vector, and an eigenvector relation for that number, the number is one of the two explicit ordered roots.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → ∀ {μ : ℝ} {v : Fin 2 → ℝ}, v ≠ 0 → G.mulVec v = μ • v → μ = Causalean.Mathlib.Analysis.lambda₁ G ∨ μ = Causalean.Mathlib.Analysis.lambda₂ G","l":90,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.eigenvectorRankOnePseudoInverse","k":"def","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real two-vector and a real matrix with two rows, the eigenvector-selected rank-one pseudoinverse formula is the transpose of the matrix times the outer product of the vector with itself, divided by the larger root of the left Gram matrix (zero if that root is zero). No condition is imposed on the vector; this is the Moore–Penrose pseudoinverse of the rank-one truncation only when the vector is a unit upper eigenvector of the left Gram matrix.","s":"{E : Type u_1} → [Fintype E] → (Fin 2 → ℝ) → Matrix (Fin 2) E ℝ → Matrix E (Fin 2) ℝ","l":59,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.eigenvectorRankOnePseudoInverse_eq_algebraic","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":2,"d":"Given a real two-row matrix with a strict positive upper left-Gram root and a unit upper left-Gram eigenvector, the eigenvector-selected and algebraic rank-one pseudoinverse formulas agree.","s":"∀ {E : Type u_1} [inst : Fintype E] {H : Matrix (Fin 2) E ℝ}, Causalean.Mathlib.Analysis.leftGram H ∈ Causalean.Mathlib.Analysis.strictGapSet → 0 < Causalean.Mathlib.Analysis.lambda₁ (Causalean.Mathlib.Analysis.leftGram H) → ∀ {u : Fin 2 → ℝ}, u ⬝ᵥ u = 1 → (Causalean.Mathlib.Analysis.leftGram H).mulVec u = Causalean.Mathlib.Analysis.lambda₁ (Causalean.Mathlib.Analysis.leftGram H) • u → Causalean.Mathlib.Analysis.eigenvectorRankOnePseudoInverse u H = Causalean.Mathlib.Analysis.algebraicRankOnePseudoInverse H","l":83,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.eigenvectorRankOneTruncation","k":"def","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real two-vector and a real matrix with two rows, the eigenvector-selected rank-one truncation multiplies the matrix on the left by the outer product of the vector with itself. No condition is imposed on the vector: this is an orthogonal rank-one projection of the rows only when the vector has unit length (the intended use is a unit upper eigenvector of the left Gram matrix); a non-unit vector rescales the result and the zero vector gives the zero matrix.","s":"{E : Type u_1} → (Fin 2 → ℝ) → Matrix (Fin 2) E ℝ → Matrix (Fin 2) E ℝ","l":54,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.eigenvectorRankOneTruncation_eq_algebraic","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":2,"d":"Given a real two-row matrix with a strict left-Gram root gap and a unit upper left-Gram eigenvector, the eigenvector-selected and algebraic rank-one truncations agree.","s":"∀ {E : Type u_1} [inst : Fintype E] {H : Matrix (Fin 2) E ℝ}, Causalean.Mathlib.Analysis.leftGram H ∈ Causalean.Mathlib.Analysis.strictGapSet → ∀ {u : Fin 2 → ℝ}, u ⬝ᵥ u = 1 → (Causalean.Mathlib.Analysis.leftGram H).mulVec u = Causalean.Mathlib.Analysis.lambda₁ (Causalean.Mathlib.Analysis.leftGram H) • u → Causalean.Mathlib.Analysis.eigenvectorRankOneTruncation u H = Causalean.Mathlib.Analysis.algebraicRankOneTruncation H","l":74,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.exists_large_subspace_norm_le_singularValues","k":"theorem","m":"Causalean.Mathlib.Analysis.SingularValueWeyl","a":"Mathlib/Analysis","t":2,"d":"Given a linear map and an in-range singular-value index, there is a subspace whose codimension is at most that index and on which the map is bounded by the indexed singular value.","s":"∀ {V : Type u_1} {W : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V] [inst_3 : NormedAddCommGroup W] [inst_4 : InnerProductSpace ℝ W] [inst_5 : FiniteDimensional ℝ W] (A : V →ₗ[ℝ] W) {j : ℕ}, j < Module.finrank ℝ V → ∃ S, Module.finrank ℝ V ≤ Module.finrank ℝ ↥S + j ∧ ∀ x ∈ S, ‖A x‖ ≤ A.singularValues j * ‖x‖","l":90,"f":"Causalean/Mathlib/Analysis/SingularValueWeyl.lean"},{"n":"Causalean.Mathlib.Analysis.exists_monomialGram_coercive","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A fixed family of distinct monomials on a cube with positive radius admits a positive uniform lower bound: its Gram quadratic form dominates the squared Euclidean norm of the coefficients.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, 0 < r → Function.Injective expo → ∃ cmin, 0 < cmin ∧ ∀ (z : Fin p → ℝ), cmin * ∑ k, z k ^ 2 ≤ ∑ k, ∑ l, z k * Causalean.Mathlib.Analysis.monomialGram expo r k l * z l","l":248,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.first_order_descent_gap","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With zero curvature, the endpoint step gives the full linear descent guaranteed by the negative initial slope.","s":"∀ {f : ℝ → ℝ} {c T : ℝ}, 0 ≤ T → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ 0) → deriv f 0 ≤ -c → f 0 - f T ≥ c * T","l":161,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.first_order_upper_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"A real function with nonpositive second derivative in an interval's interior lies below its tangent line at the left endpoint.","s":"∀ {f : ℝ → ℝ} {T : ℝ}, ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ 0) → ∀ {t : ℝ}, t ∈ Set.Icc 0 T → f t ≤ f 0 + deriv f 0 * t","l":99,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"For a square matrix dimension and two real square matrices, the Frobenius distance is the square root of the sum of squared entrywise differences between the two matrices.","s":"{p : ℕ} → Matrix (Fin p) (Fin p) ℝ → Matrix (Fin p) (Fin p) ℝ → ℝ","l":134,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_comm","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance is symmetric in its two matrix arguments.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A B = Causalean.Mathlib.Analysis.frobDist B A","l":162,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_eq_norm","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The Frobenius distance between two finite real matrices equals the Euclidean norm of their difference after vectorizing the entries.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A B = ‖(Causalean.Mathlib.Analysis.mtx p) A - (Causalean.Mathlib.Analysis.mtx p) B‖","l":149,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance is nonnegative.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ), 0 ≤ Causalean.Mathlib.Analysis.frobDist A B","l":158,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.frobDist_triangle","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Frobenius distance satisfies the triangle inequality.","s":"∀ {p : ℕ} (A B D : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.frobDist A D ≤ Causalean.Mathlib.Analysis.frobDist A B + Causalean.Mathlib.Analysis.frobDist B D","l":172,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_cos_radial_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Multiplication by cosine contributes zero after angular integration on every radial shell.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (s : ℝ) in Set.Ioc 0 r, s * g s * ∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ = 0","l":386,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_radialSet_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A cosine angular tilt has zero mass on every measurable radial subset of an upper half-disc. This is the setwise interface used to identify radial pushforwards, rather than merely their total masses.","s":"∀ (g : ℝ → ℝ) (r : ℝ) {A : Set ℝ}, MeasurableSet A → ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r} ∩ Causalean.Mathlib.Analysis.planarRadius ⁻¹' A, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) = 0","l":153,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_radial_integral","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A radial integrand on a half-disc admits the expected polar-coordinate decomposition.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) = ∫ (s : ℝ) in Set.Ioc 0 r, Real.pi * s * g s","l":304,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A cosine angular tilt has zero integral against every radial weight on an upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) = 0","l":141,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_cos_sq","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The cosine-squared angular moment converts a radial weight into the nonzero π/2 factor used to cancel the affine regression term.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * Real.cos (Causalean.Mathlib.Analysis.planarAngle z) ^ 2 = (∫ (s : ℝ) in Set.Ioc 0 r, s * g s) * (Real.pi / 2)","l":288,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"A radial weight times the Cartesian direction cosine has zero integral on an open upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * (z.1 / Causalean.Mathlib.Analysis.planarRadius z) = 0","l":199,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.halfDisc_weighted_polar_integral","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"For radial and angular weight functions g and h and a radius r, the integral of the product g(radius)·h(angle) over the open upper half-disc of radius r factors as the product of the radial integral ∫ s·g(s) ds over (0, r] and the angular integral ∫ h(θ) dθ over (0, π).","s":"∀ (g h : ℝ → ℝ) (r : ℝ), ∫ (z : ℝ × ℝ) in {z | 0 < z.2 ∧ Causalean.Mathlib.Analysis.planarRadius z ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius z) * h (Causalean.Mathlib.Analysis.planarAngle z) = (∫ (s : ℝ) in Set.Ioc 0 r, s * g s) * ∫ (θ : ℝ) in Set.Ioo 0 Real.pi, h θ","l":57,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.hasDerivAt_bernEntropy","k":"theorem","m":"Causalean.Mathlib.Analysis.BernoulliKL","a":"Mathlib/Analysis","t":2,"d":"Away from zero and one, the negative Bernoulli entropy has derivative equal to the log odds of its argument.","s":"∀ (x : ℝ), x ≠ 0 → 1 - x ≠ 0 → HasDerivAt (fun t => t * Real.log t + (1 - t) * Real.log (1 - t)) (Real.log x - Real.log (1 - x)) x","l":36,"f":"Causalean/Mathlib/Analysis/BernoulliKL.lean"},{"n":"Causalean.Mathlib.Analysis.hasDerivAt_line","k":"theorem","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":2,"d":"The line s ↦ q + s • d is differentiable with derivative d, as a HasDerivAt.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] (q d : E) (t : ℝ), HasDerivAt (fun s => q + s • d) d t","l":51,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.integral_cos_sq_zero_to_pi","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The quadratic cosine moment on a half-circle is π / 2.","s":"∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ ^ 2 = Real.pi / 2","l":50,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.integral_cos_zero_to_pi","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The cosine has zero integral on a half-circle.","s":"∫ (θ : ℝ) in Set.Ioc 0 Real.pi, Real.cos θ = 0","l":41,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.isClosed_posSemidef","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Real positive-semidefinite matrices indexed by any finite set form a closed set in the coordinatewise topology.","s":"∀ {ι : Type u_1} [Finite ι], IsClosed {A | A.PosSemidef}","l":206,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.isOpen_strictGapSet","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"The region of real two-by-two matrices with distinct explicit roots is open.","s":"IsOpen Causalean.Mathlib.Analysis.strictGapSet","l":158,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.isOpen_waldRegularSet","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":2,"d":"The regularity region of the algebraic rank-one Wald functional is open.","s":"∀ {E : Type u_1} [inst : Fintype E], IsOpen Causalean.Mathlib.Analysis.waldRegularSet","l":34,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₁","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix, its larger symmetric-formula root is half the trace plus half the nonnegative square root of the symmetric-matrix discriminant. For a symmetric matrix this is the larger eigenvalue; for a nonsymmetric matrix it need not be an eigenvalue.","s":"Matrix (Fin 2) (Fin 2) ℝ → ℝ","l":21,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₁_add_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"For a real two-by-two matrix, the two explicit roots add to its trace.","s":"∀ (G : Matrix (Fin 2) (Fin 2) ℝ), Causalean.Mathlib.Analysis.lambda₁ G + Causalean.Mathlib.Analysis.lambda₂ G = G 0 0 + G 1 1","l":57,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₁_mul_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix that is symmetric, the product of its two explicit roots equals its determinant.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → Causalean.Mathlib.Analysis.lambda₁ G * Causalean.Mathlib.Analysis.lambda₂ G = G.det","l":63,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₁_sub_lambda₂","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"For a real two-by-two matrix, the difference between its upper and lower roots is the nonnegative square root of its discriminant.","s":"∀ (G : Matrix (Fin 2) (Fin 2) ℝ), Causalean.Mathlib.Analysis.lambda₁ G - Causalean.Mathlib.Analysis.lambda₂ G = √(Causalean.Mathlib.Analysis.rootDiscriminant G)","l":39,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₂","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix, its smaller symmetric-formula root is half the trace minus half the nonnegative square root of the symmetric-matrix discriminant. For a symmetric matrix this is the smaller eigenvalue; for a nonsymmetric matrix it need not be an eigenvalue.","s":"Matrix (Fin 2) (Fin 2) ℝ → ℝ","l":25,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.lambda₂_le_lambda₁","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"For a real two-by-two matrix, its lower explicit root does not exceed its upper explicit root.","s":"∀ (G : Matrix (Fin 2) (Fin 2) ℝ), Causalean.Mathlib.Analysis.lambda₂ G ≤ Causalean.Mathlib.Analysis.lambda₁ G","l":45,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.le_singularValues_of_subspace","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given a real linear map, a candidate signal subspace, a singular-value index, a nonnegative expansion factor, a dimension condition putting that index below the signal-subspace dimension, and a lower expansion bound on the signal subspace, the indexed singular value is at least that factor.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] (T : E →ₗ[ℝ] F) (S : Submodule ℝ E) {j : ℕ} {c : ℝ}, 0 ≤ c → j < Module.finrank ℝ ↥S → (∀ x ∈ S, c * ‖x‖ ≤ ‖T x‖) → c ≤ T.singularValues j","l":29,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.least_singularValue_mul_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given an injective finite-dimensional real linear map and a vector in its domain, the map expands that vector by at least its last domain-indexed singular value.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [Nontrivial E] (T : E →ₗ[ℝ] F), Function.Injective ⇑T → ∀ (x : E), T.singularValues (Module.finrank ℝ E - 1) * ‖x‖ ≤ ‖T x‖","l":57,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.least_singularValue_mul_norm_le_adjoint_on_range","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given a full-column-rank real linear map and a vector in its column space, the adjoint expands that signal vector by at least the map's least singular value.","s":"∀ {E : Type u_1} {K : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup K] [inst_4 : InnerProductSpace ℝ K] [inst_5 : FiniteDimensional ℝ K] [Nontrivial K] (B : K →ₗ[ℝ] E), Function.Injective ⇑B → ∀ y ∈ B.range, B.singularValues (Module.finrank ℝ K - 1) * ‖y‖ ≤ ‖(LinearMap.adjoint B) y‖","l":104,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.leftGram","k":"def","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":1,"d":"Given a real matrix with two rows, its left Gram matrix records all pairwise inner products of those rows.","s":"{E : Type u_1} → [Fintype E] → Matrix (Fin 2) E ℝ → Matrix (Fin 2) (Fin 2) ℝ","l":44,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.leftGram_isHermitian","k":"theorem","m":"Causalean.Mathlib.Analysis.RankOneGramPseudoinverse","a":"Mathlib/Analysis","t":2,"d":"For a real matrix with two rows, its left Gram matrix is symmetric.","s":"∀ {E : Type u_1} [inst : Fintype E] (H : Matrix (Fin 2) E ℝ), (Causalean.Mathlib.Analysis.leftGram H).IsHermitian","l":48,"f":"Causalean/Mathlib/Analysis/RankOneGramPseudoinverse.lean"},{"n":"Causalean.Mathlib.Analysis.lipschitzWith_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping is a contraction: two numbers are never pushed further apart by being clipped to a common interval.","s":"∀ (a b : ℝ), LipschitzWith 1 (Causalean.Mathlib.Analysis.clipIcc a b)","l":59,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Given a square matrix dimension and two real endpoints, the Loewner projection maps each real square matrix to its nearest point, in Frobenius distance, in the corresponding Loewner interval when the endpoints are ordered; when they are reversed, it is the identity map.","s":"(p : ℕ) → ℝ → ℝ → Matrix (Fin p) (Fin p) ℝ → Matrix (Fin p) (Fin p) ℝ","l":261,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_eq_self","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every matrix already in a nonempty Loewner interval is unchanged by metric projection onto that interval.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ G ∈ Causalean.Mathlib.Analysis.loewnerSet p c C, Causalean.Mathlib.Analysis.loewnerProj p c C G = G","l":394,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_frobDist_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Loewner projection is a nonexpansive (nearest-point) map. For ordered interval endpoints c ≤ C, if a target matrix already lies in the Loewner interval [cI, CI], then projecting an arbitrary matrix onto that interval, in Frobenius geometry, does not increase its Frobenius distance to the in-interval target matrix.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ (G S : Matrix (Fin p) (Fin p) ℝ), S ∈ Causalean.Mathlib.Analysis.loewnerSet p c C → Causalean.Mathlib.Analysis.frobDist (Causalean.Mathlib.Analysis.loewnerProj p c C G) S ≤ Causalean.Mathlib.Analysis.frobDist G S","l":278,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerProj_mem","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"When its endpoints are ordered, Loewner projection sends every finite real matrix into the corresponding Loewner interval.","s":"∀ {p : ℕ} {c C : ℝ}, c ≤ C → ∀ (G : Matrix (Fin p) (Fin p) ℝ), Causalean.Mathlib.Analysis.loewnerProj p c C G ∈ Causalean.Mathlib.Analysis.loewnerSet p c C","l":266,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"Given a square matrix dimension and two real endpoints, the Loewner interval is the set of real square matrices for which (1) the matrix minus the lower endpoint times the identity is positive semidefinite, and (2) the upper endpoint times the identity minus the matrix is positive semidefinite.","s":"(p : ℕ) → ℝ → ℝ → Set (Matrix (Fin p) (Fin p) ℝ)","l":129,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_convex","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every Loewner interval of finite real matrices is convex.","s":"∀ (p : ℕ) (c C : ℝ), Convex ℝ (Causalean.Mathlib.Analysis.loewnerSet p c C)","l":191,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_inv_mulVec_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"The inverse of a matrix in a Loewner interval with a positive lower endpoint expands Euclidean norm by at most the reciprocal of that endpoint.","s":"∀ {p : ℕ} {c : ℝ}, 0 < c → ∀ {G : Matrix (Fin p) (Fin p) ℝ}, (G - c • 1).PosSemidef → ∀ (v : Fin p → ℝ), √(∑ k, G⁻¹.mulVec v k ^ 2) ≤ √(∑ k, v k ^ 2) / c","l":331,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_isClosed","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every Loewner interval of finite real matrices is closed in the coordinatewise product topology.","s":"∀ (p : ℕ) (c C : ℝ), IsClosed (Causalean.Mathlib.Analysis.loewnerSet p c C)","l":233,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_nonempty","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"A Loewner interval with ordered scalar endpoints is nonempty.","s":"∀ (p : ℕ) (c C : ℝ), c ≤ C → (Causalean.Mathlib.Analysis.loewnerSet p c C).Nonempty","l":178,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.loewnerSet_posDef","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Every matrix in a Loewner interval with a strictly positive lower endpoint is positive definite.","s":"∀ {p : ℕ} {c : ℝ}, 0 < c → ∀ {G : Matrix (Fin p) (Fin p) ℝ}, (G - c • 1).PosSemidef → G.PosDef","l":239,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.logRatio","k":"def","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"The log ratio of q to p is the pointwise function x ↦ log (q x / p x).","s":"(ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ","l":46,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.logRatioDerivative_fixedSign_open_C1","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":1,"d":"A nondegenerate compact interval and positive lower and derivative margins, two continuously differentiable center functions, their shared lower bounds, a chosen orientation, and a derivative margin in that orientation ensure a positive uniform C¹ neighborhood in which both perturbed functions stay positive and their log-ratio derivative retains that orientation.","s":"∀ {a b m margin : ℝ}, a < b → 0 < m → 0 < margin → ∀ {q p : ℝ → ℝ}, ContDiffOn ℝ 1 q (Set.Icc a b) → ContDiffOn ℝ 1 p (Set.Icc a b) → (∀ x ∈ Set.Icc a b, m ≤ q x) → (∀ x ∈ Set.Icc a b, m ≤ p x) → ∀ (sign : ℝ), sign = 1 ∨ sign = -1 → (∀ x ∈ Set.Icc a b, margin ≤ sign * derivWithin (Causalean.Mathlib.Analysis.logRatio q p) (Set.Icc a b) x) → ∃ ε > 0, ∀ (q' p' : ℝ → ℝ), DifferentiableOn ℝ q' (Set.Icc a b) → DifferentiableOn ℝ p' (Set.Icc a b) → Causalean.Mathlib.Analysis.C1CloseOn (Set.Icc a b) ε q' q → Causalean.Mathlib.Analysis.C1CloseOn (Set.Icc a b) ε p' p → ∀ x ∈ Set.Icc a b, 0 < q' x ∧ 0 < p' x ∧ 0 < sign * derivWithin (Causalean.Mathlib.Analysis.logRatio q' p') (Set.Icc a b) x","l":277,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.logRatioDerivative_fixedSign_open_C2","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"The same strict fixed-sign derivative property is open under uniform C² perturbations, because uniform C² control includes the required uniform C¹ control.","s":"∀ {a b m margin : ℝ}, a < b → 0 < m → 0 < margin → ∀ {q p : ℝ → ℝ}, ContDiffOn ℝ 1 q (Set.Icc a b) → ContDiffOn ℝ 1 p (Set.Icc a b) → (∀ x ∈ Set.Icc a b, m ≤ q x) → (∀ x ∈ Set.Icc a b, m ≤ p x) → ∀ (sign : ℝ), sign = 1 ∨ sign = -1 → (∀ x ∈ Set.Icc a b, margin ≤ sign * derivWithin (Causalean.Mathlib.Analysis.logRatio q p) (Set.Icc a b) x) → ∃ ε > 0, ∀ (q' p' : ℝ → ℝ), DifferentiableOn ℝ q' (Set.Icc a b) → DifferentiableOn ℝ p' (Set.Icc a b) → Causalean.Mathlib.Analysis.C2CloseOn (Set.Icc a b) ε q' q → Causalean.Mathlib.Analysis.C2CloseOn (Set.Icc a b) ε p' p → ∀ x ∈ Set.Icc a b, 0 < q' x ∧ 0 < p' x ∧ 0 < sign * derivWithin (Causalean.Mathlib.Analysis.logRatio q' p') (Set.Icc a b) x","l":408,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.matrixBorelSpace","k":"instance","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"For every nonnegative integer dimension, the Borel-space structure on real square matrices of that dimension certifies that their coordinatewise product $\\sigma$-algebra equals their Borel $\\sigma$-algebra.","s":"∀ (p : ℕ), BorelSpace (Matrix (Fin p) (Fin p) ℝ)","l":387,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.matrixMeasurableSpace","k":"instance","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"For every nonnegative integer dimension, the measurable structure on real square matrices of that dimension is the coordinatewise product $\\sigma$-algebra.","s":"(p : ℕ) → MeasurableSpace (Matrix (Fin p) (Fin p) ℝ)","l":382,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_clipIcc","k":"theorem","m":"Causalean.Mathlib.Analysis.ClipInterval","a":"Mathlib/Analysis","t":2,"d":"Clipping to a fixed interval is measurable, so clipping a random variable again yields a random variable.","s":"∀ (a b : ℝ), Measurable (Causalean.Mathlib.Analysis.clipIcc a b)","l":107,"f":"Causalean/Mathlib/Analysis/ClipInterval.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_convexProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Metric projection onto a nonempty closed convex set is measurable under the Borel sigma-algebra of the Hilbert space.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : MeasurableSpace E] [BorelSpace E] (K : Set E) (hne : K.Nonempty) (hc : IsClosed K) (hconv : Convex ℝ K), Measurable (Causalean.Mathlib.Analysis.convexProj K hne hc hconv)","l":115,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.measurable_loewnerProj","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Loewner projection is measurable under the coordinatewise Borel sigma-algebra on finite real matrix spaces.","s":"∀ (p : ℕ) [inst : MeasurableSpace (Matrix (Fin p) (Fin p) ℝ)] [BorelSpace (Matrix (Fin p) (Fin p) ℝ)] (c C : ℝ), Measurable (Causalean.Mathlib.Analysis.loewnerProj p c C)","l":292,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.mem_strictGapSet_iff_discriminant_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"For a real two-by-two matrix, having distinct ordered roots is equivalent to strict positivity of its discriminant.","s":"∀ (G : Matrix (Fin 2) (Fin 2) ℝ), G ∈ Causalean.Mathlib.Analysis.strictGapSet ↔ 0 < Causalean.Mathlib.Analysis.rootDiscriminant G","l":51,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.monomial","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"For a finite coordinate index set, an assignment of a nonnegative integer exponent to each coordinate, and a real-valued coordinate vector, the multivariate monomial is the product of the coordinates after each has been raised to its assigned exponent.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℕ) → (ι → ℝ) → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"For a coordinate dimension and a finite family size, an assignment of a multivariate exponent vector to each member of that family, and a real radius, the monomial Gram matrix has as its $(k,l)$ entry the integral of the product of the $k$th and $l$th associated monomials over the centered closed coordinate cube of that radius.","s":"{d p : ℕ} → (Fin p → Fin d → ℕ) → ℝ → Matrix (Fin p) (Fin p) ℝ","l":34,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_isHermitian","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"The monomial Gram matrix is symmetric, and hence Hermitian over the real numbers.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) (r : ℝ), (Causalean.Mathlib.Analysis.monomialGram expo r).IsHermitian","l":170,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_posDef","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"Positive-definiteness of the monomial Gram matrix. For a finite family of multivariate monomials indexed by their exponent vectors, if the cube radius r is strictly positive and the exponent vectors are pairwise distinct, then the Gram matrix of pairwise integrals of the monomials over the cube of radius r is positive-definite.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, 0 < r → Function.Injective expo → (Causalean.Mathlib.Analysis.monomialGram expo r).PosDef","l":190,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_posSemidef","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"Every coefficient vector gives a nonnegative quadratic form under the monomial Gram matrix.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ}, (Causalean.Mathlib.Analysis.monomialGram expo r).PosSemidef","l":178,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomialGram_quadForm","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"Evaluating the monomial Gram quadratic form at a coefficient vector equals integrating the square of the corresponding monomial combination over the cube.","s":"∀ {d p : ℕ} (expo : Fin p → Fin d → ℕ) {r : ℝ} (z : Fin p → ℝ), ∑ k, ∑ l, z k * Causalean.Mathlib.Analysis.monomialGram expo r k l * z l = ∫ (u : Fin d → ℝ) in {u | ∀ (j : Fin d), |u j| ≤ r}, (∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u) ^ 2","l":130,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.monomial_linearIndependent_on_cube","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"A linear combination of distinct multivariate monomials that vanishes throughout the interior of a cube with positive radius must have every coefficient equal to zero.","s":"∀ {d p : ℕ} {expo : Fin p → Fin d → ℕ} {r : ℝ}, 0 < r → Function.Injective expo → ∀ (z : Fin p → ℝ), (∀ (u : Fin d → ℝ), (∀ (j : Fin d), u j ∈ Set.Ioo (-r) r) → ∑ k, z k * Causalean.Mathlib.Analysis.monomial (expo k) u = 0) → z = 0","l":122,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.mtx","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"For a square matrix dimension, the matrix vectorization linear equivalence identifies real square matrices of that dimension with Euclidean vectors indexed by ordered pairs of row and column coordinates.","s":"(p : ℕ) → Matrix (Fin p) (Fin p) ℝ ≃ₗ[ℝ] EuclideanSpace ℝ (Fin p × Fin p)","l":138,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.mtxHomeo","k":"def","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":1,"d":"For a square matrix dimension, the matrix vectorization homeomorphism identifies real square matrices of that dimension, with their usual topology, with Euclidean vectors indexed by ordered pairs of row and column coordinates.","s":"(p : ℕ) → Matrix (Fin p) (Fin p) ℝ ≃ₜ EuclideanSpace ℝ (Fin p × Fin p)","l":144,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.mulVec_sub_norm_le","k":"theorem","m":"Causalean.Mathlib.Analysis.ConvexProjection","a":"Mathlib/Analysis","t":2,"d":"Multiplying a vector by the difference of two finite real matrices has Euclidean norm at most their Frobenius distance times the vector's Euclidean norm.","s":"∀ {p : ℕ} (A B : Matrix (Fin p) (Fin p) ℝ) (v : Fin p → ℝ), √(∑ k, (A - B).mulVec v k ^ 2) ≤ Causalean.Mathlib.Analysis.frobDist A B * √(∑ k, v k ^ 2)","l":317,"f":"Causalean/Mathlib/Analysis/ConvexProjection.lean"},{"n":"Causalean.Mathlib.Analysis.outerProjector","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":1,"d":"Given a real two-vector, its outer-product matrix has entries $v_i v_j$; it has rank at most one (exactly one for a nonzero vector) and is the orthogonal projector onto the vector's direction when the vector has unit length.","s":"(Fin 2 → ℝ) → Matrix (Fin 2) (Fin 2) ℝ","l":15,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.outerProjector_eq_of_topEigenvectors","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real symmetric two-by-two matrix with distinct explicit roots and two unit upper-root eigenvectors, their outer-product projectors coincide.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → ∀ {v w : Fin 2 → ℝ}, v ⬝ᵥ v = 1 → G.mulVec v = Causalean.Mathlib.Analysis.lambda₁ G • v → w ⬝ᵥ w = 1 → G.mulVec w = Causalean.Mathlib.Analysis.lambda₁ G • w → Causalean.Mathlib.Analysis.outerProjector v = Causalean.Mathlib.Analysis.outerProjector w","l":173,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.outerProjector_neg","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real two-vector, negating it leaves its outer-product projector unchanged.","s":"∀ (v : Fin 2 → ℝ), Causalean.Mathlib.Analysis.outerProjector (-v) = Causalean.Mathlib.Analysis.outerProjector v","l":182,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.planarAngle","k":"def","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"For a point in the real coordinate plane, the planar angle is the angular coordinate assigned by the polar-coordinate chart.","s":"ℝ × ℝ → ℝ","l":33,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarFirst_div_radius_eq_cos","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"On the open upper half-plane, the first coordinate divided by the radius is the cosine of the polar angle.","s":"∀ (z : ℝ × ℝ), 0 < z.2 → z.1 / Causalean.Mathlib.Analysis.planarRadius z = Real.cos (Causalean.Mathlib.Analysis.planarAngle z)","l":178,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarRadius","k":"def","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":1,"d":"For a point in the real coordinate plane, the planar radius is its Euclidean distance from the origin, namely $\\sqrt{x^2+y^2}$ for coordinates $(x,y)$.","s":"ℝ × ℝ → ℝ","l":18,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.planarRadius_measurable","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"The Euclidean radius on the coordinate plane is Borel measurable.","s":"Measurable Causalean.Mathlib.Analysis.planarRadius","l":26,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.rootDiscriminant","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix, the symmetric-matrix discriminant is the squared diagonal difference plus four times the squared upper off-diagonal entry. The lower off-diagonal entry is ignored, so this is the discriminant of the characteristic polynomial only for a symmetric matrix.","s":"Matrix (Fin 2) (Fin 2) ℝ → ℝ","l":17,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.rootDiscriminant_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":2,"d":"For a real two-by-two matrix, its explicit discriminant is nonnegative.","s":"∀ (G : Matrix (Fin 2) (Fin 2) ℝ), 0 ≤ Causalean.Mathlib.Analysis.rootDiscriminant G","l":33,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.secondDirDeriv","k":"def","m":"Causalean.Mathlib.Analysis.LineSecondDeriv","a":"Mathlib/Analysis","t":1,"d":"For a real normed vector space, a real-valued function on that space, a direction vector, and a base point, the second directional derivative is the directional derivative at the base point, in the given direction, of the function mapping a point to the directional derivative of the original function in that direction.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → (E → ℝ) → E → E → ℝ","l":41,"f":"Causalean/Mathlib/Analysis/LineSecondDeriv.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":1,"d":"Quadratic-model descent gap. For a real function f on [0, T] where the interval length T, the slope margin c, and the curvature bound M are all nonnegative, f is twice continuously differentiable on [0, T], f is differentiable at 0, its second derivative is bounded above by M throughout the open interval, and its derivative at 0 is at most −c, then the clipped quadratic-model step lies in [0, T], and the resulting decrease f(0) − f(step) is at least c·step − (M/2)·step².","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 ≤ M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T ∧ f 0 - f (Causalean.Mathlib.Analysis.descentStep M c T) ≥ c * Causalean.Mathlib.Analysis.descentStep M c T - M / 2 * Causalean.Mathlib.Analysis.descentStep M c T ^ 2","l":114,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap_half","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"In both zero and positive curvature regimes, the clipped quadratic-model step gives a uniform half-linear decrease.","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 ≤ M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → Causalean.Mathlib.Analysis.descentStep M c T ∈ Set.Icc 0 T ∧ f 0 - f (Causalean.Mathlib.Analysis.descentStep M c T) ≥ c / 2 * Causalean.Mathlib.Analysis.descentStep M c T","l":175,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_descent_gap_min","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"With positive curvature, the clipped quadratic-model step gives at least half of the linear descent term.","s":"∀ {f : ℝ → ℝ} {M c T : ℝ}, 0 ≤ T → 0 ≤ c → 0 < M → ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → deriv f 0 ≤ -c → f 0 - f (min T (c / M)) ≥ c / 2 * min T (c / M)","l":141,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.second_order_upper_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.SecondOrderDescent","a":"Mathlib/Analysis","t":2,"d":"A twice differentiable real function whose second derivative is bounded above on an interval's interior is no larger than its tangent at the left endpoint plus the quadratic curvature allowance.","s":"∀ {f : ℝ → ℝ} {M T : ℝ}, ContDiffOn ℝ 2 f (Set.Icc 0 T) → DifferentiableAt ℝ f 0 → (∀ t ∈ Set.Ioo 0 T, deriv (deriv f) t ≤ M) → ∀ {t : ℝ}, t ∈ Set.Icc 0 T → f t ≤ f 0 + deriv f 0 * t + M / 2 * t ^ 2","l":54,"f":"Causalean/Mathlib/Analysis/SecondOrderDescent.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_add_le_add_opNorm","k":"theorem","m":"Causalean.Mathlib.Analysis.SingularValueWeyl","a":"Mathlib/Analysis","t":2,"d":"Given a baseline linear map, an additive perturbation, and a singular-value index, the perturbation can increase that indexed singular value by no more than its operator norm.","s":"∀ {V : Type u_1} {W : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V] [inst_3 : NormedAddCommGroup W] [inst_4 : InnerProductSpace ℝ W] [inst_5 : FiniteDimensional ℝ W] (A D : V →ₗ[ℝ] W) (j : ℕ), (A + D).singularValues j ≤ A.singularValues j + ‖LinearMap.toContinuousLinearMap D‖","l":184,"f":"Causalean/Mathlib/Analysis/SingularValueWeyl.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_comp_linearIsometry_last","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given a real linear map, orthonormal coordinates for its adjoint range, the last coordinate-indexed singular value after compression equals the corresponding singular value before compression.","s":"∀ {E : Type u_1} {F : Type u_2} {K : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : NormedAddCommGroup K] [inst_7 : InnerProductSpace ℝ K] [inst_8 : FiniteDimensional ℝ K] (M : E →ₗ[ℝ] F) (V : K →ₗᵢ[ℝ] E), V.range = (LinearMap.adjoint M).range → (M ∘ₗ V.toLinearMap).singularValues (Module.finrank ℝ K - 1) = M.singularValues (Module.finrank ℝ K - 1)","l":389,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_comp_linearIsometry_of_range_eq_adjoint_range","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":1,"d":"Given a real linear map, orthonormal signal coordinates, an equality between the coordinate range and the map's adjoint range, and a singular-value index, compressing to those coordinates preserves the indexed singular value.","s":"∀ {E : Type u_1} {F : Type u_2} {K : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : NormedAddCommGroup K] [inst_7 : InnerProductSpace ℝ K] [inst_8 : FiniteDimensional ℝ K] (M : E →ₗ[ℝ] F) (V : K →ₗᵢ[ℝ] E), V.range = (LinearMap.adjoint M).range → ∀ (j : ℕ), (M ∘ₗ V.toLinearMap).singularValues j = M.singularValues j","l":295,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_le_of_large_subspace","k":"theorem","m":"Causalean.Mathlib.Analysis.SingularValueWeyl","a":"Mathlib/Analysis","t":2,"d":"Given a linear map, an in-range singular-value index, and a subspace whose codimension is at most that index, if the map is bounded by a scalar multiple of the norm on that subspace, then the indexed singular value is at most that scalar.","s":"∀ {V : Type u_1} {W : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V] [inst_3 : NormedAddCommGroup W] [inst_4 : InnerProductSpace ℝ W] [inst_5 : FiniteDimensional ℝ W] (A : V →ₗ[ℝ] W) {j : ℕ}, j < Module.finrank ℝ V → ∀ (S : Submodule ℝ V), Module.finrank ℝ V ≤ Module.finrank ℝ ↥S + j → ∀ {c : ℝ}, (∀ x ∈ S, ‖A x‖ ≤ c * ‖x‖) → A.singularValues j ≤ c","l":21,"f":"Causalean/Mathlib/Analysis/SingularValueWeyl.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_le_verticalStack_left","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":1,"d":"Given a first real linear map, a second compatible real linear map, and a singular-value index, the vertical stack's indexed singular value is at least that of the first component.","s":"∀ {E : Type u_1} {F₀ : Type u_2} {F₁ : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F₀] [inst_4 : InnerProductSpace ℝ F₀] [inst_5 : FiniteDimensional ℝ F₀] [inst_6 : NormedAddCommGroup F₁] [inst_7 : InnerProductSpace ℝ F₁] [inst_8 : FiniteDimensional ℝ F₁] (M₀ : E →ₗ[ℝ] F₀) (M₁ : E →ₗ[ℝ] F₁) (j : ℕ), M₀.singularValues j ≤ (Causalean.Mathlib.Analysis.verticalStack M₀ M₁).singularValues j","l":266,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_le_verticalStack_right","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given a first compatible real linear map, a second real linear map, and a singular-value index, the vertical stack's indexed singular value is at least that of the second component.","s":"∀ {E : Type u_1} {F₀ : Type u_2} {F₁ : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F₀] [inst_4 : InnerProductSpace ℝ F₀] [inst_5 : FiniteDimensional ℝ F₀] [inst_6 : NormedAddCommGroup F₁] [inst_7 : InnerProductSpace ℝ F₁] [inst_8 : FiniteDimensional ℝ F₁] (M₀ : E →ₗ[ℝ] F₀) (M₁ : E →ₗ[ℝ] F₁) (j : ℕ), M₁.singularValues j ≤ (Causalean.Mathlib.Analysis.verticalStack M₀ M₁).singularValues j","l":276,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.singularValues_product_adjoint_lower_bound","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":1,"d":"Given a left rectangular factor, an injective square core, a right rectangular factor, and full-column-rank assumptions for the two rectangular factors, the last signal singular value of their adjoint product is at least the product of the three least signal singular values.","s":"∀ {E : Type u_1} {F : Type u_2} {K : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : NormedAddCommGroup K] [inst_7 : InnerProductSpace ℝ K] [inst_8 : FiniteDimensional ℝ K] [Nontrivial K] (A : K →ₗ[ℝ] F) (D : K →ₗ[ℝ] K) (B : K →ₗ[ℝ] E), Function.Injective ⇑A → Function.Injective ⇑D → Function.Injective ⇑B → A.singularValues (Module.finrank ℝ K - 1) * D.singularValues (Module.finrank ℝ K - 1) * B.singularValues (Module.finrank ℝ K - 1) ≤ (A ∘ₗ D ∘ₗ LinearMap.adjoint B).singularValues (Module.finrank ℝ K - 1)","l":130,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.strictGapSet","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralRoots","a":"Mathlib/Analysis","t":1,"d":"The strict-gap region consists of real two-by-two matrices whose lower quadratic-formula root is strictly below their upper root.","s":"Set (Matrix (Fin 2) (Fin 2) ℝ)","l":29,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralRoots.lean"},{"n":"Causalean.Mathlib.Analysis.sum_sq_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":2,"d":"The sum of squared coordinates of a nonzero finite real-valued vector is strictly positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (z : ι → ℝ), z ≠ 0 → 0 < ∑ k, z k ^ 2","l":235,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.supBall","k":"def","m":"Causalean.Mathlib.Analysis.MonomialGram","a":"Mathlib/Analysis","t":1,"d":"For a coordinate index set, a real-valued center vector, and a real radius, the closed sup-norm ball is the set of real-valued vectors whose every coordinate differs from the corresponding coordinate of the center by at most that radius.","s":"{ι : Type u_1} → (ι → ℝ) → ℝ → Set (ι → ℝ)","l":21,"f":"Causalean/Mathlib/Analysis/MonomialGram.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector","k":"def","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix, the algebraic upper spectral-projector candidate is its lower-root-shifted matrix divided by the gap between its explicit roots.","s":"Matrix (Fin 2) (Fin 2) ℝ → Matrix (Fin 2) (Fin 2) ℝ","l":19,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_eq_outerProjector","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real symmetric two-by-two matrix with distinct explicit roots and a unit upper-root eigenvector, the algebraic upper projector equals that vector's outer-product projector.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → ∀ {v : Fin 2 → ℝ}, v ⬝ᵥ v = 1 → G.mulVec v = Causalean.Mathlib.Analysis.lambda₁ G • v → Causalean.Mathlib.Analysis.topProjector G = Causalean.Mathlib.Analysis.outerProjector v","l":106,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_isHermitian","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix that is symmetric, its algebraic upper projector is symmetric.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → (Causalean.Mathlib.Analysis.topProjector G).IsHermitian","l":68,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_isOrthogonalProjectorOntoTop","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":1,"d":"Given a real two-by-two matrix that is symmetric and has distinct explicit roots, its algebraic quotient is the orthogonal projector onto the upper-root eigenspace.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → Causalean.Mathlib.Analysis.IsOrthogonalProjectorOntoTop G (Causalean.Mathlib.Analysis.topProjector G)","l":99,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_mulVec_eq_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real symmetric two-by-two matrix with distinct explicit roots and a two-vector, the algebraic projector fixes that vector exactly when it is an upper-root eigenvector.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → ∀ (x : Fin 2 → ℝ), (Causalean.Mathlib.Analysis.topProjector G).mulVec x = x ↔ G.mulVec x = Causalean.Mathlib.Analysis.lambda₁ G • x","l":30,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_mulVec_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real symmetric two-by-two matrix with distinct explicit roots and a two-vector, the algebraic projector sends that vector to zero exactly when it is a lower-root eigenvector.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → ∀ (x : Fin 2 → ℝ), (Causalean.Mathlib.Analysis.topProjector G).mulVec x = 0 ↔ G.mulVec x = Causalean.Mathlib.Analysis.lambda₂ G • x","l":49,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.topProjector_mul_self","k":"theorem","m":"Causalean.Mathlib.Analysis.TwoByTwoSpectralProjector","a":"Mathlib/Analysis","t":2,"d":"Given a real two-by-two matrix that is symmetric and has distinct explicit roots, its algebraic upper projector is idempotent.","s":"∀ {G : Matrix (Fin 2) (Fin 2) ℝ}, G.IsHermitian → G ∈ Causalean.Mathlib.Analysis.strictGapSet → Causalean.Mathlib.Analysis.topProjector G * Causalean.Mathlib.Analysis.topProjector G = Causalean.Mathlib.Analysis.topProjector G","l":76,"f":"Causalean/Mathlib/Analysis/TwoByTwoSpectralProjector.lean"},{"n":"Causalean.Mathlib.Analysis.translatedClosedHalfDisc_weighted_first_div_radius_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translation preserves the zero Cartesian-cosine integral over a closed upper half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ), ∫ (z : ℝ × ℝ) in {z | 0 ≤ (z - c).2 ∧ Causalean.Mathlib.Analysis.planarRadius (z - c) ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * ((z - c).1 / Causalean.Mathlib.Analysis.planarRadius (z - c)) = 0","l":255,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.translatedHalfDisc_radialSet_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translation preserves cosine cancellation on every measurable radial subset of a half-disc.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ) {A : Set ℝ}, MeasurableSet A → ∫ (z : ℝ × ℝ) in (fun u => c + u) '' {u | 0 < u.2 ∧ Causalean.Mathlib.Analysis.planarRadius u ≤ r} ∩ {z | Causalean.Mathlib.Analysis.planarRadius (z - c) ∈ A}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * Real.cos (Causalean.Mathlib.Analysis.planarAngle (z - c)) = 0","l":420,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.translatedHalfDisc_weighted_cos_cancellation","k":"theorem","m":"Causalean.Mathlib.Analysis.HalfDiscPolar","a":"Mathlib/Analysis","t":2,"d":"Translating an upper half-disc does not change the cosine cancellation.","s":"∀ (g : ℝ → ℝ) (r : ℝ) (c : ℝ × ℝ), ∫ (z : ℝ × ℝ) in (fun u => c + u) '' {u | 0 < u.2 ∧ Causalean.Mathlib.Analysis.planarRadius u ≤ r}, g (Causalean.Mathlib.Analysis.planarRadius (z - c)) * Real.cos (Causalean.Mathlib.Analysis.planarAngle (z - c)) = 0","l":395,"f":"Causalean/Mathlib/Analysis/HalfDiscPolar.lean"},{"n":"Causalean.Mathlib.Analysis.uniformC1On_logRatio","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"Positive pairs converging uniformly in C¹ to C¹ limits on a compact interval have log ratios converging uniformly in C¹, provided the whole family shares one positive lower bound.","s":"∀ {a b m : ℝ}, a < b → 0 < m → ∀ {q p : ℕ → ℝ → ℝ} {q₀ p₀ : ℝ → ℝ}, (∀ (n : ℕ), DifferentiableOn ℝ (q n) (Set.Icc a b)) → (∀ (n : ℕ), DifferentiableOn ℝ (p n) (Set.Icc a b)) → ContDiffOn ℝ 1 q₀ (Set.Icc a b) → ContDiffOn ℝ 1 p₀ (Set.Icc a b) → Causalean.Mathlib.Analysis.UniformC1On (Set.Icc a b) q q₀ → Causalean.Mathlib.Analysis.UniformC1On (Set.Icc a b) p p₀ → (∀ (n : ℕ), ∀ x ∈ Set.Icc a b, m ≤ q n x) → (∀ (n : ℕ), ∀ x ∈ Set.Icc a b, m ≤ p n x) → (∀ x ∈ Set.Icc a b, m ≤ q₀ x) → (∀ x ∈ Set.Icc a b, m ≤ p₀ x) → Causalean.Mathlib.Analysis.UniformC1On (Set.Icc a b) (fun n => Causalean.Mathlib.Analysis.logRatio (q n) (p n)) (Causalean.Mathlib.Analysis.logRatio q₀ p₀)","l":174,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.uniformlyOn_div_of_lowerBound","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"Uniform convergence is preserved by quotients on a compact set when numerators converge to a continuous limit and denominators share a strictly positive lower bound.","s":"∀ {α : Type u_1} [inst : TopologicalSpace α] {K : Set α}, IsCompact K → ∀ {u v : ℕ → α → ℝ} {u₀ v₀ : α → ℝ} {m : ℝ}, 0 < m → Causalean.Mathlib.Analysis.UniformlyOn K u u₀ → Causalean.Mathlib.Analysis.UniformlyOn K v v₀ → ContinuousOn u₀ K → (∀ (n : ℕ), ∀ x ∈ K, m ≤ v n x) → (∀ x ∈ K, m ≤ v₀ x) → Causalean.Mathlib.Analysis.UniformlyOn K (fun n x => u n x / v n x) fun x => u₀ x / v₀ x","l":71,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.uniformlyOn_inv_of_lowerBound","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"Uniform convergence is preserved by reciprocals when every approximating denominator and its limit share one strictly positive lower bound.","s":"∀ {α : Type u_1} {K : Set α} {f : ℕ → α → ℝ} {g : α → ℝ} {m : ℝ}, 0 < m → Causalean.Mathlib.Analysis.UniformlyOn K f g → (∀ (n : ℕ), ∀ x ∈ K, m ≤ f n x) → (∀ x ∈ K, m ≤ g x) → Causalean.Mathlib.Analysis.UniformlyOn K (fun n x => (f n x)⁻¹) fun x => (g x)⁻¹","l":49,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.uniformlyOn_log_of_lowerBound","k":"theorem","m":"Causalean.Mathlib.Analysis.LogRatioStability","a":"Mathlib/Analysis","t":2,"d":"Uniform convergence is preserved by the real logarithm when all functions share a strictly positive lower bound.","s":"∀ {α : Type u_1} {K : Set α} {f : ℕ → α → ℝ} {g : α → ℝ} {m : ℝ}, 0 < m → Causalean.Mathlib.Analysis.UniformlyOn K f g → (∀ (n : ℕ), ∀ x ∈ K, m ≤ f n x) → (∀ x ∈ K, m ≤ g x) → Causalean.Mathlib.Analysis.UniformlyOn K (fun n x => Real.log (f n x)) fun x => Real.log (g x)","l":122,"f":"Causalean/Mathlib/Analysis/LogRatioStability.lean"},{"n":"Causalean.Mathlib.Analysis.verticalStack","k":"def","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":1,"d":"For a real inner-product domain space, two real inner-product codomain spaces, and two real linear maps from the common domain into those respective codomains, the vertical stack is the real linear map sending each domain vector to its ordered pair of component outputs, equipped with the Hilbert direct-sum norm.","s":"{E : Type u_1} → {F₀ : Type u_2} → {F₁ : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : NormedAddCommGroup F₀] → [inst_3 : InnerProductSpace ℝ F₀] → [inst_4 : NormedAddCommGroup F₁] → [inst_5 : InnerProductSpace ℝ F₁] → (E →ₗ[ℝ] F₀) → (E →ₗ[ℝ] F₁) → E →ₗ[ℝ] WithLp 2 (F₀ × F₁)","l":217,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.verticalStack_adjoint_comp_self","k":"theorem","m":"Causalean.Mathlib.Analysis.RectangularSignalSingularValues","a":"Mathlib/Analysis","t":2,"d":"Given two real linear maps with a common domain, the Gram operator of their vertical stack is the sum of their two component Gram operators.","s":"∀ {E : Type u_1} {F₀ : Type u_2} {F₁ : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F₀] [inst_4 : InnerProductSpace ℝ F₀] [inst_5 : FiniteDimensional ℝ F₀] [inst_6 : NormedAddCommGroup F₁] [inst_7 : InnerProductSpace ℝ F₁] [inst_8 : FiniteDimensional ℝ F₁] (M₀ : E →ₗ[ℝ] F₀) (M₁ : E →ₗ[ℝ] F₁), LinearMap.adjoint (Causalean.Mathlib.Analysis.verticalStack M₀ M₁) ∘ₗ Causalean.Mathlib.Analysis.verticalStack M₀ M₁ = LinearMap.adjoint M₀ ∘ₗ M₀ + LinearMap.adjoint M₁ ∘ₗ M₁","l":225,"f":"Causalean/Mathlib/Analysis/RectangularSignalSingularValues.lean"},{"n":"Causalean.Mathlib.Analysis.waldRegularSet","k":"def","m":"Causalean.Mathlib.Analysis.RankOneWaldSmoothness","a":"Mathlib/Analysis","t":1,"d":"The regularity region contains rank-one Wald inputs whose left Gram matrix has distinct roots and a strictly positive upper root.","s":"{E : Type u_1} → [Fintype E] → Set (Causalean.Mathlib.Analysis.WaldInput E)","l":21,"f":"Causalean/Mathlib/Analysis/RankOneWaldSmoothness.lean"},{"n":"Causalean.Mathlib.Analysis.weighted_inner_sq_le","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":1,"d":"Weighted Cauchy-Schwarz inequality on a finite support, squared form. Weighting a finite collection of index points by nonnegative weights, the square of the weighted inner product of two real-valued functions is at most the product of their weighted sums of squares.","s":"∀ {ι : Type u_1} (s : Finset ι) (w f g : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → (∑ i ∈ s, w i * (f i * g i)) ^ 2 ≤ (∑ i ∈ s, w i * f i ^ 2) * ∑ i ∈ s, w i * g i ^ 2","l":33,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Analysis.weighted_sqrt_le_sqrt_weighted","k":"theorem","m":"Causalean.Mathlib.Analysis.WeightedCauchySchwarz","a":"Mathlib/Analysis","t":2,"d":"Jensen's inequality for the square root. Averaging nonnegative values with weights that are nonnegative and sum to at most one, the weighted average of their square roots is at most the square root of their weighted average.","s":"∀ {ι : Type u_1} (s : Finset ι) (w q : ι → ℝ), (∀ i ∈ s, 0 ≤ w i) → ∀ (c : ℝ), ∑ i ∈ s, w i ≤ c → (∀ i ∈ s, 0 ≤ q i) → ∑ i ∈ s, w i * √(q i) ≤ √c * √(∑ i ∈ s, w i * q i)","l":100,"f":"Causalean/Mathlib/Analysis/WeightedCauchySchwarz.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Omega","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population with n labelled units and a requested subset size M, the uniform slice is represented by the M-element subsets of that population.","s":"ℕ → ℕ → Type","l":16,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size n and a slice size M, the slice-function space is the real Euclidean space of functions on that uniform slice.","s":"ℕ → ℕ → Type","l":21,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.card_omega","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the uniform slice has exactly the usual binomial number of points.","s":"∀ {M n : ℕ}, M ≤ n → Fintype.card (Causalean.Mathlib.Combinatorics.JohnsonKneser.Omega n M) = n.choose M","l":109,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.center","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a slice function, its centered version is obtained by subtracting its uniform mean at every slice point.","s":"{n M : ℕ} → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":38,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.constFn","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a real value, the constant slice function is the function taking that value at every slice point.","s":"{n M : ℕ} → ℝ → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":34,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size, a slice size, and a degree bound, the degree-at-most subspace is the linear span of inclusion monomials indexed by sets no larger than that bound.","s":"(n M : ℕ) → ℕ → Submodule ℝ (Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M)","l":46,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost_eq_iSup_harmonic","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"The full degree-at-most-M space is the supremum of the Johnson harmonic layers from degree zero through degree M.","s":"∀ {n M : ℕ}, Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M M = ⨆ k, Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k","l":101,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost_eq_top","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"When the requested slice size is feasible, inclusion monomials through degree M span every real function on the uniform slice.","s":"∀ {M n : ℕ}, M ≤ n → Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M M = ⊤","l":86,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost_mono","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the first degree bound does not exceed the second, every function in the first inclusion-degree subspace also belongs to the second.","s":"∀ {n M d e : ℕ}, d ≤ e → Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M d ≤ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M e","l":51,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost_succ_eq_sup_harmonic","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a degree bound, the next inclusion-degree space is the sum of the preceding space and its new Johnson harmonic layer.","s":"∀ {n M : ℕ} (d : ℕ), Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M (d + 1) = Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M d ⊔ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M (d + 1)","l":93,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.disjointIndicator","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a set of labelled units, its disjointness indicator is one at slice points disjoint from that set and zero elsewhere.","s":"{n M : ℕ} → Finset (Fin n) → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":32,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.disjointIndicator_eq_sum_powerset","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a set of labelled units, its disjointness indicator equals the alternating inclusion-exclusion sum of the inclusion monomials of its subsets.","s":"∀ {n M : ℕ} (S : Finset (Fin n)), Causalean.Mathlib.Combinatorics.JohnsonKneser.disjointIndicator S = ∑ T ∈ S.powerset, (-1) ^ T.card • Causalean.Mathlib.Combinatorics.JohnsonKneser.inclusionMonomial T","l":45,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size, a slice size, and a harmonic degree, the harmonic projection is the canonical linear orthogonal projection onto that degree's Johnson harmonic subspace.","s":"(n M : ℕ) → Fin (M + 1) → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M →ₗ[ℝ] Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":22,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_add","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a harmonic degree, the projection of the sum equals the sum of the projections of the first slice function and the second slice function.","s":"∀ {M n : ℕ} (k : Fin (M + 1)) (f g : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) (f + g) = (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f + (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) g","l":27,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_eq_self_iff","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a harmonic degree and a slice function, projection leaves the function unchanged exactly when it already lies in that harmonic subspace.","s":"∀ {M n : ℕ} (k : Fin (M + 1)) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f = f ↔ f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k","l":111,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_idem","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a harmonic degree and a slice function, applying the harmonic projection twice gives the same component as applying it once.","s":"∀ {M n : ℕ} (k : Fin (M + 1)) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) ((Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f) = (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f","l":116,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_mem","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a harmonic degree and a slice function, the projected component belongs to that degree's Johnson harmonic subspace.","s":"∀ {M n : ℕ} (k : Fin (M + 1)) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k","l":106,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_pairwise_orthogonal","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible and the two harmonic degrees are distinct, the corresponding projected components of the given function are orthogonal under the uniform slice inner product, for the slice function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M) {j k : Fin (M + 1)}, j ≠ k → Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner ((Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M j) f) ((Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f) = 0","l":199,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_smul","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a harmonic degree, a real scalar, and a slice function, projecting after scalar multiplication equals scalar multiplication after projection.","s":"∀ {M n : ℕ} (k : Fin (M + 1)) (c : ℝ) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) (c • f) = c • (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f","l":33,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection_zero_eq_mean","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the degree-zero projection of the given slice function is the constant function at its uniform mean, for the slice function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M ⟨0, Nat.zero_lt_succ M⟩) f = Causalean.Mathlib.Combinatorics.JohnsonKneser.constFn (Causalean.Mathlib.Combinatorics.JohnsonKneser.mean f)","l":151,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.inclusionMonomial","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a set of labelled units, its inclusion monomial is one on slice points containing that set and zero elsewhere.","s":"{n M : ℕ} → Finset (Fin n) → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":42,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.instDecidableEqOmega","k":"instance","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"","s":"(n M : ℕ) → DecidableEq (Causalean.Mathlib.Combinatorics.JohnsonKneser.Omega n M)","l":19,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.instFintypeOmega","k":"instance","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"","s":"(n M : ℕ) → Fintype (Causalean.Mathlib.Combinatorics.JohnsonKneser.Omega n M)","l":19,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size and a slice size, the Johnson harmonic subspaces are the degree-zero inclusion layer at degree zero and the new orthogonal inclusion-degree layer at each positive degree.","s":"(n M : ℕ) → ℕ → Submodule ℝ (Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M)","l":17,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic_pairwise_orthogonal","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible and the two harmonic degrees are distinct, functions in those two Johnson harmonic subspaces are orthogonal under the uniform slice inner product, for the first function's membership evidence and the second function's membership evidence.","s":"∀ {M n : ℕ}, M ≤ n → ∀ {j k : Fin (M + 1)}, j ≠ k → ∀ {f g : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M}, f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑j → g ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k → Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner f g = 0","l":131,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size and a slice size, the unnormalized Kneser adjacency operator is specified by summing a function over all equally sized subsets disjoint from the argument, preserving addition, and commuting with real scalar multiplication.","s":"(n M : ℕ) → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M →ₗ[ℝ] Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":15,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_eigen","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"When two disjoint slice-sized subsets can fit in the population, the selected harmonic degree, and the given function belongs to that degree's Johnson harmonic subspace, unnormalized Kneser adjacency acts by its classical signed binomial eigenvalue, for the slice function.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (k : Fin (M + 1)), ∀ f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k, (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) f = ((-1) ^ ↑k * ↑((n - M - ↑k).choose (M - ↑k))) • f","l":279,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_harmonicProjection","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the selected harmonic degree, and the given slice function, Kneser adjacency acts on its projected harmonic component by the classical signed binomial eigenvalue.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (k : Fin (M + 1)) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) ((Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f) = ((-1) ^ ↑k * ↑((n - M - ↑k).choose (M - ↑k))) • (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f","l":347,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_inclusionMonomial_eq","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population and the indexing set is no larger than the slice size, Kneser adjacency maps the given inclusion monomial to its disjointness indicator times the number of compatible completions, for the indexing set.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (S : Finset (Fin n)), S.card ≤ M → (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) (Causalean.Mathlib.Combinatorics.JohnsonKneser.inclusionMonomial S) = ↑((n - M - S.card).choose (M - S.card)) • Causalean.Mathlib.Combinatorics.JohnsonKneser.disjointIndicator S","l":73,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_inclusionMonomial_mod_lower","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the degree is positive, the indexing set has exactly that degree, and the degree does not exceed the slice size, Kneser adjacency differs from its classical degree eigenvalue times that monomial only by a lower-degree function.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ {d : ℕ}, 0 < d → ∀ (S : Finset (Fin n)), S.card = d → d ≤ M → (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) (Causalean.Mathlib.Combinatorics.JohnsonKneser.inclusionMonomial S) - ((-1) ^ d * ↑((n - M - d).choose (M - d))) • Causalean.Mathlib.Combinatorics.JohnsonKneser.inclusionMonomial S ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M (d - 1)","l":154,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_mem_degreeAtMost","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the degree bound does not exceed the slice size, and the given function belongs to that inclusion-degree subspace, Kneser adjacency remains in the same inclusion-degree subspace, for the slice function.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ {d : ℕ}, d ≤ M → ∀ f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M d, (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M d","l":188,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency_sub_eigen_mem_lower","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the degree is positive, the degree does not exceed the slice size, and the given function has inclusion degree at most that degree, subtracting the classical degree eigenvalue leaves a function one degree lower, for the slice function.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ {d : ℕ}, 0 < d → d ≤ M → ∀ f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M d, (Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) f - ((-1) ^ d * ↑((n - M - d).choose (M - d))) • f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M (d - 1)","l":214,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.mean","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a slice function, its uniform mean is the average of its values over the slice.","s":"{n M : ℕ} → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M → ℝ","l":30,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.mean_constFn","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the uniform mean of the given constant slice function equals its constant value, for the real value.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (c : ℝ), Causalean.Mathlib.Combinatorics.JohnsonKneser.mean (Causalean.Mathlib.Combinatorics.JohnsonKneser.constFn c) = c","l":131,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.mem_degreeAtMost_zero_iff","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, a slice function belongs to the degree-zero subspace exactly when it is constant, for the given slice function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.degreeAtMost n M 0 ↔ ∃ c, f = Causalean.Mathlib.Combinatorics.JohnsonKneser.constFn c","l":56,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneserAdjacency","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a population size and a slice size, the normalized Kneser adjacency operator is the unnormalized disjointness sum divided by the number of disjoint neighbors.","s":"(n M : ℕ) → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M →ₗ[ℝ] Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M","l":355,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneserAdjacency_eigen","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the selected harmonic degree, and the given function belongs to that degree's Johnson harmonic subspace, normalized Kneser adjacency acts by the signed falling-factorial eigenvalue, for the slice function.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (k : Fin (M + 1)), ∀ f ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k, (Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneserAdjacency n M) f = ((-1) ^ ↑k * (↑(M.descFactorial ↑k) / ↑((n - M).descFactorial ↑k))) • f","l":393,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneserAdjacency_harmonicProjection","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population, the selected harmonic degree, and the given slice function, normalized Kneser adjacency acts on its projected harmonic component by the signed falling-factorial eigenvalue.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (k : Fin (M + 1)) (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), (Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneserAdjacency n M) ((Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f) = ((-1) ^ ↑k * (↑(M.descFactorial ↑k) / ↑((n - M).descFactorial ↑k))) • (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f","l":407,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.normalizedKneser_eigenvalue_eq","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When two disjoint slice-sized subsets can fit in the population and the selected harmonic degree, the normalized Kneser eigenvalue magnitude equals the corresponding ratio of falling factorials.","s":"∀ {M n : ℕ}, 2 * M ≤ n → ∀ (k : Fin (M + 1)), (↑((n - M).choose M))⁻¹ * ↑((n - M - ↑k).choose (M - ↑k)) = ↑(M.descFactorial ↑k) / ↑((n - M).descFactorial ↑k)","l":369,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner","k":"def","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"For a first slice function and a second slice function, the uniform-slice inner product is the average of their pointwise products over all slice points.","s":"{n M : ℕ} → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M → Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M → ℝ","l":25,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner_eq","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Basic","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the uniform inner product of the two given slice functions equals the ordinary function-space inner product divided by the number of slice points, for the first function and the second function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f g : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner f g = (↑(n.choose M))⁻¹ * inner ℝ f g","l":125,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Basic.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner_kneserAdjacency","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Kneser","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"For a first slice function and a second slice function, Kneser adjacency is self-adjoint under the uniform slice inner product.","s":"∀ {n M : ℕ} (f g : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner ((Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) f) g = Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner f ((Causalean.Mathlib.Combinatorics.JohnsonKneser.kneserAdjacency n M) g)","l":248,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Kneser.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner_residual_eq_zero","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the residual after projection is orthogonal under the uniform slice inner product to the given harmonic function, for a harmonic degree, a slice function, a comparison function, and evidence that the comparison function is in that harmonic subspace.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (k : Fin (M + 1)) (f g : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), g ∈ Causalean.Mathlib.Combinatorics.JohnsonKneser.johnsonHarmonic n M ↑k → Causalean.Mathlib.Combinatorics.JohnsonKneser.sliceInner (f - (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f) g = 0","l":122,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sum_harmonicProjection_eq","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":2,"d":"When the requested slice size is feasible, the given slice function equals the finite sum of its harmonic projections from degree zero through degree M, for the slice function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), ∑ k, (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f = f","l":139,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.Combinatorics.JohnsonKneser.sum_positive_harmonicProjection_eq_center","k":"theorem","m":"Causalean.Mathlib.Combinatorics.JohnsonKneser.Harmonics","a":"Mathlib/Combinatorics/JohnsonKneser","t":1,"d":"When the requested slice size is feasible, centering the given slice function equals the sum of its positive-degree harmonic projections, for the slice function.","s":"∀ {M n : ℕ}, M ≤ n → ∀ (f : Causalean.Mathlib.Combinatorics.JohnsonKneser.SliceFn n M), ∑ k with 0 < ↑k, (Causalean.Mathlib.Combinatorics.JohnsonKneser.harmonicProjection n M k) f = Causalean.Mathlib.Combinatorics.JohnsonKneser.center f","l":172,"f":"Causalean/Mathlib/Combinatorics/JohnsonKneser/Harmonics.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_eq_of_inner_ae","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":1,"d":"CompProd assembly from an a.e. inner equality. Consider two Markov-style kernels KL, KR from a base space to an outer space, each built by mixing the section at a base point of an inner kernel — fL, respectively fR, on the product of the base space and an intermediate space — against a fixed intermediate measure μ: KL a equals the section fL(a,·) composed with μ and KR a equals the section fR(a,·) composed with μ. If the inner kernels fL and fR agree almost everywhere with respect to the composition product of the base measure ν with the constant-μ kernel, then the composition products ν ⊗ₘ KL and ν ⊗ₘ KR are equal.","s":"∀ {α : Type u_1} {γ : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace β] (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] (μ : MeasureTheory.Measure γ) [MeasureTheory.SFinite μ] (KL KR : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel KL] [ProbabilityTheory.IsSFiniteKernel KR] (fL fR : ProbabilityTheory.Kernel (α × γ) β), (∀ (a : α), KL a = μ.bind ⇑(fL.sectR a)) → (∀ (a : α), KR a = μ.bind ⇑(fR.sectR a)) → (∀ᵐ (p : α × γ) ∂ν.compProd (ProbabilityTheory.Kernel.const α μ), fL p = fR p) → ν.compProd KL = ν.compProd KR","l":77,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_eq_of_inner_ae_kernel","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":1,"d":"CompProd assembly from an a.e. inner equality, indexed-integrator form. Consider two Markov-style kernels KL, KR from a base space to an outer space, each built by mixing the section at a base point of an inner kernel — fL, respectively fR, on the product of the base space and an intermediate space — against a fixed intermediate kernel κ evaluated at that point: KL a equals the section fL(a,·) composed with κ a and KR a equals the section fR(a,·) composed with κ a. If the inner kernels fL and fR agree almost everywhere with respect to the composition product of the base measure ν and κ, then the composition products ν ⊗ₘ KL and ν ⊗ₘ KR are equal.","s":"∀ {α : Type u_1} {γ : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace β] (ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsSFiniteKernel κ] (KL KR : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel KL] [ProbabilityTheory.IsSFiniteKernel KR] (fL fR : ProbabilityTheory.Kernel (α × γ) β), (∀ (a : α), KL a = (κ a).bind ⇑(fL.sectR a)) → (∀ (a : α), KR a = (κ a).bind ⇑(fR.sectR a)) → (∀ᵐ (p : α × γ) ∂ν.compProd κ, fL p = fR p) → ν.compProd KL = ν.compProd KR","l":33,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.CompProdAssembly.compProd_map_snd_apply","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.CompProdAssembly","a":"Mathlib/Probability/Kernel","t":2,"d":"Snd-marginal of a composition product, pointwise (disintegration backbone).","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] (κ₁ : ProbabilityTheory.Kernel α β) (κ₂ : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel κ₁] [ProbabilityTheory.IsSFiniteKernel κ₂] (a : α), ((κ₁.compProd κ₂).map Prod.snd) a = (κ₁ a).bind ⇑(κ₂.sectR a)","l":98,"f":"Causalean/Mathlib/Probability/Kernel/CompProdAssembly.lean"},{"n":"Causalean.Mathlib.FWL.fittedValue","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a finite number of regressors, a tuple of regressor vectors, and a vector of real coefficients, the fitted value is the sum of each regressor multiplied by its corresponding coefficient.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → {K : ℕ} → (Fin K → V) → (Fin K → ℝ) → V","l":153,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fittedValue_sub","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Fitted values are linear in the coefficient vector.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] {K : ℕ} (X : Fin K → V) (β γ : Fin K → ℝ), (Causalean.Mathlib.FWL.fittedValue X fun j => β j - γ j) = Causalean.Mathlib.FWL.fittedValue X β - Causalean.Mathlib.FWL.fittedValue X γ","l":159,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a subspace that admits an orthogonal projection, a finite number of regressors, a tuple of regressor vectors, and an outcome vector, the Frisch–Waugh–Lovell coefficient vector is the inverse residualized Gram matrix multiplied by the vector of inner products between residualized regressors and the outcome.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → Fin K → ℝ","l":212,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_eq_of_original_minimizer","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Any minimizer of the original least-squares problem has coefficient block equal to the FWL coefficient. This is the uniqueness part of the standard FWL statement for the X-block.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : V}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y","l":423,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_original_minimizes","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Standard finite-dimensional Frisch–Waugh–Lovell theorem. On a real inner-product space with a nuisance subspace H admitting an orthogonal projection, and a finite tuple of regressors X, if the residualized Gram matrix of X has nonzero determinant, i.e. is invertible and a candidate nuisance vector h lies in H, then the original least-squares objective — evaluated at the FWL coefficient together with the orthogonal-projection nuisance term of its raw residual — is at most the original objective at any other coefficient vector β and nuisance term h.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))) ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":400,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.fwlCoef_residualized_minimizes","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"The FWL coefficient minimizes the residualized least-squares objective.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualizedObjective H X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) ≤ Causalean.Mathlib.FWL.residualizedObjective H X Y β","l":334,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a subspace that admits an orthogonal projection, a finite number of regressors, and a tuple of regressor vectors, the residualized Gram matrix has as its $(j,k)$ entry the inner product of the orthogonal residuals of the $j$th and $k$th regressors.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → Matrix (Fin K) (Fin K) ℝ","l":187,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Each entry of the residualized Gram matrix is the inner product of two residualized regressors.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (j k : Fin K), Causalean.Mathlib.FWL.gramResid H X j k = inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) (Causalean.Mathlib.FWL.residualize H (X k))","l":194,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.gramResid_mulVec_fwlCoef","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualized normal equations (FWL coefficient form). If Q_{XX} = gramResid H X is invertible (equivalently, the residualized regressors are linearly independent), the FWL coefficient fwlCoef H X Y = Q_{XX}⁻¹ *ᵥ (fun j => ⟨X̃ j, Y⟩) satisfies the residualized normal equations Q_{XX} *ᵥ fwlCoef H X Y = (fun j => ⟨X̃ j, Y⟩).","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → (Causalean.Mathlib.FWL.gramResid H X).mulVec (Causalean.Mathlib.FWL.fwlCoef H X Y) = Causalean.Mathlib.FWL.residInnerVec H X Y","l":220,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.inner_residualize_of_mem","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residual is orthogonal to every element of H. This is the defining property of the orthogonal residual.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V) {w : V}, w ∈ H → inner ℝ (Causalean.Mathlib.FWL.residualize H v) w = 0","l":119,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.inner_residualizedRegressor_fwlResidual","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"The residual left by fwlCoef is orthogonal to every residualized regressor. This is the normal-equation form used in the completing-square argument.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (j : Fin K), inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) (Causalean.Mathlib.FWL.residualize H Y - Causalean.Mathlib.FWL.fittedValue (fun k => Causalean.Mathlib.FWL.residualize H (X k)) (Causalean.Mathlib.FWL.fwlCoef H X Y)) = 0","l":259,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a finite number of regressors, a tuple of regressor vectors, an outcome vector, a vector of real coefficients, and a nuisance vector, the original least-squares objective is the squared norm of the outcome minus its fitted value and minus the nuisance vector.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → {K : ℕ} → (Fin K → V) → V → (Fin K → ℝ) → V → ℝ","l":252,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective_eq_residualizedObjective_add_projectionError","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Pythagorean split of the original objective into the residualized objective plus the squared distance from h to the projection of the raw regression residual onto H.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h = Causalean.Mathlib.FWL.residualizedObjective H X Y β + inner ℝ (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β) - h) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β) - h)","l":342,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.originalObjective_projection_eq_residualizedObjective","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"At the nuisance projection, the original objective equals the residualized objective.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.originalObjective X Y β (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X β)) = Causalean.Mathlib.FWL.residualizedObjective H X Y β","l":389,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.original_minimizer_eq_fwlCoef_projection","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"Uniqueness of the original least-squares minimizer. If the residualized Gram matrix of X is invertible, a candidate nuisance vector h lies in H, and the pair (β, h) minimizes the original least-squares objective over all coefficient vectors and nuisance terms in H, then β equals the FWL coefficient and h equals the orthogonal-projection nuisance term of the FWL coefficient's raw residual.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : V}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : V}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y ∧ h = Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))","l":474,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a subspace that admits an orthogonal projection, a finite number of regressors, a tuple of regressor vectors, and an outcome vector, the residualized inner-product vector assigns to each regressor index the inner product of that regressor's orthogonal residual with the outcome.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → Fin K → ℝ","l":200,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Each entry of the residualized right-hand side is the inner product of a residualized regressor with the outcome.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (j : Fin K), Causalean.Mathlib.FWL.residInnerVec H X Y j = inner ℝ (Causalean.Mathlib.FWL.residualize H (X j)) Y","l":207,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residInnerVec_eq_residualize_right","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Symmetric residualization for the FWL right-hand side. The residualized inner products ⟨X̃ j, Y⟩ agree with ⟨X̃ j, Ỹ⟩, so the FWL coefficient depends only on the residualized response.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), Causalean.Mathlib.FWL.residInnerVec H X Y = Causalean.Mathlib.FWL.residInnerVec H X (Causalean.Mathlib.FWL.residualize H Y)","l":234,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a subspace that admits an orthogonal projection, and a vector, the orthogonal residual is the vector minus its orthogonal projection onto that subspace.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → V → V","l":93,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizeL","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space and a subspace that admits an orthogonal projection, the residual-maker linear map sends each vector to its orthogonal residual after removal of its projection onto that subspace.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → V →L[ℝ] V","l":106,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizeL_apply","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Applying the residual-maker linear map gives the residualized vector.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V), (Causalean.Mathlib.FWL.residualizeL H) v = Causalean.Mathlib.FWL.residualize H v","l":114,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_def","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization is the original vector minus its orthogonal projection onto the nuisance subspace.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v : V), Causalean.Mathlib.FWL.residualize H v = v - Submodule.orthogonalProjectionFn v","l":101,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_fittedValue","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization commutes with forming a fitted value.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualize H (Causalean.Mathlib.FWL.fittedValue X β) = Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) β","l":165,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_inner_swap_right","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Symmetric residualization in the right slot. Since M_H is self-adjoint and idempotent, ⟨X̃, Y⟩ = ⟨X̃, Ỹ⟩.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] (v w : V), inner ℝ (Causalean.Mathlib.FWL.residualize H v) w = inner ℝ (Causalean.Mathlib.FWL.residualize H v) (Causalean.Mathlib.FWL.residualize H w)","l":127,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualize_regressionResidual","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Residualization of the raw regression residual is the residualized outcome minus the fitted value of the residualized regressors.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualize H (Y - Causalean.Mathlib.FWL.fittedValue X β) = Causalean.Mathlib.FWL.residualize H Y - Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) β","l":174,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective","k":"def","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":1,"d":"For a real inner-product space, a subspace that admits an orthogonal projection, a finite number of regressors, a tuple of regressor vectors, an outcome vector, and a vector of real coefficients, the residualized least-squares objective is the squared norm of the orthogonal residual of the outcome minus the fitted value formed from the orthogonal residuals of the regressors.","s":"{V : Type u_1} → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace ℝ V] → (H : Submodule ℝ V) → [H.HasOrthogonalProjection] → {K : ℕ} → (Fin K → V) → V → (Fin K → ℝ) → ℝ","l":242,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective_eq_fwlCoef_add_error","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"Completing-square identity for the residualized objective around the FWL coefficient.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ), Causalean.Mathlib.FWL.residualizedObjective H X Y β = Causalean.Mathlib.FWL.residualizedObjective H X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) + inner ℝ (Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) fun j => Causalean.Mathlib.FWL.fwlCoef H X Y j - β j) (Causalean.Mathlib.FWL.fittedValue (fun j => Causalean.Mathlib.FWL.residualize H (X j)) fun j => Causalean.Mathlib.FWL.fwlCoef H X Y j - β j)","l":285,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.FWL.residualizedObjective_le_originalObjective","k":"theorem","m":"Causalean.Mathlib.FWL","a":"Mathlib","t":2,"d":"For fixed β, the residualized objective is the minimum of the original objective over the nuisance subspace H.","s":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (H : Submodule ℝ V) [inst_2 : H.HasOrthogonalProjection] {K : ℕ} (X : Fin K → V) (Y : V) (β : Fin K → ℝ) {h : V}, h ∈ H → Causalean.Mathlib.FWL.residualizedObjective H X Y β ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":380,"f":"Causalean/Mathlib/FWL.lean"},{"n":"Causalean.Mathlib.GraphMapProd.instIsMarkovKernelMechanismKernel","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":2,"d":"The mechanism kernel is Markov when the exogenous-noise law is a probability measure and the mechanism is measurable.","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.IsProbabilityMeasure νₗ] {Φ : β × γ → δ}, Measurable Φ → ProbabilityTheory.IsMarkovKernel (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ)","l":59,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.map_graph_prod_eq_compProd","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":1,"d":"Graph push-forward of a product measure is a composition product. For an s-finite measure α on a first factor, an s-finite measure νₗ on a second factor, and a measurable mechanism map Φ combining the two factors into a third space, pushing the product measure α.prod νₗ forward along the graph map (o, l) ↦ (o, Φ (o, l)) equals the composition product of α with the mechanism kernel that sends each value o of the first coordinate to the pushforward of νₗ through the slice l ↦ Φ (o, l).","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (α : MeasureTheory.Measure β) [MeasureTheory.SFinite α] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.SFinite νₗ] {Φ : β × γ → δ}, Measurable Φ → MeasureTheory.Measure.map (fun p => (p.1, Φ p)) (α.prod νₗ) = α.compProd (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ)","l":72,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.mechanismKernel","k":"def","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":1,"d":"For three measurable spaces serving respectively as predictor, exogenous-noise, and response spaces, a measure on the exogenous-noise space, and a mechanism mapping a predictor--noise pair to a response, the structural-mechanism kernel assigns to each predictor value the push-forward of the noise measure through the corresponding slice of the mechanism.","s":"{β : Type u_1} → {γ : Type u_2} → {δ : Type u_3} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSpace γ] → [inst_2 : MeasurableSpace δ] → MeasureTheory.Measure γ → (β × γ → δ) → ProbabilityTheory.Kernel β δ","l":34,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.GraphMapProd.mechanismKernel_apply","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.GraphMapProd","a":"Mathlib/Probability/Kernel","t":2,"d":"Pointwise value of mechanismKernel: κ o = νₗ.map (fun l => Φ (o, l)).","s":"∀ {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace δ] (νₗ : MeasureTheory.Measure γ) [MeasureTheory.SFinite νₗ] {Φ : β × γ → δ}, Measurable Φ → ∀ (o : β), (Causalean.Mathlib.GraphMapProd.mechanismKernel νₗ Φ) o = MeasureTheory.Measure.map (fun l => Φ (o, l)) νₗ","l":47,"f":"Causalean/Mathlib/Probability/Kernel/GraphMapProd.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.absolutelyContinuous_of_strictlyPositivePMF","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two strictly positive finite probability mass functions, the first induced measure is absolutely continuous with respect to the second.","s":"∀ {W : Type u_2} [Fintype W] [inst : MeasurableSpace W] [MeasurableSingletonClass W] {p q : PMF W}, (∀ (w : W), 0 < p w) → (∀ (w : W), 0 < q w) → p.toMeasure.AbsolutelyContinuous q.toMeasure","l":140,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.appendSymbol","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a finite word and a final symbol, append the symbol to the word.","s":"{Ω : Type u_1} → {k : ℕ} → (Fin k → Ω) → Ω → Fin (k + 1) → Ω","l":23,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.appendSymbol_eq_snoc","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Appending a symbol to a finite word agrees with Mathlib's tuple append Fin.snoc.","s":"∀ {Ω : Type u_1} [Fintype Ω] [Nonempty Ω] [inst : MeasurableSpace Ω] [MeasurableSingletonClass Ω] {k : ℕ} (u : Fin k → Ω) (x : Ω), Causalean.Mathlib.InformationTheory.FiniteWordChainRule.appendSymbol u x = Fin.snoc u x","l":27,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.chainKL","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"The chronological conditional-KL sum recursively adds the shared-prefix conditional divergence of two finite-word laws at each word length.","s":"{Ω : Type u_1} → [Fintype Ω] → [Nonempty Ω] → [inst : MeasurableSpace Ω] → [MeasurableSingletonClass Ω] → (k : ℕ) → PMF (Fin k → Ω) → PMF (Fin k → Ω) → ENNReal","l":267,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.conditionalKL_eq_sum","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two finite-word laws whose conditional last-symbol laws satisfy pointwise absolute continuity, their shared-prefix conditional KL is the prefix-probability- weighted sum of pointwise conditional divergences.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p q : PMF (Fin (k + 1) → Ω)), (∀ (u : Fin k → Ω), (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure.AbsolutelyContinuous (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure) → InformationTheory.klDiv ((Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure.compProd (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel p)) ((Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure.compProd (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel q)) = ∑ u, (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u * InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure","l":221,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.klDiv_eq_chainKL","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a word length and two finite-word laws, their KL divergence equals the chronological sum of their history-dependent one-step conditional increments.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] (k : ℕ) (p q : PMF (Fin k → Ω)), InformationTheory.klDiv p.toMeasure q.toMeasure = Causalean.Mathlib.InformationTheory.FiniteWordChainRule.chainKL k p q","l":278,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.klDiv_lastCoordinate_eq_add","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two finite-word laws, one chronological extension splits their KL divergence into the prefix KL and a shared-prefix conditional-kernel KL.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p q : PMF (Fin (k + 1) → Ω)), InformationTheory.klDiv p.toMeasure q.toMeasure = InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF q).toMeasure + InformationTheory.klDiv ((Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure.compProd (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel p)) ((Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure.compProd (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel q))","l":202,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.klDiv_lastCoordinate_eq_add_sum","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two finite-word laws whose conditional last-symbol laws satisfy pointwise absolute continuity, one chronological extension is the prefix KL plus the explicit average of pointwise conditional divergences.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p q : PMF (Fin (k + 1) → Ω)), (∀ (u : Fin k → Ω), (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure.AbsolutelyContinuous (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure) → InformationTheory.klDiv p.toMeasure q.toMeasure = InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF q).toMeasure + ∑ u, (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u * InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure","l":242,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.klDiv_lastCoordinate_eq_add_sum_of_fullSupport","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two finite-word laws that have strictly positive mass at every word, one chronological extension is the prefix KL plus the explicit average of pointwise conditional divergences.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p q : PMF (Fin (k + 1) → Ω)), (∀ (w : Fin (k + 1) → Ω), 0 < p w) → (∀ (w : Fin (k + 1) → Ω), 0 < q w) → InformationTheory.klDiv p.toMeasure q.toMeasure = InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF q).toMeasure + ∑ u, (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u * InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure","l":254,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.lastCoordinateSplit","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a prefix length, split a nonempty finite word into its strict prefix and last symbol.","s":"{Ω : Type u_1} → (k : ℕ) → (Fin (k + 1) → Ω) ≃ (Fin k → Ω) × Ω","l":34,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.lastCoordinateSplit_compProd","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given a finite-word law, transport by the last-coordinate split equals the composition of its prefix marginal and conditional last-symbol kernel.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p : PMF (Fin (k + 1) → Ω)), MeasureTheory.Measure.map (⇑(Causalean.Mathlib.InformationTheory.FiniteWordChainRule.lastCoordinateSplit k)) p.toMeasure = (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p).toMeasure.compProd (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel p)","l":168,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a finite-word law, its conditional last-symbol laws form a kernel on strict prefixes.","s":"{Ω : Type u_1} → [Fintype Ω] → [Nonempty Ω] → [inst : MeasurableSpace Ω] → [MeasurableSingletonClass Ω] → {k : ℕ} → PMF (Fin (k + 1) → Ω) → ProbabilityTheory.Kernel (Fin k → Ω) Ω","l":90,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel_isMarkov","k":"instance","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"For a finite-word law, the conditional last-symbol kernel is Markov.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p : PMF (Fin (k + 1) → Ω)), ProbabilityTheory.IsMarkovKernel (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolKernel p)","l":96,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a finite-word law and a strict prefix, the conditional law of the last symbol is the normalized extension mass, totalized by a point mass on null prefixes.","s":"{Ω : Type u_1} → [Fintype Ω] → [Nonempty Ω] → [inst : MeasurableSpace Ω] → [MeasurableSingletonClass Ω] → {k : ℕ} → PMF (Fin (k + 1) → Ω) → (Fin k → Ω) → PMF Ω","l":72,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given two finite-word laws that are strictly positive everywhere, then for every strict prefix, the first conditional last-symbol law is absolutely continuous with respect to the second.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} {p q : PMF (Fin (k + 1) → Ω)}, (∀ (w : Fin (k + 1) → Ω), 0 < p w) → (∀ (w : Fin (k + 1) → Ω), 0 < q w) → ∀ (u : Fin k → Ω), (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u).toMeasure.AbsolutelyContinuous (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF q u).toMeasure","l":158,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF_fullSupport","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"If a finite-word law has strictly positive mass at every word, then for every strict prefix, its totalized conditional last-symbol law has strictly positive mass at every symbol.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} {p : PMF (Fin (k + 1) → Ω)}, (∀ (w : Fin (k + 1) → Ω), 0 < p w) → ∀ (u : Fin k → Ω) (x : Ω), 0 < (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u) x","l":130,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF","k":"def","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":1,"d":"Given a probability mass function on nonempty finite words, its strict-prefix marginal sums over the final symbol.","s":"{Ω : Type u_1} → [Fintype Ω] → {k : ℕ} → PMF (Fin (k + 1) → Ω) → PMF (Fin k → Ω)","l":48,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF_apply","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"For a finite-word law and a strict prefix, the prefix mass is the sum of the masses of all one-symbol extensions.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [Nonempty Ω] [inst_2 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] {k : ℕ} (p : PMF (Fin (k + 1) → Ω)) (u : Fin k → Ω), (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u = ∑ x, p (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.appendSymbol u x)","l":65,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF_fullSupport","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"If a finite-word law has strictly positive mass at every word, its strict-prefix marginal also has strictly positive mass everywhere.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [Nonempty Ω] [inst_2 : MeasurableSpace Ω] [MeasurableSingletonClass Ω] {k : ℕ} {p : PMF (Fin (k + 1) → Ω)}, (∀ (w : Fin (k + 1) → Ω), 0 < p w) → ∀ (u : Fin k → Ω), 0 < (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u","l":120,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefix_mul_nextSymbol","k":"theorem","m":"Causalean.Mathlib.InformationTheory.FiniteWordChainRule","a":"Mathlib/InformationTheory","t":2,"d":"Given a finite-word law, a strict prefix, and a final symbol, prefix mass times conditional final-symbol mass recovers the joint word mass, including on null prefixes.","s":"∀ {Ω : Type u_1} [inst : Fintype Ω] [inst_1 : Nonempty Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : MeasurableSingletonClass Ω] {k : ℕ} (p : PMF (Fin (k + 1) → Ω)) (u : Fin k → Ω) (x : Ω), (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.prefixPMF p) u * (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.nextSymbolPMF p u) x = p (Causalean.Mathlib.InformationTheory.FiniteWordChainRule.appendSymbol u x)","l":103,"f":"Causalean/Mathlib/InformationTheory/FiniteWordChainRule.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.abs_klRemainder_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Dominated cubic remainder bound. For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ deviates from its second-order Taylor approximation (h²/2) · ∫ s² dμ by at most C³ · |h|³:","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 / 2 → |(InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal - h ^ 2 / 2 * ∫ (y : Z), s y ^ 2 ∂μ| ≤ C ^ 3 * |h| ^ 3","l":105,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.abs_tiltRemainder_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.CubicRemainder","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Cubic remainder bound for (1 + x) · log (1 + x). For a real number x with |x| ≤ 1/2, the second-order Taylor remainder of (1 + x) · log (1 + x) at 0 is bounded by |x|³:","s":"∀ {x : ℝ}, |x| ≤ 1 / 2 → |(1 + x) * Real.log (1 + x) - x - x ^ 2 / 2| ≤ |x| ^ 3","l":131,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/CubicRemainder.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.integral_tiltDensity","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"The tilt density integrates to 1: ∫ (1 + h · s) dμ = 1 + h · ∫ s = 1 for a probability measure μ and a mean-zero score s.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {h : ℝ}, MeasureTheory.Integrable s μ → ∫ (y : Z), s y ∂μ = 0 → ∫ (y : Z), 1 + h * s y ∂μ = 1","l":74,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.isProbabilityMeasure_tiltMeasure","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"The linear tilt is a probability measure for small |h|. For a probability measure μ and a measurable score function s that is bounded by a constant C (|s y| ≤ C for every y) and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the linearly tilted measure tiltMeasure μ s h is again a probability measure.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h)","l":93,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.klDiv_tiltMeasure_toReal_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"KL of the linear tilt as an x log x density integral. For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, if the tilt strength satisfies |h| · C ≤ 1, then the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ, as a real number, equals the μ-integral of (1 + h·s)·log(1 + h·s):","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C h : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → |h| * C ≤ 1 → (InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal = ∫ (y : Z), (1 + h * s y) * Real.log (1 + h * s y) ∂μ","l":55,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.klDiv_tilt_expansion","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.KLExpansion","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"Second-order KL expansion of the linear density tilt (headline result). For a probability measure μ and a measurable score function s that is bounded by a constant C and has mean zero under μ, the Kullback–Leibler divergence of the linearly tilted measure tiltMeasure μ s h from μ, as a function of the tilt strength h, agrees with the quadratic approximation (h²/2) · ∫ s² dμ up to an error that is little-o of h² as h → 0:","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {μ : MeasureTheory.Measure Z} [MeasureTheory.IsProbabilityMeasure μ] {s : Z → ℝ} {C : ℝ}, Measurable s → (∀ (y : Z), |s y| ≤ C) → ∫ (y : Z), s y ∂μ = 0 → (fun h => (InformationTheory.klDiv (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h) μ).toReal - h ^ 2 / 2 * ∫ (y : Z), s y ^ 2 ∂μ) =o[nhds 0] fun h => h ^ 2","l":209,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/KLExpansion.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltDensity_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"For a bounded score |s y| ≤ C and tilt strength with |h| · C ≤ 1, the tilt density 1 + h · s y is nonnegative: |h · s y| ≤ |h| · C ≤ 1.","s":"∀ {Z : Type u_1} {s : Z → ℝ} {C h : ℝ}, (∀ (y : Z), |s y| ≤ C) → |h| * C ≤ 1 → ∀ (y : Z), 0 ≤ 1 + h * s y","l":55,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure","k":"def","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":1,"d":"On a measurable sample space, given a base measure, a real-valued score function, and a real tilt strength, the linear density tilt is the measure having density $\\max\\{1+h s(z),0\\}$ with respect to the base measure.","s":"{Z : Type u_1} → [inst : MeasurableSpace Z] → MeasureTheory.Measure Z → (Z → ℝ) → ℝ → MeasureTheory.Measure Z","l":42,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.Basic","a":"Mathlib/InformationTheory/KlDensityTiltExpansion","t":2,"d":"The tilt is absolutely continuous with respect to the base measure μ (any withDensity measure is).","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] (μ : MeasureTheory.Measure Z) (s : Z → ℝ) (h : ℝ), (Causalean.Mathlib.InformationTheory.KlDensityTiltExpansion.tiltMeasure μ s h).AbsolutelyContinuous μ","l":48,"f":"Causalean/Mathlib/InformationTheory/KlDensityTiltExpansion/Basic.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_eq_of_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"For measurable spaces B and Ω, a finite base measure m, finite kernels κ, η from B to Ω, and a measurable projection proj : Ω → B whose graph {(b, ω) | b = proj ω} is a measurable subset of B × Ω, suppose κ-almost every output, for m-almost every base point b, lands in the fibre proj⁻¹{b}, likewise for η, and κ b is absolutely continuous with respect to η b for m-almost every b. Then the Kullback–Leibler divergence between the bind of m with κ and the bind of m with η equals the m-average, over the base point b, of the Kullback–Leibler divergence between κ b and η b.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] [MeasurableSpace.CountableOrCountablyGenerated B Ω] (m : MeasureTheory.Measure B) [MeasureTheory.IsFiniteMeasure m] (κ η : ProbabilityTheory.Kernel B Ω) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (proj : Ω → B), Measurable proj → MeasurableSet {p | p.1 = proj p.2} → (∀ᵐ (b : B) ∂m, (κ b) {ω | proj ω = b}ᶜ = 0) → (∀ᵐ (b : B) ∂m, (η b) {ω | proj ω = b}ᶜ = 0) → (∀ᵐ (b : B) ∂m, (κ b).AbsolutelyContinuous (η b)) → InformationTheory.klDiv (m.bind ⇑κ) (m.bind ⇑η) = ∫⁻ (b : B), InformationTheory.klDiv (κ b) (η b) ∂m","l":203,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"Passing two finite input laws μ and ν through the same randomized observation channel κ, the Kullback–Leibler divergence between the channel's output laws is no larger than the divergence between the original input laws, including when the channel is non-injective or the original divergence is infinite. Mathlib now provides the same data-processing inequality as InformationTheory.klDiv_comp_right_le.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.bind ⇑κ) (ν.bind ⇑κ) ≤ InformationTheory.klDiv μ ν","l":462,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_bind_le_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Passing two probability laws through a shared Markov channel cannot increase their Kullback--Leibler divergence; this is the probability-law specialization of the finite-measure data-processing inequality.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.bind ⇑κ) (ν.bind ⇑κ) ≤ InformationTheory.klDiv μ ν","l":475,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_compProd_left","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Adding an output drawn from the same Markov kernel preserves the Kullback--Leibler divergence between two finite input laws because the joint observation still retains the input coordinate.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ], InformationTheory.klDiv (μ.compProd κ) (ν.compProd κ) = InformationTheory.klDiv μ ν","l":434,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_compProd_right_of_forall_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":1,"d":"For a countably-generated pair of measurable spaces, a finite base measure μ, and finite kernels κ, η out of the base, if κ b is absolutely continuous with respect to η b for μ-almost every base point b, then the Kullback–Leibler divergence between the composition products μ ⊗ₘ κ and μ ⊗ₘ η equals the μ-average, over the base point, of the Kullback–Leibler divergence between κ and η at that base point.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasurableSpace.CountableOrCountablyGenerated α β] [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → InformationTheory.klDiv (μ.compProd κ) (μ.compProd η) = ∫⁻ (a : α), InformationTheory.klDiv (κ a) (η a) ∂μ","l":86,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_map_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"A common measurable observation rule cannot increase the Kullback--Leibler divergence between two finite input laws, even when the rule merges distinct inputs.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] {f : α → β}, Measurable f → InformationTheory.klDiv (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map f ν) ≤ InformationTheory.klDiv μ ν","l":342,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.klDiv_map_measurableEmbedding","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"KL is invariant under a measurable embedding.","s":"∀ {α : Type u_1} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ}, MeasurableEmbedding f → ∀ [MeasureTheory.IsFiniteMeasure μ] {ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map f ν) = InformationTheory.klDiv μ ν","l":122,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.map_bind_eq_compProd_of_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"If a kernel is supported almost everywhere on outputs that record their base coordinate, then mapping its bound measure to the recorded base-output pair gives the corresponding composition-product measure.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] (m : MeasureTheory.Measure B) [MeasureTheory.SFinite m] (κ : ProbabilityTheory.Kernel B Ω) [ProbabilityTheory.IsSFiniteKernel κ] (proj : Ω → B), Measurable proj → (∀ᵐ (b : B) ∂m, (κ b) {ω | proj ω = b}ᶜ = 0) → MeasureTheory.Measure.map (fun ω => (proj ω, ω)) (m.bind ⇑κ) = m.compProd κ","l":173,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.measurableEmbedding_base_recording","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"If a measurable map has a measurable graph, pairing each observation with its map value produces a measurable embedding into the corresponding product space.","s":"∀ {B : Type u_4} {Ω : Type u_5} [inst : MeasurableSpace B] [inst_1 : MeasurableSpace Ω] (proj : Ω → B), Measurable proj → MeasurableSet {p | p.1 = proj p.2} → MeasurableEmbedding fun ω => (proj ω, ω)","l":153,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.Measure.rnDeriv_compProd_right_of_forall_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.KLBind","a":"Mathlib/InformationTheory","t":2,"d":"Radon--Nikodym derivative of a shared-base composition product.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":52,"f":"Causalean/Mathlib/InformationTheory/KLBind.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.ae_prod_fst_of_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"A measurable property holding almost everywhere under a measure also holds for the first coordinate almost everywhere under its product with a probability measure.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν] {p : α → Prop}, MeasurableSet {x | p x} → (∀ᵐ (x : α) ∂μ, p x) → ∀ᵐ (z : α × β) ∂μ.prod ν, p z.1","l":59,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.ae_prod_snd_of_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"A measurable property holding almost everywhere under a measure also holds for the second coordinate almost everywhere under its product with a probability measure.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SFinite ν] {p : β → Prop}, MeasurableSet {y | p y} → (∀ᵐ (y : β) ∂ν, p y) → ∀ᵐ (z : α × β) ∂μ.prod ν, p z.2","l":69,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.klDiv_prod_toReal_add","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"For product probability laws with integrable component log-likelihood ratios, the real-valued KL divergence of the product equals the sum of the component KL divergences.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.IsProbabilityMeasure ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.IsProbabilityMeasure ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (MeasureTheory.llr μ₁ ν₁) μ₁ → MeasureTheory.Integrable (MeasureTheory.llr μ₂ ν₂) μ₂ → (InformationTheory.klDiv (μ₁.prod μ₂) (ν₁.prod ν₂)).toReal = (InformationTheory.klDiv μ₁ ν₁).toReal + (InformationTheory.klDiv μ₂ ν₂).toReal","l":241,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_integrable_of_map_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Pushing two measures through a measurable relabelling preserves integrability of their log-likelihood ratio, allowing KL side conditions to transfer between equivalent sample spaces.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (e : α ≃ᵐ β) (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν)) (MeasureTheory.Measure.map (⇑e) μ) → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ","l":145,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_prod_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"When each component law is absolutely continuous with respect to its reference law, the log-likelihood ratio of their product laws is almost surely the sum of the two component log-likelihood ratios.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.SigmaFinite ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.llr (μ₁.prod μ₂) (ν₁.prod ν₂) =ᵐ[μ₁.prod μ₂] fun z => MeasureTheory.llr μ₁ ν₁ z.1 + MeasureTheory.llr μ₂ ν₂ z.2","l":79,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.llr_prod_integrable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Integrable component log-likelihood ratios imply that the log-likelihood ratio of the corresponding product laws is integrable.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.SigmaFinite ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (MeasureTheory.llr μ₁ ν₁) μ₁ → MeasureTheory.Integrable (MeasureTheory.llr μ₂ ν₂) μ₂ → MeasureTheory.Integrable (MeasureTheory.llr (μ₁.prod μ₂) (ν₁.prod ν₂)) (μ₁.prod μ₂)","l":124,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLPrivate.productKL_tensorization_toReal_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Under absolute-continuity and integrability conditions for every finite product, the real-valued Kullback–Leibler divergence of two n-fold product laws is n times the one-law divergence.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → (∀ (k : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)) → (∀ (k : ℕ), MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)) (MeasureTheory.Measure.pi fun x => μ)) → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal = ↑n * (InformationTheory.klDiv μ ν).toReal","l":288,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound","k":"def","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable observation space, a sample size, and two measures on that space, the product-KL tensorisation bound asserts that the Kullback--Leibler divergence between their $n$-fold product measures is finite, their one-observation Kullback--Leibler divergence is finite, and the real-valued product divergence is at most $n$ times the real-valued one-observation divergence.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → ℕ → MeasureTheory.Measure α → MeasureTheory.Measure α → Prop","l":357,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.apply","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Unpack a supplied product-KL tensorisation bound.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal ≤ ↑n * (InformationTheory.klDiv μ ν).toReal","l":411,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.one_ne_top","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"The one-observation KL divergence in a supplied tensorisation bound is finite.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → InformationTheory.klDiv μ ν ≠ ⊤","l":430,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound.product_ne_top","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"The product KL divergence in a supplied tensorisation bound is finite.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {n : ℕ} {μ ν : MeasureTheory.Measure α}, Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν → InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) ≠ ⊤","l":421,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued success-weight function, and a real-valued statistic, the clipped statistic-level success parameter is the statistic-level success parameter truncated below at $1/4$ and above at $3/4$.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → ℝ → ℝ","l":302,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_abs_sub_le_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A localized setwise bound on success-weighted statistic masses yields the corresponding almost-everywhere bound on conditional Bernoulli parameters.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ), Measurable p → Measurable p' → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → ∀ᵐ (r : ℝ) ∂MeasureTheory.Measure.map stat nu, |Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat r - Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu' p' stat r| ≤ E.indicator (fun x => D) r","l":480,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_ae_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat =ᵐ[MeasureTheory.Measure.map stat nu] Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat","l":325,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_measurable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) (p stat : A → ℝ), Measurable (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat)","l":309,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_mem_Icc","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) (p stat : A → ℝ) (r : ℝ), Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat r ∈ Set.Icc (1 / 4) (3 / 4)","l":317,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable input space, a real-valued function of the input, and the hypothesis that this function is measurable, the common-statistic Bernoulli kernel assigns to every input the two-point Bernoulli law on the real line with success weight given by that function at the input. It is a probability measure at inputs where that value lies between zero and one (see commonStatisticBernoulliKernel_isMarkovKernel); elsewhere the weights are clipped to be nonnegative and need not sum to one.","s":"{S : Type u_1} → [inst : MeasurableSpace S] → (p : S → ℝ) → Measurable p → ProbabilityTheory.Kernel S ℝ","l":38,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel_isMarkovKernel","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Pointwise unit-interval parameters make the common-statistic Bernoulli kernel Markov.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (p : S → ℝ) (hp : Measurable p), (∀ (r : S), 0 ≤ p r) → (∀ (r : S), p r ≤ 1) → ProbabilityTheory.IsMarkovKernel (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)","l":47,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel_setLIntegral_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Integrating Bernoulli kernels over two base sets gives the same outcome measure when the base masses and success-weighted masses agree.","s":"∀ {A : Type u_1} {B : Type u_2} [inst : MeasurableSpace A] [inst_1 : MeasurableSpace B] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (nu' : MeasureTheory.Measure B) [MeasureTheory.IsFiniteMeasure nu'] (p : A → ℝ) (p' : B → ℝ) (hp : Measurable p) (hp' : Measurable p'), (∀ (x : A), 0 ≤ p x) → (∀ (x : A), p x ≤ 1) → (∀ (x : B), 0 ≤ p' x) → (∀ (x : B), p' x ≤ 1) → ∀ {D : Set A} {D' : Set B}, nu D = nu' D' → ∫ (x : A) in D, p x ∂nu = ∫ (x : B) in D', p' x ∂nu' → ∀ (E : Set ℝ), ∫⁻ (x : A) in D, ((Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp) x) E ∂nu = ∫⁻ (x : B) in D', ((Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp') x) E ∂nu'","l":337,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulliOutcome_klDiv_le_of_localized_parameter","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Swapping a common statistic behind its Bernoulli outcome preserves the localized KL estimate, giving the (outcome, statistic) coordinate order used by signed observations.","s":"∀ (m : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure m] (p q : ℝ → ℝ) (hp : Measurable p) (hq : Measurable q), (∀ (r : ℝ), 1 / 4 ≤ p r) → (∀ (r : ℝ), p r ≤ 3 / 4) → (∀ (r : ℝ), 1 / 4 ≤ q r) → (∀ (r : ℝ), q r ≤ 3 / 4) → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ᵐ (r : ℝ) ∂m, |p r - q r| ≤ E.indicator (fun x => D) r) → InformationTheory.klDiv (MeasureTheory.Measure.map Prod.swap (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))) (MeasureTheory.Measure.map Prod.swap (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq))) ≤ ENNReal.ofReal (4 * D ^ 2) * m E","l":120,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.commonStatisticBernoulli_klDiv_le_of_localized_parameter","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A common statistic with conditionally Bernoulli outcomes has KL bounded by the squared change in its success parameter, integrated only over the statistic region where that parameter can change. This is the generic disintegration step used by the signed hard-cell comparison.","s":"∀ (m : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure m] (p q : ℝ → ℝ) (hp : Measurable p) (hq : Measurable q), (∀ (r : ℝ), 1 / 4 ≤ p r) → (∀ (r : ℝ), p r ≤ 3 / 4) → (∀ (r : ℝ), 1 / 4 ≤ q r) → (∀ (r : ℝ), q r ≤ 3 / 4) → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ᵐ (r : ℝ) ∂m, |p r - q r| ≤ E.indicator (fun x => D) r) → InformationTheory.klDiv (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)) (m.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel q hq)) ≤ ENNReal.ofReal (4 * D ^ 2) * m E","l":58,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.condEntropy","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"For finite outcome and conditioning alphabets and a joint real-valued mass function on their product, the conditional Shannon entropy is the Shannon entropy of the joint mass function minus the Shannon entropy of its conditioning-variable marginal.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → (α × β → ℝ) → ℝ","l":50,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.condEntropy_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"Conditional entropy unfolds to total joint entropy minus the entropy of the conditioning marginal.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] (p : α × β → ℝ), Causalean.Mathlib.InformationTheory.condEntropy p = Causalean.Mathlib.InformationTheory.entropy p - Causalean.Mathlib.InformationTheory.entropy (Causalean.Mathlib.InformationTheory.yMarginal p)","l":69,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.correctMass_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The correct-decision mass equals 1 − errorProb: splitting the total mass = 1 into the correct cells (x = decode y) and the error cells gives ∑_{x = decode y} p (x, y) = 1 − errorProb p decode.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, ∑ xy, p xy = 1 → ∀ (decode : β → α), (∑ xy, if xy.1 = decode xy.2 then p xy else 0) = 1 - Causalean.Mathlib.InformationTheory.errorProb p decode","l":189,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy","k":"def","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"For a finite alphabet and a real-valued mass function on it, the Shannon entropy in nats is $\\sum_i -p(i)\\log p(i)$, with the standard zero-continuous convention for the summands.","s":"{α : Type u_1} → [Fintype α] → (α → ℝ) → ℝ","l":41,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_const_eq_log_card","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"Sharpness of the maximum-entropy bound. The uniform probability mass function on a nonempty finite alphabet — assigning every outcome probability equal to the reciprocal of the alphabet's cardinality — attains entropy exactly equal to the natural logarithm of that cardinality.","s":"∀ {α : Type u_1} [inst : Fintype α] [Nonempty α], (Causalean.Mathlib.InformationTheory.entropy fun x => (↑(Fintype.card α))⁻¹) = Real.log ↑(Fintype.card α)","l":142,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"The finite-alphabet entropy definition unfolds to the sum of Real.negMulLog over the alphabet.","s":"∀ {α : Type u_1} [inst : Fintype α] (p : α → ℝ), Causalean.Mathlib.InformationTheory.entropy p = ∑ i, (p i).negMulLog","l":47,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_le_crossEntropy","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"Gibbs / cross-entropy inequality. For a nonnegative mass function p and a nonnegative reference mass function g on a finite type γ, if the total mass of g is at most the total mass of p (∑ g ≤ ∑ p) and g dominates the support of p (p i ≠ 0 → 0 < g i), then the Shannon entropy of p is bounded by the cross-entropy of p relative to g: entropy p ≤ −∑ i, p i * Real.log (g i).","s":"∀ {γ : Type u_3} [inst : Fintype γ] {p g : γ → ℝ}, (∀ (i : γ), 0 ≤ p i) → (∀ (i : γ), 0 ≤ g i) → ∑ i, g i ≤ ∑ i, p i → (∀ (i : γ), p i ≠ 0 → 0 < g i) → Causalean.Mathlib.InformationTheory.entropy p ≤ -∑ i, p i * Real.log (g i)","l":105,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_le_log_card","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":1,"d":"Maximum-entropy (Gibbs) bound. For a nonnegative function p on a finite alphabet α that sums to one (∑ i, p i = 1, i.e. p is a probability mass function) with n = Fintype.card α, the Shannon entropy of p is at most log n: entropy p ≤ Real.log (Fintype.card α).","s":"∀ {α : Type u_1} [inst : Fintype α] {p : α → ℝ}, (∀ (i : α), 0 ≤ p i) → ∑ i, p i = 1 → Causalean.Mathlib.InformationTheory.entropy p ≤ Real.log ↑(Fintype.card α)","l":89,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.entropy_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"Entropy is nonnegative for any sub-probability vector: if 0 ≤ p i ≤ 1 for every i, then 0 ≤ entropy p. Each summand Real.negMulLog (p i) is nonnegative on [0,1] (Real.negMulLog_nonneg), so the finite sum is nonnegative.","s":"∀ {α : Type u_1} [inst : Fintype α] {p : α → ℝ}, (∀ (i : α), 0 ≤ p i) → (∀ (i : α), p i ≤ 1) → 0 ≤ Causalean.Mathlib.InformationTheory.entropy p","l":51,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"For finite outcome and observation alphabets, with equality decidable for outcomes, a joint real-valued mass function on their product, and a deterministic decoder from observations to outcomes, the decoder's error mass is the sum of the joint masses of all outcome--observation pairs for which the decoder's output differs from the outcome.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → [DecidableEq α] → (α × β → ℝ) → (β → α) → ℝ","l":55,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The decoder error probability unfolds to the sum of the joint masses on incorrect decoding cells.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] (p : α × β → ℝ) (decode : β → α), Causalean.Mathlib.InformationTheory.errorProb p decode = ∑ xy, if xy.1 = decode xy.2 then 0 else p xy","l":74,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_le_one","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The error probability is at most one (it is a sub-sum of the total mass = 1).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → ∀ (decode : β → α), Causalean.Mathlib.InformationTheory.errorProb p decode ≤ 1","l":178,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.errorProb_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The error probability is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (decode : β → α), 0 ≤ Causalean.Mathlib.InformationTheory.errorProb p decode","l":168,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef","k":"def","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"For finite outcome and observation alphabets, with equality decidable for outcomes, a joint real-valued mass function on their product, a deterministic decoder from observations to outcomes, and an outcome--observation pair, the Fano reference mass at that pair equals the marginal mass of its observation multiplied by $1-P_e$ when its outcome is the decoded outcome, and by $P_e/(|\\mathcal A|-1)$ otherwise, where $P_e$ is the decoder's error mass and $|\\mathcal A|$ is the number of outcome values.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → [DecidableEq α] → (α × β → ℝ) → (β → α) → α × β → ℝ","l":41,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_ac","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution dominates the support of p: p xy ≠ 0 → 0 < fanoRef xy. A nonzero p (x, y) forces yMarginal p y > 0; on a correct cell the complementary mass 1 − Pe ≥ p (x,y) > 0 and on an error cell Pe ≥ p (x,y) > 0, so the inner weight is positive. This is the absolute-continuity hypothesis of the Gibbs inequality.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α) (xy : α × β), p xy ≠ 0 → 0 < Causalean.Mathlib.InformationTheory.fanoRef p decode xy","l":122,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] (p : α × β → ℝ) (decode : β → α) (xy : α × β), Causalean.Mathlib.InformationTheory.fanoRef p decode xy = Causalean.Mathlib.InformationTheory.yMarginal p xy.2 * if xy.1 = decode xy.2 then 1 - Causalean.Mathlib.InformationTheory.errorProb p decode else Causalean.Mathlib.InformationTheory.errorProb p decode / (↑(Fintype.card α) - 1)","l":49,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → ∀ (decode : β → α) (xy : α × β), 0 ≤ Causalean.Mathlib.InformationTheory.fanoRef p decode xy","l":55,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fanoRef_sum_eq_one","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The Fano reference distribution is a probability mass function: ∑ xy, fanoRef = 1. On each column the inner weights sum to (1 − Pe) + (card α − 1) · Pe/(card α − 1) = 1, so the total is ∑ y, yMarginal p y = 1.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), ∑ xy, Causalean.Mathlib.InformationTheory.fanoRef p decode xy = 1","l":68,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fano_error_lower_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"Fano error lower bound (the standard weakened corollary). For a nonnegative function p on α × β that sums to one, with at least two symbols in the alphabet α (2 ≤ Fintype.card α) and a decoder decode : β → α, the error probability Pe = errorProb p decode is bounded below: Pe ≥ (condEntropy p − Real.log 2) / Real.log (Fintype.card α).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), (Causalean.Mathlib.InformationTheory.condEntropy p - Real.log 2) / Real.log ↑(Fintype.card α) ≤ Causalean.Mathlib.InformationTheory.errorProb p decode","l":306,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.fano_inequality","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":1,"d":"Fano's inequality (Cover & Thomas, Thm 2.10.1). For a nonnegative function p on α × β that sums to one, i.e. a joint probability mass function, with at least two symbols in the alphabet α (2 ≤ Fintype.card α), and a decoder decode : β → α, the conditional entropy of p is bounded by the binary entropy of the error probability Pe = errorProb p decode plus Pe times the log of one less than the alphabet size: condEntropy p ≤ Real.binEntropy Pe + Pe * Real.log (Fintype.card α − 1).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), Causalean.Mathlib.InformationTheory.condEntropy p ≤ Real.binEntropy (Causalean.Mathlib.InformationTheory.errorProb p decode) + Causalean.Mathlib.InformationTheory.errorProb p decode * Real.log (↑(Fintype.card α) - 1)","l":282,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.gaussianKL_eq","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":1,"d":"Equal-variance Gaussian KL divergence (closed form). For means m₀, m₁ and a strictly positive common variance v, the Kullback–Leibler divergence between two real Gaussians of equal variance v and respective means m₀ and m₁ equals (m₀ - m₁)² / (2v):","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), 0 < v → InformationTheory.klDiv (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) = ENNReal.ofReal ((m₀ - m₁) ^ 2 / (2 * ↑v))","l":176,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.gaussianReal_ac_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Absolute continuity of nondegenerate Gaussians. For v₀ ≠ 0 and v₁ ≠ 0, gaussianReal m₀ v₀ is absolutely continuous with respect to gaussianReal m₁ v₁: both are volume.withDensity of a strictly positive density, so each is mutually absolutely continuous with Lebesgue measure, and absolute continuity is transitive.","s":"∀ (m₀ m₁ : ℝ) {v₀ v₁ : NNReal}, v₀ ≠ 0 → v₁ ≠ 0 → (ProbabilityTheory.gaussianReal m₀ v₀).AbsolutelyContinuous (ProbabilityTheory.gaussianReal m₁ v₁)","l":56,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.integrable_llr_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Integrability of the Gaussian log-likelihood ratio. For v ≠ 0, the affine log-likelihood ratio x ↦ (m₀ - m₁) * (2*x - m₀ - m₁) / (2*v) is integrable with respect to gaussianReal m₀ v, because the Gaussian has a finite first moment.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → MeasureTheory.Integrable (MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v)) (ProbabilityTheory.gaussianReal m₀ v)","l":123,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.integral_llr_gaussianReal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Integral of the Gaussian log-likelihood ratio. For v ≠ 0, ∫ llr (gaussianReal m₀ v) (gaussianReal m₁ v) ∂(gaussianReal m₀ v) = (m₀ - m₁)^2 / (2*v), obtained by integrating the affine a.e. form against ∫ x ∂gaussianReal m₀ v = m₀.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → ∫ (x : ℝ), MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) x ∂ProbabilityTheory.gaussianReal m₀ v = (m₀ - m₁) ^ 2 / (2 * ↑v)","l":145,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.le_yMarginal","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"A joint mass is dominated by its β-marginal: p (x, y) ≤ yMarginal p y for nonnegative p.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (x : α) (y : β), p (x, y) ≤ Causalean.Mathlib.InformationTheory.yMarginal p y","l":161,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.llr_gaussianReal_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Log-likelihood ratio of equal-variance Gaussians. For v ≠ 0, the log-likelihood ratio llr (gaussianReal m₀ v) (gaussianReal m₁ v) is a.e. (with respect to gaussianReal m₀ v) equal to the affine function x ↦ (m₀ - m₁) * (2*x - m₀ - m₁) / (2*v). At equal variance the (√(2πv))⁻¹ normalising constants cancel in the density ratio, so the log-ratio reduces to ((x - m₁)^2 - (x - m₀)^2) / (2*v) = (m₀ - m₁)*(2*x - m₀ - m₁)/(2*v).","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → MeasureTheory.llr (ProbabilityTheory.gaussianReal m₀ v) (ProbabilityTheory.gaussianReal m₁ v) =ᵐ[ProbabilityTheory.gaussianReal m₀ v] fun x => (m₀ - m₁) * (2 * x - m₀ - m₁) / (2 * ↑v)","l":93,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.negMulLog_add_mul_log_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"Per-coordinate Gibbs lever. For 0 ≤ x and 0 ≤ g, with g positive whenever x is nonzero (absolute continuity), the cross-entropy summand is controlled: Real.negMulLog x + x * Real.log g ≤ g - x.","s":"∀ {x g : ℝ}, 0 ≤ x → 0 ≤ g → (x ≠ 0 → 0 < g) → x.negMulLog + x * Real.log g ≤ g - x","l":79,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.negMulLog_sub_mul_log_le","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Entropy","a":"Mathlib/InformationTheory","t":2,"d":"Per-coordinate Gibbs lever. At any positive real scale c, the entropy summand minus x · log c is bounded by 1/c - x: Real.negMulLog x - x * Real.log c ≤ c⁻¹ - x.","s":"∀ {c : ℝ}, 0 < c → ∀ {x : ℝ}, 0 ≤ x → x.negMulLog - x * Real.log c ≤ c⁻¹ - x","l":61,"f":"Causalean/Mathlib/InformationTheory/Entropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.neg_crossEntropy_fanoRef","k":"theorem","m":"Causalean.Mathlib.InformationTheory.Fano","a":"Mathlib/InformationTheory","t":2,"d":"The cross-entropy of p against the Fano reference splits, via the chain rule for log on the support of p, into the marginal entropy plus the binary-entropy/error terms: −∑ xy, p xy · log (fanoRef p decode xy) = entropy (yMarginal p) + Real.binEntropy Pe + Pe · Real.log (card α − 1), where Pe = errorProb p decode. This is the key algebraic computation behind Fano.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : DecidableEq α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∑ xy, p xy = 1 → 2 ≤ Fintype.card α → ∀ (decode : β → α), -∑ xy, p xy * Real.log (Causalean.Mathlib.InformationTheory.fanoRef p decode xy) = Causalean.Mathlib.InformationTheory.entropy (Causalean.Mathlib.InformationTheory.yMarginal p) + Real.binEntropy (Causalean.Mathlib.InformationTheory.errorProb p decode) + Causalean.Mathlib.InformationTheory.errorProb p decode * Real.log (↑(Fintype.card α) - 1)","l":168,"f":"Causalean/Mathlib/InformationTheory/Fano.lean"},{"n":"Causalean.Mathlib.InformationTheory.one_add_mul_one_sub_mem_Icc","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"A Bernoulli parameter in the middle half of the unit interval has variance between zero and one quarter, so adding unit noise gives variance between one and five quarters.","s":"∀ {p : ℝ}, p ∈ Set.Icc (1 / 4) (3 / 4) → 1 + p * (1 - p) ∈ Set.Icc 1 (5 / 4)","l":25,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.pi_iid_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Public: absolute continuity of i.i.d. finite products from the one-sample hypothesis μ ≪ ν for sigma-finite laws. (Thin wrapper over the private induction.)","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)","l":437,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.pi_iid_llr_integrable","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Public: log-likelihood-ratio integrability for i.i.d. finite products from the one-sample hypotheses μ ≪ ν and Integrable (llr μ ν) μ. Combined with pi_iid_absolutelyContinuous this certifies klDiv (pi μ) (pi ν) ≠ ⊤ (via InformationTheory.klDiv_ne_top).","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → ∀ (n : ℕ), MeasureTheory.Integrable (MeasureTheory.llr (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)) (MeasureTheory.Measure.pi fun x => μ)","l":445,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.productKL_tensorization","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":1,"d":"For a sample size n and probability measures μ, ν on a measurable space α, if μ is absolutely continuous with respect to ν and the log-likelihood ratio of μ against ν is μ-integrable, then both the KL divergence between the n-fold product of μ and the n-fold product of ν, and the one-observation KL divergence between μ and ν, are finite, and the real-valued product KL divergence is at most n times the real-valued one-observation KL divergence: this is product-KL tensorisation packaged in the Le Cam interface.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → Causalean.Mathlib.InformationTheory.ProductKLTensorizationBound n μ ν","l":388,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.productKL_tensorization_of_finite","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ProductKLLeCam","a":"Mathlib/InformationTheory","t":2,"d":"Finite-branch product-KL tensorisation for i.i.d. finite products.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (n : ℕ) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)).toReal = ↑n * (InformationTheory.klDiv μ ν).toReal","l":371,"f":"Causalean/Mathlib/InformationTheory/ProductKLLeCam.lean"},{"n":"Causalean.Mathlib.InformationTheory.rnDeriv_toReal_gaussianReal_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.GaussianKL","a":"Mathlib/InformationTheory","t":2,"d":"Radon–Nikodym ratio of equal-variance Gaussians, as a real number. For v ≠ 0, the real part of the Radon–Nikodym derivative ∂(gaussianReal m₀ v)/∂(gaussianReal m₁ v) is a.e. (with respect to gaussianReal m₀ v) the pointwise ratio of the two Gaussian densities gaussianPDFReal m₀ v x / gaussianPDFReal m₁ v x.","s":"∀ {v : NNReal} (m₀ m₁ : ℝ), v ≠ 0 → (fun x => ((ProbabilityTheory.gaussianReal m₀ v).rnDeriv (ProbabilityTheory.gaussianReal m₁ v) x).toReal) =ᵐ[ProbabilityTheory.gaussianReal m₀ v] fun x => ProbabilityTheory.gaussianPDFReal m₀ v x / ProbabilityTheory.gaussianPDFReal m₁ v x","l":66,"f":"Causalean/Mathlib/InformationTheory/GaussianKL.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcomeLaw_eq_map_swap_compProd","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Compressing the base coordinate to a statistic turns a Bernoulli mixture into a Bernoulli composition product over the statistic marginal.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ) (hp : Measurable p), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp)) = MeasureTheory.Measure.map Prod.swap ((MeasureTheory.Measure.map stat nu).compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter nu p stat) (Causalean.Mathlib.InformationTheory.clippedStatisticSuccessParameter_measurable nu p stat)))","l":401,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcome_klDiv_le_of_localized_success_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable space A, two finite measures nu and nu' on it, and success-probability functions p, p' on A, suppose p, p', and a statistic stat are all measurable, p takes values in [1/4, 3/4] and p' likewise takes values in [1/4, 3/4], and stat pushes nu and nu' forward to the same marginal law. Given a nonnegative discrepancy bound D and a measurable exceptional set E such that for every measurable set B of statistic values, the setwise success-mass discrepancy |∫_{stat∈B} p dnu − ∫_{stat∈B} p' dnu'| is at most D times the stat-pushforward mass of nu on B ∩ E, then the Kullback–Leibler divergence between the compressed Bernoulli-outcome laws obtained by pairing the outcome with stat under nu and under nu' is at most 4·D² times the stat-pushforward mass of E under nu.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ) (hp : Measurable p) (hp' : Measurable p'), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → InformationTheory.klDiv (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))) (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu'.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp'))) ≤ ENNReal.ofReal (4 * D ^ 2) * (MeasureTheory.Measure.map stat nu) E","l":586,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticBernoulliOutcome_restrict_compl_eq_of_localized_success_bound","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Common statistic marginals and a localized setwise success-mass bound also imply exact agreement of the compressed outcome laws away from the exceptional statistic set.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu nu' : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure nu'] (p p' stat : A → ℝ) (hp : Measurable p) (hp' : Measurable p'), Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → (∀ (x : A), 1 / 4 ≤ p' x) → (∀ (x : A), p' x ≤ 3 / 4) → MeasureTheory.Measure.map stat nu = MeasureTheory.Measure.map stat nu' → ∀ {D : ℝ}, 0 ≤ D → ∀ {E : Set ℝ}, MeasurableSet E → (∀ (B : Set ℝ), MeasurableSet B → |∫ (x : A) in {x | stat x ∈ B}, p x ∂nu - ∫ (x : A) in {x | stat x ∈ B}, p' x ∂nu'| ≤ D * ((MeasureTheory.Measure.map stat nu) (B ∩ E)).toReal) → (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p hp))).restrict {z | z.2 ∉ E} = (MeasureTheory.Measure.map (fun z => (z.2, stat z.1)) (nu'.compProd (Causalean.Mathlib.InformationTheory.commonStatisticBernoulliKernel p' hp'))).restrict {z | z.2 ∉ E}","l":629,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessMeasure","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued success-weight function, and a real-valued statistic, the success-weighted statistic law is the pushforward along the statistic of the base measure weighted by $\\max\\{p(x),0\\}$. No measurability of the statistic is assumed: if it is not almost-everywhere measurable the pushforward is the zero measure by convention, so this is the statistic law only for a measurable statistic.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → MeasureTheory.Measure ℝ","l":150,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessMeasure_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"The success-weighted statistic law is dominated by the statistic marginal when the pointwise success probability is at most one.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable stat → (∀ (x : A), p x ≤ 1) → (Causalean.Mathlib.InformationTheory.statisticSuccessMeasure nu p stat).AbsolutelyContinuous (MeasureTheory.Measure.map stat nu)","l":161,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter","k":"def","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued success-weight function, and a real-valued statistic, the statistic-level success parameter is the Radon--Nikodym derivative of the success-weighted statistic law with respect to the statistic's pushforward law under the base measure, converted to a real number. Each pushforward is the zero measure when the statistic is not almost-everywhere measurable for its own source measure (the weighted measure for the numerator, the base measure for the denominator), so this is the conditional success parameter only for a measurable statistic.","s":"{A : Type u_1} → [inst : MeasurableSpace A] → MeasureTheory.Measure A → (A → ℝ) → (A → ℝ) → ℝ → ℝ","l":155,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter_mem_Icc_ae","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Middle-half pointwise bounds pass to the conditional statistic parameter almost everywhere.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 1 / 4 ≤ p x) → (∀ (x : A), p x ≤ 3 / 4) → ∀ᵐ (r : ℝ) ∂MeasureTheory.Measure.map stat nu, Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat r ∈ Set.Icc (1 / 4) (3 / 4)","l":221,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.statisticSuccessParameter_setIntegral","k":"theorem","m":"Causalean.Mathlib.InformationTheory.CommonStatisticBernoulli","a":"Mathlib/InformationTheory","t":2,"d":"Set integrals of the conditional parameter recover success-weighted integrals on statistic preimages.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] (nu : MeasureTheory.Measure A) [MeasureTheory.IsFiniteMeasure nu] (p stat : A → ℝ), Measurable p → Measurable stat → (∀ (x : A), 0 ≤ p x) → (∀ (x : A), p x ≤ 1) → ∀ (B : Set ℝ), MeasurableSet B → ∫ (r : ℝ) in B, Causalean.Mathlib.InformationTheory.statisticSuccessParameter nu p stat r ∂MeasureTheory.Measure.map stat nu = ∫ (x : A) in {x | stat x ∈ B}, p x ∂nu","l":179,"f":"Causalean/Mathlib/InformationTheory/CommonStatisticBernoulli.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal","k":"def","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":1,"d":"For a finite outcome alphabet and an arbitrary conditioning-value space, a joint real-valued mass function on their product, and a chosen conditioning value, the marginal mass at that conditioning value is the finite sum of the joint masses over all outcome values.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → (α × β → ℝ) → β → ℝ","l":47,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_def","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal is the finite sum of joint masses over the α coordinate at the chosen value of β.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] (p : α × β → ℝ) (y : β), Causalean.Mathlib.InformationTheory.yMarginal p y = ∑ x, p (x, y)","l":63,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_nonneg","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal of a nonnegative mass function is nonnegative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] {p : α × β → ℝ}, (∀ (xy : α × β), 0 ≤ p xy) → ∀ (y : β), 0 ≤ Causalean.Mathlib.InformationTheory.yMarginal p y","l":154,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.InformationTheory.yMarginal_sum","k":"theorem","m":"Causalean.Mathlib.InformationTheory.ConditionalEntropy","a":"Mathlib/InformationTheory","t":2,"d":"The β-marginal of a pmf is itself a pmf summing to one: if ∑ xy, p xy = 1 then ∑ y, yMarginal p y = 1.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] {p : α × β → ℝ}, ∑ xy, p xy = 1 → ∑ y, Causalean.Mathlib.InformationTheory.yMarginal p y = 1","l":145,"f":"Causalean/Mathlib/InformationTheory/ConditionalEntropy.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"A Euclidean real vector indexed by a finite coordinate type, called a finite real coordinate vector, is given by the Euclidean space on that coordinate type.","s":"(ι : Type u_2) → [Fintype ι] → Type u_2","l":23,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.IsIrreducible.eigenvector_pos","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Positivity","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"An irreducible finite matrix and a nonnegative nonzero eigenvector ensure that every coordinate is strictly positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {A : Matrix ι ι ℝ}, A.IsIrreducible → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, (∀ (i : ι), 0 ≤ x.ofLp i) → x ≠ 0 → ∀ {ρ : ℝ}, A.mulVec x.ofLp = ρ • x.ofLp → ∀ (i : ι), 0 < x.ofLp i","l":34,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Positivity.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.IsIrreducible.unit_eigenvector_pos","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Positivity","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An irreducible finite matrix and a normalized nonnegative eigenvector ensure that every coordinate is strictly positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {A : Matrix ι ι ℝ}, A.IsIrreducible → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, (∀ (i : ι), 0 ≤ x.ofLp i) → ‖x‖ = 1 → ∀ {ρ : ℝ}, A.mulVec x.ofLp = ρ • x.ofLp → ∀ (i : ι), 0 < x.ofLp i","l":57,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Positivity.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The Euclidean vector obtained from a finite real vector by taking the absolute value of every coordinate is given coordinate by coordinate.","s":"{ι : Type u_1} → [inst : Fintype ι] → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι","l":26,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec_eigenvector_of_isMaxOn","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"On a nonempty finite coordinate space, an entrywise nonnegative symmetric real matrix, a unit-sphere vector, and its Rayleigh-form maximality ensure that the coordinatewise absolute vector is an eigenvector at the top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → (∀ (i j : ι), 0 ≤ A i j) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, x ∈ Metric.sphere 0 1 → IsMaxOn (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A) (Metric.sphere 0 1) x → A.mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A • (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp","l":71,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec_isMaxOn","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An entrywise nonnegative finite real matrix, a unit-sphere vector, and its Rayleigh-form maximality ensure that coordinatewise absolute value is another unit-sphere maximizer.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ), (∀ (i j : ι), 0 ≤ A i j) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, x ∈ Metric.sphere 0 1 → IsMaxOn (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A) (Metric.sphere 0 1) x → IsMaxOn (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A) (Metric.sphere 0 1) (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x)","l":61,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec_mem_sphere","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A radius and a Euclidean coordinate vector on the sphere of that radius satisfy that coordinatewise absolute value remains on the same sphere.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {r : ℝ} {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, x ∈ Metric.sphere 0 r → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x ∈ Metric.sphere 0 r","l":45,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec_nonneg","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"With a Euclidean coordinate vector and a coordinate, the corresponding coordinatewise absolute value is nonnegative.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι) (i : ι), 0 ≤ (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp i","l":26,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec_preserves_top_eigenvector","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, an entrywise nonnegative symmetric real matrix and a normalized top eigenvector ensure that coordinatewise absolute value is a normalized nonnegative top eigenvector with the same top value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → (∀ (i j : ι), 0 ≤ A i j) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, ‖x‖ = 1 → A.mulVec x.ofLp = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A • x.ofLp → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A → ‖Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x‖ = 1 ∧ (∀ (i : ι), 0 ≤ (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp i) ∧ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A ∧ A.mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A • (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x).ofLp","l":104,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The greatest quadratic Rayleigh-form value among the coordinate vectors whose squared coordinates sum to one for a finite real matrix, called the coordinate-sphere top Rayleigh value, is given by a supremum. For an empty coordinate type no such vector exists and the supremum of the empty set is zero by convention.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → ℝ","l":38,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue_eq_iSupRayleighValue","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, a real matrix has its coordinate unit-sphere top value equal to Mathlib's supremum Rayleigh quotient.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue A = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.iSupRayleighValue A","l":93,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue_eq_sphereRayleighValue","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"With a finite real matrix, the coordinate and Euclidean unit-sphere top Rayleigh values agree.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue A = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A","l":55,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.exists_unit_eigenvector_sphereRayleighValue","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"On a nonempty finite coordinate space, a real symmetric matrix has a unit eigenvector at its Euclidean-sphere top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → ∃ x, ‖x‖ = 1 ∧ A.mulVec x.ofLp = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A • x.ofLp ∧ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A","l":123,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.exists_unit_isMaxOn_rayleighForm","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, a real symmetric matrix has a unit vector attaining its Euclidean-sphere top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → ∃ x, ‖x‖ = 1 ∧ IsMaxOn (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A) (Metric.sphere 0 1) x ∧ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A","l":100,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.finite_positive_perron_eigenvector","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Main","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"A real symmetric irreducible finite matrix has a strictly positive unit eigenvector whose eigenvalue is simultaneously the coordinate-sphere, Euclidean-sphere, and nonzero-vector top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → A.IsIrreducible → ∃ v ρ, ‖v‖ = 1 ∧ (∀ (i : ι), 0 < v.ofLp i) ∧ A.mulVec v.ofLp = ρ • v.ofLp ∧ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A v = ρ ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue A ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.iSupRayleighValue A","l":20,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Main.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.finite_positive_perron_eigenvector_on_restriction","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Main","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A symmetric finite matrix, a nonempty finite coordinate set, an irreducible restricted block, and block closure from outside rows into that set ensure that the block’s strictly positive unit Perron vector zero-extends to a global eigenvector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), A.IsSymm → ∀ (s : Finset ι) [Nonempty ↥s], (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).IsIrreducible → (∀ (i j : ι), i ∉ s → j ∈ s → A i j = 0) → ∃ v ρ, ‖v‖ = 1 ∧ (∀ (i : ↥s), 0 < v.ofLp i) ∧ (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).mulVec v.ofLp = ρ • v.ofLp ∧ A.mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s v).ofLp = ρ • (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s v).ofLp ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s) ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.coordinateSphereRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s) ∧ ρ = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.iSupRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s)","l":43,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Main.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.iSupRayleighValue","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"Mathlib's supremum Rayleigh quotient for a finite real matrix, called the nonzero-vector top Rayleigh value, is given by the matrix's Euclidean linear map. For an empty coordinate type there are no nonzero vectors and the empty supremum is zero by convention.","s":"{ι : Type u_1} → [Fintype ι] → [DecidableEq ι] → Matrix ι ι ℝ → ℝ","l":43,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.mulVec_zeroExtendVec_of_closed","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite matrix, a finite coordinate set, and block closure from outside rows into the set ensure that applying the original matrix to a zero extension equals the zero extension of the restricted action.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (A : Matrix ι ι ℝ) (s : Finset ι), (∀ (i j : ι), i ∉ s → j ∈ s → A i j = 0) → ∀ (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s), A.mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x).ofLp = (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s (WithLp.toLp 2 ((Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).mulVec x.ofLp))).ofLp","l":269,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.norm_absVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"With a Euclidean coordinate vector, coordinatewise absolute value preserves its Euclidean norm.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι), ‖Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x‖ = ‖x‖","l":21,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.norm_zeroExtendVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite coordinate set and a vector on its subtype satisfy that zero extension preserves Euclidean norm.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (s : Finset ι) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s), ‖Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x‖ = ‖x‖","l":45,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The quadratic form associated with a finite real matrix and a Euclidean coordinate vector, called its Rayleigh form, is given by the vector-matrix-vector quadratic sum.","s":"{ι : Type u_1} → [inst : Fintype ι] → Matrix ι ι ℝ → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι → ℝ","l":30,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm_absVec_eq_of_isMaxOn","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An entrywise nonnegative finite real matrix, a unit-sphere vector, and its Rayleigh-form maximality ensure that taking coordinatewise absolute values preserves the Rayleigh-form value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ), (∀ (i j : ι), 0 ≤ A i j) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, x ∈ Metric.sphere 0 1 → IsMaxOn (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A) (Metric.sphere 0 1) x → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x","l":51,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm_le_absVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.AbsoluteValue","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An entrywise nonnegative finite real matrix and a Euclidean coordinate vector satisfy that taking coordinatewise absolute values cannot lower the Rayleigh form.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ), (∀ (i j : ι), 0 ≤ A i j) → ∀ (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x ≤ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.absVec x)","l":30,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/AbsoluteValue.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm_restrictVec_of_zero_off","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite matrix, a finite coordinate set, and a vector that vanishes outside that set ensure that restriction preserves its Rayleigh form.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ) (s : Finset ι) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι), (∀ i ∉ s, x.ofLp i = 0) → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s) (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec s x) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x","l":308,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm_zeroExtendMatrix_zeroExtendVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite coordinate set, a subtype matrix, and a subtype vector satisfy that zero extension preserves their Rayleigh form.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (s : Finset ι) (B : Matrix ↥s ↥s ℝ) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendMatrix s B) (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm B x","l":83,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The principal submatrix of a real square matrix on a finite set of coordinates, called its restricted matrix, is given by selecting those rows and columns.","s":"{ι : Type u_1} → Matrix ι ι ℝ → (s : Finset ι) → Matrix ↥s ↥s ℝ","l":21,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix_isSymm","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A symmetric finite matrix and a finite coordinate set ensure that the principal restricted matrix remains symmetric.","s":"∀ {ι : Type u_1} [Fintype ι] [DecidableEq ι] {A : Matrix ι ι ℝ}, A.IsSymm → ∀ (s : Finset ι), (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).IsSymm","l":240,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix_mulVec_restrictVec_of_zero_off","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite matrix, a finite coordinate set, and a vector that vanishes outside the set ensure that restriction commutes with applying the matrix.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ) (s : Finset ι) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι), (∀ i ∉ s, x.ofLp i = 0) → (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec s x).ofLp = (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec s (WithLp.toLp 2 (A.mulVec x.ofLp))).ofLp","l":253,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix_nonneg","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An entrywise nonnegative finite matrix and a finite coordinate set ensure that the principal restricted matrix remains entrywise nonnegative.","s":"∀ {ι : Type u_1} [Fintype ι] [DecidableEq ι] {A : Matrix ι ι ℝ}, (∀ (i j : ι), 0 ≤ A i j) → ∀ (s : Finset ι) (i j : ↥s), 0 ≤ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s i j","l":246,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The Euclidean vector obtained by restricting a finite real vector to a finite coordinate set, called its restricted vector, is given by retaining those coordinates.","s":"{ι : Type u_1} → [inst : Fintype ι] → (s : Finset ι) → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s","l":25,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec_zeroExtendVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite coordinate set and a vector on its subtype satisfy that restricting its zero extension recovers the original subtype vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (s : Finset ι) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictVec s (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x) = x","l":39,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The greatest quadratic Rayleigh-form value among the Euclidean unit vectors for a finite real matrix, called the Euclidean-sphere top Rayleigh value, is given by a supremum. For an empty coordinate type there are no unit vectors and the supremum of the empty set is zero by convention.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → ℝ","l":34,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue_eq_iSupRayleighValue","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Basic","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, a real matrix has its Euclidean unit-sphere top value equal to Mathlib's supremum Rayleigh quotient.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ), Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.iSupRayleighValue A","l":74,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Basic.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue_restrictMatrix_eq_of_supported_maximizer","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, a finite matrix, a nonempty finite coordinate set, and a global unit maximizer supported on that set ensure that the restricted and global top Rayleigh values agree.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ) (s : Finset ι) [Nonempty ↥s] {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, ‖x‖ = 1 → (∀ i ∉ s, x.ofLp i = 0) → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.rayleighForm A x = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A","l":346,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue_restrictMatrix_le","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"On a nonempty finite coordinate space, a finite matrix and a nonempty finite coordinate set satisfy that the restricted top Rayleigh value is at most the global top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] [Nonempty ι] (A : Matrix ι ι ℝ) (s : Finset ι) [Nonempty ↥s], Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s) ≤ Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue A","l":326,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue_zeroExtendMatrix","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"On a nonempty finite coordinate space, a nonempty finite coordinate set and an entrywise nonnegative subtype matrix satisfy that zero extension preserves its top Rayleigh value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nonempty ι] (s : Finset ι) [Nonempty ↥s] (B : Matrix ↥s ↥s ℝ), (∀ (i j : ↥s), 0 ≤ B i j) → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendMatrix s B) = Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.sphereRayleighValue B","l":190,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendMatrix","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The matrix obtained by extending a matrix on a finite coordinate subtype by zero outside that coordinate set, called its zero extension, is given entry by entry.","s":"{ι : Type u_1} → [DecidableEq ι] → (s : Finset ι) → Matrix ↥s ↥s ℝ → Matrix ι ι ℝ","l":33,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendMatrix_mulVec_zeroExtendVec","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite coordinate set, a subtype matrix, and a subtype vector satisfy that applying the zero-extended matrix to the zero-extended vector equals the zero extension of the subtype action.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (s : Finset ι) (B : Matrix ↥s ↥s ℝ) (x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s), (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendMatrix s B).mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x).ofLp = (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s (WithLp.toLp 2 (B.mulVec x.ofLp))).ofLp","l":62,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec","k":"def","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":1,"d":"The Euclidean vector obtained by extending a vector on a finite coordinate subtype by zero outside that finite coordinate set, called its zero extension, is given coordinate by coordinate.","s":"{ι : Type u_1} → [inst : Fintype ι] → [DecidableEq ι] → (s : Finset ι) → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s → Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι","l":29,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec_eigenvector_of_closed","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Restriction","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"A finite matrix, a finite coordinate set, block closure from outside rows into the set, and a restricted eigen-equation ensure that zero extension satisfies the corresponding global eigen-equation.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (A : Matrix ι ι ℝ) (s : Finset ι), (∀ (i j : ι), i ∉ s → j ∈ s → A i j = 0) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ↥s} {ρ : ℝ}, (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.restrictMatrix A s).mulVec x.ofLp = ρ • x.ofLp → A.mulVec (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x).ofLp = ρ • (Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zeroExtendVec s x).ofLp","l":297,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Restriction.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.zero_coordinate_propagates_across_positive_entry","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.Positivity","a":"Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector","t":2,"d":"An entrywise nonnegative matrix, a nonnegative eigenvector, a row coordinate where it vanishes, and a strictly positive matrix entry from that row ensure that the eigenvector also vanishes at the entry’s target coordinate.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] (A : Matrix ι ι ℝ), (∀ (i j : ι), 0 ≤ A i j) → ∀ {x : Causalean.Mathlib.LinearAlgebra.FinitePerronFrobeniusPositiveEigenvector.EVec ι}, (∀ (i : ι), 0 ≤ x.ofLp i) → ∀ {ρ : ℝ}, A.mulVec x.ofLp = ρ • x.ofLp → ∀ {i j : ι}, x.ofLp i = 0 → 0 < A i j → x.ofLp j = 0","l":19,"f":"Causalean/Mathlib/LinearAlgebra/FinitePerronFrobeniusPositiveEigenvector/Positivity.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.IsUpperTri","k":"def","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"For a matrix \\(U\\) whose rows and columns are indexed by an ordered set, the upper-triangularity property holds precisely when every entry in a row strictly below its column is zero.","s":"{ι : Type u_1} → {K : Type u_2} → [LT ι] → [Zero K] → Matrix ι ι K → Prop","l":31,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.affineBinaryPower","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given an element \\(z\\) of a multiplicative monoid and a nonnegative integer \\(k\\), the affine power vector is the vector indexed by the integers from zero through \\(k\\) whose entry at \\(r\\) is \\(z^r\\).","s":"{K : Type u_1} → [Monoid K] → K → (k : ℕ) → Fin (k + 1) → K","l":30,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.blockVandermondeWitnessWeights","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a positive integer \\(N\\), the block-Vandermonde witness weights assign weight one only to index zero in block zero and to every nonzero index in block \\(N-1\\), and assign weight zero in all other cases. The first block is chosen as block zero, the second as block \\(N-1\\), and the stated case assignment defines the weights.","s":"{K : Type u_1} → [Zero K] → [One K] → (N : ℕ) → 1 ≤ N → Fin (N + 1) → Fin N → K","l":53,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.cholesky_exists","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"For a real positive-definite d × d matrix M, there exists an upper-triangular matrix U (zero below the diagonal) with strictly positive diagonal entries such that M factors as Uᵀ · U.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ}, M.PosDef → ∃ U, (∀ (i j : Fin d), j < i → U i j = 0) ∧ (∀ (i : Fin d), 0 < U i i) ∧ M = U.transpose * U","l":109,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.cholesky_unique","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":1,"d":"For a linearly ordered finite index type ι and a linearly ordered field K, and two matrices U, V indexed by ι × ι and valued in K, if U is upper-triangular, i.e. zero below the diagonal, with strictly positive diagonal entries, V is likewise upper-triangular with strictly positive diagonal entries, and U and V have the same Gram matrix, Uᵀ · U = Vᵀ · V, then U equals V.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : LinearOrder ι] [inst_2 : Field K] [inst_3 : LinearOrder K] [IsStrictOrderedRing K] {U V : Matrix ι ι K}, (∀ (i j : ι), j < i → U i j = 0) → (∀ (i : ι), 0 < U i i) → (∀ (i j : ι), j < i → V i j = 0) → (∀ (i : ι), 0 < V i i) → U.transpose * U = V.transpose * V → U = V","l":200,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.coeff_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Reading a doubled coefficient polynomial at an encoded exponent recovers the corresponding coefficient.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin n → K) (i : Fin n ⊕ Fin n), (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v).coeff (Causalean.Mathlib.LinearAlgebra.doubledExponent i) = v i","l":71,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.coeff_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Reading a pinned coefficient polynomial at an encoded exponent recovers the corresponding coefficient.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin (n - 1) → K) (i : Fin n ⊕ Fin (n - 1)), (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v).coeff (Causalean.Mathlib.LinearAlgebra.pinnedExponent i) = v i","l":283,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.confluentVandermonde","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), a semiring of coefficients, and \\(n\\) node values, the confluent Vandermonde matrix is the square matrix with rows indexed by monomial degrees from zero through \\(2n-1\\), value columns at every node, and first-derivative columns at every node.","s":"{n : ℕ} → {K : Type u_1} → [Semiring K] → (Fin n → K) → Matrix (Fin n ⊕ Fin n) (Fin n ⊕ Fin n) K","l":110,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.det_confluentVandermonde_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a positive number of nodes n (1 ≤ n), a field K, and pairwise distinct nodes s : Fin n → K, the confluent Vandermonde determinant at s is nonzero.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → (Causalean.Mathlib.LinearAlgebra.confluentVandermonde s).det ≠ 0","l":187,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.det_pinnedConfluentVandermonde_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a positive number of nodes n (1 ≤ n), a field K, and pairwise distinct nodes s : Fin n → K, the pinned confluent Vandermonde determinant at s is nonzero: one node contributes only value evaluation and every other node contributes both value and first-derivative evaluation.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → (Causalean.Mathlib.LinearAlgebra.pinnedConfluentVandermonde s).det ≠ 0","l":425,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), a semiring of coefficients, and a coefficient vector indexed by two blocks of \\(n\\) positions, the doubled coefficient polynomial is the polynomial whose coefficient at each encoded exponent is the corresponding entry of that vector.","s":"{n : ℕ} → {K : Type u_1} → [inst : Semiring K] → (Fin n ⊕ Fin n → K) → Polynomial K","l":63,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A doubled coefficient polynomial vanishes exactly when its coefficient vector vanishes.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin n → K), Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v = 0 ↔ v = 0","l":87,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial_eq_zero_of_eval_derivative","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A polynomial of degree below 2n whose value and derivative vanish at n distinct points is zero.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → ∀ (v : Fin n ⊕ Fin n → K), (∀ (i : Fin n), Polynomial.eval (s i) (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v) = 0) → (∀ (i : Fin n), Polynomial.eval (s i) (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v)) = 0) → v = 0","l":149,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), the doubled exponent assigns exponent \\(i\\) to the \\(i\\)-th index in the first block of \\(n\\) indices and exponent \\(n+i\\) to the \\(i\\)-th index in the second block. The first-block assignment and the second-block assignment together define the encoding.","s":"{n : ℕ} → Fin n ⊕ Fin n → ℕ","l":29,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The doubled exponent encoding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.doubledExponent","l":37,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.doubledExponent_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Every doubled exponent is strictly below 2n.","s":"∀ {n : ℕ} (i : Fin n ⊕ Fin n), Causalean.Mathlib.LinearAlgebra.doubledExponent i < 2 * n","l":55,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis","k":"def","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), a commutative semiring of coefficients, \\(n\\) node values, and a nonnegative integer \\(r\\), the endpoint-order synthesis map sends \\(n+1\\) coefficients to the first \\(r+1\\) moment sums of the node coefficients, with the terminal coefficient added only to the moment of order \\(r\\).","s":"{n : ℕ} → {K : Type u_1} → [inst : CommSemiring K] → (Fin n → K) → (r : ℕ) → (Fin (n + 1) → K) →ₗ[K] Fin (r + 1) → K","l":28,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"With distinct node values, moments through an order at least as large as the number of nodes uniquely determine every coordinate supplied to the moment-synthesis map.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : CommRing K] [IsDomain K] (s : Fin n → K), Function.Injective s → n ≤ r → Function.Injective ⇑(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r)","l":104,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_ker_finrank","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":1,"d":"For natural numbers n, r, a field K, and pairwise distinct nodes s : Fin n → K, the dimension of the kernel of the order-r endpoint synthesis map equals n - r: below the square threshold (r < n) the kernel has dimension n - r, and at or above it (r ≥ n) the kernel is zero, matching n - r = 0 under truncated subtraction.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : Field K] (s : Fin n → K), Function.Injective s → Module.finrank K ↥(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r).ker = n - r","l":127,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_single","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"Applying the moment-synthesis map to the standard basis vector for a node produces the vector of that node's powers across the requested moment orders.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : CommSemiring K] (s : Fin n → K) (j : Fin n), (Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r) (Pi.single j.castSucc 1) = fun a => s j ^ ↑a","l":47,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis_surjective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.VandermondeSynthesis","a":"Mathlib/LinearAlgebra","t":2,"d":"With distinct node values, the finite moment-synthesis map can reproduce every target vector of moments up through an order smaller than the number of nodes.","s":"∀ {n r : ℕ} {K : Type u_1} [inst : Field K] (s : Fin n → K), Function.Injective s → r < n → Function.Surjective ⇑(Causalean.Mathlib.LinearAlgebra.endpointOrderSynthesis s r)","l":62,"f":"Causalean/Mathlib/LinearAlgebra/VandermondeSynthesis.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eq_of_genPerm_triangular_unitDiag","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"Generalized-permutation reduction. For matrices C, C' over Fin n × Fin n valued in K, if C has unit diagonal and C' has unit diagonal, C is lower triangular in some causal order σ (C i j = 0 when σ i < σ j), and C' is obtained from C by a generalized permutation with permutation τ and scalings d, i.e. C' i j = d i · C (τ i) j for all i, j, then C = C': the unit diagonal plus triangularity force the underlying permutation to be the identity and every scaling to be one. (Formerly Discovery.LiNGAM.lingam_reduction.)","s":"∀ {n : ℕ} {K : Type u_1} [inst : MulZeroOneClass K] [Nontrivial K] {C C' : Matrix (Fin n) (Fin n) K}, (∀ (i : Fin n), C i i = 1) → (∀ (i : Fin n), C' i i = 1) → ∀ {σ : Equiv.Perm (Fin n)}, (∀ (i j : Fin n), σ i < σ j → C i j = 0) → ∀ {τ : Equiv.Perm (Fin n)} {d : Fin n → K}, (∀ (i j : Fin n), C' i j = d i * C (τ i) j) → C = C'","l":125,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_derivative_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating the derivative of a polynomial whose coefficients are indexed by two blocks of monomial powers gives the finite sum of coefficients weighted by their monomial exponents and the corresponding reduced powers.","s":"∀ {n : ℕ} {K : Type u_1} [inst : CommSemiring K] (v : Fin n ⊕ Fin n → K) (x : K), Polynomial.eval x (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v)) = ∑ a, ↑(Causalean.Mathlib.LinearAlgebra.doubledExponent a) * x ^ (Causalean.Mathlib.LinearAlgebra.doubledExponent a - 1) * v a","l":134,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_derivative_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating the derivative of a pinned coefficient polynomial at a point equals the sum of its coefficients weighted by the corresponding derivative monomial values at that point.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K] (v : Fin n ⊕ Fin (n - 1) → K) (x : K), Polynomial.eval x (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v)) = ∑ a, ↑(Causalean.Mathlib.LinearAlgebra.pinnedExponent a) * x ^ (Causalean.Mathlib.LinearAlgebra.pinnedExponent a - 1) * v a","l":348,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_doubledCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating a polynomial whose coefficients are indexed by two blocks of monomial powers gives the finite sum of those coefficients weighted by the corresponding powers of the evaluation point.","s":"∀ {n : ℕ} {K : Type u_1} [inst : CommSemiring K] (v : Fin n ⊕ Fin n → K) (x : K), Polynomial.eval x (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v) = ∑ a, x ^ Causalean.Mathlib.LinearAlgebra.doubledExponent a * v a","l":120,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.eval_pinnedCoefficientPolynomial","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Evaluating a pinned coefficient polynomial at a point equals the sum of its coefficients, each weighted by that point raised to the coefficient's associated pinned exponent.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K] (v : Fin n ⊕ Fin (n - 1) → K) (x : K), Polynomial.eval x (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v) = ∑ a, x ^ Causalean.Mathlib.LinearAlgebra.pinnedExponent a * v a","l":335,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.genPerm_of_det_ne_zero_of_colSupport","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"For a finite index type ι and a commutative ring K, and a square matrix W over ι × ι valued in K with nonzero determinant, if every column of W has at most one non-zero entry (for any two distinct rows i ≠ k, at least one of W i j, W k j vanishes at column j), then W is a generalized permutation matrix: there are a permutation τ of ι and non-zero scalings d with W i j = if j = τ i then d i else 0.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : DecidableEq ι] [inst_2 : CommRing K] {W : Matrix ι ι K}, W.det ≠ 0 → (∀ (j i k : ι), i ≠ k → W i j = 0 ∨ W k j = 0) → ∃ τ d, (∀ (i : ι), d i ≠ 0) ∧ ∀ (i j : ι), W i j = if j = τ i then d i else 0","l":32,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.isUpperTri_iff_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The entrywise upper-triangular predicate is exactly Mathlib's block-triangular predicate for the identity order.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : LT ι] [inst_1 : Zero K] {U : Matrix ι ι K}, Causalean.Mathlib.LinearAlgebra.IsUpperTri U ↔ U.BlockTriangular id","l":39,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_diagEntries_pos","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The diagonal entries of the LDL decomposition of a real positive-definite matrix are strictly positive.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), 0 < LDL.diagEntries hM i","l":68,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lowerInv_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lowerInv is lower-triangular in the BlockTriangular toDual sense.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef), (LDL.lowerInv hM).BlockTriangular ⇑OrderDual.toDual","l":81,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lowerInv_diag_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"The Gram-Schmidt lower-inverse matrix is unitriangular: its diagonal entries are 1.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), LDL.lowerInv hM i i = 1","l":52,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lower_blockTriangular","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lower (the inverse of LDL.lowerInv) is lower-triangular.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef), (LDL.lower hM).BlockTriangular ⇑OrderDual.toDual","l":87,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.ldl_lower_diag_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"LDL.lower is unitriangular: its diagonal entries are 1.","s":"∀ {d : ℕ} {M : Matrix (Fin d) (Fin d) ℝ} (hM : M.PosDef) (i : Fin d), LDL.lower hM i i = 1","l":93,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.natDegree_doubledCoefficientPolynomial_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A doubled coefficient polynomial has degree below 2n when n is positive.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K], 1 ≤ n → ∀ {v : Fin n ⊕ Fin n → K}, (Causalean.Mathlib.LinearAlgebra.doubledCoefficientPolynomial v).natDegree < 2 * n","l":99,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.natDegree_pinnedCoefficientPolynomial_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A pinned coefficient polynomial has degree below 2n - 1.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K], 1 ≤ n → ∀ {v : Fin n ⊕ Fin (n - 1) → K}, (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v).natDegree < 2 * n - 1","l":312,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.orthogonal_upperTri_pos_diag_eq_one","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.Cholesky","a":"Mathlib/LinearAlgebra","t":2,"d":"An orthogonal (Wᵀ * W = 1) upper-triangular matrix with strictly positive diagonal is the identity matrix.","s":"∀ {ι : Type u_1} {K : Type u_2} [inst : Fintype ι] [inst_1 : LinearOrder ι] [inst_2 : Field K] [inst_3 : LinearOrder K] [IsStrictOrderedRing K] {W : Matrix ι ι K}, W.transpose * W = 1 → (∀ (i j : ι), j < i → W i j = 0) → (∀ (i : ι), 0 < W i i) → W = 1","l":155,"f":"Causalean/Mathlib/LinearAlgebra/Cholesky.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.perm_uniqueness","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.MonomialMatrix","a":"Mathlib/LinearAlgebra","t":1,"d":"Permutation uniqueness for lower-triangular matrices (LiNGAM Appendix A, Lemma 1). For a matrix M over Fin n × Fin n that is lower-triangular (M i j = 0 whenever i < j) with non-zero diagonal entries, and permutations σ, τ of Fin n, the row/column-permuted matrix (i ↦ M (σ i) (τ i)) has a non-zero diagonal at every i if and only if σ = τ.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Zero K] {M : Matrix (Fin n) (Fin n) K}, (∀ (i j : Fin n), i < j → M i j = 0) → (∀ (i : Fin n), M i i ≠ 0) → ∀ {σ τ : Equiv.Perm (Fin n)}, (∀ (i : Fin n), M (σ i) (τ i) ≠ 0) ↔ σ = τ","l":99,"f":"Causalean/Mathlib/LinearAlgebra/MonomialMatrix.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), a semiring of coefficients, and a coefficient vector indexed by \\(n\\) value positions and \\(n-1\\) derivative positions, the pinned coefficient polynomial is the polynomial whose coefficient at each pinned exponent is the corresponding vector entry.","s":"{n : ℕ} → {K : Type u_1} → [inst : Semiring K] → (Fin n ⊕ Fin (n - 1) → K) → Polynomial K","l":275,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A pinned coefficient polynomial vanishes exactly when its coefficient vector vanishes.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Semiring K] (v : Fin n ⊕ Fin (n - 1) → K), Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v = 0 ↔ v = 0","l":298,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial_eq_zero_of_eval_derivative","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A polynomial in the pinned coefficient model is zero when it vanishes at all nodes and its derivative vanishes at every nondistinguished node.","s":"∀ {n : ℕ} {K : Type u_1} [inst : Field K], 1 ≤ n → ∀ (s : Fin n → K), Function.Injective s → ∀ (v : Fin n ⊕ Fin (n - 1) → K), (∀ (i : Fin n), Polynomial.eval (s i) (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v) = 0) → (∀ (i : Fin (n - 1)), Polynomial.eval (s (Causalean.Mathlib.LinearAlgebra.pinnedSucc i)) (Polynomial.derivative (Causalean.Mathlib.LinearAlgebra.pinnedCoefficientPolynomial v)) = 0) → v = 0","l":362,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedConfluentVandermonde","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), a semiring of coefficients, and \\(n\\) node values, the pinned confluent Vandermonde matrix records values at all nodes and first derivatives at nodes one through \\(n-1\\), omitting the derivative at node zero.","s":"{n : ℕ} → {K : Type u_1} → [Semiring K] → (Fin n → K) → Matrix (Fin n ⊕ Fin (n - 1)) (Fin n ⊕ Fin (n - 1)) K","l":323,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\), the pinned exponent assigns exponent \\(i\\) to the \\(i\\)-th index in its block of \\(n\\) simple-node positions and exponent \\(n+i\\) to the \\(i\\)-th index in its block of \\(n-1\\) derivative positions. The simple-node assignment and the derivative-position assignment together define the encoding.","s":"{n : ℕ} → Fin n ⊕ Fin (n - 1) → ℕ","l":220,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The pinned exponent encoding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.pinnedExponent","l":249,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedExponent_lt","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"Every pinned exponent is strictly below 2n - 1.","s":"∀ {n : ℕ} (i : Fin n ⊕ Fin (n - 1)), Causalean.Mathlib.LinearAlgebra.pinnedExponent i < 2 * n - 1","l":266,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc","k":"def","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(n\\) and an index \\(i<n-1\\), the pinned successor index is the index \\(i+1<n\\).","s":"{n : ℕ} → Fin (n - 1) → Fin n","l":228,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc_injective","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"The positive-node embedding is injective.","s":"∀ {n : ℕ}, Function.Injective Causalean.Mathlib.LinearAlgebra.pinnedSucc","l":233,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.pinnedSucc_ne_zero","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.ConfluentVandermonde","a":"Mathlib/LinearAlgebra","t":2,"d":"A positive-node index never equals the distinguished zero node.","s":"∀ {n : ℕ} (hn : 1 ≤ n) (i : Fin (n - 1)), Causalean.Mathlib.LinearAlgebra.pinnedSucc i ≠ ⟨0, hn⟩","l":242,"f":"Causalean/Mathlib/LinearAlgebra/ConfluentVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.stackedContraction","k":"def","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"Given a nonnegative integer \\(N\\), \\(N+1\\) slope values, a weight assigned to each slope and each of the \\(N\\) blocks, and a coefficient attached to each slope, the stacked contraction maps a block index \\(k<N\\) and an exponent \\(r\\leq k\\) to the sum, over all slopes, of the weight times the coefficient times the \\(r\\)-th power of that slope.","s":"{K : Type u_1} → [Semiring K] → (N : ℕ) → (Fin (N + 1) → K) → (Fin (N + 1) → Fin N → K) → (Fin (N + 1) → K) → (k : Fin N) → Fin (↑k + 1) → K","l":39,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.LinearAlgebra.stacked_contraction_injective_of_generic_weights","k":"theorem","m":"Causalean.Mathlib.LinearAlgebra.StackedVandermonde","a":"Mathlib/LinearAlgebra","t":1,"d":"For a commutative integral domain K, at least three indices N + 1 (2 ≤ N), and pairwise distinct slope values slopes : Fin (N + 1) → K, there exist weights Fin (N + 1) → Fin N → K for which the stacked contraction is injective. The first block detects coordinate 0, while the last block is a square Vandermonde system on coordinates 1,…,N.","s":"∀ {K : Type u_1} [inst : CommRing K] [IsDomain K] {N : ℕ}, 2 ≤ N → ∀ (slopes : Fin (N + 1) → K), Function.Injective slopes → ∃ weights, Function.Injective (Causalean.Mathlib.LinearAlgebra.stackedContraction N slopes weights)","l":69,"f":"Causalean/Mathlib/LinearAlgebra/StackedVandermonde.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn","k":"def","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":1,"d":"A finite coordinate set and an assignment-valued function determine the property that the function is unchanged whenever two assignments agree on that set.","s":"{V : Type u_1} → {X : V → Type u_2} → {Y : Type u_3} → Finset V → (((i : V) → X i) → Y) → Prop","l":29,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn.combine","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"Two functions depending on the same coordinate set, combined through a fixed binary operation, still depend only on that set.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [(i : V) → MeasurableSpace (X i)] {Y : Type u_3} {Z : Type u_4} {W : Type u_5} {S : Finset V} {f : ((i : V) → X i) → Y} {g : ((i : V) → X i) → Z}, Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S f → Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S g → ∀ (op : Y → Z → W), Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S fun x => op (f x) (g x)","l":47,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn.coordinateExtension_projection","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A function depending only on a coordinate set, together with an anchor assignment and a full assignment, has the same value after restricting and then extending that assignment.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(i : V) → MeasurableSpace (X i)] {Y : Type u_3} {S : Finset V} {f : ((i : V) → X i) → Y}, Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S f → ∀ (x₀ x : (i : V) → X i), f (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateExtension S x₀ (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S x)) = f x","l":97,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn.mono","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A function depending on a smaller coordinate set and that set's inclusion in a larger one also depends only on the larger set.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [(i : V) → MeasurableSpace (X i)] {Y : Type u_3} {S T : Finset V} {f : ((i : V) → X i) → Y}, Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S f → S ⊆ T → Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn T f","l":34,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateExtension","k":"def","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":1,"d":"A finite coordinate set, an anchor assignment, and a restricted assignment determine the full assignment that uses the restriction on the set and the anchor elsewhere.","s":"{V : Type u_1} → [DecidableEq V] → {X : V → Type u_2} → (S : Finset V) → ((i : V) → X i) → ((i : ↥S) → X ↑i) → (i : V) → X i","l":70,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection","k":"def","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":1,"d":"A finite coordinate set and a full assignment determine its restriction to those coordinates.","s":"{V : Type u_1} → {X : V → Type u_2} → (S : Finset V) → ((i : V) → X i) → (i : ↥S) → X ↑i","l":57,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection_extension","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A finite coordinate set, an anchor assignment, and a restricted assignment are recovered unchanged after extension followed by restriction.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(i : V) → MeasurableSpace (X i)] (S : Finset V) (x₀ : (i : V) → X i) (z : (i : ↥S) → X ↑i), Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateExtension S x₀ z) = z","l":88,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.dependsOn_apply","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A chosen coordinate can be read from an assignment using only that coordinate.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [(i : V) → MeasurableSpace (X i)] (i : V), Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn {i} fun x => x i","l":41,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.map_comp_eq_of_map_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A measurable source map, a measurable common downstream map, and equality of the source laws imply equality after the common downstream mapping.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [inst : (i : V) → MeasurableSpace (X i)] {Y : Type u_3} {Z : Type u_4} [inst_1 : MeasurableSpace Y] [inst_2 : MeasurableSpace Z] {μ ν : MeasureTheory.Measure ((i : V) → X i)} {f : ((i : V) → X i) → Y} {mix : Y → Z}, Measurable f → Measurable mix → MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map f ν → MeasureTheory.Measure.map (mix ∘ f) μ = MeasureTheory.Measure.map (mix ∘ f) ν","l":136,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.map_coordinateProjection_withDensity_eq_of_lmarginal_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":1,"d":"A finite coordinate set, measurable observational density, measurable comparison density, and equality after integrating out all complementary coordinates imply equality of the projected measures induced by those densities.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [inst_1 : (i : V) → MeasurableSpace (X i)] [inst_2 : Fintype V] {μ : (i : V) → MeasureTheory.Measure (X i)} [∀ (i : V), MeasureTheory.SigmaFinite (μ i)] (S : Finset V) {f g : ((i : V) → X i) → ENNReal}, Measurable f → Measurable g → ∫⋯∫⁻_Finset.univ \\ S, f ∂μ = ∫⋯∫⁻_Finset.univ \\ S, g ∂μ → MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S) ((MeasureTheory.Measure.pi μ).withDensity f) = MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S) ((MeasureTheory.Measure.pi μ).withDensity g)","l":150,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.map_eq_of_map_coordinateProjection_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"An anchor assignment, a measurable outcome map, its dependence only on a finite coordinate set, and equality of the two corresponding coordinate marginals imply equality of the outcome laws under the two measures.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [inst : (i : V) → MeasurableSpace (X i)] {Y : Type u_3} [inst_1 : MeasurableSpace Y] {S : Finset V} {μ ν : MeasureTheory.Measure ((i : V) → X i)} {f : ((i : V) → X i) → Y} (x₀ : (i : V) → X i), Measurable f → Causalean.Mathlib.MeasureTheory.FiniteCoordinate.DependsOn S f → MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S) μ = MeasureTheory.Measure.map (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S) ν → MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map f ν","l":107,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.measurable_coordinateExtension","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A finite coordinate set and an anchor assignment have a measurable extension map from restricted to full assignments.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [inst_1 : (i : V) → MeasurableSpace (X i)] (S : Finset V) (x₀ : (i : V) → X i), Measurable (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateExtension S x₀)","l":76,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.FiniteCoordinate.measurable_coordinateProjection","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteCoordinateDependence","a":"Mathlib/MeasureTheory","t":2,"d":"A finite coordinate set has a measurable assignment-restriction map.","s":"∀ {V : Type u_1} [DecidableEq V] {X : V → Type u_2} [inst : (i : V) → MeasurableSpace (X i)] (S : Finset V), Measurable (Causalean.Mathlib.MeasureTheory.FiniteCoordinate.coordinateProjection S)","l":62,"f":"Causalean/Mathlib/MeasureTheory/FiniteCoordinateDependence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.IsExtremePoint","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"Given a set \\(C\\) of measures on the real line and a measure \\(\\mu\\) on the real line, the extreme-point property holds when \\(\\mu\\) belongs to \\(C\\) and, for every pair of measures in \\(C\\) and every mixing weight strictly between zero and one, equality of \\(\\mu\\) to their weighted mixture implies that the two measures are equal.","s":"Set (MeasureTheory.Measure ℝ) → MeasureTheory.Measure ℝ → Prop","l":67,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.MemLp.of_measurable_of_integral_sq_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MemLp","a":"Mathlib/MeasureTheory","t":1,"d":"If a normed-vector-space-valued function f is almost-everywhere strongly measurable with respect to Q and the squared norm ‖f x‖^2 is integrable against Q, then f belongs to $L^2(Q)$.","s":"∀ {X : Type u_1} {F : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup F] {Q : MeasureTheory.Measure X} {f : X → F}, MeasureTheory.AEStronglyMeasurable f Q → MeasureTheory.Integrable (fun x => ‖f x‖ ^ 2) Q → MeasureTheory.MemLp f 2 Q","l":22,"f":"Causalean/Mathlib/MeasureTheory/MemLp.lean"},{"n":"Causalean.Mathlib.MeasureTheory.MomentSlice","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"Given real numbers \\(a\\), \\(b\\), and \\(s\\), the moment slice is the set of probability measures on the real line that [are supported on the closed interval from \\(a\\) to \\(b\\)](step:1,step:2), have mean zero, and have second moment \\(s\\).","s":"ℝ → ℝ → ℝ → Set (MeasureTheory.Measure ℝ)","l":60,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.SupportMeasurableOn","k":"def","m":"Causalean.Mathlib.MeasureTheory.SupportRnDerivTransport","a":"Mathlib/MeasureTheory","t":1,"d":"A map and a set determine the proposition that the map is measurable on that set, by testing its restriction to the set's subtype.","s":"{X : Type u_1} → {Y : Type u_2} → [MeasurableSpace X] → [MeasurableSpace Y] → (X → Y) → Set X → Prop","l":23,"f":"Causalean/Mathlib/MeasureTheory/SupportRnDerivTransport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.abs_le_one_ae_of_setIntegral_le_measure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SetIntegralRecovery","a":"Mathlib/MeasureTheory","t":2,"d":"A measurable function is bounded in absolute value by one almost everywhere if the absolute value of its integral over every measurable event is at most that event's measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (mu : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure mu] (f : Ω → ℝ), Measurable f → (∀ (A : Set Ω), MeasurableSet A → |∫ (x : Ω) in A, f x ∂mu| ≤ (mu A).toReal) → ∀ᵐ (x : Ω) ∂mu, |f x| ≤ 1","l":48,"f":"Causalean/Mathlib/MeasureTheory/SetIntegralRecovery.lean"},{"n":"Causalean.Mathlib.MeasureTheory.aemeasurable_of_supportMeasurableOn","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupportRnDerivTransport","a":"Mathlib/MeasureTheory","t":2,"d":"A measurable support set, a measure concentrated on it, and a map measurable there give a map that is almost-everywhere measurable for that measure.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] [MeasurableSingletonClass Y] [Nonempty Y] {m : MeasureTheory.Measure X} {S : Set X} {f : X → Y}, MeasurableSet S → m Sᶜ = 0 → Causalean.Mathlib.MeasureTheory.SupportMeasurableOn f S → AEMeasurable f m","l":29,"f":"Causalean/Mathlib/MeasureTheory/SupportRnDerivTransport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.bddAbove_image_of_bound","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"A uniform upper bound F ω π ≤ C over the index class S makes the image (F ω) '' S bounded above (for every ω). Companion BddAbove fact accompanying the integrability lemma.","s":"∀ {Ω : Type u_1} {ι : Type u_2} (S : Set ι) (F : Ω → ι → ℝ) (C : ℝ), (∀ (ω : Ω), ∀ π ∈ S, F ω π ≤ C) → ∀ (ω : Ω), BddAbove ((fun π => F ω π) '' S)","l":107,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.borelMeasurable_compact_argmin_selector","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CompactArgminSelection","a":"Mathlib/MeasureTheory","t":1,"d":"A jointly Borel real objective with continuous action sections on a fixed nonempty compact action set in finite-dimensional Euclidean space admits a Borel measurable, feasible rule that minimizes the objective at every parameter.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] {ι : Type u_2} [Fintype ι] (K : Set (EuclideanSpace ℝ ι)), IsCompact K → K.Nonempty → ∀ (f : X × EuclideanSpace ℝ ι → ℝ), Measurable f → (∀ (x : X), ContinuousOn (fun y => f (x, y)) K) → ∃ π, Measurable π ∧ ∀ (x : X), π x ∈ K ∧ ∀ y ∈ K, f (x, π x) ≤ f (x, y)","l":115,"f":"Causalean/Mathlib/MeasureTheory/CompactArgminSelection.lean"},{"n":"Causalean.Mathlib.MeasureTheory.borelMeasurable_nearestPoint_selector","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CompactArgminSelection","a":"Mathlib/MeasureTheory","t":1,"d":"Every nonempty compact subset of a finite-dimensional Euclidean space admits a total Borel measurable nearest-point rule whose chosen point is feasible and realizes the exact distance infimum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (K : Set (EuclideanSpace ℝ ι)), IsCompact K → K.Nonempty → ∃ π, Measurable π ∧ ∀ (x : EuclideanSpace ℝ ι), π x ∈ K ∧ dist x (π x) = Metric.infDist x K","l":326,"f":"Causalean/Mathlib/MeasureTheory/CompactArgminSelection.lean"},{"n":"Causalean.Mathlib.MeasureTheory.card_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"Richter–Rogosinski support bound (finite-atom case). Consider the discrete probability measure μ = ∑_{x∈T} w x · δ_x carried by a finite set T ⊆ ℝ. If every atom weight w x is strictly positive for x ∈ T, every atom lies in the interval [a, b], and μ is an extreme point of the moment slice — the probability measures on [a, b] with mean 0 and second moment s, then T has at most three elements: μ is supported on at most three atoms.","s":"∀ {a b s : ℝ} {T : Finset ℝ} {w : ℝ → ℝ}, (∀ x ∈ T, 0 < w x) → (∀ x ∈ T, x ∈ Set.Icc a b) → Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) → T.card ≤ 3","l":181,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.condExp_eq_of_integral_preimage_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpPreimage","a":"Mathlib/MeasureTheory","t":1,"d":"For a finite sampling measure, a measurable design map, an integrable outcome and candidate regression, a candidate regression that is almost-everywhere strongly measurable with respect to the design σ-algebra, and matching outcome and candidate-regression integrals on every measurable design event, the candidate regression is a version of the outcome's conditional expectation given the design.","s":"∀ {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsFiniteMeasure mu] (design : Omega → D), Measurable design → ∀ (Y m : Omega → ℝ), MeasureTheory.Integrable Y mu → MeasureTheory.Integrable m mu → MeasureTheory.AEStronglyMeasurable m mu → (∀ (S : Set D), MeasurableSet S → ∫ (omega : Omega) in design ⁻¹' S, Y omega ∂mu = ∫ (omega : Omega) in design ⁻¹' S, m omega ∂mu) → mu[Y | MeasurableSpace.comap design inferInstance] =ᵐ[mu] m","l":20,"f":"Causalean/Mathlib/MeasureTheory/CondExpPreimage.lean"},{"n":"Causalean.Mathlib.MeasureTheory.continuous_setIntegral_of_continuous_uniformOn","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.Integral.UniformConvergence","a":"Mathlib/MeasureTheory/Integral","t":1,"d":"For a measure, a finite-measure integration set, and an family of functions, each integrable on that set that varies continuously in the topology of uniform convergence on that set, the corresponding set integrals vary continuously.","s":"∀ {X : Type u_1} {Y : Type u_2} {E : Type u_3} [inst : TopologicalSpace X] [inst_1 : MeasurableSpace Y] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (mu : MeasureTheory.Measure Y) (K : Set Y), mu K < ⊤ → ∀ (F : X → Y → E), (Continuous fun x => (UniformOnFun.ofFun {K}) (F x)) → (∀ (x : X), MeasureTheory.IntegrableOn (F x) K mu) → Continuous fun x => ∫ (y : Y) in K, F x y ∂mu","l":22,"f":"Causalean/Mathlib/MeasureTheory/Integral/UniformConvergence.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure","k":"def","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":1,"d":"Given a measurable sample space, a finite set \\(T\\) of points in that space, and a real-valued weight function, the discrete measure is the sum of the Dirac measures at the points of \\(T\\), each multiplied by the nonnegative part of its weight.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → Finset α → (α → ℝ) → MeasureTheory.Measure α","l":53,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_apply_compl_of_subset","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A discrete measure whose atoms all lie in a set gives zero mass to that set's complement.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {T : Finset α} {w : α → ℝ} {K : Set α}, (∀ x ∈ T, x ∈ K) → (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) Kᶜ = 0","l":104,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_midpoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A discrete measure whose atom weights are the pointwise average of two nonnegative weightings is the midpoint of the two corresponding discrete measures.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {T : Finset α} {w wp wm : α → ℝ}, (∀ x ∈ T, 0 ≤ wp x) → (∀ x ∈ T, 0 ≤ wm x) → (∀ x ∈ T, w x = 1 / 2 * wp x + 1 / 2 * wm x) → Causalean.Mathlib.MeasureTheory.discreteMeasure T w = (1 / 2) • Causalean.Mathlib.MeasureTheory.discreteMeasure T wp + (1 / 2) • Causalean.Mathlib.MeasureTheory.discreteMeasure T wm","l":159,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.discreteMeasure_singleton","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"The mass assigned by a finite discrete measure to an atom in its support is the corresponding atom weight, coerced to ℝ≥0∞.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {T : Finset α} {w : α → ℝ} {x₀ : α}, x₀ ∈ T → (Causalean.Mathlib.MeasureTheory.discreteMeasure T w) {x₀} = ENNReal.ofReal (w x₀)","l":144,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_cardSupportLe_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice is carried by a finite set of at most three points.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∃ T, T.card ≤ 3 ∧ μ (↑T)ᶜ = 0","l":715,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_isMinOn_momentSlice","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Attainment of the minimum over a moment slice. On a compact Hausdorff space Ω, the set of probability measures pinned by two bounded-continuous moment constraints ∫ g₁ = c₁, ∫ g₂ = c₂ is weak-* compact, so any bounded-continuous objective ∫ f attains its minimum over that (nonempty) slice. The number of constraints is immaterial; the two-moment case is stated to match the mean/second-moment slice.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : TopologicalSpace Ω] [T2Space Ω] [BorelSpace Ω] [CompactSpace Ω] (g₁ g₂ f : BoundedContinuousFunction Ω ℝ) (c₁ c₂ : ℝ), {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}.Nonempty → ∃ μ ∈ {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}, ∀ ν ∈ {μ | ∫ (x : Ω), g₁ x ∂↑μ = c₁ ∧ ∫ (x : Ω), g₂ x ∂↑μ = c₂}, ∫ (x : Ω), f x ∂↑μ ≤ ∫ (x : Ω), f x ∂↑ν","l":281,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.exists_moment_perturbation","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Three-moment perturbation. On any finite set T ⊆ ℝ of more than three points there is a nonzero real weighting δ whose total mass, first moment and second moment all vanish. This is rank–nullity: three linear functionals (∑ δ, ∑ δ·x, ∑ δ·x²) on a space of dimension > 3 have a nonzero common kernel.","s":"∀ {T : Finset ℝ}, 3 < T.card → ∃ δ, ∑ x ∈ T, δ x = 0 ∧ ∑ x ∈ T, δ x * x = 0 ∧ ∑ x ∈ T, δ x * x ^ 2 = 0 ∧ ∃ x ∈ T, δ x ≠ 0","l":77,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integrable_of_finite_atomic_support","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":2,"d":"Every strongly measurable normed-vector-valued function is integrable under a finite measure concentrated on finitely many measurable points.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_2} [Finite ι] (cell : ι ↪ 𝒳), (∀ (i : ι), MeasurableSet {cell i}) → μ (Set.range ⇑cell) = μ Set.univ → ∀ {E : Type u_3} [inst_3 : NormedAddCommGroup E] (f : 𝒳 → E), MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable f μ","l":91,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integrable_sSup_image_of_countable_dense","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":1,"d":"On a finite measure μ, if the bound C is nonnegative, the skeleton index set D is countable, each coordinate map ω ↦ F ω π is measurable for π in the skeleton D, the pointwise supremum of F ω over the index class S equals its supremum over the countable skeleton D, for every ω, and F ω π is bounded in absolute value by C uniformly over π ∈ S, then the pointwise supremum ω ↦ sSup ((F ω) '' S) is integrable against μ. This is the entry point that discharges the Bochner side conditions of a localized empirical-process supremum in one call.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (S D : Set ι) (F : Ω → ι → ℝ) (C : ℝ), 0 ≤ C → D.Countable → (∀ π ∈ D, Measurable fun ω => F ω π) → (∀ (ω : Ω), sSup ((fun π => F ω π) '' S) = sSup ((fun π => F ω π) '' D)) → (∀ (ω : Ω), ∀ π ∈ S, |F ω π| ≤ C) → MeasureTheory.Integrable (fun ω => sSup ((fun π => F ω π) '' S)) μ","l":116,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"If κ is a measurable family of measures, one per point of the base space (a measurable kernel) and f is Bochner-integrable against the measure m.bind κ obtained by mixing κ over the base measure m, then the Bochner integral of f against m.bind κ equals the iterated integral: first integrate f against κ a for each base point a, then integrate the result against m. This is the Bochner analogue of MeasureTheory.Measure.lintegral_bind.","s":"∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {f : β → E}, Measurable κ → MeasureTheory.Integrable f (m.bind κ) → ∫ (x : β), f x ∂m.bind κ = ∫ (a : α), ∫ (x : β), f x ∂κ a ∂m","l":39,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_bind_map","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel a ↦ (κ₁ a).bind (fun b => (κ₂ a b).map (g a b)) is itself measurable, and f is Bochner-integrable against the measure obtained by mixing κ₁ over the base measure m and, within each fibre, mixing the pushforward of κ₂ under g. Then the Bochner integral of f against that triply-nested mixed measure equals the threefold iterated integral of the pulled-back integrand (a, b, c) ↦ f (g a b c), integrated successively against κ₂ a b, κ₁ a, and m. This is the single bridge for a bind-then-bind-then-map integrand, which neither integral_bind nor integral_bind_map covers in one step.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {E : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {f : δ → E}, (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → MeasureTheory.Integrable f (m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (∫ (z : δ), f z ∂m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = ∫ (a : α), ∫ (b : β), ∫ (c : γ), f (g a b c) ∂κ₂ a b ∂κ₁ a ∂m","l":98,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_bind_map_proj","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose every fibre κ₁ a is a probability measure, every inner fibre κ₂ a b is a probability measure, each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel is measurable, the projection π is measurable, π undoes g by recovering the base point: π (g a b c) = a for all a, b, c, and f is Bochner-integrable against the base measure m. Then integrating the pulled-back function f ∘ π against the triply-nested mixed measure equals integrating f directly against the base measure m. This is the one-step bridge for marginalising a nested Giry-monad construction back onto the coordinate that the innermost pushforward carries through; the fibrewise probability-mass-one hypotheses are what make the two inner integrals of the constant f a evaluate to f a (no assumption on m is needed).","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {E : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {π : δ → α} {f : α → E}, (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ₁ a)) → (∀ (a : α) (b : β), MeasureTheory.IsProbabilityMeasure (κ₂ a b)) → (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → Measurable π → (∀ (a : α) (b : β) (c : γ), π (g a b c) = a) → MeasureTheory.Integrable f m → (∫ (z : δ), f (π z) ∂m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = ∫ (a : α), f a ∂m","l":205,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_map","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose each map g a is measurable, the kernel sending a base point a to the pushforward measure (κ a).map (g a) is itself measurable, and f is Bochner-integrable against the mixed measure m.bind (fun a => (κ a).map (g a)). Then the Bochner integral of f against that mixed measure equals the iterated integral of the pulled-back integrand a ↦ ∫ x, f (g a x) ∂κ a against the base measure m. This packages a single application of integral_bind with the fibrewise MeasureTheory.integral_map, supplying the bridge needed to expand a nested bind-then-map Bochner integral.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {E : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {g : α → β → γ} {f : γ → E}, (∀ (a : α), Measurable (g a)) → (Measurable fun a => MeasureTheory.Measure.map (g a) (κ a)) → MeasureTheory.Integrable f (m.bind fun a => MeasureTheory.Measure.map (g a) (κ a)) → (∫ (z : γ), f z ∂m.bind fun a => MeasureTheory.Measure.map (g a) (κ a)) = ∫ (a : α), ∫ (x : β), f (g a x) ∂κ a ∂m","l":61,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_bind_of_ae_eq_const","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose κ is a measurable family of measures, every fibre κ a is a probability measure, the integrand f agrees κ a-almost everywhere with a constant f' a on that fibre, for every base point a, and f is Bochner-integrable against the mixed measure m.bind κ. Then the Bochner integral of f against m.bind κ equals the integral of the fibrewise constant f' against the base measure m. This is the bridge for the situation where the integrand only depends on a coordinate that is constant within each inner kernel, so the inner integral evaluates to that constant and the bind reduces to ∫ a, f' a ∂m.","s":"∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {m : MeasureTheory.Measure α} {κ : α → MeasureTheory.Measure β} {f : β → E} {f' : α → E}, Measurable κ → (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ a)) → (∀ (a : α), ∀ᵐ (y : β) ∂κ a, f y = f' a) → MeasureTheory.Integrable f (m.bind κ) → ∫ (y : β), f y ∂m.bind κ = ∫ (a : α), f' a ∂m","l":136,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_cellConst","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":2,"d":"Cell-weight aggregation. For a finite measure, the integral of a cell-constant function c (H ω) is the cell-weighted sum ∑ h, c h · μ(H⁻¹{h}). Special case of integral_cellConst_mul with f ≡ 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] [MeasureTheory.IsFiniteMeasure μ] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ (c : ι → ℝ), ∫ (ω : Ω), c (H ω) ∂μ = ∑ h, c h * (μ (H ⁻¹' {h})).toReal","l":138,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_cellConst_mul","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":1,"d":"Cell-constant pull-out. If H is a finite-valued map on the sample space whose fibers H⁻¹{h} are all measurable and f is integrable with respect to the ambient measure, then, for any weighting c of the fiber values, the integral of the ω-dependent product c (H ω) • f ω decomposes as the sum, over the values h of H, of c h scaling the set integral of f over the fiber H⁻¹{h}. No measurability of c is needed: on each fiber c (H ω) is the constant c h.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ (c : ι → ℝ) {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {f : Ω → E}, MeasureTheory.Integrable f μ → ∫ (ω : Ω), c (H ω) • f ω ∂μ = ∑ h, c h • ∫ (ω : Ω) in H ⁻¹' {h}, f ω ∂μ","l":97,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_discreteMeasure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"Integral against a discrete measure is the weighted sum of the integrand over the atoms. Every function into a real normed vector space is integrable because the measure has finite support.","s":"∀ {α : Type u_1} {E : Type u_2} [inst : MeasurableSpace α] [MeasurableSingletonClass α] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [CompleteSpace E] {T : Finset α} {w : α → ℝ}, (∀ x ∈ T, 0 ≤ w x) → ∀ (f : α → E), ∫ (x : α), f x ∂Causalean.Mathlib.MeasureTheory.discreteMeasure T w = ∑ x ∈ T, w x • f x","l":129,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.integral_eq_sum_setIntegral_fiber","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionIntegral","a":"Mathlib/MeasureTheory","t":1,"d":"If H is a finite-valued map on the sample space whose fibers H⁻¹{h} are all measurable and f is integrable with respect to the ambient measure, then the integral of f splits as the sum, over the finitely many values h of H, of the set integrals of f restricted to the fiber H⁻¹{h}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {H : Ω → ι}, (∀ (h : ι), MeasurableSet (H ⁻¹' {h})) → ∀ {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {f : Ω → E}, MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ h, ∫ (ω : Ω) in H ⁻¹' {h}, f ω ∂μ","l":79,"f":"Causalean/Mathlib/MeasureTheory/PartitionIntegral.lean"},{"n":"Causalean.Mathlib.MeasureTheory.isAtomic_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice is a positive discrete measure supported on at most three points of the interval.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∃ T w, (∀ x ∈ T, 0 < w x) ∧ (∀ x ∈ T, x ∈ Set.Icc a b) ∧ T.card ≤ 3 ∧ μ = Causalean.Mathlib.MeasureTheory.discreteMeasure T w","l":639,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.isProbabilityMeasure_discreteMeasure","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"A finite discrete measure is a probability measure when all atom weights are nonnegative and their sum is one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {T : Finset α} {w : α → ℝ}, (∀ x ∈ T, 0 ≤ w x) → ∑ x ∈ T, w x = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.MeasureTheory.discreteMeasure T w)","l":117,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.map_bind_bind_map_proj","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.IntegralBind","a":"Mathlib/MeasureTheory","t":1,"d":"Suppose every fibre κ₁ a is a probability measure, every inner fibre κ₂ a b is a probability measure, each map g a b is measurable, for every base point a the pushforward kernel b ↦ (κ₂ a b).map (g a b) is measurable, the resulting doubly-nested mixed kernel is measurable, the projection π is measurable, and π undoes g by recovering the base point: π (g a b c) = a for all a, b, c. Then pushing the triply-nested mixed measure forward along π returns exactly the base measure m. This is the underlying measure identity behind integral_bind_bind_map_proj, stated without any integrability or integrand hypotheses: the two inner probability fibres each contribute total mass one over a fixed base point, so transporting back along π returns m unchanged (no hypothesis on m is needed). It is the bridge for marginalising a nested Giry-monad construction onto its reattached coordinate when only measurability of the eventual integrand is available.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] {m : MeasureTheory.Measure α} {κ₁ : α → MeasureTheory.Measure β} {κ₂ : α → β → MeasureTheory.Measure γ} {g : α → β → γ → δ} {π : δ → α}, (∀ (a : α), MeasureTheory.IsProbabilityMeasure (κ₁ a)) → (∀ (a : α) (b : β), MeasureTheory.IsProbabilityMeasure (κ₂ a b)) → (∀ (a : α) (b : β), Measurable (g a b)) → (∀ (a : α), Measurable fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → (Measurable fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) → Measurable π → (∀ (a : α) (b : β) (c : γ), π (g a b c) = a) → MeasureTheory.Measure.map π (m.bind fun a => (κ₁ a).bind fun b => MeasureTheory.Measure.map (g a b) (κ₂ a b)) = m","l":163,"f":"Causalean/Mathlib/MeasureTheory/IntegralBind.lean"},{"n":"Causalean.Mathlib.MeasureTheory.map_toReal_rnDeriv_eq_map_toReal_rnDeriv_map_of_support_equiv","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupportRnDerivTransport","a":"Mathlib/MeasureTheory","t":1,"d":"Finite numerator and denominator measures with absolute continuity, measurable source and target supports, concentration on the source support, and forward and reverse maps measurable on their supports, whose images stay in the matching supports and are mutual inverses there, give an unchanged pushforward law for the real-valued canonical Radon--Nikodym ratio.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] [StandardBorelSpace X] [StandardBorelSpace Y] (m n : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure m] [MeasureTheory.IsFiniteMeasure n], m.AbsolutelyContinuous n → ∀ (S : Set X) (T : Set Y), MeasurableSet S → MeasurableSet T → m Sᶜ = 0 → n Sᶜ = 0 → ∀ (f : X → Y) (g : Y → X), Causalean.Mathlib.MeasureTheory.SupportMeasurableOn f S → Causalean.Mathlib.MeasureTheory.SupportMeasurableOn g T → (∀ x ∈ S, f x ∈ T) → (∀ y ∈ T, g y ∈ S) → (∀ x ∈ S, g (f x) = x) → (∀ y ∈ T, f (g y) = y) → MeasureTheory.Measure.map (fun x => (m.rnDeriv n x).toReal) n = MeasureTheory.Measure.map (fun y => ((MeasureTheory.Measure.map f m).rnDeriv (MeasureTheory.Measure.map f n) y).toReal) (MeasureTheory.Measure.map f n)","l":129,"f":"Causalean/Mathlib/MeasureTheory/SupportRnDerivTransport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measurable_sSup_image_of_countable_dense","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"The pointwise supremum ω ↦ sSup ((F ω) '' S) of a real process over an index class S is measurable, provided S has a countable skeleton D on which the process is measurable in ω, and the supremum over S agrees pointwise with the supremum over D. Paper-agnostic generalization of the policy-class skeleton measurability step.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (S D : Set ι) (F : Ω → ι → ℝ), D.Countable → (∀ π ∈ D, Measurable fun ω => F ω π) → (∀ (ω : Ω), sSup ((fun π => F ω π) '' S) = sSup ((fun π => F ω π) '' D)) → Measurable fun ω => sSup ((fun π => F ω π) '' S)","l":68,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measure_eq_fin_sum_smul_dirac_of_range","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":1,"d":"Let μ be a finite measure and cell an injective family of finitely many points of the sample space, indexed by a finite type ι. If every singleton {cell i} is measurable and μ assigns its full mass to the range of cell, i.e. μ puts no mass outside these finitely many points, then μ equals the sum, over the index i, of the point mass μ {cell i} scaling the Dirac measure at cell i.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] (cell : ι ↪ 𝒳), (∀ (i : ι), MeasurableSet {cell i}) → μ (Set.range ⇑cell) = μ Set.univ → μ = ∑ i, μ {cell i} • MeasureTheory.Measure.dirac (cell i)","l":23,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.measure_eq_withDensity_of_toReal_setIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SetIntegralRecovery","a":"Mathlib/MeasureTheory","t":1,"d":"If a weight function w is integrable with respect to a reference measure μ and a finite measure ν's mass on every measurable event A equals, as a real number, the integral of w over A against μ, then ν is obtained from μ by weighting with the nonnegative part of w: ν = μ.withDensity (fun x => ENNReal.ofReal (w x)).","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure ν] {w : α → ℝ}, MeasureTheory.Integrable w μ → (∀ (A : Set α), MeasurableSet A → (ν A).toReal = ∫ (x : α) in A, w x ∂μ) → ν = μ.withDensity fun x => ENNReal.ofReal (w x)","l":27,"f":"Causalean/Mathlib/MeasureTheory/SetIntegralRecovery.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus","k":"def","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":1,"d":"For a real multivariate polynomial, its real zero locus is the set of real assignments at which the polynomial evaluates to zero.","s":"{σ : Type u_1} → MvPolynomial σ ℝ → Set (σ → ℝ)","l":146,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus_finset_prod","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":1,"d":"For a finite set of polynomial indices and an indexed family of real multivariate polynomials, the zero locus of their product equals the union of their zero loci.","s":"∀ {σ : Type u_1} {κ : Type u_2} (s : Finset κ) (p : κ → MvPolynomial σ ℝ), Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (∏ i ∈ s, p i) = ⋃ i ∈ s, Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (p i)","l":151,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus_fintype_prod","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"For a finite-type-indexed family of real multivariate polynomials, the zero locus of the product over all indices equals the union of all factor zero loci.","s":"∀ {σ : Type u_1} {κ : Type u_2} [inst : Fintype κ] (p : κ → MvPolynomial σ ℝ), Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (∏ i, p i) = ⋃ i, Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (p i)","l":170,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus_subtype_prod","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"Given a finite set of indices and a polynomial for every member of its finite subtype, the zero locus of the subtype product equals the union of the subtype-indexed zero loci.","s":"∀ {σ : Type u_1} {κ : Type u_2} (s : Finset κ) (p : ↥s → MvPolynomial σ ℝ), Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (∏ i, p i) = ⋃ i, Causalean.Mathlib.MeasureTheory.mvPolynomialZeroLocus (p i)","l":185,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomial_finset_prod_ne_zero","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"If every factor in a finite indexed family of real multivariate polynomials is nonzero, then their finite product is nonzero.","s":"∀ {σ : Type u_1} {κ : Type u_2} (s : Finset κ) (p : κ → MvPolynomial σ ℝ), (∀ i ∈ s, p i ≠ 0) → ∏ i ∈ s, p i ≠ 0","l":162,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomial_fintype_prod_ne_zero","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"If every member of a finite-type-indexed family of real multivariate polynomials is nonzero, then the product over the whole index type is nonzero.","s":"∀ {σ : Type u_1} {κ : Type u_2} [inst : Fintype κ] (p : κ → MvPolynomial σ ℝ), (∀ (i : κ), p i ≠ 0) → ∏ i, p i ≠ 0","l":177,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.mvPolynomial_subtype_prod_ne_zero","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"Given a finite set of indices, a polynomial for every member of its finite subtype, and a proof that every factor is nonzero, the product of the subtype-indexed factors is nonzero.","s":"∀ {σ : Type u_1} {κ : Type u_2} (s : Finset κ) (p : ↥s → MvPolynomial σ ℝ), (∀ (i : ↥s), p i ≠ 0) → ∏ i, p i ≠ 0","l":193,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.not_four_distinct_in_support","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice cannot have four distinct support points. Four separated neighborhoods would give a nonzero signed perturbation preserving mass, mean, and second moment, so the measure would be the midpoint of two different slice members.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → ∀ (x : Fin 4 → ℝ), Function.Injective x → (∀ (i : Fin 4), x i ∈ μ.support) → False","l":310,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.partition_restrict_absolutelyContinuous","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionRnDeriv","a":"Mathlib/MeasureTheory","t":1,"d":"Absolute continuity from a finite proportional partition. Suppose each cell s i of a finite family is measurable, the cells are pairwise disjoint, and the cells cover the whole ambient space — together, (s i) is a finite measurable partition — and suppose on every cell the numerator measure μ restricted to that cell equals the denominator measure ν restricted to the same cell, scaled by the constant r i. Then μ is absolutely continuous with respect to ν. The global density is the simple function whose value on cell i is r i, so μ = ν.withDensity d.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [Finite ι] (μ ν : MeasureTheory.Measure Ω) (s : ι → Set Ω) (r : ι → ENNReal), (∀ (i : ι), MeasurableSet (s i)) → Pairwise (Function.onFun Disjoint s) → ⋃ i, s i = Set.univ → (∀ (i : ι), μ.restrict (s i) = r i • ν.restrict (s i)) → μ.AbsolutelyContinuous ν","l":28,"f":"Causalean/Mathlib/MeasureTheory/PartitionRnDeriv.lean"},{"n":"Causalean.Mathlib.MeasureTheory.partition_restrict_integrable_pow_rnDeriv","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PartitionRnDeriv","a":"Mathlib/MeasureTheory","t":1,"d":"Power-deviation integrability from a finite proportional partition. Under the same partition hypotheses as partition_restrict_absolutelyContinuous — the cells s i are measurable, pairwise disjoint, and cover the ambient space — and again assuming on every cell the numerator measure μ restricted to that cell equals the denominator measure ν restricted to the same cell, scaled by the constant r i, with ν finite, then for any natural number n, the n-th power of the deviation of the Radon–Nikodym derivative dμ/dν from 1 is integrable against ν. On each cell the derivative equals r i, so the function is a finite simple function and the integral is a finite sum of per-cell constants.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [Finite ι] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure ν] (s : ι → Set Ω) (r : ι → ENNReal) (n : ℕ), (∀ (i : ι), MeasurableSet (s i)) → Pairwise (Function.onFun Disjoint s) → ⋃ i, s i = Set.univ → (∀ (i : ι), μ.restrict (s i) = r i • ν.restrict (s i)) → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ n) ν","l":83,"f":"Causalean/Mathlib/MeasureTheory/PartitionRnDeriv.lean"},{"n":"Causalean.Mathlib.MeasureTheory.property_at_of_ae_of_singleton_pos","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FiniteAtomicMeasure","a":"Mathlib/MeasureTheory","t":2,"d":"An almost-sure property holds at every point to which the measure assigns nonzero mass.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) (p : 𝒳 → Prop) {x : 𝒳}, (∀ᵐ (y : 𝒳) ∂μ, p y) → μ {x} ≠ 0 → p x","l":78,"f":"Causalean/Mathlib/MeasureTheory/FiniteAtomicMeasure.lean"},{"n":"Causalean.Mathlib.MeasureTheory.rnDeriv_map_of_support_equiv","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupportRnDerivTransport","a":"Mathlib/MeasureTheory","t":2,"d":"Finite numerator and denominator measures with absolute continuity, measurable source and target supports, concentration on the source support, and forward and reverse maps measurable on their supports, whose images stay in the matching supports and are mutual inverses there, give a canonical Radon--Nikodym derivative preserved after the forward map.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] [StandardBorelSpace X] [StandardBorelSpace Y] (m n : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure m] [MeasureTheory.IsFiniteMeasure n], m.AbsolutelyContinuous n → ∀ (S : Set X) (T : Set Y), MeasurableSet S → MeasurableSet T → m Sᶜ = 0 → n Sᶜ = 0 → ∀ (f : X → Y) (g : Y → X), Causalean.Mathlib.MeasureTheory.SupportMeasurableOn f S → Causalean.Mathlib.MeasureTheory.SupportMeasurableOn g T → (∀ x ∈ S, f x ∈ T) → (∀ y ∈ T, g y ∈ S) → (∀ x ∈ S, g (f x) = x) → (∀ y ∈ T, f (g y) = y) → (fun x => (MeasureTheory.Measure.map f m).rnDeriv (MeasureTheory.Measure.map f n) (f x)) =ᵐ[n] m.rnDeriv n","l":44,"f":"Causalean/Mathlib/MeasureTheory/SupportRnDerivTransport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.sSup_image_eq_of_dense_tendsto","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.SupCountableDense","a":"Mathlib/MeasureTheory","t":2,"d":"If F is bounded above on S, D ⊆ S, and every x ∈ S is the limit along a D-valued sequence of F-values (F (seq j) → F x), then the supremum of F over S coincides with the supremum over the countable skeleton D. This is the density-side input that turns a supremum over an uncountable class into a supremum over a countable skeleton (used both for measurability and for evaluating the supremum).","s":"∀ {ι : Type u_1} (F : ι → ℝ) (S D : Set ι), D ⊆ S → BddAbove (F '' S) → (∀ x ∈ S, ∃ seq, (∀ (j : ℕ), seq j ∈ D) ∧ Filter.Tendsto (fun j => F (seq j)) Filter.atTop (nhds (F x))) → sSup (F '' S) = sSup (F '' D)","l":41,"f":"Causalean/Mathlib/MeasureTheory/SupCountableDense.lean"},{"n":"Causalean.Mathlib.MeasureTheory.support_finite_ncard_le_three_of_isExtremePoint","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.MomentSliceSupport","a":"Mathlib/MeasureTheory","t":2,"d":"An extreme probability measure in the two-moment slice has a finite topological support with at most three points.","s":"∀ {a b s : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Mathlib.MeasureTheory.IsExtremePoint (Causalean.Mathlib.MeasureTheory.MomentSlice a b s) μ → μ.support.Finite ∧ μ.support.ncard ≤ 3","l":603,"f":"Causalean/Mathlib/MeasureTheory/MomentSliceSupport.lean"},{"n":"Causalean.Mathlib.MeasureTheory.unitIntervalVolumeIsOpenPosMeasure","k":"instance","m":"Causalean.Mathlib.MeasureTheory.UnitInterval.OpenPos","a":"Mathlib/MeasureTheory/UnitInterval","t":2,"d":"Lebesgue volume on the closed unit interval is positive on every nonempty relatively open set.","s":"MeasureTheory.volume.IsOpenPosMeasure","l":25,"f":"Causalean/Mathlib/MeasureTheory/UnitInterval/OpenPos.lean"},{"n":"Causalean.Mathlib.MeasureTheory.volume_mk_zeroLocus_mvPolynomial","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":2,"d":"For a real polynomial in finitely many numbered variables that is nonzero, its zero locus has Lebesgue measure zero.","s":"∀ {d : ℕ} (P : MvPolynomial (Fin d) ℝ), P ≠ 0 → MeasureTheory.volume {x | (MvPolynomial.eval x) P = 0} = 0","l":28,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.MeasureTheory.volume_zeroLocus_mvPolynomial_finite","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.PolynomialZeroLocus","a":"Mathlib/MeasureTheory","t":1,"d":"For a real polynomial indexed by an arbitrary finite variable type that is nonzero, its zero locus has Lebesgue measure zero.","s":"∀ {α : Type u_1} [inst : Fintype α] (P : MvPolynomial α ℝ), P ≠ 0 → MeasureTheory.volume {x | (MvPolynomial.eval x) P = 0} = 0","l":111,"f":"Causalean/Mathlib/MeasureTheory/PolynomialZeroLocus.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.max_two_split","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"Split a clipped two-term offset across its summands: max 0 (a − 2r) ≤ max 0 (b − r) + max 0 (c − r) when a ≤ b + c.","s":"∀ (a b c r : ℝ), a ≤ b + c → max 0 (a - 2 * r) ≤ max 0 (b - r) + max 0 (c - r)","l":154,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant","k":"def","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Given a real number \\(\\theta\\), the quarter-offset peeling constant is \\((1-\\theta)(4\\theta)^{\\theta/(1-\\theta)}\\).","s":"ℝ → ℝ","l":22,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC","k":"def","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Given a real offset coefficient \\(c\\) and a real number \\(\\theta\\), the general-offset peeling constant is \\((1-\\theta)(\\theta/c)^{\\theta/(1-\\theta)}).","s":"ℝ → ℝ → ℝ","l":27,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The general offset-peeling constant is nonnegative when the offset is positive and the exponent lies between zero and one.","s":"∀ (c θ : ℝ), 0 ≤ c → 0 ≤ θ → θ ≤ 1 → 0 ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ","l":91,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant_nonneg","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The offset-peeling constant for the quarter-offset case is nonnegative when the exponent lies between zero and one.","s":"∀ (θ : ℝ), 0 ≤ θ → θ ≤ 1 → 0 ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ","l":120,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":1,"d":"Deterministic Young/AM-GM peeling for the quarter-offset. For an exponent strictly between zero and one and nonnegative coefficients a and t, the positive part of a·t^θ − t/4 is at most the offset-peeling constant times a raised to the power 1/(1−θ).","s":"∀ (θ a t : ℝ), 0 < θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - 1 / 4 * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ * a ^ (1 / (1 - θ))","l":128,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_coeff","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"Generalized offset peeling at coefficient c: for 0 < c, 0 < θ < 1, max 0 (a·t^θ − c·t) ≤ offsetPeelingConstantC c θ · a^{1/(1−θ)} (weighted AM-GM).","s":"∀ (c θ a t : ℝ), 0 < c → 0 < θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - c * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ * a ^ (1 / (1 - θ))","l":35,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_coeff_nonneg_theta","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The general offset-peeling bound remains valid at exponent zero by the same constant formula.","s":"∀ (c θ a t : ℝ), 0 < c → 0 ≤ θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - c * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC c θ * a ^ (1 / (1 - θ))","l":101,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.OffsetPeeling.offset_peeling_nonneg_theta","k":"theorem","m":"Causalean.Mathlib.Analysis.OffsetPeeling","a":"Mathlib/Analysis","t":2,"d":"The quarter-offset peeling bound remains valid at exponent zero by continuity of the constant formula.","s":"∀ (θ a t : ℝ), 0 ≤ θ → θ < 1 → 0 ≤ a → 0 ≤ t → max 0 (a * t ^ θ - 1 / 4 * t) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstant θ * a ^ (1 / (1 - θ))","l":143,"f":"Causalean/Mathlib/Analysis/OffsetPeeling.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn","k":"structure","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite number of coordinates, an affine real-valued function is specified by one coefficient for each coordinate and one constant term.","s":"ℕ → Type","l":21,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn.continuous_eval","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine function, its evaluation map is continuous.","s":"∀ {n : ℕ} (f : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n), Continuous f.eval","l":59,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn.eval","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine function and a coordinate vector, its evaluated real value is given by the finite coefficient-weighted sum plus the constant.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n → (Fin n → ℝ) → ℝ","l":33,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn.eval_affineCombination","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine function, two coordinate vectors, and a real weight, evaluation at their affine combination equals the same affine combination of their evaluations.","s":"∀ {n : ℕ} (f : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n) (x y : Fin n → ℝ) (t : ℝ), f.eval ((1 - t) • x + t • y) = (1 - t) * f.eval x + t * f.eval y","l":40,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite number of coordinates, an affine constraint system is given by a finite list of marked affine constraints.","s":"ℕ → Type","l":85,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint","k":"structure","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite number of coordinates, a marked affine constraint contains its normalized affine left-hand side and its weak-or-strict mark.","s":"ℕ → Type","l":75,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint.strictHolds","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a marked affine constraint and a coordinate vector, the original constraint-satisfaction condition uses the comparison selected by the constraint's mark.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint n → (Fin n → ℝ) → Prop","l":91,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint.weakHolds","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a marked affine constraint and a coordinate vector, the weak constraint-satisfaction condition is given by the nonpositive affine evaluation.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint n → (Fin n → ℝ) → Prop","l":100,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.ConstraintKind","k":"inductive","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"A constraint kind records whether its affine inequality is weak or strict.","s":"Type","l":66,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint","k":"structure","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite number of coordinates, a checked polynomial constraint contains its polynomial left-hand side, its weak-or-strict mark, and a certificate that it is affine.","s":"ℕ → Type","l":104,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint.toConstraint","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a checked polynomial constraint, its affine constraint compilation is given by the compiled affine function with the original mark retained.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n → Causalean.Mathlib.Optimization.AffineSignCellClosure.Constraint n","l":117,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.affineFnOfMvPolynomial","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a real multivariate polynomial, its compiled affine function is given by the coefficients of its degree-one monomials and its constant coefficient.","s":"{n : ℕ} → MvPolynomial (Fin n) ℝ → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n","l":21,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.affineFnOfMvPolynomial_eval","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a real multivariate polynomial, a proof that its total degree is at most one, and a coordinate vector, evaluating its compiled affine function equals evaluating the polynomial.","s":"∀ {n : ℕ} (p : MvPolynomial (Fin n) ℝ), p.totalDegree ≤ 1 → ∀ (x : Fin n → ℝ), (Causalean.Mathlib.Optimization.AffineSignCellClosure.affineFnOfMvPolynomial p).eval x = (MvPolynomial.eval x) p","l":63,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.affineSystemOfPolynomials","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite list of checked polynomial constraints, its compiled affine system is given by compiling every listed constraint.","s":"{n : ℕ} → List (Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n) → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n","l":125,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.bddAbove_commonSlackSet","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine constraint system, its feasible slack set is bounded above.","s":"∀ {n : ℕ} (Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n), BddAbove (Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackSet Γ)","l":44,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.bddAbove_image_weakCell_of_strictCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a nonempty strict cell, a continuous real objective, and an upper bound for its strict-cell image, the weak-cell image is bounded above.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {φ : (Fin n → ℝ) → ℝ}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Continuous φ → BddAbove (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → BddAbove (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":37,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.bddBelow_image_weakCell_of_strictCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a nonempty strict cell, a continuous real objective, and a lower bound for its strict-cell image, the weak-cell image is bounded below.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {φ : (Fin n → ℝ) → ℝ}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Continuous φ → BddBelow (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → BddBelow (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":27,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.closure_image_strictCell_eq_closure_image_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a nonempty strict cell and a continuous real objective, the closures of its strict-cell and weak-cell image sets are equal.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {φ : (Fin n → ℝ) → ℝ}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Continuous φ → closure (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = closure (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":19,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.closure_strictCell_eq_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a nonempty strict cell, the closure of that strict cell equals its weak relaxation.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → closure (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ","l":86,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.closure_strictCell_mixedExample","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Example","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"The closure of the example's half-open strict interval equals its closed weak interval.","s":"closure (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample) = Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample","l":43,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Example.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackFeasible","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, a coordinate vector, and a real slack, the common-slack feasibility condition is given by the slack being nonnegative, the slack being at most one, and every weak constraint evaluating to at most zero while every strict constraint evaluates to at most the negative slack.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n → (Fin n → ℝ) → ℝ → Prop","l":20,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackSet","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its feasible slack set is given by the slacks for which some coordinate vector is common-slack feasible.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n → Set ℝ","l":30,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its common-slack value is given by the supremum of its feasible slack set.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n → ℝ","l":37,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue_nil","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a number of coordinates, the empty system's common-slack value equals one.","s":"∀ (n : ℕ), Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue [] = 1","l":147,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue_pos_of_witness","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a positive slack and a common-slack feasible point, the common-slack value is positive.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x : Fin n → ℝ} {δ : ℝ}, 0 < δ → Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackFeasible Γ x δ → 0 < Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue Γ","l":109,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.convex_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine constraint system, its weak relaxation is convex.","s":"∀ {n : ℕ} (Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n), Convex ℝ (Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":153,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.exists_strictCell_sequence_tendsto","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a weakly feasible point and a strictly feasible point, there is a sequence of strictly feasible points converging to the weakly feasible point.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x x₀ : Fin n → ℝ}, x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ → x₀ ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ → ∃ u, (∀ (k : ℕ), u k ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) ∧ Filter.Tendsto u Filter.atTop (nhds x)","l":63,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.half_mem_strictCell_mixedExample","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Example","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"The constant one-half coordinate vector belongs to the example's strict cell.","s":"(fun x => 1 / 2) ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample","l":37,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Example.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.instDecidableEqConstraintKind","k":"instance","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"","s":"DecidableEq Causalean.Mathlib.Optimization.AffineSignCellClosure.ConstraintKind","l":73,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.instReprConstraintKind","k":"instance","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"","s":"Repr Causalean.Mathlib.Optimization.AffineSignCellClosure.ConstraintKind","l":73,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.isClosed_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its weak relaxation is closed.","s":"∀ {n : ℕ} (Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n), IsClosed (Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":143,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.mem_strictCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a point, membership in the strict cell is equivalent to satisfying every listed constraint with its original comparison.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x : Fin n → ℝ}, x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ ↔ ∀ c ∈ Γ, c.strictHolds x","l":123,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.mem_strictCell_mixedExample","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Example","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a one-dimensional coordinate vector, it belongs to the example's strict cell exactly when its coordinate lies above zero and at most one.","s":"∀ (x : Fin 1 → ℝ), x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample ↔ 0 < x 0 ∧ x 0 ≤ 1","l":25,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Example.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.mem_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a point, membership in the weak relaxation is equivalent to weakly satisfying every listed constraint.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x : Fin n → ℝ}, x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ ↔ ∀ c ∈ Γ, c.weakHolds x","l":128,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.mem_weakCell_mixedExample","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Example","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a one-dimensional coordinate vector, it belongs to the example's weak cell exactly when its coordinate lies between zero and one inclusively.","s":"∀ (x : Fin 1 → ℝ), x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample ↔ 0 ≤ x 0 ∧ x 0 ≤ 1","l":31,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Example.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.mixedExample","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Example","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"The mixed one-dimensional example system is given by the strict lower inequality -x < 0 and the weak upper inequality x - 1 ≤ 0.","s":"Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem 1","l":19,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Example.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.polynomialStrictCell","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite list of checked polynomial constraints, its direct strict polynomial cell is given by satisfaction of each original weak-or-strict polynomial comparison.","s":"{n : ℕ} → List (Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n) → Set (Fin n → ℝ)","l":132,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.polynomialWeakCell","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a finite list of checked polynomial constraints, its direct weak polynomial cell is given by weak satisfaction of every polynomial constraint.","s":"{n : ℕ} → List (Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n) → Set (Fin n → ℝ)","l":141,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.positiveSlackWitness_of_commonSlackValue_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a positive common-slack value, some feasible slack is positive.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n}, 0 < Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue Γ → ∃ δ, 0 < δ ∧ δ ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackSet Γ","l":120,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.positiveSlackWitness_of_strictPoint","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a strictly feasible point, there is a positive common slack for that same point.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x : Fin n → ℝ}, x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ → ∃ δ, 0 < δ ∧ Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackFeasible Γ x δ","l":52,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.sInf_affineEval_strictCell_eq_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine objective, a nonempty strict cell, and a lower bound for the strict-cell objective image, the affine objective has the same infimum on the strict cell and weak relaxation.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} (f : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n), (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → BddBelow (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → sInf (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = sInf (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":81,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.sInf_image_strictCell_eq_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a nonempty strict cell, a continuous real objective, and a lower bound for its strict-cell image, the strict and weak image sets have the same infimum.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {φ : (Fin n → ℝ) → ℝ}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Continuous φ → BddBelow (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → sInf (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = sInf (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":47,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.sSup_affineEval_strictCell_eq_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine objective, a nonempty strict cell, and an upper bound for the strict-cell objective image, the affine objective has the same supremum on the strict cell and weak relaxation.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} (f : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineFn n), (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → BddAbove (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → sSup (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = sSup (f.eval '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":90,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.sSup_image_strictCell_eq_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Extrema","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a nonempty strict cell, a continuous real objective, and an upper bound for its strict-cell image, the strict and weak image sets have the same supremum.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {φ : (Fin n → ℝ) → ℝ}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Continuous φ → BddAbove (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) → sSup (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ) = sSup (φ '' Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ)","l":64,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Extrema.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.segmentPoint","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given a weak endpoint, a distinguished strict endpoint, and a real weight, the segment point is given by placing weight ε on the strict endpoint.","s":"{n : ℕ} → (Fin n → ℝ) → (Fin n → ℝ) → ℝ → Fin n → ℝ","l":20,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.segmentPoint_mem_strictCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a weakly feasible point, a strictly feasible point, a positive segment weight, and a weight at most one, the corresponding segment point is strictly feasible.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x x₀ : Fin n → ℝ}, x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ → x₀ ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ → ∀ {ε : ℝ}, 0 < ε → ε ≤ 1 → Causalean.Mathlib.Optimization.AffineSignCellClosure.segmentPoint x x₀ ε ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ","l":37,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its strict cell is given by the points satisfying every listed constraint using its original comparison.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n → Set (Fin n → ℝ)","l":109,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell_affineSystemOfPolynomials","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a finite checked polynomial system, compilation preserves its strict cell.","s":"∀ {n : ℕ} (Γ : List (Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n)), Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell (Causalean.Mathlib.Optimization.AffineSignCellClosure.affineSystemOfPolynomials Γ) = Causalean.Mathlib.Optimization.AffineSignCellClosure.polynomialStrictCell Γ","l":148,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell_subset_weakCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given an affine constraint system, every strictly feasible point belongs to its weak relaxation.","s":"∀ {n : ℕ} (Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n), Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ ⊆ Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ","l":133,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.strictFeasible_iff_commonSlackValue_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its strict cell is nonempty exactly when its bounded common-slack value is positive.","s":"∀ {n : ℕ} (Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n), (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty ↔ 0 < Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackValue Γ","l":135,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.strictPoint_of_positiveSlack","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.CommonSlack","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a positive slack and a common-slack feasible point, that point belongs to the strict cell.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n} {x : Fin n → ℝ} {δ : ℝ}, 0 < δ → Causalean.Mathlib.Optimization.AffineSignCellClosure.commonSlackFeasible Γ x δ → x ∈ Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ","l":95,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/CommonSlack.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.tendsto_segmentPoint_zero","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a weak endpoint and a distinguished strict endpoint, their segment points converge to the weak endpoint as the strict-endpoint weight tends to zero.","s":"∀ {n : ℕ} (x x₀ : Fin n → ℝ), Filter.Tendsto (Causalean.Mathlib.Optimization.AffineSignCellClosure.segmentPoint x x₀) (nhds 0) (nhds x)","l":27,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell","k":"def","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Basic","a":"Mathlib/Optimization/AffineSignCellClosure","t":1,"d":"Given an affine constraint system, its weak relaxation is given by the points weakly satisfying every listed constraint.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n → Set (Fin n → ℝ)","l":116,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Basic.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell_affineSystemOfPolynomials","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Polynomial","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a finite checked polynomial system, compilation preserves its weak cell.","s":"∀ {n : ℕ} (Γ : List (Causalean.Mathlib.Optimization.AffineSignCellClosure.PolynomialConstraint n)), Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell (Causalean.Mathlib.Optimization.AffineSignCellClosure.affineSystemOfPolynomials Γ) = Causalean.Mathlib.Optimization.AffineSignCellClosure.polynomialWeakCell Γ","l":166,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Polynomial.lean"},{"n":"Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell_subset_closure_strictCell","k":"theorem","m":"Causalean.Mathlib.Optimization.AffineSignCellClosure.Closure","a":"Mathlib/Optimization/AffineSignCellClosure","t":2,"d":"Given a nonempty strict cell, every point of the weak relaxation belongs to the closure of the strict cell.","s":"∀ {n : ℕ} {Γ : Causalean.Mathlib.Optimization.AffineSignCellClosure.AffineSystem n}, (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ).Nonempty → Causalean.Mathlib.Optimization.AffineSignCellClosure.weakCell Γ ⊆ closure (Causalean.Mathlib.Optimization.AffineSignCellClosure.strictCell Γ)","l":77,"f":"Causalean/Mathlib/Optimization/AffineSignCellClosure/Closure.lean"},{"n":"Causalean.Mathlib.Optimization.InSimplex","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"For a real total mass and a three-coordinate allocation, the scaled three-point simplex membership condition holds exactly when every coordinate is nonnegative and the three coordinates sum to the total mass.","s":"ℝ → (Fin 3 → ℝ) → Prop","l":33,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.InTruncSimplex","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"For a total mass and truncation threshold and a three-coordinate allocation, the truncated-simplex membership condition holds exactly when the allocation belongs to the scaled three-point simplex and its second and third coordinates sum to at least the threshold.","s":"ℝ → ℝ → (Fin 3 → ℝ) → Prop","l":20,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.IsAdmissibleSupport","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"Given three linear coefficients and three weights, a scale, a set of selected coordinates, and a real multiplier, the admissible-support condition holds exactly when the selected set is nonempty, its summed squared multiplier gaps divided by their weights equal the squared scale, every selected coefficient is strictly below the multiplier, and every unselected coefficient is at least the multiplier.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → Finset (Fin 3) → ℝ → Prop","l":41,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq","k":"structure","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"A linear inequality in ordinary variables x and a retained objective coordinate t, written a · x + s * t ≤ b.","s":"ℕ → Type","l":46,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq.Holds","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a number of ordinary coordinates, an augmented rational inequality, a rational ordinary-variable vector, and a retained rational objective coordinate, the inequality-satisfaction condition holds exactly when the dot product of the ordinary coefficients and vector plus the objective coefficient times the retained coordinate is no greater than the right-hand side.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n → (Fin n → ℚ) → ℚ → Prop","l":56,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq.cancel","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given two augmented rational inequalities with one more ordinary coordinate, the cancellation inequality has each remaining coefficient, its objective coefficient, and its right-hand side equal to the negative first coefficient of the second inequality times the corresponding quantity of the first, plus the first coefficient of the first inequality times the corresponding quantity of the second.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq (n + 1) → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq (n + 1) → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n","l":77,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq.tail","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given an augmented rational inequality with one more ordinary coordinate, the tail inequality removes its first ordinary coefficient while retaining its objective coefficient and right-hand side.","s":"{n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq (n + 1) → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n","l":66,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.CertifiedIneq","k":"structure","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"An augmented inequality together with rational combination weights over a fixed finite family of original rows. The structure does not require the weights to be nonnegative; nonnegativity, which makes the derived row a valid consequence of the original ones, is tracked separately by the elimination construction. The coefficient identity is stated through a common projection of the original variables; this makes repeated elimination independent of coordinate-reassociation bookkeeping.","s":"{κ : Type u_2} → [Fintype κ] → {N n : ℕ} → (κ → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq N) → ((Fin N → ℚ) → Fin n → ℚ) → Type u_2","l":440,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.FinalCertifiedIneq","k":"structure","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"A fully eliminated inequality with nonnegative provenance on the original rows and with every original-variable coefficient certified to cancel.","s":"{κ : Type u_2} → [Fintype κ] → {N : ℕ} → (κ → Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq N) → Type u_2","l":613,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.Program","k":"structure","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"A rational LP in inequality form: minimize c · x subject to A i · x ≤ b i, with the constraints indexed by an arbitrary type (finiteness is assumed by the algorithms and theorems that need it).","s":"Type u_1 → ℕ → Type u_1","l":260,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.Program.DualFeasible","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a finite set of constraint labels and number of ordinary variables, a finite rational linear program, and rational multipliers for its constraints, the dual-feasibility condition holds exactly when every multiplier is nonnegative and, for every ordinary coordinate, the multiplier-weighted sum of constraint coefficients equals the negative objective coefficient.","s":"{ι : Type u_1} → [Fintype ι] → {n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.Program ι n → (ι → ℚ) → Prop","l":754,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.Program.PrimalFeasible","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a constraint-label set, a number of ordinary variables, a rational linear program, and a rational vector of those variables, the primal-feasibility condition holds exactly when, for every constraint, the dot product of its coefficient row with the vector is no greater than its right-hand side.","s":"{ι : Type u_1} → {n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.Program ι n → (Fin n → ℚ) → Prop","l":273,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.Program.dualObjective","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a finite set of constraint labels and number of ordinary variables, a finite rational linear program, and rational constraint multipliers, the dual objective value is the negative of the multiplier-weighted sum of the constraint right-hand sides.","s":"{ι : Type u_1} → [Fintype ι] → {n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.Program ι n → (ι → ℚ) → ℚ","l":766,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.Program.objective","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a constraint-label set, a number of ordinary variables, a rational linear program, and a rational vector of those variables, the primal objective value is the dot product of the program's objective coefficient vector with that variable vector.","s":"{ι : Type u_1} → {n : ℕ} → Causalean.Mathlib.Optimization.RationalLP.Program ι n → (Fin n → ℚ) → ℚ","l":282,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.eliminateAll","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"For a nonnegative number of ordinary coordinates and a list of augmented rational inequalities in that many coordinates, the complete Fourier--Motzkin elimination output is the original list when there are no ordinary coordinates and otherwise is obtained by one first-coordinate elimination followed by complete elimination of the remaining coordinates.","s":"(n : ℕ) → List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n) → List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq 0)","l":226,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.eliminateAll_iff","k":"theorem","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":2,"d":"Eliminating all ordinary variables preserves exactly the feasible values of the retained objective coordinate.","s":"∀ {n : ℕ} {rows : List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n)} {t : ℚ}, (∃ x, ∀ r ∈ rows, r.Holds x t) ↔ ∀ r ∈ Causalean.Mathlib.Optimization.RationalLP.eliminateAll n rows, r.Holds Fin.elim0 t","l":238,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.eliminateOne","k":"def","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Given a list of augmented rational inequalities with one more ordinary coordinate, the one-step Fourier--Motzkin elimination output contains the tails of all rows whose first coefficient is zero and, for every row with negative first coefficient and every row with positive first coefficient, their cancellation inequality.","s":"{n : ℕ} → List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq (n + 1)) → List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq n)","l":91,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.eliminateOne_iff","k":"theorem","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":2,"d":"One Fourier–Motzkin elimination step preserves exactly the feasible values of the retained objective coordinate.","s":"∀ {n : ℕ} {rows : List (Causalean.Mathlib.Optimization.RationalLP.AugmentedIneq (n + 1))} {x : Fin n → ℚ} {t : ℚ}, (∃ z, ∀ r ∈ rows, r.Holds (Fin.cons z x) t) ↔ ∀ r ∈ Causalean.Mathlib.Optimization.RationalLP.eliminateOne rows, r.Holds x t","l":216,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.exists_rational_optimal_primal_dual","k":"theorem","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":1,"d":"Exact rational LP attainment and strong duality. If a finite rational linear program in inequality form has a feasible rational point and its objective is bounded below on feasible rational points, then it has a rational optimal primal point and rational nonnegative dual multipliers with the same objective value.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [DecidableEq ι] {n : ℕ} (P : Causalean.Mathlib.Optimization.RationalLP.Program ι n), (∃ x, P.PrimalFeasible x) → (∃ l, ∀ (x : Fin n → ℚ), P.PrimalFeasible x → l ≤ P.objective x) → ∃ xStar yStar, P.PrimalFeasible xStar ∧ P.DualFeasible yStar ∧ P.objective xStar = P.dualObjective yStar ∧ ∀ (x : Fin n → ℚ), P.PrimalFeasible x → P.objective xStar ≤ P.objective x","l":776,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.RationalLP.exists_rational_primal_optimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.RationalLP","a":"Mathlib/Optimization","t":2,"d":"If a finite rational linear program has a feasible rational point and its objective is bounded below on feasible rational points, then it has a rational primal optimizer.","s":"∀ {ι : Type u_1} [Fintype ι] [DecidableEq ι] {n : ℕ} (P : Causalean.Mathlib.Optimization.RationalLP.Program ι n), (∃ x, P.PrimalFeasible x) → (∃ l, ∀ (x : Fin n → ℚ), P.PrimalFeasible x → l ≤ P.objective x) → ∃ xStar, P.PrimalFeasible xStar ∧ ∀ (x : Fin n → ℚ), P.PrimalFeasible x → P.objective xStar ≤ P.objective x","l":411,"f":"Causalean/Mathlib/Optimization/RationalLP.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"Given a total mass, three linear coefficients and three weights, a selected coordinate set, and a multiplier, the active-set allocation assigns each selected coordinate the total mass times its multiplier gap divided by its weight and by the sum of those ratios, and assigns every unselected coordinate zero.","s":"ℝ → (Fin 3 → ℝ) → (Fin 3 → ℝ) → Finset (Fin 3) → ℝ → Fin 3 → ℝ","l":58,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_isMinimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationMinimizers","a":"Mathlib/Optimization","t":1,"d":"KKT admissible ⟹ global minimizer (κ > 0). For a positive total mass M, positive coordinate weights β, a positive SOCP scale κ, and a KKT-admissible support/multiplier pair (S, λ): a nonempty index set S with $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i = \\kappa^2$, strict activity $\\lambda>\\alpha_i$ on S, and inactivity $\\lambda\\le\\alpha_j$ off S, the induced active-set point lies in the simplex $\\{t\\ge0:\\sum t_i=M\\}$ and globally minimizes the objective $\\sum\\alpha_it_i+\\kappa\\sqrt{\\sum\\beta_it_i^2}$ over that simplex.","s":"∀ (M : ℝ), 0 < M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 < kappa → ∀ (S : Finset (Fin 3)) (lam : ℝ), Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s","l":30,"f":"Causalean/Mathlib/Optimization/SimplexTruncationMinimizers.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_mem","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set point lies in the simplex Δ_M.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 ≤ M → (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam)","l":45,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_normSq","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"Weighted squared norm of the active-set point: Σ βᵢ tᵢ² = (Mκ)² / D², using the admissibility identity Σ_{i∈S}(λ−αᵢ)²/βᵢ = κ².","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → ∑ i ∈ S, (lam - α i) ^ 2 / β i = kappa ^ 2 → ∑ i, β i * Causalean.Mathlib.Optimization.activeSetPoint M α β S lam i ^ 2 = (M * kappa) ^ 2 / (∑ h ∈ S, (lam - α h) / β h) ^ 2","l":70,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_pos_iff","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set point has support exactly S: on S its coordinate is positive, off S it is 0.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 < M → (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → ∀ (i : Fin 3), 0 < Causalean.Mathlib.Optimization.activeSetPoint M α β S lam i ↔ i ∈ S","l":30,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_strict_min","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":1,"d":"Strict minimality of the active-set point (κ > 0). For a positive total mass M, positive coordinate weights β, a positive SOCP scale κ, and a KKT-admissible support/multiplier pair (S, λ), every other point of the simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ has strictly larger objective value than the induced active-set point.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 < M → (∀ (i : Fin 3), 0 < β i) → 0 < kappa → Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → s ≠ Causalean.Mathlib.Optimization.activeSetPoint M α β S lam → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) < Causalean.Mathlib.Optimization.wsObj α β kappa s","l":156,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSetPoint_value","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":1,"d":"Objective value of the active-set point. For a nonnegative total mass M, positive weights on the support S, a positive SOCP scale κ, a nonempty support S, strict activity $\\lambda > \\alpha_i$ on S, and the admissibility identity $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i=\\kappa^2$, the objective wsObj evaluated at the induced active-set point equals the closed form $M\\lambda$.","s":"∀ (M : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ) (S : Finset (Fin 3)) (lam : ℝ), 0 ≤ M → (∀ i ∈ S, 0 < β i) → 0 < kappa → S.Nonempty → (∀ i ∈ S, α i < lam) → ∑ i ∈ S, (lam - α i) ^ 2 / β i = kappa ^ 2 → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) = M * lam","l":94,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.activeSet_denom_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexKKT","a":"Mathlib/Optimization","t":2,"d":"The active-set denominator D = Σ_{h∈S} (λ−αₕ)/βₕ is strictly positive: S is nonempty and each summand is positive (λ > αₕ on S, βₕ > 0).","s":"∀ (α β : Fin 3 → ℝ) (S : Finset (Fin 3)) (lam : ℝ), (∀ i ∈ S, 0 < β i) → S.Nonempty → (∀ i ∈ S, α i < lam) → 0 < ∑ h ∈ S, (lam - α h) / β h","l":22,"f":"Causalean/Mathlib/Optimization/WeightedSimplexKKT.lean"},{"n":"Causalean.Mathlib.Optimization.continuous_kktThreshold","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold function varies continuously with the multiplier because the weights are fixed real coefficients.","s":"∀ (α β : Fin 3 → ℝ), Continuous (Causalean.Mathlib.Optimization.kktThreshold α β)","l":31,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.eq_truncSegPoint_of_simplex_face","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":2,"d":"A three-coordinate vector with total mass M whose last two coordinates sum to d is the corresponding point on the truncation segment, indexed by its second coordinate.","s":"∀ (M d : ℝ) (x : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M x → x 1 + x 2 = d → x = Causalean.Mathlib.Optimization.truncSegPoint M d (x 1)","l":59,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.exists_admissible","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":1,"d":"Existence of an admissible support/multiplier pair (κ > 0). For positive coordinate weights β and a positive SOCP scale κ, there exists a nonempty index set S and a multiplier λ such that $\\sum_{i\\in S}(\\lambda-\\alpha_i)^2/\\beta_i=\\kappa^2$, with $\\lambda>\\alpha_i$ on S and $\\lambda\\le\\alpha_j$ off S.","s":"∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 < kappa → ∃ S lam, Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam","l":82,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.exposedMinFace","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"Given a total mass and three linear coefficients, the exposed minimum face is the set of three-coordinate allocations that belong to the scaled simplex and have every nonzero coordinate's coefficient no larger than every coefficient.","s":"ℝ → (Fin 3 → ℝ) → Set (Fin 3 → ℝ)","l":71,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.exposedMinFace_isMinimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationMinimizers","a":"Mathlib/Optimization","t":1,"d":"Exposed face ⟹ global minimizer (κ = 0). If t_rel lies in the simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ and every coordinate at which it is nonzero attains the minimum value of α, then t_rel globally minimizes the purely linear objective $\\sum \\alpha_i t_i$ — wsObj at $\\kappa = 0$ — over the whole simplex.","s":"∀ (M : ℝ) (α β t_rel : Fin 3 → ℝ), t_rel ∈ Causalean.Mathlib.Optimization.exposedMinFace M α → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β 0 t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β 0 s","l":275,"f":"Causalean/Mathlib/Optimization/SimplexTruncationMinimizers.lean"},{"n":"Causalean.Mathlib.Optimization.kappa_zero_face","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexFace","a":"Mathlib/Optimization","t":1,"d":"The κ = 0 minimizer set is the exposed α-minimizing face. For a simplex budget M, a linear weight vector α, and a candidate point t, t lies in the simplex of budget M and minimizes the linear objective Σᵢ αᵢtᵢ over that simplex if and only if t belongs to the exposed α-minimizing face of the simplex.","s":"∀ (M : ℝ) (α t : Fin 3 → ℝ), (Causalean.Mathlib.Optimization.InSimplex M t ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → ∑ i, α i * t i ≤ ∑ i, α i * s i) ↔ t ∈ Causalean.Mathlib.Optimization.exposedMinFace M α","l":23,"f":"Causalean/Mathlib/Optimization/WeightedSimplexFace.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold","k":"def","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":1,"d":"Given three linear coefficients and three weights and a real multiplier, the active-set threshold value is the sum over the three coordinates of the squared positive part of the multiplier minus the coefficient, divided by the corresponding weight.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → ℝ","l":21,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold_eq_zero_of_le","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold is zero when the multiplier is no larger than every coefficient.","s":"∀ (α β : Fin 3 → ℝ) (lam : ℝ), (∀ (i : Fin 3), lam ≤ α i) → Causalean.Mathlib.Optimization.kktThreshold α β lam = 0","l":48,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.kktThreshold_nonneg","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"The active-set threshold is nonnegative when every coordinate has a nonnegative weight, because it sums squared multiplier gaps divided by those weights.","s":"∀ (α β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → ∀ (lam : ℝ), 0 ≤ Causalean.Mathlib.Optimization.kktThreshold α β lam","l":39,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.support_sum_eq_kktThreshold","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexExists","a":"Mathlib/Optimization","t":2,"d":"Summing squared multiplier gaps only over coordinates whose coefficients lie below the multiplier gives exactly the active-set threshold.","s":"∀ (α β : Fin 3 → ℝ) (lam : ℝ), ∑ i with α i < lam, (lam - α i) ^ 2 / β i = Causalean.Mathlib.Optimization.kktThreshold α β lam","l":58,"f":"Causalean/Mathlib/Optimization/WeightedSimplexExists.lean"},{"n":"Causalean.Mathlib.Optimization.truncSegPoint","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"For a total mass, threshold, and segment coordinate, the truncation-boundary point is the three-coordinate allocation whose first coordinate is the total mass minus the threshold, whose second coordinate is the segment coordinate, and whose third coordinate is the threshold minus that coordinate.","s":"ℝ → ℝ → ℝ → Fin 3 → ℝ","l":29,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_cs_sqrt","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"For a nonnegative baseline component, the inner product of two three-dimensional boundary vectors is no greater than the product of their Euclidean norms.","s":"∀ {A d s σ : ℝ}, 0 ≤ A → A + s * σ + (d - s) * (d - σ) ≤ √(A + s ^ 2 + (d - s) ^ 2) * √(A + σ ^ 2 + (d - σ) ^ 2)","l":29,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_reduction","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":1,"d":"Boundary reduction onto the truncation face. Fix nonnegative coordinate weights β and a nonnegative SOCP scale κ. If t_rel lies in the three-coordinate simplex $\\{t \\ge 0 : \\sum t_i = M\\}$ and globally minimizes the objective $\\sum \\alpha_i t_i + \\kappa\\sqrt{\\sum \\beta_i t_i^2}$ over that simplex, while t_rel fails the parity cut defining the truncated sub-simplex, since its last two coordinates sum to strictly less than $d$, then for every point t of the truncated sub-simplex $\\{t \\in \\Delta_M : t_y + t_z \\ge d\\}$, there is some $\\sigma$ between $0$ and $d$ such that the boundary point $(M-d,\\sigma,d-\\sigma)$ of the truncation face attains an objective value no larger than the objective at t.","s":"∀ (M d kappa : ℝ) (α β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → 0 ≤ kappa → ∀ (t_rel : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M t_rel → (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) → t_rel 1 + t_rel 2 < d → ∀ (t : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d t → ∃ σ, 0 ≤ σ ∧ σ ≤ d ∧ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa t","l":109,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.truncSeg_selector_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":1,"d":"The selector minimizes the boundary slice. Suppose the truncation width $d$ is positive, the first-coordinate weight $\\beta_0$ is nonnegative while the other two weights both equal 1, and the SOCP scale $\\kappa$ is nonnegative. Then for every offset $\\sigma$ between $0$ and $d$, the objective wsObj evaluated at the boundary point truncSegPoint M d applied to the selector truncSelector M d α β κ is at most its value at the boundary point for $\\sigma$.","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ), 0 < d → 0 ≤ β 0 → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → ∀ (σ : ℝ), 0 ≤ σ → σ ≤ d → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ)","l":295,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector","k":"def","m":"Causalean.Mathlib.Optimization.SimplexTruncationDefs","a":"Mathlib/Optimization","t":1,"d":"Given a total mass and threshold, three linear coefficients and three weights, and a scale, the boundary-segment selector first sets the difference of the second and third coefficients to $δ$ as its first intermediate quantity and the weighted squared first coordinate to $A$ as its second intermediate quantity; it then returns zero when $δ$ is at least $κ d/\\sqrt{A+d^2}$, returns $d$ when $δ$ is at most its negative, and otherwise returns $(d-δ\\sqrt{(A+d^2/2)/(\\kappa^2-δ^2/2)})/2$.","s":"ℝ → ℝ → (Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → ℝ","l":37,"f":"Causalean/Mathlib/Optimization/SimplexTruncationDefs.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_interior_den_pos","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"Under the selector's strict interior guard inequalities, a positive interval width, and nonnegative baseline and scale, the squared-scale denominator minus half the squared offset is positive.","s":"∀ {A d kappa δ : ℝ}, 0 < d → 0 ≤ A → 0 ≤ kappa → ¬kappa * d / √(A + d ^ 2) ≤ δ → ¬δ ≤ -(kappa * d / √(A + d ^ 2)) → 0 < kappa ^ 2 - δ ^ 2 / 2","l":95,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_interior_sq_bound","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"If the truncation offset lies strictly between the two interior guard bounds, then its squared size times the radicand is strictly smaller than the squared truncation scale times the squared interval width.","s":"∀ {A d kappa δ : ℝ}, 0 < d → 0 ≤ A → 0 ≤ kappa → ¬kappa * d / √(A + d ^ 2) ≤ δ → ¬δ ≤ -(kappa * d / √(A + d ^ 2)) → δ ^ 2 * (A + d ^ 2) < kappa ^ 2 * d ^ 2","l":67,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.truncSelector_mem","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":2,"d":"The selector lands in [0,d]. The endpoint/interior selector truncSelector satisfies 0 ≤ s⋆ ≤ d: the two endpoint branches give 0 and d directly, and in the interior branch the guard failures |δ| < κ d / √(A + d²) force δ²(A + d²) < κ² d², whence |s⋆ − d/2| < d/2.","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa : ℝ), 0 < d → 0 ≤ β 0 → 0 ≤ kappa → 0 ≤ Causalean.Mathlib.Optimization.truncSelector M d α β kappa ∧ Causalean.Mathlib.Optimization.truncSelector M d α β kappa ≤ d","l":276,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Optimization.trunc_from_minimizer","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncation","a":"Mathlib/Optimization","t":1,"d":"Truncation dichotomy from a relaxed minimizer. Fix a truncation threshold d at most the total simplex mass M, a linear weighting α, coordinate weights β with its zeroth entry nonnegative and its remaining two entries fixed equal to 1, and a nonnegative regularization parameter κ. Given a point t_rel of the simplex Δ_M that globally minimizes the weighted objective wsObj over Δ_M, then the constrained problem over the truncated simplex K_d = {t ∈ Δ_M : t₁ + t₂ ≥ d} splits into two cases: if t_rel already satisfies the truncation constraint, it remains a global minimizer over K_d; otherwise, the face selector point truncSegPoint M d sStar is feasible for K_d and is a global minimizer over K_d. This is the κ-agnostic core shared by the κ > 0 and κ = 0 branches of the headline lemma.","s":"∀ (M d : ℝ), d ≤ M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), 0 ≤ β 0 → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → ∀ (t_rel : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M t_rel → (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) → (Causalean.Mathlib.Optimization.InTruncSimplex M d t_rel → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ t_rel 1 + t_rel 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s)","l":31,"f":"Causalean/Mathlib/Optimization/SimplexTruncation.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_cs_simplex_strict","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexCS","a":"Mathlib/Optimization","t":1,"d":"Strict weighted Cauchy–Schwarz on the simplex slice. For a nonzero total mass M and positive coordinate weights β, if the coordinates of $s$ sum to $M$, the coordinates of $t$ sum to $M$, and $s$ and $t$ are distinct vectors, then the weighted inner product $\\sum \\beta_i s_i t_i$ is strictly less than the product of the weighted Euclidean norms $\\sqrt{\\sum \\beta_i s_i^2} \\cdot \\sqrt{\\sum \\beta_i t_i^2}$.","s":"∀ (M : ℝ), M ≠ 0 → ∀ (β s t : Fin 3 → ℝ), (∀ (i : Fin 3), 0 < β i) → ∑ i, s i = M → ∑ i, t i = M → s ≠ t → ∑ i, β i * (s i * t i) < √(∑ i, β i * s i ^ 2) * √(∑ i, β i * t i ^ 2)","l":39,"f":"Causalean/Mathlib/Optimization/WeightedSimplexCS.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_cs_sq","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexCS","a":"Mathlib/Optimization","t":1,"d":"Weighted Cauchy–Schwarz (squared form). For finitely many coordinates weighted by nonnegative masses β, the square of the weighted inner product $\\sum \\beta_i s_i t_i$ of two vectors $s$ and $t$ is at most the product of their weighted sums of squares $(\\sum \\beta_i s_i^2)(\\sum \\beta_i t_i^2)$.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (β s t : ι → ℝ), (∀ (i : ι), 0 ≤ β i) → (∑ i, β i * (s i * t i)) ^ 2 ≤ (∑ i, β i * s i ^ 2) * ∑ i, β i * t i ^ 2","l":26,"f":"Causalean/Mathlib/Optimization/WeightedSimplexCS.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_simplex_active_set","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexActiveSet","a":"Mathlib/Optimization","t":1,"d":"Weighted-simplex active-set solution. Fix a positive total simplex mass M, linear weights α, coordinate weights β that are all strictly positive, and a nonnegative regularization parameter κ, and consider minimizing the second-order-cone objective Σ αᵢtᵢ + κ·√(Σβᵢtᵢ²) over the three-point simplex Δ_M of total mass M. Then the following two facts hold: whenever κ is strictly positive, there is a *unique* admissible support/multiplier pair (S, λ), its induced active-set point lies in Δ_M and is the *unique* global minimizer of the objective, with optimal value the closed form M · λ; and whenever κ equals zero, a point of Δ_M minimizes the objective exactly when it lies on the exposed α-minimizing face. This encodes the displayed KKT coordinate formula, the uniqueness of the admissible support, the value formula, and the κ = 0 face clause.","s":"∀ (M : ℝ), 0 < M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → 0 ≤ kappa → (0 < kappa → ∃! p, Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa p.1 p.2 ∧ Causalean.Mathlib.Optimization.InSimplex M (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) ∧ (∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → s ≠ Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2 → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) < Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β p.1 p.2) = M * p.2) ∧ (kappa = 0 → ∀ (t : Fin 3 → ℝ), (Causalean.Mathlib.Optimization.InSimplex M t ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InSimplex M s → Causalean.Mathlib.Optimization.wsObj α β kappa t ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ↔ t ∈ Causalean.Mathlib.Optimization.exposedMinFace M α)","l":29,"f":"Causalean/Mathlib/Optimization/SimplexActiveSet.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_simplex_truncation","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncation","a":"Mathlib/Optimization","t":1,"d":"Weighted-simplex truncation. In the notation of weighted_simplex_active_set, fix a positive total simplex mass M, a truncation threshold d at most M, a linear weighting α, coordinate weights β that are everywhere strictly positive with its last two entries fixed equal to 1, and a nonnegative regularization parameter κ. Then the global minimizer of the weighted-simplex objective over the truncated simplex K_d = {t ∈ Δ_M : t₁ + t₂ ≥ d} is given, case by case on the sign of κ, by the active-set/face-selector construction below.","s":"∀ (M d : ℝ), 0 < M → d ≤ M → ∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 < β i) → β 1 = 1 → β 2 = 1 → 0 ≤ kappa → (0 < kappa → ∀ (S : Finset (Fin 3)) (lam : ℝ), Causalean.Mathlib.Optimization.IsAdmissibleSupport α β kappa S lam → (Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.activeSetPoint M α β S lam) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ Causalean.Mathlib.Optimization.activeSetPoint M α β S lam 1 + Causalean.Mathlib.Optimization.activeSetPoint M α β S lam 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s)) ∧ (kappa = 0 → ∀ t_rel ∈ Causalean.Mathlib.Optimization.exposedMinFace M α, (Causalean.Mathlib.Optimization.InTruncSimplex M d t_rel → ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa t_rel ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s) ∧ (¬d ≤ t_rel 1 + t_rel 2 → Causalean.Mathlib.Optimization.InTruncSimplex M d (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ∧ ∀ (s : Fin 3 → ℝ), Causalean.Mathlib.Optimization.InTruncSimplex M d s → Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d (Causalean.Mathlib.Optimization.truncSelector M d α β kappa)) ≤ Causalean.Mathlib.Optimization.wsObj α β kappa s))","l":95,"f":"Causalean/Mathlib/Optimization/SimplexTruncation.lean"},{"n":"Causalean.Mathlib.Optimization.weighted_sqrt_segment_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":2,"d":"The weighted Euclidean root-mean-square of a convex combination is no larger than the same convex combination of the two weighted root-mean-squares.","s":"∀ (β : Fin 3 → ℝ), (∀ (i : Fin 3), 0 ≤ β i) → ∀ (u v : Fin 3 → ℝ) (θ : ℝ), 0 ≤ θ → θ ≤ 1 → √(∑ i, β i * ((1 - θ) * u i + θ * v i) ^ 2) ≤ (1 - θ) * √(∑ i, β i * u i ^ 2) + θ * √(∑ i, β i * v i ^ 2)","l":33,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj","k":"def","m":"Causalean.Mathlib.Optimization.SimplexActiveSetDefs","a":"Mathlib/Optimization","t":1,"d":"Given three linear coefficients and three weights, a real scale, and a three-coordinate allocation, the weighted-simplex objective value is the linear weighted sum of the allocation plus the scale times the square root of the weighted sum of squared coordinates.","s":"(Fin 3 → ℝ) → (Fin 3 → ℝ) → ℝ → (Fin 3 → ℝ) → ℝ","l":23,"f":"Causalean/Mathlib/Optimization/SimplexActiveSetDefs.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_kappa_zero","k":"theorem","m":"Causalean.Mathlib.Optimization.WeightedSimplexFace","a":"Mathlib/Optimization","t":2,"d":"At κ = 0 the weighted-simplex objective collapses to the linear form Σ αᵢ tᵢ.","s":"∀ (α β t : Fin 3 → ℝ), Causalean.Mathlib.Optimization.wsObj α β 0 t = ∑ i, α i * t i","l":17,"f":"Causalean/Mathlib/Optimization/WeightedSimplexFace.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_segment_le","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationConvex","a":"Mathlib/Optimization","t":1,"d":"Two-point convexity of the weighted-simplex objective. For nonnegative coordinate weights β and a nonnegative SOCP scale κ, if θ lies between 0 and 1 then the objective $\\sum \\alpha_i t_i + \\kappa\\sqrt{\\sum \\beta_i t_i^2}$ evaluated at the convex combination $(1-\\theta)u + \\theta v$ of two points $u,v$ is at most the same convex combination of the objective values at $u$ and at $v$.","s":"∀ (α β : Fin 3 → ℝ) (kappa : ℝ), (∀ (i : Fin 3), 0 ≤ β i) → 0 ≤ kappa → ∀ (u v : Fin 3 → ℝ) (θ : ℝ), 0 ≤ θ → θ ≤ 1 → (Causalean.Mathlib.Optimization.wsObj α β kappa fun i => (1 - θ) * u i + θ * v i) ≤ (1 - θ) * Causalean.Mathlib.Optimization.wsObj α β kappa u + θ * Causalean.Mathlib.Optimization.wsObj α β kappa v","l":74,"f":"Causalean/Mathlib/Optimization/SimplexTruncationConvex.lean"},{"n":"Causalean.Mathlib.Optimization.wsObj_truncSeg_eq","k":"theorem","m":"Causalean.Mathlib.Optimization.SimplexTruncationSlice","a":"Mathlib/Optimization","t":1,"d":"Objective on the truncation face. For a total budget M and a truncation level d, a linear weight vector α and a quadratic weight vector β, and a curvature coefficient kappa and a face coordinate σ, evaluating the weighted-simplex objective at the truncation-face point (M−d, σ, d−σ) gives the explicit one-dimensional form α₀(M−d) + α₁σ + α₂(d−σ) + κ√(β₀(M−d)² + β₁σ² + β₂(d−σ)²).","s":"∀ (M d : ℝ) (α β : Fin 3 → ℝ) (kappa σ : ℝ), Causalean.Mathlib.Optimization.wsObj α β kappa (Causalean.Mathlib.Optimization.truncSegPoint M d σ) = α 0 * (M - d) + α 1 * σ + α 2 * (d - σ) + kappa * √(β 0 * (M - d) ^ 2 + β 1 * σ ^ 2 + β 2 * (d - σ) ^ 2)","l":261,"f":"Causalean/Mathlib/Optimization/SimplexTruncationSlice.lean"},{"n":"Causalean.Mathlib.Probability.BoundedTestFactorization","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, two measurable value spaces, a measure, and two random elements, the bounded-test factorization condition holds exactly when, for every pair of measurable bounded real-valued test functions on the two value spaces, the integral of their product after applying the two random elements equals the product of their separate integrals.","s":"{Ω : Type u_1} → {S : Type u_2} → {T : Type u_3} → [inst : MeasurableSpace Ω] → [MeasurableSpace S] → [MeasurableSpace T] → MeasureTheory.Measure Ω → (Ω → S) → (Ω → T) → Prop","l":99,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A bounded reward certificate gives entrywise rational enclosures and an exact rational uniform absolute bound on all their endpoints.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → Type u_1","l":23,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate.bound_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The endpoint check in a bounded reward certificate implies the semantic absolute bound for every true reward coordinate.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {reward : ι → ℝ} (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (i : ι), |reward i| ≤ ↑r.bound","l":167,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedChunk","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A certified chunk records a bounded partial interval sum of a finite interval family, with the selected maximum chunk size, its indices, its reported bound, a size proof, and a local exact-sum refinement proof.","s":"{ι : Type u_1} → [DecidableEq ι] → (ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → Type u_1","l":280,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A certified real kernel packages a stochastic matrix together with an entrywise rational interval table that contains it.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → Type u_1","l":69,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.CentralCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Central","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"Finite certificate data for central evaluation: a series degree and a certified enclosure of 1 / sqrt (2π).","s":"Type","l":17,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Central.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.ExpCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A finite exponential certificate is raw caller-supplied range-reduction, Taylor-degree, and rational-enclosure data; expCheck validates it.","s":"ℚ → Type","l":119,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.ExpCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A finite rational exponential certificate whose executable check succeeds contains the real exponential at its certified argument.","s":"∀ {q : ℚ} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.ExpCertificate q), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expCheck q c = true → c.enclosure.Contains (Real.exp ↑q)","l":140,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A fine endpoint certificate packages its data with successful evaluation of the public executable high-precision checker; range and width are checked, not separately trusted as caller-provided proof fields.","s":"ℚ → Type","l":77,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate.sound_range_and_width","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"A fine certificate simultaneously gives CDF containment, the supported endpoint range, and the requested 10⁻¹² rational width guarantee.","s":"∀ {q : ℚ} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate q), c.enclosure.Contains (Causalean.Mathlib.stdNormalCDF ↑q) ∧ |q| ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.supportedEndpointBound ∧ c.enclosure.width ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.targetWidth","l":98,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate.toNormalCDFCertificate","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"Forgetting the fine limits yields an ordinary checked CDF certificate.","s":"{q : ℚ} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate q → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate q","l":88,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.MethodCertificate","k":"inductive","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The top-level certificate method is either a central-series certificate or a tail certificate tied to the endpoint whose exponential it encloses.","s":"ℚ → Type","l":19,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A proof-producing endpoint certificate packages caller data with the successful result of the executable checker.","s":"ℚ → Type","l":62,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Every proof-producing endpoint certificate soundly encloses the exact CDF.","s":"∀ {q : ℚ} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate q), c.enclosure.Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":72,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A normalization certificate is raw finite schedule and caller-enclosure data; normalizationCheck validates all side conditions.","s":"Type","l":39,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A normalization certificate whose executable check succeeds contains the standard-normal density scale.","s":"∀ (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationCertificate), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationCheck c = true → c.enclosure.Contains (1 / √(2 * Real.pi))","l":78,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationSchedule","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A normalization schedule is the finite precision datum used for the Machin-π and Newton square-root computations.","s":"Type","l":17,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A tail certificate combines rational normalization and exponential certificates for the density at the absolute endpoint.","s":"ℚ → Type","l":232,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Central","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The central checker validates normalization, the cutoff, the exact alternating-tail condition, and refinement into the caller-supplied interval.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.CentralCertificate → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Bool","l":37,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Central.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCheck_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Central","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"When the central-series checker accepts the supplied certificate and interval, that interval contains the standard-normal CDF at the rational endpoint.","s":"∀ {q : ℚ} {c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.CentralCertificate} {reported : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCheck q c reported = true → reported.Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":46,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Central.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCutoff","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The central-series/tail boundary is the exact rational number eight.","s":"ℚ","l":24,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Central","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The signed central evaluator uses CDF symmetry when the endpoint is negative.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.CentralCertificate → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":31,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Central.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralMagnitudeInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Central","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The positive-magnitude central evaluator returns 1/2 plus the product of the normalization enclosure and alternating integral enclosure.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.CentralCertificate → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":25,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Central.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.central_or_tail","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Every endpoint lies either in the central regime or strictly in the tail regime; this is the exhaustive case split used by the top-level checker.","s":"∀ (q : ℚ), |q| ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCutoff ∨ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCutoff < |q|","l":47,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expCertificateInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The exact interval computed from the caller's split and Taylor degree.","s":"{q : ℚ} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.ExpCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":129,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The executable exponential checker validates positive range reduction, a unit reduced argument, and exact refinement into the reported interval.","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.ExpCertificate q → Bool","l":133,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The finite Taylor interval for exp z, using Causalean's exact rational partial sum and explicit remainder radius.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":18,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"On the unit interval, the finite rational Taylor interval encloses the real exponential.","s":"∀ (z : ℚ), |z| ≤ 1 → ∀ (degree : ℕ), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval z degree).Contains (Real.exp ↑z)","l":24,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval_width","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The width of the raw reduced exponential interval is exactly twice its explicit rational Taylor remainder radius.","s":"∀ (z : ℚ) (degree : ℕ), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval z degree).width = 2 * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.Transcendental.expRemainder z degree","l":54,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval_width_le_geometric","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"From degree one onward, the unit-range exponential interval has width at most 3 / 2^degree, explicitly witnessing logarithmic degree in inverse target width.","s":"∀ (z : ℚ), |z| ≤ 1 → ∀ (degree : ℕ), 1 ≤ degree → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expTaylorInterval z degree).width ≤ 3 / 2 ^ degree","l":78,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.fineNormalCDFCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The executable high-precision checker accepts exactly when the analytic checker accepts and the caller's endpoint and reported width satisfy the exact rational limits |q| ≤ 193/5 and width ≤ 10⁻¹².","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.MethodCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Bool","l":43,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.fineNormalCDFCheck_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Soundness of the executable high-precision checker: an accepted cell contains the exact CDF value and satisfies both advertised rational limits.","s":"∀ {q : ℚ} {method : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.MethodCertificate q} {reported : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.fineNormalCDFCheck q method reported = true → reported.Contains (Causalean.Mathlib.stdNormalCDF ↑q) ∧ |q| ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.supportedEndpointBound ∧ reported.width ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.targetWidth","l":51,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.fineNormalCDFDifference_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Subtracting two fine checked endpoint cells contains their exact CDF difference while retaining executable validation of each endpoint's range and 10⁻¹² width bound.","s":"∀ {a b : ℚ} (ca : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate a) (cb : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.FineNormalCDFCertificate b), (cb.enclosure.sub ca.enclosure).Contains (Causalean.Mathlib.stdNormalCDF ↑b - Causalean.Mathlib.stdNormalCDF ↑a)","l":115,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The alternating enclosure is the rational interval whose endpoints are two consecutive Gaussian-integral partial sums.","s":"ℚ → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":109,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"At a rational endpoint that is nonnegative, with a partial-sum index satisfying the exact decreasing-tail condition, two consecutive rational Gaussian-series sums enclose the unnormalised Gaussian integral.","s":"∀ (x : ℚ), 0 ≤ x → ∀ (n : ℕ), x ^ 2 ≤ 2 * ↑(n + 2) → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralInterval x n).Contains (∫ (t : ℝ) in 0..↑x, Real.exp (-t ^ 2 / 2))","l":203,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The magnitude of the kth integrated Gaussian-series term at x, namely x to the power 2k+1 divided by 2^k k! (2k+1). It is nonnegative for nonnegative x, which is how it is used; for negative x the same formula is negative.","s":"ℝ → ℕ → ℝ","l":23,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude_antitone_from","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"If x² ≤ 2(n+2), the coefficient magnitudes decrease at every index after the partial sum boundary n.","s":"∀ (x : ℝ), 0 ≤ x → ∀ (n : ℕ), x ^ 2 ≤ 2 * ↑(n + 2) → ∀ (k : ℕ), n + 1 ≤ k → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude x (k + 1) ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude x k","l":125,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"For a nonnegative endpoint, all unsigned Gaussian-series coefficients are nonnegative.","s":"∀ (x : ℝ), 0 ≤ x → ∀ (k : ℕ), 0 ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude x k","l":116,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralPartial","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The exact rational Gaussian-integral partial sum through degree index n.","s":"ℚ → ℕ → ℚ","l":99,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The first omitted coefficient magnitude after partial sum index n.","s":"ℚ → ℕ → ℚ","l":103,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder_geometric","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Consequently, adding j terms beyond index 64 reduces the central-series remainder by at least the rational factor 2^j; this records logarithmic precision scaling for the checker.","s":"∀ (x : ℚ), 0 ≤ x → x ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCutoff → ∀ (j : ℕ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder x (64 + j) ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder x 64 / 2 ^ j","l":335,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder_geometric_step","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"On the full central range 0 ≤ x ≤ 8, once index 64 is reached each next remainder is at most half the preceding remainder.","s":"∀ (x : ℚ), 0 ≤ x → x ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.centralCutoff → ∀ (n : ℕ), 64 ≤ n → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder x (n + 1) ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralRemainder x n / 2","l":290,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralTerm","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The exact rational kth signed term in the integrated Gaussian power series at the rational endpoint x.","s":"ℚ → ℕ → ℚ","l":93,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegral_hasSum","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The signed integrated Gaussian coefficients sum to the exact Gaussian integral on every finite nonnegative interval.","s":"∀ (x : ℝ), 0 ≤ x → HasSum (fun k => (-1) ^ k * Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude x k) (∫ (t : ℝ) in 0..x, Real.exp (-t ^ 2 / 2))","l":165,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandMajorant_summable","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The endpoint majorants for the Gaussian integrand's power series are summable for every fixed nonnegative endpoint.","s":"∀ (x : ℝ), 0 ≤ x → Summable fun k => x ^ (2 * k) / (2 ^ k * ↑k.factorial)","l":81,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The kth function in the power series for exp (-t²/2).","s":"ℕ → ℝ → ℝ","l":30,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm_integral","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Integrating one Gaussian power-series term from zero to a nonnegative endpoint gives the corresponding signed integrated-series coefficient.","s":"∀ (x : ℝ), 0 ≤ x → ∀ (k : ℕ), ∫ (t : ℝ) in 0..x, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm k t = (-1) ^ k * Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegralMagnitude x k","l":49,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm_norm_le","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"On [0,x], the norm of one Gaussian-series function is bounded by the same unsigned monomial evaluated at the nonnegative right endpoint.","s":"∀ (x : ℝ), 0 ≤ x → ∀ (k : ℕ) {t : ℝ}, t ∈ Set.Icc 0 x → ‖Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm k t‖ ≤ x ^ (2 * k) / (2 ^ k * ↑k.factorial)","l":66,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrand_hasSum","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.PowerSeries","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"At every real argument, the Gaussian integrand is the sum of its classical everywhere-convergent power series.","s":"∀ (t : ℝ), HasSum (fun k => Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.gaussianIntegrandTerm k t) (Real.exp (-t ^ 2 / 2))","l":35,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/PowerSeries.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.hasDerivAt_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The derivative of the standard-normal density is -x · φ(x).","s":"∀ (x : ℝ), HasDerivAt Causalean.Mathlib.stdNormalPDF (-x * Causalean.Mathlib.stdNormalPDF x) x","l":26,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsRatio_bounds","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The classical two-sided Mills inequality for the standard-normal upper tail at a positive real argument.","s":"∀ (x : ℝ), 0 < x → x / (x ^ 2 + 1) * Causalean.Mathlib.stdNormalPDF x ≤ 1 - Causalean.Mathlib.stdNormalCDF x ∧ 1 - Causalean.Mathlib.stdNormalCDF x ≤ Causalean.Mathlib.stdNormalPDF x / x","l":223,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsRatio_lower","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The classical lower Mills inequality bounds the standard-normal upper tail below by xφ(x)/(x²+1) at every positive argument.","s":"∀ (x : ℝ), 0 < x → x / (x ^ 2 + 1) * Causalean.Mathlib.stdNormalPDF x ≤ 1 - Causalean.Mathlib.stdNormalCDF x","l":97,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsRatio_upper","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The classical upper Mills inequality bounds the standard-normal upper tail by φ(x)/x at every positive argument.","s":"∀ (x : ℝ), 0 < x → 1 - Causalean.Mathlib.stdNormalCDF x ≤ Causalean.Mathlib.stdNormalPDF x / x","l":54,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsTailInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"A rational Mills tail enclosure is the hull of the lower-factor product and upper-factor product with the certified density interval, the factors being x/(x²+1) and 1/x. It encloses the upper tail only for positive x, as required by its soundness theorem; at x = 0 both factors are zero by the division convention.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":257,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsTailInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"A density enclosure and positivity of x imply that the rational Mills interval contains the exact standard-normal upper-tail probability.","s":"∀ (x : ℚ), 0 < x → ∀ (density : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval), density.Contains (Causalean.Mathlib.stdNormalPDF ↑x) → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.millsTailInterval x density).Contains (1 - Causalean.Mathlib.stdNormalCDF ↑x)","l":265,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalCDFCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The top-level executable checker dispatches to the selected method and validates the caller-supplied rational enclosure.","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.MethodCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Bool","l":27,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalCDFCheck_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"When the selected top-level certificate check succeeds, the caller-supplied rational interval contains the exact standard-normal CDF at its endpoint.","s":"∀ {q : ℚ} {method : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.MethodCertificate q} {reported : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalCDFCheck q method reported = true → reported.Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":35,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalCDFDifference_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Checker","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Subtracting two checked endpoint enclosures contains the corresponding CDF difference, the form used by Gaussian threshold-cell transition probabilities.","s":"∀ {a b : ℚ} (ca : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate a) (cb : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalCDFCertificate b), (cb.enclosure.sub ca.enclosure).Contains (Causalean.Mathlib.stdNormalCDF ↑b - Causalean.Mathlib.stdNormalCDF ↑a)","l":106,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Checker.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The executable normalization checker validates every guarded interval operation and the exact rational refinement into the caller's enclosure.","s":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationCertificate → Bool","l":47,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The raw normalization interval is the reciprocal square-root enclosure when its exact rational guards succeed, and a harmless fallback otherwise.","s":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationSchedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":23,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"Every raw normalization interval contains the real number 1 / sqrt (2π).","s":"∀ (s : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationSchedule), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationScheduleCheck s = true → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationInterval s).Contains (1 / √(2 * Real.pi))","l":54,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationScheduleCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Normalization","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The executable schedule checker validates that the computed 2π interval is nonnegative and that its square-root enclosure excludes zero.","s":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.NormalizationSchedule → Bool","l":31,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Normalization.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.one_sub_stdNormalCDF_eq_integral_Ioi","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The standard-normal upper-tail probability equals the integral of its density over the ray strictly above the endpoint.","s":"∀ (x : ℝ), 1 - Causalean.Mathlib.stdNormalCDF x = ∫ (t : ℝ) in Set.Ioi x, Causalean.Mathlib.stdNormalPDF t","l":47,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.reflectInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"Reflecting an interval I across one half produces the interval 1 - I.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":27,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.reflectInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"If an interval contains x, its reflection contains 1 - x.","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains x → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.reflectInterval I).Contains (1 - x)","l":31,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.reflectInterval_stdNormalCDF_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"An enclosure of Φ(x) reflects to an enclosure of Φ(-x).","s":"∀ {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ℝ}, I.Contains (Causalean.Mathlib.stdNormalCDF x) → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.reflectInterval I).Contains (Causalean.Mathlib.stdNormalCDF (-x))","l":40,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.stdNormalCDF_eq_half_add_scale_mul_integral","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"At a rational endpoint with nonnegative value, the standard-normal CDF is one half plus the density normalization constant times the unnormalised Gaussian integral from zero to that endpoint.","s":"∀ (q : ℚ), 0 ≤ q → Causalean.Mathlib.stdNormalCDF ↑q = 1 / 2 + 1 / √(2 * Real.pi) * ∫ (t : ℝ) in 0..↑q, Real.exp (-t ^ 2 / 2)","l":53,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.stdNormalPDF_pos","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The standard-normal density is strictly positive at every real argument.","s":"∀ (x : ℝ), 0 < Causalean.Mathlib.stdNormalPDF x","l":20,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.supportedEndpointBound","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The reusable checker is required to support endpoints of absolute value at most 193 / 5.","s":"ℚ","l":20,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The tail checker validates normalization and exponential subcertificates, the strict cutoff, and exact refinement into the caller-supplied interval.","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Bool","l":303,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailCheck_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"When the Mills-tail checker accepts the supplied certificate and interval, that interval contains the standard-normal CDF at the rational endpoint.","s":"∀ {q : ℚ} {c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate q} {reported : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailCheck q c reported = true → reported.Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":312,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailDensityInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The certified rational interval for the density φ(|q|).","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":240,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailDensityInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The product of the certified normalization and exponential intervals contains the exact standard-normal density at the absolute endpoint.","s":"∀ (q : ℚ) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate q), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.normalizationCheck c.normalization = true → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.expCheck (-|q| ^ 2 / 2) c.exponential = true → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailDensityInterval q c).Contains (Causalean.Mathlib.stdNormalPDF ↑|q|)","l":244,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.tailInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Tail","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The signed tail evaluator returns the upper-tail interval directly for a negative endpoint and reflects it for a nonnegative endpoint.","s":"(q : ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.TailCertificate q → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":297,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Tail.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.targetWidth","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Basic","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":1,"d":"The requested maximum width of a high-precision caller cell is exactly 10⁻¹², represented as a rational rather than a floating-point number.","s":"ℚ","l":16,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Basic.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.two_pow_le_factorial_succ","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedNormalCDFEnclosure.Exponential","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure","t":2,"d":"The factorial (n+1)! dominates 2^n; this is the discrete estimate used to turn Taylor factorial decay into a geometric precision bound.","s":"∀ (n : ℕ), 2 ^ n ≤ (n + 1).factorial","l":64,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/CertifiedNormalCDFEnclosure/Exponential.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedDotCertificate","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given left interval-vector factors, right interval-vector factors, a chunk-size bound, and an output interval, a chunked dot-product certificate is the corresponding chunked certificate for outward interval products.","s":"{ι : Type u_1} → [Fintype ι] → [DecidableEq ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Type u_1","l":364,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedDotCertificate.contains","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given a chunked dot-product certificate, left-vector containment, and right-vector containment, the reported interval contains the corresponding real dot product.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {left right : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {chunkSize : ℕ} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedDotCertificate left right chunkSize output) {x y : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector left x → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector right y → output.Contains (∑ i, x i * y i)","l":376,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedDotCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a chunked dot-product certificate is supplied, the library interval dot product refines its reported coordinate interval.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {left right : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {chunkSize : ℕ} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedDotCertificate left right chunkSize output), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalDot left right).Subinterval output","l":369,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A chunked finite-iterate certificate stores a recurrence trace for an interval transition matrix from an exact rational initial distribution, using a number of approximation steps, a positive chunk-size bound and its positivity proof, the interval rows, initial coordinate checks, and bounded coordinate recurrence checks.","s":"{ι : Type u_1} → [inst : Fintype ι] → [DecidableEq ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι → Type u_1","l":324,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate.adapter_rows","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a chunked finite-iterate certificate is supplied, its adapter has exactly the same table, terminal row, and successor row.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate K p0), c.toFiniteIterateCertificate.table = c.table ∧ c.toFiniteIterateCertificate.terminal = c.table (Fin.last c.steps).castSucc ∧ c.toFiniteIterateCertificate.next = c.table (Fin.last (c.steps + 1))","l":364,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate.table_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given a chunked finite-iterate certificate, a real transition matrix enclosed by its interval matrix, and a selected trace row, that row contains the corresponding exact Markov iterate.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate K p0), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → ∀ (k : Fin (c.steps + 2)), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector (c.table k) (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate P p0.toReal ↑k)","l":354,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate.toFiniteIterateCertificate","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given a chunked finite-iterate certificate, the corresponding established finite-iterate certificate is the same trace with each opaque coordinate proof composed into its recurrence field.","s":"{ι : Type u_1} → [inst : Fintype ι] → [inst_1 : DecidableEq ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate K p0 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0","l":341,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedSumCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A chunked-sum certificate assembles a finite interval family using a selected chunk-size bound into a reported output interval, through bounded chunk certificates, their pairwise disjointness, their exhaustive coverage, and a checked binary assembly.","s":"{ι : Type u_1} → [Fintype ι] → [DecidableEq ι] → (ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Type u_1","l":292,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedSumCertificate.contains","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given a chunked-sum certificate and coordinatewise interval containment of real summands, the reported interval contains their full real sum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {terms : ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {chunkSize : ℕ} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedSumCertificate terms chunkSize output) {values : ι → ℝ}, (∀ (i : ι), (terms i).Contains (values i)) → output.Contains (∑ i, values i)","l":356,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedSumCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"When a chunked-sum certificate is supplied, the complete interval sum refines its reported output interval.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {terms : ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {chunkSize : ℕ} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedSumCertificate terms chunkSize output), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalSum terms).Subinterval output","l":335,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"An interval matrix contains a real matrix when it contains every entry.","s":"{ι : Type u_1} → {κ : Type u_2} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Matrix ι κ ℝ → Prop","l":71,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix.mono","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Coordinatewise refinement preserves containment of a real matrix.","s":"∀ {ι : Type u_1} {κ : Type u_2} {I J : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ} {A : Matrix ι κ ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.MatrixSubinterval I J → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix I A → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix J A","l":169,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"An interval vector contains a real vector when it contains every coordinate.","s":"{ι : Type u_1} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → (ι → ℝ) → Prop","l":67,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector.mono","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Coordinatewise refinement preserves containment of a real vector.","s":"∀ {ι : Type u_1} {I J : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {x : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.VectorSubinterval I J → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector I x → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector J x","l":162,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A square matrix contracts probability vectors in ℓ¹ by a supplied coefficient: the one-step images of any two probability vectors are at most the coefficient times their ℓ¹ distance apart. Row-stochasticity of the matrix is not part of this predicate and is assumed separately.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → ℝ → Prop","l":54,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CoordinateRecurrenceCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A coordinate recurrence certificate records, for an interval transition matrix, a current interval vector, a next interval vector, and a chunk-size bound, one chunked dot-product certificate for each output coordinate.","s":"{ι : Type u_1} → [Fintype ι] → [DecidableEq ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → ℕ → Type u_1","l":307,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CoordinateRecurrenceCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a coordinate recurrence certificate is supplied, the interval row-vector recurrence refines the reported next interval vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {current next : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {chunkSize : ℕ} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CoordinateRecurrenceCertificate K current next chunkSize), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.VectorSubinterval (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalVecMul current K) next","l":315,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A finite iterate certificate stores steps + 2 interval vectors and checks each recurrence inclusion using exact rational interval arithmetic.","s":"{ι : Type u_1} → [inst : Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι → Type u_1","l":108,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate.next","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The final stored interval vector encloses one further Markov step and is used to certify the finite-iterate residual.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι","l":132,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate.next_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The last stored row contains the true iterate one step after the selected terminal row.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector c.next (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate P p0.toReal (c.steps + 1))","l":293,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate.table_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Every row of a checked finite interval recurrence contains the corresponding true Markov iterate for any real kernel enclosed by the certified table.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → ∀ (k : Fin (c.steps + 2)), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector (c.table k) (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate P p0.toReal ↑k)","l":270,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate.terminal","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The penultimate stored interval vector is the finite-iterate approximation used for the stationary enclosure.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι","l":125,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate.terminal_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The terminal stored row contains the true iterate at the certificate's selected step.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector c.terminal (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate P p0.toReal c.steps)","l":285,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteMatrixCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A finite-matrix certificate records independently checked scalar enclosures for each entry of an exact rational matrix in a rational interval matrix, through one certificate per entry.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → [Fintype κ] → Matrix ι κ ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Prop","l":216,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteMatrixCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a finite-matrix certificate is supplied, the reported interval matrix contains the real matrix obtained from its rational entries.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype ι] [inst_1 : Fintype κ] {q : Matrix ι κ ℚ} {I : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteMatrixCertificate q I → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix I fun i j => ↑(q i j)","l":222,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteVectorCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A finite-vector certificate records independently checked scalar enclosures for each coordinate of an exact rational vector in a rational interval vector, through one certificate per coordinate.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℚ) → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Prop","l":203,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteVectorCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a finite-vector certificate is supplied, the reported interval vector contains the real vector obtained from its rational coordinates.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {q : ι → ℚ} {I : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteVectorCertificate q I → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector I fun i => ↑(q i)","l":209,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalFoldCertificate","k":"inductive","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"An interval-fold certificate records a proof-producing binary assembly of a list of intervals into one output interval, with the empty-list case and the one-more-interval case checked separately.","s":"List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":230,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalFoldCertificate.contains_sum","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given an interval-fold certificate and coordinatewise evidence that each input interval contains its corresponding real value, the reported interval contains the sum of those real values.","s":"∀ {xs : List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {values : List ℝ} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalFoldCertificate xs output → List.Forall₂ (fun I x => I.Contains x) xs values → output.Contains values.sum","l":254,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalFoldCertificate.refines_sum","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When an interval-fold certificate is supplied, the ordinary interval sum of its input list refines its reported output interval.","s":"∀ {xs : List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {output : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalFoldCertificate xs output → xs.sum.Subinterval output","l":242,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"An interval matrix assigns one exact rational interval to each row and column.","s":"Type u_1 → Type u_2 → Type (max u_1 u_2)","l":64,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"An interval vector assigns one exact rational interval to each coordinate.","s":"Type u_1 → Type u_1","l":61,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A real vector is a probability vector when its entries are nonnegative and sum to one. This is definitionally membership in Mathlib's standard simplex (see isProbabilityVector_iff_mem_stdSimplex).","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → Prop","l":18,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A stationary distribution is a probability vector fixed by one Markov step.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → (ι → ℝ) → Prop","l":45,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A real square matrix is row-stochastic when every entry is nonnegative and every row sums to one. This is the same condition as membership in Mathlib's Matrix.rowStochastic (see isStochasticMatrix_iff_mem_rowStochastic), stated without its decidable-equality infrastructure.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → Prop","l":24,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix.iterate_probability","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Every finite iterate of a probability vector under a stochastic matrix is a probability vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P → ∀ {p0 : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p0 → ∀ (n : ℕ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate P p0 n)","l":170,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix.step_probability","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"A stochastic matrix sends every probability row vector to a probability row vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P → ∀ {p : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep p P)","l":148,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.MatrixSubinterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Coordinatewise rational interval refinement for matrices.","s":"{ι : Type u_1} → {κ : Type u_2} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Prop","l":79,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Minorizes","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A Doeblin minorization says every row of a square matrix dominates the same probability vector scaled by a common mass ε between zero and one. Row-stochasticity of the matrix is not part of this predicate and is assumed separately.","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → ℝ → (ι → ℝ) → Prop","l":62,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A standard-normal CDF certificate consists of a checked internal schedule, a caller-facing rational interval, and an exact proof that the internally computed interval refines the caller-facing one.","s":"ℚ → Type","l":94,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given a checked rational certificate at a rational endpoint, the certificate's reported interval contains the true standard-normal cumulative probability at that endpoint.","s":"∀ {q : ℚ} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFCertificate q), c.enclosure.Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":295,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A normal-CDF schedule records a positive mesh and the exact rational side conditions needed to invert the computed enclosure of sqrt (2π).","s":"Type","l":42,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A rational minorization certificate is checked solely by exact rational nonnegativity, normalization, and entrywise lower-endpoint comparisons.","s":"{ι : Type u_1} → [Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι → Type u_1","l":93,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"An exact rational minorization certificate soundly minorizes every real kernel contained in its interval table.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate K), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Minorizes P (↑c.epsilon) c.nu.toReal","l":180,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A rational probability-vector certificate contains exact nonnegativity and sum checks.","s":"(ι : Type u_1) → [Fintype ι] → Type u_1","l":79,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector.isProbability","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"An exact rational probability-vector certificate denotes a real probability vector.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (p : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p.toReal","l":139,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector.toReal","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The real probability vector denoted by an exact rational certificate.","s":"{ι : Type u_1} → [inst : Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι → ι → ℝ","l":88,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ScalarIntervalCertificate","k":"structure","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"A scalar interval certificate records that an exact rational value lies in a reported rational interval, through a checked point-interval refinement.","s":"ℚ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Prop","l":193,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ScalarIntervalCertificate.sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When a scalar interval certificate is supplied, its reported interval contains the certificate's rational value as a real number.","s":"∀ {q : ℚ} {I : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ScalarIntervalCertificate q I → I.Contains ↑q","l":198,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.VectorSubinterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Coordinatewise rational interval refinement for vectors.","s":"{ι : Type u_1} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Prop","l":75,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.abs_rewardExpectation_sub_le","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"A bounded reward changes by at most its uniform absolute bound times the ℓ¹ distance between two mass vectors.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {p q reward : ι → ℝ} {B : ℝ}, 0 ≤ B → (∀ (i : ι), |reward i| ≤ B) → |Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation p reward - Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation q reward| ≤ B * Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance p q","l":140,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.contractsL1_of_minorization","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Doeblin minorization by mass ε yields ℓ¹ contraction coefficient 1 - ε.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {ε : ℝ} {ν : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Minorizes P ε ν → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P (1 - ε)","l":197,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.contractsL1_of_rationalMinorization","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given a row-stochastic real transition matrix, its containment in an exact rational interval table, and a checked rational minorization certificate for that table, the kernel contracts finite probability vectors in total variation with coefficient one minus the certified minorization mass.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix K P → ∀ (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate K), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P (1 - ↑c.epsilon)","l":262,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.existsUnique_stationary_and_reward_enclosed","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"For a nonempty state space, the checked recurrence and contraction both produce a unique stationary distribution and certify its reward expectation.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [Nonempty ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernel.intervals p0) (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (rho : ℚ) (hrho0 : 0 ≤ rho) (hrho1 : rho < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rho → ∃! π, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π ∧ (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval c r rho hrho0 hrho1).Contains (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation π reward)","l":280,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.existsUnique_stationary_of_contractsL1","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Existence","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"On a nonempty finite state space, given a row-stochastic transition matrix, a nonnegative contraction coefficient, the fact that the coefficient is strictly below one, and a total-variation contraction bound, there is exactly one stationary probability distribution.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [Nonempty ι] {P : Matrix ι ι ℝ} {rho : ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P → 0 ≤ rho → rho < 1 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P rho → ∃! π, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π","l":32,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Existence.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.finiteIterateResidualBound","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The rational residual bound sums the maximum absolute endpoint of the difference between the terminal interval row and its certified successor.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → ℚ","l":37,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.finiteIterateResidualBound_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The exact rational finite-iterate residual bound is nonnegative.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0), 0 ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.finiteIterateResidualBound c","l":63,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.instAddCommMonoidRatInterval","k":"instance","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Rational intervals form an additive commutative monoid under outward interval addition, with the point interval at zero as identity.","s":"AddCommMonoid Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":19,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalDot","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The interval dot product sums outward products of corresponding coordinates.","s":"{ι : Type u_1} → [Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":87,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalDot_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"An interval dot product contains the real dot product of any two enclosed vectors.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {I J : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {x y : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector I x → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector J y → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalDot I J).Contains (∑ i, x i * y i)","l":126,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalExpectation","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The interval reward expectation is the interval dot product of mass and reward vectors.","s":"{ι : Type u_1} → [Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":107,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMatrixMul","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Interval matrix multiplication computes every entry by an interval dot product.","s":"{ι : Type u_1} → {κ : Type u_2} → {υ : Type u_3} → [Fintype κ] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix κ υ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι υ","l":102,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMatrixMul_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"When the first rational interval matrix encloses a real matrix and the second rational interval matrix encloses a real matrix, their interval matrix product encloses the corresponding real matrix product.","s":"∀ {ι : Type u_1} {κ : Type u_2} {υ : Type u_3} [inst : Fintype κ] {A : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ} {B : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix κ υ} {M : Matrix ι κ ℝ} {N : Matrix κ υ ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix A M → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix B N → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMatrixMul A B) (M * N)","l":153,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMulVec","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Applying an interval matrix to an interval column vector uses interval dot products rowwise.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype κ] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector κ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι","l":92,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMulVec_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Interval matrix-vector multiplication contains the corresponding real matrix-vector product.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype κ] {A : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ} {x : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector κ} {M : Matrix ι κ ℝ} {v : κ → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix A M → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector x v → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalMulVec A x) (M.mulVec v)","l":135,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalSum","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The finite interval sum is the ordinary finite sum using outward interval addition.","s":"{ι : Type u_1} → [Fintype ι] → (ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval) → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":83,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalSum_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Finite outward interval addition contains the sum of any coordinatewise enclosed real family.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {I : ι → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x : ι → ℝ}, (∀ (i : ι), (I i).Contains (x i)) → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalSum I).Contains (∑ i, x i)","l":112,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalVecMul","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Applying an interval row vector to an interval matrix uses interval dot products columnwise.","s":"{ι : Type u_1} → {κ : Type u_2} → [Fintype ι] → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector κ","l":97,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalVecMul_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Interval row-vector multiplication contains the corresponding real row-vector action.","s":"∀ {ι : Type u_1} {κ : Type u_2} [inst : Fintype ι] {x : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι} {A : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι κ} {v : ι → ℝ} {M : Matrix ι κ ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector x v → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsMatrix A M → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.intervalVecMul x A) (Matrix.vecMul v M)","l":144,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.isProbabilityVector_iff_mem_stdSimplex","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Existence","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"A real vector is a probability vector exactly when it lies in the standard simplex.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (p : ι → ℝ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p ↔ p ∈ stdSimplex ℝ ι","l":18,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Existence.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.isStochasticMatrix_iff_mem_rowStochastic","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Existence","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"For a real square matrix indexed by a finite state space, being row-stochastic, with nonnegative entries and every row summing to one, is equivalent to membership in Mathlib's submonoid of row-stochastic matrices.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (P : Matrix ι ι ℝ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStochasticMatrix P ↔ P ∈ Matrix.rowStochastic ℝ ι","l":24,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Existence.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The finite ℓ¹ distance is the sum of absolute coordinate differences.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → (ι → ℝ) → ℝ","l":50,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance_le_finiteIterateResidualBound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Two real vectors enclosed by the terminal and successor rows have ℓ¹ distance at most the exact rational residual bound.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0) {x y : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector c.terminal x → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector c.next y → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance x y ≤ ↑(Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.finiteIterateResidualBound c)","l":100,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance_stationary_le_residual","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"An a posteriori contraction estimate bounds the distance from an approximate probability vector to any stationary distribution by its one-step residual divided by 1 - rho.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {p π : ι → ℝ} {rho residual : ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P rho → 0 ≤ rho → rho < 1 → 0 ≤ residual → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance p (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep p P) ≤ residual → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.l1Distance p π ≤ residual / (1 - rho)","l":112,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.listSum_subinterval","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When each interval in one list refines the corresponding interval in another list, the sum of the first list refines the sum of the second.","s":"∀ {xs ys : List Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, List.Forall₂ Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval.Subinterval xs ys → xs.sum.Subinterval ys.sum","l":272,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.lt_of_disjoint_enclosures","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Comparison","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"If two rational intervals contain two real values and the first upper endpoint is below the second lower endpoint, then the first value is strictly smaller.","s":"∀ {I J : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval} {x y : ℝ}, I.Contains x → J.Contains y → I.hi < J.lo → x < y","l":23,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Comparison.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovIterate","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The successive row vectors obtained from an initial row vector by repeated multiplication by a fixed square matrix. They are the successive distributions of a Markov chain when the start is a probability vector and the matrix is row-stochastic (see IsStochasticMatrix.iterate_probability).","s":"{ι : Type u_1} → [Fintype ι] → Matrix ι ι ℝ → (ι → ℝ) → ℕ → ι → ℝ","l":37,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteKernel","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"One Markov step multiplies a row vector by a square matrix. No condition is imposed; it is a step of a Markov chain when the vector is a probability vector and the matrix is row-stochastic.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → Matrix ι ι ℝ → ι → ℝ","l":31,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/FiniteKernel.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.nonnegativeNormalIntegralInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"For a nonnegative rational endpoint, the computed interval encloses the integral of the standard-normal density from zero to that endpoint.","s":"(q : ℚ) → 0 ≤ q → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":79,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.nonnegativeNormalIntegralInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The nonnegative-endpoint quadrature interval contains the exact integral of the standard-normal density from zero to the endpoint.","s":"∀ (q : ℚ) (hq : 0 ≤ q) (s : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.nonnegativeNormalIntegralInterval q hq s).Contains (∫ (u : ℝ) in 0..1, ↑q * Causalean.Mathlib.stdNormalPDF (↑q * u))","l":191,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalCDFDifference_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Subtracting two checked CDF enclosures contains the corresponding difference of standard-normal CDF values, as used for threshold-cell probabilities.","s":"∀ {a b : ℚ} (ca : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFCertificate a) (cb : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFCertificate b), (cb.enclosure.sub ca.enclosure).Contains (Causalean.Mathlib.stdNormalCDF ↑b - Causalean.Mathlib.stdNormalCDF ↑a)","l":300,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalCDFInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The internally computed rational enclosure of the standard-normal CDF at any rational endpoint, using symmetry for negative endpoints.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":87,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalCDFInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The internally evaluated interval soundly contains the standard-normal CDF at its rational endpoint; no external numerical approximation is assumed.","s":"∀ (q : ℚ) (s : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalCDFInterval q s).Contains (Causalean.Mathlib.stdNormalCDF ↑q)","l":268,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityComplexNode","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The complex rectangle obtained from a real Gaussian quadrature node by adjoining the point interval at zero as imaginary coordinate.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.ComplexRatInterval","l":73,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityNode","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"For a rational endpoint q, a schedule, and a natural node index k, this interval encloses the rescaled Gaussian integrand at the point q k / mesh used for CDF quadrature. The quadrature applies it to nonnegative endpoints and node indices up to the mesh size; no such restriction is imposed here.","s":"ℚ → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule → ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":63,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityNode_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Every computed Gaussian quadrature node contains the corresponding value of the rescaled standard-normal density.","s":"∀ (q : ℚ), 0 ≤ q → ∀ (s : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule) (k : ℕ), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityNode q s k).Contains (↑q * Causalean.Mathlib.stdNormalPDF (↑q * Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.CircleMesh.meshPoint s.mesh k))","l":116,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityScaleInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The certified rational enclosure of the standard-normal density's scale factor 1 / sqrt (2π) for a schedule.","s":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":57,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityScaleInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The executable density-scale interval contains the mathematical constant 1 / sqrt (2π).","s":"∀ (s : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDFSchedule), (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.normalDensityScaleInterval s).Contains (1 / √(2 * Real.pi))","l":105,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rescaledNormalDensity_lipschitz","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"The rescaled standard-normal density on the unit interval is Lipschitz with the exact rational constant q³ for every nonnegative rational q.","s":"∀ (q : ℚ), 0 ≤ q → ∀ u ∈ Set.Icc 0 1, ∀ v ∈ Set.Icc 0 1, |↑q * Causalean.Mathlib.stdNormalPDF (↑q * u) - ↑q * Causalean.Mathlib.stdNormalPDF (↑q * v)| ≤ ↑(q ^ 3) * |u - v|","l":133,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The expectation of a finite reward vector under a real mass vector.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → (ι → ℝ) → ℝ","l":18,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.scalarSubintervalCheck","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given an inner rational interval and an outer rational interval, the scalar subinterval check is the exact endpoint comparison that accepts exactly when the inner interval lies inside the outer interval.","s":"Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval → Bool","l":183,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.scalarSubintervalCheck_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"When the scalar subinterval check accepts, the inner interval is contained in the outer interval.","s":"∀ {inner outer : Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.scalarSubintervalCheck inner outer = true → inner.Subinterval outer","l":187,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Comparison","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The stationary bias of a reward process relative to a benchmark is the reward expectation under a supplied weight vector minus that benchmark. No condition is imposed on the vector; it is the bias of the stationary reward when the vector is the stationary distribution.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → (ι → ℝ) → ℝ → ℝ","l":16,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Comparison.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias_lt_of_certified_intervals","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Comparison","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"For two certified policy kernels, disjoint stationary-expectation intervals imply a strict comparison of their stationary biases relative to any common benchmark.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P Q : Matrix ι ι ℝ} {p0P p0Q : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernelP : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (kernelQ : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel Q) (iterateP : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernelP.intervals p0P) (iterateQ : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernelQ.intervals p0Q) (rewardCert : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (rhoP rhoQ : ℚ) (hrhoP0 : 0 ≤ rhoP) (hrhoP1 : rhoP < 1) (hrhoQ0 : 0 ≤ rhoQ) (hrhoQ1 : rhoQ < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rhoP → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 Q ↑rhoQ → ∀ {πP πQ : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P πP → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary Q πQ → ∀ {benchmark : ℝ}, (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval iterateP rewardCert rhoP hrhoP0 hrhoP1).hi < (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval iterateQ rewardCert rhoQ hrhoQ0 hrhoQ1).lo → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias πP reward benchmark < Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias πQ reward benchmark","l":72,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Comparison.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias_lt_of_expectation_lt","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Comparison","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Strict ordering of two stationary reward expectations is preserved after subtracting the same real benchmark, yielding a strict stationary-bias comparison.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {πP πQ reward : ι → ℝ} {benchmark : ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation πP reward < Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation πQ reward → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias πP reward benchmark < Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryBias πQ reward benchmark","l":34,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Comparison.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryDistributionInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The certified stationary-distribution enclosure expands each terminal coordinate interval by the common a posteriori ℓ¹ radius.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → (rho : ℚ) → 0 ≤ rho → rho < 1 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalVector ι","l":80,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryDistributionInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"A checked finite recurrence and a strict ℓ¹ contraction enclose every stationary distribution coordinate in the computed rational expansion of the terminal interval row.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernel.intervals p0) (rho : ℚ) (hrho0 : 0 ≤ rho) (hrho1 : rho < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rho → ∀ {π : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContainsVector (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryDistributionInterval c rho hrho0 hrho1) π","l":198,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryErrorRadius","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The a posteriori stationary radius is the residual bound divided by one minus the supplied exact rational contraction coefficient. It is a valid radius only for a coefficient strictly below one, as assumed by the enclosure results; at coefficient one the division by zero gives zero by convention.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → ℚ → ℚ","l":44,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryErrorRadius_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"A nonnegative contraction coefficient strictly below one gives a nonnegative a posteriori stationary error radius.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0) (rho : ℚ), 0 ≤ rho → rho < 1 → 0 ≤ Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryErrorRadius c rho","l":71,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryExpectation_lt_of_certified_intervals","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Comparison","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given the first certified policy kernel, the second certified policy kernel, their checked finite interval recurrences, a certified bounded reward vector, their rational contraction coefficients, the first coefficient's nonnegativity and strict upper bound, the second coefficient's nonnegativity and strict upper bound, the two contraction guarantees, their stationary distributions, and strictly separated computed reward intervals, the first stationary reward expectation is strictly smaller than the second.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P Q : Matrix ι ι ℝ} {p0P p0Q : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernelP : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (kernelQ : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel Q) (iterateP : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernelP.intervals p0P) (iterateQ : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernelQ.intervals p0Q) (rewardCert : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (rhoP rhoQ : ℚ) (hrhoP0 : 0 ≤ rhoP) (hrhoP1 : rhoP < 1) (hrhoQ0 : 0 ≤ rhoQ) (hrhoQ1 : rhoQ < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rhoP → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 Q ↑rhoQ → ∀ {πP πQ : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P πP → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary Q πQ → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval iterateP rewardCert rhoP hrhoP0 hrhoP1).hi < (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval iterateQ rewardCert rhoQ hrhoQ0 hrhoQ1).lo → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation πP reward < Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation πQ reward","l":43,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Comparison.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The stationary reward interval is the terminal interval expectation widened by the reward bound times the a posteriori stationary radius.","s":"{ι : Type u_1} → [inst : Fintype ι] → {K : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IntervalMatrix ι ι} → {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} → {reward : ι → ℝ} → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate K p0 → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward → (rho : ℚ) → 0 ≤ rho → rho < 1 → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":89,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval_sound","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given a certified finite transition kernel, a checked finite interval recurrence, a certified bounded reward vector, a rational contraction coefficient, its nonnegativity, its strict upper bound by one, the corresponding contraction guarantee, and a stationary distribution, the computed rational interval contains that distribution's stationary reward expectation.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernel.intervals p0) (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (rho : ℚ) (hrho0 : 0 ≤ rho) (hrho1 : rho < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rho → ∀ {π : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval c r rho hrho0 hrho1).Contains (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation π reward)","l":225,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval_sound_of_chunked","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"Given a certified real kernel, a chunked finite-iterate certificate, a bounded reward certificate, a rational contraction coefficient that is nonnegative and strictly below one, the contraction property, and a stationary distribution, the stationary-reward interval obtained through the adapter contains the stationary reward expectation.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate kernel.intervals p0) (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (rho : ℚ) (hrho0 : 0 ≤ rho) (hrho1 : rho < 1), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ContractsL1 P ↑rho → ∀ {π : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval c.toFiniteIterateCertificate r rho hrho0 hrho1).Contains (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation π reward)","l":374,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval_sound_of_minorization","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"An exact rational minorization certificate supplies the contraction needed to certify the stationary reward interval with coefficient 1 - epsilon.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (minor : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate kernel.intervals) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.FiniteIterateCertificate kernel.intervals p0) (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (hepsilon_pos : 0 < minor.epsilon) {π : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval c r (1 - minor.epsilon) (sub_nonneg.mpr minor.epsilon_le_one) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.Common.add_congr (Mathlib.Tactic.Ring.Common.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0))) (Mathlib.Tactic.Ring.Common.atom_pf minor.epsilon rfl (Eq.mpr (id (congrArg (fun _a => minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1 = minor.epsilon ^ Nat.rawCast 1 * _a) (Eq.symm rfl))) (Eq.refl (minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1)))) (Mathlib.Tactic.Ring.Common.sub_pf (Mathlib.Tactic.Ring.Common.neg_add (Mathlib.Tactic.Ring.Common.neg_mul minor.epsilon (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_neg (Eq.refl Neg.neg) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.Common.neg_zero) (Mathlib.Tactic.Ring.Common.add_pf_zero_add (minor.epsilon ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.Common.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.Common.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.Common.atom_pf minor.epsilon rfl (Eq.mpr (id (congrArg (fun _a => minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1 = minor.epsilon ^ Nat.rawCast 1 * _a) (Eq.symm rfl))) (Eq.refl (minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1)))) 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(Mathlib.Meta.NormNum.isInt_neg (Eq.refl Neg.neg) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) Mathlib.Tactic.Ring.Common.neg_zero)) (Mathlib.Tactic.Ring.Common.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.Common.add_pf_zero_add (minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.Common.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.Common.add_overlap_pf_zero minor.epsilon (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.Common.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt hepsilon_pos) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))).Contains (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation π reward)","l":261,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval_sound_of_minorization_chunked","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Stationary","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given a certified real kernel, a rational minorization certificate, a chunked finite-iterate certificate, a bounded reward certificate, positive minorization mass, and a stationary distribution, the stationary-reward interval at coefficient one minus the minorization mass contains the stationary reward expectation.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] {P : Matrix ι ι ℝ} {p0 : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalProbabilityVector ι} {reward : ι → ℝ} (kernel : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.CertifiedKernel P) (minor : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.RationalMinorizationCertificate kernel.intervals) (c : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.ChunkedFiniteIterateCertificate kernel.intervals p0) (r : Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.BoundedRewardCertificate reward) (hepsilon_pos : 0 < minor.epsilon) {π : ι → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsStationary P π → (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stationaryRewardInterval c.toFiniteIterateCertificate r (1 - minor.epsilon) (sub_nonneg.mpr minor.epsilon_le_one) (lt_of_not_ge fun a => 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(Mathlib.Tactic.Ring.Common.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.Common.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.Common.atom_pf minor.epsilon rfl (Eq.mpr (id (congrArg (fun _a => minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1 = minor.epsilon ^ Nat.rawCast 1 * _a) (Eq.symm rfl))) (Eq.refl (minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1)))) (Mathlib.Tactic.Ring.Common.sub_pf (Mathlib.Tactic.Ring.Common.neg_add (Mathlib.Tactic.Ring.Common.neg_mul minor.epsilon (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_neg (Eq.refl Neg.neg) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.Common.neg_zero) (Mathlib.Tactic.Ring.Common.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.Common.add_pf_zero_add (minor.epsilon ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.Common.sub_pf (Mathlib.Tactic.Ring.Common.neg_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_neg (Eq.refl Neg.neg) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.Common.neg_add (Mathlib.Tactic.Ring.Common.neg_mul minor.epsilon (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_neg (Eq.refl Neg.neg) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) Mathlib.Tactic.Ring.Common.neg_zero)) (Mathlib.Tactic.Ring.Common.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.Common.add_pf_zero_add (minor.epsilon ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.Common.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.Common.add_overlap_pf_zero minor.epsilon (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.Common.add_pf_zero_add 0))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt hepsilon_pos) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))).Contains (Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.rewardExpectation π reward)","l":389,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Stationary.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.stdNormalCDF_eq_half_add_rescaled_integral","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"For a nonnegative real endpoint, its standard-normal CDF is one half plus the integral of the rescaled density over the unit interval.","s":"∀ (q : ℚ), 0 ≤ q → Causalean.Mathlib.stdNormalCDF ↑q = 1 / 2 + ∫ (u : ℝ) in 0..1, ↑q * Causalean.Mathlib.stdNormalPDF (↑q * u)","l":214,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.sum_eq_sum_chunks","k":"theorem","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.Interval","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":2,"d":"Given a finite additive family, a list of index chunks, pairwise disjoint chunks, and chunks covering every index, the full finite sum equals the list sum of the chunk sums.","s":"∀ {ι : Type u_1} {M : Type u_2} [inst : Fintype ι] [inst_1 : DecidableEq ι] [inst_2 : AddCommMonoid M] (f : ι → M) (chunks : List (Finset ι)), List.Pairwise (fun s t => Disjoint s t) chunks → List.foldr (fun s acc => s ∪ acc) ∅ chunks = Finset.univ → ∑ i, f i = (List.map (fun s => ∑ i ∈ s, f i) chunks).sum","l":306,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/Interval.lean"},{"n":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.twoPiInterval","k":"def","m":"Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.NormalCDF","a":"Mathlib/Probability/CertifiedFiniteMarkovExpectation","t":1,"d":"The interval enclosing 2π at a chosen exact arithmetic fuel level.","s":"ℕ → Causalean.Mathlib.Analysis.CertifiedContourIntervalArithmetic.RatInterval","l":37,"f":"Causalean/Mathlib/Probability/CertifiedFiniteMarkovExpectation/NormalCDF.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.clt_of_l2_approx","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Converging-together corollary for the standard normal (the CLT diagonal step). The specialization of tendsto_inDistribution_of_l2_approx to the standard normal limit law: let $(S_n)$ be a sequence of real random variables, one on each probability space in a sequence, and for every row index $m$ let $(T_{m,n})_n$ be the $m$-th approximating triangular row on the same spaces, with every $S_n$ square-integrable and every $T_{m,n}$ square-integrable. If for every fixed row $m$ the law of $T_{m,n}$ converges weakly to the standard normal as $n \\to \\infty$, i.e. row $m$ obeys its own standard-normal central limit theorem, and for every tolerance $\\varepsilon > 0$ some row $M$ makes the limit superior over $n$ of $E[(S_n - T_{M,n})^2]$ at most $\\varepsilon$, so row $M$ approximates $S_n$ in $L^2$ in the iterated-limsup sense, then the law of $S_n$ converges weakly to the standard normal as $n \\to \\infty$.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [inst_1 : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (S : (n : ℕ) → Ω n → ℝ) (T : ℕ → (n : ℕ) → Ω n → ℝ) (hS_sq : ∀ (n : ℕ), MeasureTheory.MemLp (S n) 2 (μ n)) (hT_sq : ∀ (m n : ℕ), MeasureTheory.MemLp (T m n) 2 (μ n)), (∀ (m : ℕ), Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (T m n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hT_sq m n))))) Filter.atTop (nhds (Causalean.Mathlib.Probability.ConvergingTogether.lawPM (ProbabilityTheory.gaussianReal 0 1) inferInstance))) → (∀ (ε : ℝ), 0 < ε → ∃ M, Filter.limsup (fun n => ENNReal.ofReal (∫ (ω : Ω n), (S n ω - T M n ω) ^ 2 ∂μ n)) Filter.atTop ≤ ENNReal.ofReal ε) → Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (S n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hS_sq n))))) Filter.atTop (nhds (Causalean.Mathlib.Probability.ConvergingTogether.lawPM (ProbabilityTheory.gaussianReal 0 1) inferInstance))","l":176,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.integral_abs_le_sqrt_integral_sq","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"On a probability space, the expected norm of a square-integrable variable is at most the square root of its expected squared norm.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {E : Type u_2} [inst_2 : NormedAddCommGroup E] (f : Ω → E), MeasureTheory.MemLp f 2 μ → ∫ (ω : Ω), ‖f ω‖ ∂μ ≤ √(∫ (ω : Ω), ‖f ω‖ ^ 2 ∂μ)","l":131,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.lawPM","k":"def","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Given a measure on the real line and a proof that it has total mass one, the bundled real probability law is that measure regarded as a probability law on the real line.","s":"(m : MeasureTheory.Measure ℝ) → MeasureTheory.IsProbabilityMeasure m → MeasureTheory.ProbabilityMeasure ℝ","l":47,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.lawPM_coe","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"","s":"∀ (m : MeasureTheory.Measure ℝ) (h : MeasureTheory.IsProbabilityMeasure m), ↑(Causalean.Mathlib.Probability.ConvergingTogether.lawPM m h) = m","l":57,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_cexp_mul_I_sub_cexp_mul_I_le","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"Pointwise Lipschitz bound for the unit-circle exponential. For real arguments a b, the chord between the points e^{ia} and e^{ib} on the unit circle is no longer than the arc, i.e. ‖exp (a·I) − exp (b·I)‖ ≤ |a − b|. This is the elementary input to the characteristic-function approximation bound.","s":"∀ (a b : ℝ), ‖Complex.exp (↑a * Complex.I) - Complex.exp (↑b * Complex.I)‖ ≤ |a - b|","l":38,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_charFun_sub_le","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":2,"d":"Almost-everywhere measurable real variables with an integrable difference have characteristic functions whose distance is at most the frequency magnitude times their expected absolute difference.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {S T : Ω → ℝ}, AEMeasurable S μ → AEMeasurable T μ → MeasureTheory.Integrable (fun ω => S ω - T ω) μ → ∀ (t : ℝ), ‖MeasureTheory.charFun (MeasureTheory.Measure.map S μ) t - MeasureTheory.charFun (MeasureTheory.Measure.map T μ) t‖ ≤ |t| * ∫ (ω : Ω), |S ω - T ω| ∂μ","l":65,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.norm_charFun_sub_le_L2","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.CharFunBound","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"Characteristic-function approximation bound (L² form). Let $\\mu$ be a probability measure and let $S, T$ be real random variables on the same space, with $S$ almost-everywhere measurable and $T$ almost-everywhere measurable; assume further that their difference $S - T$ is square-integrable under $\\mu$. Then for every real frequency $t$, the characteristic functions of the laws of $S$ and $T$ differ at $t$ by at most $|t|$ times the $L^2$ norm of $S - T$: $\\|{\\rm charFun}(\\mu \\circ S^{-1})(t) - {\\rm charFun}(\\mu \\circ T^{-1})(t)\\| \\le |t| \\cdot \\sqrt{\\int (S - T)^2 \\, d\\mu}$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {S T : Ω → ℝ}, AEMeasurable S μ → AEMeasurable T μ → MeasureTheory.MemLp (fun ω => S ω - T ω) 2 μ → ∀ (t : ℝ), ‖MeasureTheory.charFun (MeasureTheory.Measure.map S μ) t - MeasureTheory.charFun (MeasureTheory.Measure.map T μ) t‖ ≤ |t| * √(∫ (ω : Ω), (S ω - T ω) ^ 2 ∂μ)","l":166,"f":"Causalean/Mathlib/Probability/ConvergingTogether/CharFunBound.lean"},{"n":"Causalean.Mathlib.Probability.ConvergingTogether.tendsto_inDistribution_of_l2_approx","k":"theorem","m":"Causalean.Mathlib.Probability.ConvergingTogether.Diagonal","a":"Mathlib/Probability/ConvergingTogether","t":1,"d":"The converging-together theorem (Billingsley Thm 3.2). Let $G$ be a limit probability law on the reals, let $(S_n)$ be a sequence of real random variables, one on each probability space in a sequence, and for every row index $m$ let $(T_{m,n})_n$ be the $m$-th approximating triangular row of real random variables on the same spaces, with every $S_n$ square-integrable and every $T_{m,n}$ square-integrable. Suppose for every fixed row $m$ the law of $T_{m,n}$ converges weakly to $G$ as $n \\to \\infty$, and for every tolerance $\\varepsilon > 0$ some row $M$ makes the limit superior over $n$ of $E[(S_n - T_{M,n})^2]$ at most $\\varepsilon$, i.e. row $M$ approximates $S_n$ in $L^2$ uniformly enough in the iterated-limsup sense. Then the law of $S_n$ converges weakly to $G$ as $n \\to \\infty$.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [inst_1 : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (G : MeasureTheory.ProbabilityMeasure ℝ) (S : (n : ℕ) → Ω n → ℝ) (T : ℕ → (n : ℕ) → Ω n → ℝ) (hS_sq : ∀ (n : ℕ), MeasureTheory.MemLp (S n) 2 (μ n)) (hT_sq : ∀ (m n : ℕ), MeasureTheory.MemLp (T m n) 2 (μ n)), (∀ (m : ℕ), Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (T m n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hT_sq m n))))) Filter.atTop (nhds G)) → (∀ (ε : ℝ), 0 < ε → ∃ M, Filter.limsup (fun n => ENNReal.ofReal (∫ (ω : Ω n), (S n ω - T M n ω) ^ 2 ∂μ n)) Filter.atTop ≤ ENNReal.ofReal ε) → Filter.Tendsto (fun n => Causalean.Mathlib.Probability.ConvergingTogether.lawPM (MeasureTheory.Measure.map (S n) (μ n)) (MeasureTheory.Measure.isProbabilityMeasure_map (MeasureTheory.AEStronglyMeasurable.aemeasurable (MeasureTheory.MemLp.aestronglyMeasurable (hS_sq n))))) Filter.atTop (nhds G)","l":60,"f":"Causalean/Mathlib/Probability/ConvergingTogether/Diagonal.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition","k":"structure","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"A finite measurable partition of the sample space X into cells indexed by ι, represented by a classifier assigning each observation to its cell — the cells are the fibres of this map — where that classifier is measurable. The structure itself does not require the index type to be finite or singletons of it to be measurable; results that need finitely many cells or measurable individual cells assume those conditions separately.","s":"(X : Type u_3) → (ι : Type u_4) → [MeasurableSpace X] → [MeasurableSpace ι] → Type (max u_3 u_4)","l":22,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellIndices","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition, a cell index, and a finite sample of observation--real-mark pairs, the cell indices are precisely the original sample positions whose observations belong to that cell.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Finset (Fin s.count)","l":166,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.CellLaws","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"On its count-n fibre, the jth cell experiment is the Poisson mass at n times n independent draws from the normalised restriction and mark law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (j : ι) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure (lam * p.cellMass P j)) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => (p.cellObservationLaw P j).prod R)","l":64,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/CellLaws.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellMass","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition, a measure on the observation space, and a cell index, the cell mass is the measure of that cell, represented as a nonnegative real number. A cell of infinite measure is sent to zero by this conversion, so it is the cell's mass for finite (in particular probability) measures, as used throughout.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → MeasureTheory.Measure X → ι → NNReal","l":112,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition, a probability measure on the observation space, and a cell index, the within-cell observation law is the normalized restriction of that probability measure to the cell when the cell has positive probability, and is the original probability measure when the cell has probability zero.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → ι → MeasureTheory.Measure X","l":138,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw_apply_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"A positive-mass within-cell law assigns probability one to its own cell.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P] (j : ι), P (p.cellSet j) ≠ 0 → (p.cellObservationLaw P j) (p.cellSet j) = 1","l":156,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellObservationLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Let the observation space and the cell-index space each be equipped with a $\\sigma$-algebra. For a finite measurable partition of the observation space indexed by the cell-index space, a probability measure on the observation space, and a cell index, the assertion that the associated within-cell observation law is a probability measure holds, including when the selected cell has probability zero.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : MeasurableSpace ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P] (j : ι), MeasureTheory.IsProbabilityMeasure (p.cellObservationLaw P j)","l":144,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.cellSet","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition and a cell index, the cell set is the set of observations assigned to that index by the partition.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → Set X","l":90,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.disjoint_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Distinct classifier cells are disjoint.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) {i j : ι}, i ≠ j → Disjoint (p.cellSet i) (p.cellSet j)","l":99,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.fixedPartitionEmbed","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a nonnegative integer count for every cell index and for each cell index, a tuple of points having its specified count, the fixed partition embedding assigns to every cell index the finite sample with that count and that tuple of points.","s":"{ι : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace Y] → (c : ι → ℕ) → ((j : ι) → Fin (c j) → Y) → ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample Y","l":321,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.gatherWord","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given an assignment of each of $n$ positions to a cell index and an $n$-tuple of points, the cell-wise regrouping assigns to each cell the tuple of points at positions assigned to that cell, ordered by their original positions.","s":"{ι : Type u_2} → [Fintype ι] → [inst : MeasurableSpace ι] → [MeasurableSingletonClass ι] → {Y : Type u_3} → [MeasurableSpace Y] → {n : ℕ} → (w : Fin n → ι) → (Fin n → Y) → (j : ι) → Fin (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.wordHistogram✝ w j) → Y","l":231,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.iUnion_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"The union of all classifier cells is the whole observation space.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι), ⋃ j, p.cellSet j = Set.univ","l":106,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.map_restrictCell_count_finiteMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.CellLaws","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Cell counts are Poisson. Under the marked Poisson sample law with base probability measure P, mark distribution R, and nonnegative intensity lam, the number of marked observations landing in cell j of the finite measurable partition p, viewed as a random variable, is Poisson distributed with mean equal to lam times the P-mass of cell j.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (j : ι), MeasureTheory.Measure.map (fun s => (p.restrictCell j s).count) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure (lam * p.cellMass P j)","l":20,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/CellLaws.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.map_restrictPartition_finiteMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Partition splitting. Under the marked Poisson sample law with base probability measure P, mark distribution R, and nonnegative intensity lam, restricting the sample to each cell of the finite measurable partition p yields, jointly across cells, the product of independent marked Poisson sample laws, one per cell j, each with base measure p.cellObservationLaw P j, mark distribution R, and intensity lam times the P-mass of cell j.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)","l":756,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurableSet_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Every classifier cell is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (j : ι), MeasurableSet (p.cellSet j)","l":94,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_fixedPartitionEmbed","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Embedding one fixed-size point tuple per cell into one finite sample per cell is a measurable map.","s":"∀ {ι : Type u_2} [Fintype ι] [inst : MeasurableSpace ι] [MeasurableSingletonClass ι] {Y : Type u_3} [inst : MeasurableSpace Y] (c : ι → ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.fixedPartitionEmbed c)","l":327,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_gatherWord","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Splitting","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Regrouping an n-tuple of points by the cell word w is a measurable map.","s":"∀ {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace ι] [inst_2 : MeasurableSingletonClass ι] {Y : Type u_3} [inst_3 : MeasurableSpace Y] {n : ℕ} (w : Fin n → ι), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.gatherWord w)","l":243,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Splitting.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_restrictCell","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Restriction to one measurable cell is a measurable map on finite marked sequences.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (j : ι), Measurable (p.restrictCell j)","l":184,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.measurable_restrictPartition","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"For a finite measurable partition p of the sample space into cells indexed by ι, the map sending a finite marked sequence to its family of restrictions to every cell simultaneously is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι), Measurable p.restrictPartition","l":259,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a family of subsets of the observation space indexed by a finite index set, each of which is measurable, which are pairwise disjoint, and whose union is the whole observation space, the partition constructed from these sets assigns every observation to the unique index of the set containing it.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → [inst_2 : MeasurableSpace ι] → (A : ι → Set X) → (∀ (j : ι), MeasurableSet (A j)) → (Pairwise fun i j => Disjoint (A i) (A j)) → ⋃ j, A j = Set.univ → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι","l":40,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets_cellSet","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"The classifier constructed from a disjoint measurable cover has exactly the supplied sets as its fibres.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (A : ι → Set X) (hA : ∀ (j : ι), MeasurableSet (A j)) (hdis : Pairwise fun i j => Disjoint (A i) (A j)) (hcover : ⋃ j, A j = Set.univ) (j : ι), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.ofSets A hA hdis hcover).cell ⁻¹' {j} = A j","l":64,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.restrictCell","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition, a cell index, and a finite sample of observation--real-mark pairs, the restriction to that cell retains exactly the pairs whose observations belong to the cell, in their original relative order.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":172,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.restrictPartition","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":1,"d":"Given a measurable classifier partition and a finite sample of observation--real-mark pairs, the partition-wise restriction assigns to every cell index the sample obtained by retaining exactly the pairs in that cell, in their original relative order.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [inst_1 : MeasurableSpace ι] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":179,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition.sum_cellMass","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Partition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Partition","t":2,"d":"Cell masses sum to one under a probability law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P], ∑ j, p.cellMass P j = 1","l":119,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Partition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"For an observation space, a finite sample is a nonnegative integer sample size together with one observation for each position below that size.","s":"(X : Type u_2) → [MeasurableSpace X] → Type (max 0 u_2)","l":37,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The number of observations in a finite sequence.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℕ","l":40,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.points","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"The coordinates of a finite sequence at its dependent finite index type.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X) → Fin s.count → X","l":43,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a probability measure for observations, a probability measure for real-valued marks, and a nonnegative Poisson mean, the canonical marked Poisson sample law is the distribution obtained from the finite marked Poisson sample law by arranging each realized sample in increasing order of its marks, breaking ties by original position.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":308,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Let the observation space be equipped with a $\\sigma$-algebra. For a probability measure on the observation space, a probability measure on real-valued marks, and a nonnegative Poisson mean, the assertion that the canonical mark-ordered Poisson sample law is a probability measure holds.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam)","l":315,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure lam","l":359,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a fallback observation, a nonnegative integer prefix length, and a finite sample of observation--real-mark pairs, the canonical prefix observations are the first $n$ observations when the sample has at least $n$ pairs, and otherwise are the constant $n$-tuple of the fallback observation.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → (n : ℕ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Fin n → X","l":127,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.cappedPrefix","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"For a fixed array and a requested length, the capped prefix is the genuine prefix in the right summand when the request fits and the distinguished left summand on overflow, so overflow is never confused with an empty observation.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → (Fin n → X) → ℕ → Unit ⊕ Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":203,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.cellObservationLaw_eq_of_restrict_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Equal restrictions and equal cell masses give equal normalized cell laws.","s":"∀ {X : Type u_1} {ι : Type u_2} [inst : MeasurableSpace X] [Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (μ ν : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] [inst_5 : MeasureTheory.IsProbabilityMeasure ν] (j : ι), μ (p.cellSet j) ≠ 0 → μ (p.cellSet j) = ν (p.cellSet j) → μ.restrict (p.cellSet j) = ν.restrict (p.cellSet j) → p.cellObservationLaw μ j = p.cellObservationLaw ν j","l":55,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fiberPoints","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a nonnegative integer count for every index and a family of finite samples whose member at each index has exactly the specified count, the fixed-count component points are the corresponding family of point tuples.","s":"{ι : Type u_2} → {Y : Type u_3} → [inst : MeasurableSpace Y] → (c : ι → ℕ) → ↑(Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.countFiber✝ c) → (j : ι) → Fin (c j) → Y","l":126,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a probability measure for observations, a probability measure for real-valued marks, and a nonnegative Poisson mean, the finite marked Poisson sample law is the finite Poisson sample law whose independent observation--mark pairs have the product of those two measures as their common distribution.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":147,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Let the observation space be equipped with a $\\sigma$-algebra. For a probability measure on the observation space, a probability measure on real-valued marks, and a nonnegative Poisson mean, the assertion that the finite marked Poisson sample law is a probability measure holds.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam)","l":153,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count of a finite marked Poisson sample has scalar Poisson law with mean lam.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) = ProbabilityTheory.poissonMeasure lam","l":161,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_marks_pairwise_distinct","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Atomless independent marks are pairwise distinct with probability one in the finite marked Poisson sample.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam) {s | ∀ (i j : Fin s.count), i ≠ j → (s.points i).2 ≠ (s.points j).2} = 1","l":760,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Fix a nonnegative Poisson rate lam. For an observation law P and an independent mark law R, restricting the finite marked Poisson sample law to the event that the observed count equals n yields exactly poissonMeasure lam {n} times the pushforward, under the fixed-size embedding, of n independent draws from the product measure P.prod R.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure lam) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => P.prod R)","l":170,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a finite intensity measure on the observation space, a fallback probability measure on that space, a probability measure for real-valued marks, and a nonnegative scalar intensity, the finite-measure marked Poisson law is the finite marked Poisson sample law with observation distribution equal to the normalized intensity measure and Poisson mean equal to the scalar intensity times the total intensity mass.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → (P₀ : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P₀] → (R : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure R] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ))","l":369,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For every measurable observation space, finite intensity measure on that space, fallback probability measure on that space, probability distribution for real-valued marks, and nonnegative scalar intensity, the resulting finite-measure marked Poisson law is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν P₀ R lam)","l":378,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count in the finite-measure marked Poisson experiment is Poisson with mean lam times the total mass of the intensity measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν P₀ R lam) = ProbabilityTheory.poissonMeasure (lam * Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMass ν)","l":387,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw_probability_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"When the intensity measure is already a probability law, the finite-measure Poisson wrapper agrees with the ordinary marked-Poisson sample law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (P0 : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P0] (R : MeasureTheory.Measure ℝ) [inst_3 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw P P0 R lam = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam","l":370,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMass","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a finite measure on the observation space, the finite-measure mass is its total mass on the whole observation space, represented as a nonnegative real number.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → NNReal","l":329,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a probability measure on the observation space and a nonnegative Poisson mean, the finite Poisson sample law is the distribution obtained by drawing a Poisson count with that mean and an independent infinite sequence of independent observations from that probability measure, then retaining the initial segment selected by the count.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → NNReal → MeasureTheory.Measure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X)","l":101,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Let the observation space be equipped with a $\\sigma$-algebra. For a probability measure on that space and a nonnegative Poisson mean, the assertion that the finite Poisson sample law is a probability measure holds.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam)","l":106,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count of a finite Poisson sample has scalar Poisson law with the requested mean.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam) = ProbabilityTheory.poissonMeasure lam","l":112,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw_restrict_count_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"On the fibre where the count equals n, the finite Poisson law is the Poisson mass at n times the embedded n-fold product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal) (n : ℕ), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' {n}) = (ProbabilityTheory.poissonMeasure lam) {n} • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n) (MeasureTheory.Measure.pi fun x => P)","l":122,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a map from one observation space to another and a finite sample in the first space, the mapped finite sample has the same size and applies the map to every observation.","s":"{X : Type u_1} → {Y : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (X → Y) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample Y","l":27,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap_fixedSizeEmbed","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Mapping commutes with fixed-size embedding.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (f : X → Y) (n : ℕ) (x : Fin n → X), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap f (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n x) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n fun i => f (x i)","l":49,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a fallback observation and a finite sample, the padded stream representation is the pair consisting of its size and an infinite stream that agrees with the sample at positions below that size and equals the fallback observation thereafter.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℕ × (ℕ → X)","l":251,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream_measurable","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0)","l":256,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream_range","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X), Set.range (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0) = {z | ∀ (k : ℕ), z.1 ≤ k → z.2 k = x0}","l":285,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSample_standardBorelSpace","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For every nonempty standard Borel observation space equipped with its measurable structure, the space of finite samples from that observation space is a standard Borel space.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] [Nonempty X], StandardBorelSpace (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X)","l":305,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedCountSuperpose","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a nonnegative integer count for every member of a finite index set and a tuple of points of the specified count for each member, the fixed-count superposition is the finite sample obtained by concatenating all those tuples.","s":"{ι : Type u_2} → [Fintype ι] → {Y : Type u_3} → [inst : MeasurableSpace Y] → (c : ι → ℕ) → ((j : ι) → Fin (c j) → Y) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample Y","l":105,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a nonnegative integer sample size and a tuple of observations indexed by its positions, the fixed-size embedding is the finite sample having that size and those observations.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (n : ℕ) → (Fin n → X) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":46,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a probability measure on a measurable sample space, the law of an infinite independent and identically distributed stream is the product probability measure on infinite sequences whose every coordinate has that measure as its marginal.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → MeasureTheory.Measure (ℕ → X)","l":20,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For every measurable sample space and probability measure on that space, the law of the infinite independent and identically distributed stream is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw P)","l":27,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw_map_finPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Every finite prefix of length n of an infinite stream whose coordinates are i.i.d. with common law P has exactly the corresponding n-fold product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (n : ℕ), MeasureTheory.Measure.map (fun z i => z ↑i) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.iidStreamLaw P) = MeasureTheory.Measure.pi fun x => P","l":33,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.klDiv_finiteMeasureMarkedPoissonLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Fix a nonnegative Poisson rate lam and suppose the finite intensity measures ν₀ and ν₁ have equal total mass. Then, for a common baseline probability measure P₀ and mark law R, the KL divergence between the finite-measure marked Poisson experiments generated by ν₀ and by ν₁ equals lam times the KL divergence between ν₀ and ν₁.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (ν₀ ν₁ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsFiniteMeasure ν₀] [inst_3 : MeasureTheory.IsFiniteMeasure ν₁] (P₀ : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal), ν₀ Set.univ = ν₁ Set.univ → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₀ P₀ R lam) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₁ P₀ R lam) = ↑lam * InformationTheory.klDiv ν₀ ν₁","l":398,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.klDiv_finiteMeasureMarkedPoissonLaw_le","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"A one-point finite-measure KL bound transfers directly to the corresponding marked Poisson experiments after multiplication by the scalar intensity.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (ν₀ ν₁ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsFiniteMeasure ν₀] [inst_3 : MeasureTheory.IsFiniteMeasure ν₁] (P₀ : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P₀] (R : MeasureTheory.Measure ℝ) [inst_5 : MeasureTheory.IsProbabilityMeasure R] (lam : NNReal) (B : ENNReal), ν₀ Set.univ = ν₁ Set.univ → InformationTheory.klDiv ν₀ ν₁ ≤ B → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₀ P₀ R lam) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw ν₁ P₀ R lam) ≤ ↑lam * B","l":457,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_canonicalPrefixObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"On the successful-count event, the canonical marked-Poisson configuration's first n observations have the unnormalised product law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) • MeasureTheory.Measure.pi fun x => P","l":152,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_restrictPartition_canonicalMarkedPoissonSampleLaw","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Restricting the canonical global configuration gives exactly the product of independent canonical cell configurations.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam) = MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)","l":780,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_retainedObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"On the event that at least n points exist, retaining the n smallest atomless independent marks and forgetting marks gives the event probability times the exact product law P^n.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) • MeasureTheory.Measure.pi fun x => P","l":205,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_superposeByMarks_canonicalCellLaws","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Superposition recovers the canonical global law. Under nonnegative intensity lam, and independently for each cell j of the finite measurable partition p, draw a canonical marked Poisson configuration with base measure p.cellObservationLaw P j, atomless mark distribution R (so marks are almost surely distinct), and intensity lam times the P-mass of cell j; merging these per-cell configurations by increasing mark (mark-ordered superposition) has exactly the law of the canonical marked Poisson configuration with base measure P, mark distribution R, and intensity lam.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks (MeasureTheory.Measure.pi fun j => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw (p.cellObservationLaw P j) R (lam * p.cellMass P j)) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam","l":840,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_superposeByMarks_map_restrictPartition","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Restriction followed by mark-ordered superposition is the identity in law on the canonical marked Poisson configuration.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] [StandardBorelSpace X] (p : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteMeasurablePartition X ι) (P : MeasureTheory.Measure X) [inst_5 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_6 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal), MeasureTheory.Measure.map Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks (MeasureTheory.Measure.map p.restrictPartition (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam)) = Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalMarkedPoissonSampleLaw P R lam","l":815,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.map_totalizedPrefix_restrict_nonoverflow","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"For an observation probability law, a Poisson mean, for a fixed sample size, and an arbitrary overflow totalization, the totalized prefix of that many iid observations and an independent Poisson count, restricted to nonoverflow, has exactly the finite Poisson sample law restricted to counts at most n.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) (overflow : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), MeasureTheory.Measure.map (fun z => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.totalizedPrefix overflow z.1 z.2) (((MeasureTheory.Measure.pi fun x => P).prod (ProbabilityTheory.poissonMeasure lambda)).restrict (Prod.snd ⁻¹' Set.Iic n)) = (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Iic n)","l":299,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKey_decode","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"A key belonging to the mark-ordering set is recovered by decoding its stored original index and rebuilding its mark-and-index pair.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) {z : Lex (ℝ × Fin s.count)}, z ∈ Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys s → toLex ((s.points (ofLex z).2).2, (ofLex z).2) = z","l":199,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a finite sample of observation--real-mark pairs, the mark-ordering keys are the finite set of lexicographically ordered pairs consisting of each real mark and its original sample position; the position distinguishes pairs with equal marks.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Finset (Lex (ℝ × Fin s.count))","l":183,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys_card","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The number of lexicographic mark-and-index keys equals the number of marked observations in the finite sample.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedKeys s).card = s.count","l":189,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.markedPoissonKL_le_two_mul_of_piKL","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Consider a sample size n that is at least 1 and a nonnegative KL budget B, and suppose the KL divergence between n independent identically distributed draws from P and from Q is at most B. Then the KL divergence between the marked Poisson experiments with mean count 2n, mark law R, and intensity measures P and Q respectively (both built over the same baseline P) is at most 2B.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [StandardBorelSpace X] (P Q : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P] [inst_3 : MeasureTheory.IsProbabilityMeasure Q] (R : MeasureTheory.Measure ℝ) [inst_4 : MeasureTheory.IsProbabilityMeasure R] (n : ℕ), 1 ≤ n → ∀ {B : ℝ}, 0 ≤ B → InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q) ≤ ENNReal.ofReal B → InformationTheory.klDiv (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw P P R (2 * ↑n)) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMeasureMarkedPoissonLaw Q P R (2 * ↑n)) ≤ ENNReal.ofReal (2 * B)","l":69,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_canonicalPrefixObservations","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Reading a fixed prefix from a canonical finite configuration is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x₀ : X) (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.canonicalPrefixObservations x₀ n)","l":133,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_cappedPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For fixed array length, the option-valued capped prefix is jointly measurable in the array and requested count, with the left summand recording every overflow outcome.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (n : ℕ), Measurable fun z => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.cappedPrefix z.1 z.2","l":226,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_fiberPoints","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Reading off the points of every component sample on the fixed-count event is a measurable map.","s":"∀ {ι : Type u_2} [Fintype ι] [inst : MeasurableSpace ι] [MeasurableSingletonClass ι] {Y : Type u_3} [inst : MeasurableSpace Y] (c : ι → ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fiberPoints c)","l":131,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_finiteSampleMap","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Pointwise mapping of dependent finite samples is measurable.","s":"∀ {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (f : X → Y), Measurable f → Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSampleMap f)","l":32,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_finiteSample_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Reading the count of a finite sequence is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count","l":60,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_fixedCountSuperpose","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Concatenating a family of point tuples of fixed sizes into one finite sample is a measurable map.","s":"∀ {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace ι] [MeasurableSingletonClass ι] {Y : Type u_3} [inst_3 : MeasurableSpace Y] (c : ι → ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedCountSuperpose c)","l":113,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_fixedSizeEmbed","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Embedding a fixed-size tuple into the finite-sequence space is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.fixedSizeEmbed n)","l":53,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_orderByMarks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The mark-ordering map is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks","l":225,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_pointsOfCount","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Reading the n points off a finite sample of known size n is a measurable map into the space of n-tuples.","s":"∀ {Y : Type u_3} [inst : MeasurableSpace Y] (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.pointsOfCount n)","l":59,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_prefixOfLE","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For a fixed admissible length, taking that prefix is a measurable map of the fixed array.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {n m : ℕ} (h : m ≤ n), Measurable fun x => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.prefixOfLE x m h","l":217,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_prefixPointOr","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Reading off the k-th point of a finite sample, with a fallback value when the sample is too short, is a measurable map.","s":"∀ {Y : Type u_2} [inst : MeasurableSpace Y] (y₀ : Y) (n : ℕ) (k : Fin n), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.prefixPointOr y₀ n k)","l":38,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_retainedObservations","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Retaining and forgetting marks is a measurable map to an n-tuple.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x₀ : X) (n : ℕ), Measurable (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n)","l":55,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_streamToFiniteSample","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Truncating a count-and-stream outcome to its selected prefix is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample","l":76,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_superpose","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Finite superposition is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose","l":157,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_superposeByMarks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Mark-ordered finite superposition is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι], Measurable Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks","l":328,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.measurable_totalizedPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For a specified overflow sample, the map from a fixed array and count to its totalized capped prefix is measurable.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {n : ℕ} (overflow : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), Measurable fun z => Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.totalizedPrefix overflow z.1 z.2","l":242,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a finite measure on the observation space and a fallback probability measure on that space, the normalized finite measure is the fallback probability measure when the finite measure is zero, and otherwise is that finite measure divided by its total mass.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (ν : MeasureTheory.Measure X) → [MeasureTheory.IsFiniteMeasure ν] → (P₀ : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P₀] → MeasureTheory.Measure X","l":333,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.KL","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For every measurable observation space, finite measure on that space, and probability measure on the same space, the normalized finite measure is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (ν : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsFiniteMeasure ν] (P₀ : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure P₀], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalizedFiniteMeasure ν P₀)","l":339,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/KL.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.normalized_map_retainedObservations_restrict_count_ge","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Conditioning on enough points gives i.i.d. draws. Fix a nonnegative intensity lam and suppose the Poisson(lam) probability of observing at least n points is nonzero. Under the marked Poisson sample law with base probability measure P, mark distribution R, and intensity lam, condition on the event that the sample count is at least n, retain the n mark-smallest points and forget their marks (x₀ is an irrelevant filler value used only outside this event); the resulting normalised law equals the product of n independent copies of P.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (R : MeasureTheory.Measure ℝ) [inst_2 : MeasureTheory.IsProbabilityMeasure R] [MeasureTheory.NullSingletonClass R] (lam : NNReal) (x₀ : X) (n : ℕ), (ProbabilityTheory.poissonMeasure lam) (Set.Ici n) ≠ 0 → ((ProbabilityTheory.poissonMeasure lam) (Set.Ici n))⁻¹ • MeasureTheory.Measure.map (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations x₀ n) ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteMarkedPoissonSampleLaw P R lam).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Ici n)) = MeasureTheory.Measure.pi fun x => P","l":245,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a finite sample of observation--real-mark pairs, the mark-ordered sample contains the same pairs arranged in increasing order of their real marks, with equal marks ordered by original sample position.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":211,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Ordering by marks preserves the sample count.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).count = s.count","l":220,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_monotone_marks","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"Marks in the ordered sequence are nondecreasing.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) {a b : Fin s.count}, a ≤ b → ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).points (Fin.cast (Eq.symm (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count s)) a)).2 ≤ ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks s).points (Fin.cast (Eq.symm (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.orderByMarks_count s)) b)).2","l":860,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.pointsOfCount","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given a nonnegative integer and a finite sample known to have exactly that many points, the points at the specified count are the resulting tuple of its $n$ points.","s":"{Y : Type u_3} → [inst : MeasurableSpace Y] → (n : ℕ) → { s // s.count = n } → Fin n → Y","l":54,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a probability measure on a measurable sample space and a nonnegative Poisson rate, the joint count-and-stream law is the product law of a Poisson count with that rate and an independent infinite stream whose coordinates are independent with the given common distribution.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → NNReal → MeasureTheory.Measure (ℕ × (ℕ → X))","l":68,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"For every measurable sample space, probability measure on that space, and nonnegative Poisson rate, the joint law of an independent Poisson count and an independent and identically distributed stream is a probability measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam)","l":76,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_map_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The count coordinate has the requested scalar Poisson law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.Measure.map Prod.fst (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam) = ProbabilityTheory.poissonMeasure lam","l":82,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw_map_finPrefix","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.IIDPoisson","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"Every finite stream prefix remains an exact product sample after pairing the stream with an independent Poisson count.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal) (n : ℕ), MeasureTheory.Measure.map (fun z i => z.2 ↑i) (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonIIDStreamLaw P lam) = MeasureTheory.Measure.pi fun x => P","l":89,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/IIDPoisson.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poissonMeasure_singleton_eq_poissonPMF","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The Poisson law with mean r gives the single count n exactly the Poisson probability mass at n.","s":"∀ (r : NNReal) (n : ℕ), (ProbabilityTheory.poissonMeasure r) {n} = (ProbabilityTheory.poissonPMF r) n","l":21,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.poisson_two_n_lower_tail","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"The lower tail used in de-Poissonization is exponentially small.","s":"∀ (n : ℕ), (ProbabilityTheory.poissonMeasure (2 * ↑n)) {k | k < n} ≤ ENNReal.ofReal (Real.exp (-↑n * (1 - Real.log 2)))","l":188,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.prefixOfLE","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a fixed array, a requested prefix length, and a proof that the request fits in the array, the finite sample consisting of exactly that prefix retains the first m coordinates.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → (Fin n → X) → (m : ℕ) → m ≤ n → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":197,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.prefixPointOr","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"For a fallback sample value, a nonnegative integer prefix length, a position among the first $n$ positions, and a finite sample, the prefix point with fallback is the sample point at that position when the sample contains at least $n$ points, and is otherwise the fallback value.","s":"{Y : Type u_2} → [inst : MeasurableSpace Y] → Y → (n : ℕ) → Fin n → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample Y → Y","l":30,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.retainedObservations","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Retention","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"For a fallback sample value, a nonnegative integer prefix length, and a finite sample of value--mark pairs, the retained-observations vector consists of the values attached to the first $n$ sample points after ordering by their marks when the sample contains at least $n$ points, and otherwise consists entirely of the fallback value.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → X → (n : ℕ) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ) → Fin n → X","l":19,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Retention.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"Given a pair consisting of a nonnegative integer and an infinite observation stream, the associated finite sample has that integer as its size and retains exactly the corresponding initial segment of the stream.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → ℕ × (ℕ → X) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":49,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample_paddedStream","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Depoissonization","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (x0 : X) (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.streamToFiniteSample (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finiteSamplePaddedStream x0 s) = s","l":274,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Depoissonization.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given one finite sample of observation--real-mark pairs for every member of a finite index set, the superposed finite sample contains all of their pairs, enumerated through the disjoint union of their coordinate positions.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → (ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":23,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superposeByMarks","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":1,"d":"Given one finite sample of observation--real-mark pairs for every member of a finite index set, the mark-ordered superposition is the superposition of those samples arranged in increasing order of real mark, with ties broken by original position.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {ι : Type u_2} → [Fintype ι] → (ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)) → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)","l":323,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose_count","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Superposition.Canonical","a":"Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition","t":2,"d":"The count after finite superposition is the sum of the cell counts.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : MeasurableSpace ι] [MeasurableSingletonClass ι] (q : ι → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample (X × ℝ)), (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.superpose q).count = ∑ j, (q j).count","l":33,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Superposition/Canonical.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.totalizedPrefix","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.Basic","a":"Mathlib/Probability/FiniteMarkedPoissonPartition","t":1,"d":"For an overflow finite sample, a fixed array, and a requested length, the totalized capped prefix agrees with the genuine prefix off overflow and uses the specified value on overflow.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → (Fin n → X) → ℕ → Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X","l":210,"f":"Causalean/Mathlib/Probability/FiniteMarkedPoissonPartition/Basic.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":1,"d":"Given an upper bound and a real-valued function, the function has pairwise oscillation at most that bound when every two values differ by at most it.","s":"{S : Type u_1} → ℝ → (S → ℝ) → Prop","l":37,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound.mono","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"If a function has oscillation at most A and A is at most B, the function has oscillation at most B.","s":"∀ {S : Type u_1} {A B : ℝ} {f : S → ℝ}, Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound A f → A ≤ B → Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound B f","l":193,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.abs_sum_sub_mul_le_oscillation_halfL1","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"Given two finite probability vectors and a function with oscillation at most B, the difference of their expectations is at most B times half their ℓ¹ distance.","s":"∀ {S : Type u_1} [inst : Fintype S] [Nonempty S] {p q : S → ℝ}, Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector q → ∀ {B : ℝ} {f : S → ℝ}, Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound B f → |∑ s, (p s - q s) * f s| ≤ B * (1 / 2 * ∑ s, |p s - q s|)","l":48,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.halfL1_indicator_sub_indicator_le_one","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"For two states, half the ℓ¹ distance between their point masses is at most one.","s":"∀ {S : Type u_1} [inst : Fintype S] [inst_1 : DecidableEq S] (x y : S), 1 / 2 * ∑ z, |(if z = x then 1 else 0) - if z = y then 1 else 0| ≤ 1","l":102,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.isProbabilityVector_indicator","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"For a state, its point mass is a finite probability vector.","s":"∀ {S : Type u_1} [inst : Fintype S] [inst_1 : DecidableEq S] (x : S), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector fun y => if y = x then 1 else 0","l":94,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperator","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":1,"d":"Given a finite transition matrix, a real-valued function, and a starting state, the backward Markov operator is the transition-weighted sum of the function.","s":"{S : Type u_1} → [Fintype S] → (S → S → ℝ) → (S → ℝ) → S → ℝ","l":17,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperatorIter","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":1,"d":"Given a finite transition matrix, the iterated backward Markov operator applies the matrix a specified number of times to a real-valued function.","s":"{S : Type u_1} → [Fintype S] → (S → S → ℝ) → ℕ → (S → ℝ) → S → ℝ","l":23,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperatorIter_succ","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"For a finite transition matrix, a step count, and a function, the successor iterate is one backward Markov step after the preceding iterate.","s":"∀ {S : Type u_1} [inst : Fintype S] (P : S → S → ℝ) (n : ℕ) (f : S → ℝ), Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperatorIter P (n + 1) f = Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperator P (Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperatorIter P n f)","l":30,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.nonemptyOfProbabilityVector","k":"def","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":1,"d":"Given a real vector on a finite carrier that is a probability vector, the carrier is nonempty.","s":"∀ {S : Type u_1} [inst : Fintype S] (p : S → ℝ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Nonempty S","l":199,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.oscillationBound_markovOperator","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"Given a finite stochastic matrix, a nonnegative contraction coefficient, half-ℓ¹ contraction of its forward action, and a function with oscillation at most B, one backward Markov step has oscillation at most alpha * B.","s":"∀ {S : Type u_1} [inst : Fintype S] [Nonempty S] (P : S → S → ℝ), (∀ (s : S), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector (P s)) → ∀ {alpha B : ℝ}, 0 ≤ alpha → (∀ (p q : S → ℝ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector q → 1 / 2 * ∑ s, |Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep p P s - Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep q P s| ≤ alpha * (1 / 2 * ∑ s, |p s - q s|)) → ∀ {f : S → ℝ}, Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound B f → Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound (alpha * B) (Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperator P f)","l":131,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.oscillationBound_markovOperatorIter","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":1,"d":"Given a finite stochastic matrix, a nonnegative contraction coefficient, half-ℓ¹ contraction of its forward action, an initial oscillation bound, and a number of steps, the iterated backward operator's oscillation is bounded by the coefficient raised to the number of steps times the initial bound, which decays geometrically when the coefficient is below one.","s":"∀ {S : Type u_1} [inst : Fintype S] [Nonempty S] (P : S → S → ℝ), (∀ (s : S), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector (P s)) → ∀ {alpha B : ℝ}, 0 ≤ alpha → (∀ (p q : S → ℝ), Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector p → Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.IsProbabilityVector q → 1 / 2 * ∑ s, |Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep p P s - Causalean.Mathlib.Probability.CertifiedFiniteMarkovExpectation.markovStep q P s| ≤ alpha * (1 / 2 * ∑ s, |p s - q s|)) → ∀ {f : S → ℝ}, Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound B f → ∀ (n : ℕ), Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound (alpha ^ n * B) (Causalean.Mathlib.Probability.FiniteMarkovOscillation.markovOperatorIter P n f)","l":174,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.FiniteMarkovOscillation.oscillationBound_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteMarkovOscillation","a":"Mathlib/Probability","t":2,"d":"If a function has an oscillation bound, that bound is nonnegative.","s":"∀ {S : Type u_1} [Nonempty S] {B : ℝ} {f : S → ℝ}, Causalean.Mathlib.Probability.FiniteMarkovOscillation.OscillationBound B f → 0 ≤ B","l":42,"f":"Causalean/Mathlib/Probability/FiniteMarkovOscillation.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.SupportedOnUnitInterval","k":"def","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":1,"d":"For a real-valued kernel, the unit-interval support property says that each parameter-specific fiber measure assigns mass one to the closed interval from zero to one. The kernel is not required to be Markov here; for a Markov kernel, as in the realization theorems, this says each fiber is a probability distribution supported on that interval.","s":"{S : Type u_1} → [inst : MeasurableSpace S] → ProbabilityTheory.Kernel S ℝ → Prop","l":27,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.ae_quantileRealization_mem_unitInterval","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"For a real-valued Markov kernel and a parameter value, the realization section lies in the closed unit interval almost everywhere under the uniform law.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) (s : S), ∀ᵐ (u : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc 0 1), Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization κ (s, u) ∈ Set.Icc 0 1","l":252,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.kernelUnitQuantile","k":"def","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":1,"d":"For a real-valued kernel, a parameter value, and a uniform level in the closed unit interval, the unit-interval-valued parameterized quantile is the supremum of the points x of the unit interval whose fiber mass on the closed interval from zero to x is strictly smaller than the supplied level. The kernel is not required to be Markov here; the quantile interpretation holds for Markov kernels, as assumed by the realization theorems.","s":"{S : Type u_1} → [inst : MeasurableSpace S] → ProbabilityTheory.Kernel S ℝ → S → ↑unitInterval → ↑unitInterval","l":36,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.map_kernelUnitQuantile","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"Given a real-valued Markov kernel, the condition that each fiber is supported on the closed unit interval, and a parameter value, pushing canonical uniform unit-interval volume through the fiber's quantile map exactly recovers that kernel fiber.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) [ProbabilityTheory.IsMarkovKernel κ], Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.SupportedOnUnitInterval κ → ∀ (s : S), MeasureTheory.Measure.map (fun u => ↑(Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.kernelUnitQuantile κ s u)) MeasureTheory.volume = κ s","l":130,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.map_quantileRealization","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":1,"d":"Given a real-valued Markov kernel, the condition that each fiber is supported on the closed unit interval, and a parameter value, pushing Lebesgue measure restricted to the closed unit interval through the real quantile section exactly recovers that kernel fiber.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) [ProbabilityTheory.IsMarkovKernel κ], Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.SupportedOnUnitInterval κ → ∀ (s : S), MeasureTheory.Measure.map (fun u => Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization κ (s, u)) (MeasureTheory.volume.restrict (Set.Icc 0 1)) = κ s","l":233,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.measurable_kernelUnitQuantile","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"For a real-valued Markov kernel that is Markov, the unit-interval generalized inverse is jointly measurable in the kernel parameter and uniform level.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) [ProbabilityTheory.IsMarkovKernel κ], Measurable (Function.uncurry (Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.kernelUnitQuantile κ))","l":66,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.measurable_quantileRealization","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"For a real-valued Markov kernel, the real-valued quantile realization is jointly measurable in its parameter and real randomization input.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) [ProbabilityTheory.IsMarkovKernel κ], Measurable (Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization κ)","l":106,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.measurable_quantileRealization_section","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"For a real-valued Markov kernel and a fixed parameter value, the resulting real-valued function of the randomization input is measurable.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) [ProbabilityTheory.IsMarkovKernel κ] (s : S), Measurable fun u => Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization κ (s, u)","l":120,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization","k":"def","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":1,"d":"For a real-valued kernel and a parameter--randomization pair, the real-valued quantile realization evaluates the unit-interval quantile after clamping the randomization coordinate to the closed unit interval.","s":"{S : Type u_1} → [inst : MeasurableSpace S] → ProbabilityTheory.Kernel S ℝ → S × ℝ → ℝ","l":48,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization_mem_unitInterval","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization","a":"Mathlib/Probability/Kernel","t":2,"d":"For a real-valued Markov kernel and a parameter--randomization pair, the corresponding realization value belongs to the closed unit interval.","s":"∀ {S : Type u_1} [inst : MeasurableSpace S] (κ : ProbabilityTheory.Kernel S ℝ) (p : S × ℝ), Causalean.Mathlib.Probability.Kernel.ParameterizedKernelQuantileRealization.quantileRealization κ p ∈ Set.Icc 0 1","l":58,"f":"Causalean/Mathlib/Probability/Kernel/ParameterizedKernelQuantileRealization.lean"},{"n":"Causalean.Mathlib.Probability.NormalizedRestrictedBoundedTestFactorization","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, two measurable value spaces, a measure, a cell, and two random elements, the normalized restricted bounded-test factorization condition is bounded-test factorization of those random elements under the normalized restriction of the measure to the cell.","s":"{Ω : Type u_1} → {S : Type u_2} → {T : Type u_3} → [inst : MeasurableSpace Ω] → [MeasurableSpace S] → [MeasurableSpace T] → MeasureTheory.Measure Ω → Set Ω → (Ω → S) → (Ω → T) → Prop","l":117,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.ENNReal_tsum_lt_eq_tsum_fin","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"For a nonnegative doubly indexed series, summing its strict upper triangle by endpoints equals summing its finite rows.","s":"∀ (a : ℕ → ℕ → ENNReal), (∑' (m : ℕ) (n : ℕ), if n < m then a n m else 0) = ∑' (m : ℕ) (n : Fin m), a (↑n) m","l":66,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.ENNReal_tsum_upperTriangle_gap","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"For a nonnegative doubly indexed series, the strict upper triangle has equal sums under the gap and endpoint parametrizations.","s":"∀ (a : ℕ → ℕ → ENNReal), ∑' (n : ℕ) (k : ℕ), a n (n + k + 1) = ∑' (m : ℕ) (n : Fin m), a (↑n) m","l":88,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A real sequence at an index has a forward add-one increment, given by the next value minus the current value.","s":"(ℕ → ℝ) → ℕ → ℝ","l":26,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.coordinateAddOne","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A coordinate, real-valued statistic, and count-vector observation determine the statistic's add-one increment in that coordinate, given by increasing the selected count by one and subtracting the original statistic.","s":"{iota : Type u_1} → [DecidableEq iota] → iota → ((iota → ℕ) → ℝ) → (iota → ℕ) → ℝ","l":33,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.coordinateReplace","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A coordinate, dependent-product observation, and replacement value determine the observation with that coordinate replaced, given by updating exactly the selected coordinate.","s":"{iota : Type u_1} → [DecidableEq iota] → {X : iota → Type u_2} → (i : iota) → ((j : iota) → X j) → X i → (j : iota) → X j","l":27,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.finiteSupport_addOne","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"The add-one increment of a real sequence with finite support also has finite support.","s":"∀ (f : ℕ → ℝ), (Function.support f).Finite → (Function.support (Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f)).Finite","l":112,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.memLp_poissonMeasure_of_finiteSupport","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, a real sequence with finite support is square-integrable under the Poisson law.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), (Function.support f).Finite → MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure lambda)","l":96,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneFst","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A first array coordinate, second array coordinate, real-valued nested-array statistic, and nested count-pair observation determine the increment from adding one to that cell's first count, given by updating the selected first count and subtracting the original statistic.","s":"{iota : Type u_1} → {kappa : Type u_2} → [DecidableEq iota] → [DecidableEq kappa] → iota → kappa → ((iota → kappa → ℕ × ℕ) → ℝ) → (iota → kappa → ℕ × ℕ) → ℝ","l":619,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneSnd","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A first array coordinate, second array coordinate, real-valued nested-array statistic, and nested count-pair observation determine the increment from adding one to that cell's second count, given by updating the selected second count and subtracting the original statistic.","s":"{iota : Type u_1} → {kappa : Type u_2} → [DecidableEq iota] → [DecidableEq kappa] → iota → kappa → ((iota → kappa → ℕ × ℕ) → ℝ) → (iota → kappa → ℕ × ℕ) → ℝ","l":628,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"Two arrays of nonnegative Poisson rates determine the nested product law of independent Poisson-count pairs, given by independent two-count product laws in every array cell.","s":"{iota : Type u_1} → {kappa : Type u_2} → [Fintype iota] → [Fintype kappa] → (iota → kappa → NNReal) → (iota → kappa → NNReal) → MeasureTheory.Measure (iota → kappa → ℕ × ℕ)","l":611,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoisson_addOne_poincare","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"For two arrays of nonnegative Poisson rates, a real-valued nested-array statistic that is square-integrable, with first-count add-one increments square-integrable and second-count add-one increments square-integrable, has variance bounded by the sum of the two rate-weighted expected squared increments in every cell.","s":"∀ {iota : Type u_1} {kappa : Type u_2} [inst : Fintype iota] [inst_1 : DecidableEq iota] [inst_2 : Fintype kappa] [inst_3 : DecidableEq kappa] (lambda₁ lambda₂ : iota → kappa → NNReal) (F : (iota → kappa → ℕ × ℕ) → ℝ), MeasureTheory.MemLp F 2 (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂) → (∀ (i : iota) (j : kappa), MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneFst i j F) 2 (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂)) → (∀ (i : iota) (j : kappa), MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneSnd i j F) 2 (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂)) → ProbabilityTheory.variance F (Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂) ≤ ∑ i, ∑ j, (↑(lambda₁ i j) * ∫ (x : iota → kappa → ℕ × ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneFst i j F x ^ 2 ∂Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂ + ↑(lambda₂ i j) * ∫ (x : iota → kappa → ℕ × ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairAddOneSnd i j F x ^ 2 ∂Causalean.Mathlib.Probability.PoissonAddOnePoincare.nestedPairedPoissonMeasure lambda₁ lambda₂)","l":663,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A finite collection of nonnegative Poisson rates determines the product law of independent Poisson counts, given by the coordinatewise product of their Poisson laws.","s":"{iota : Type u_1} → [Fintype iota] → (iota → NNReal) → MeasureTheory.Measure (iota → ℕ)","l":39,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi_addOne_poincare","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"For a finite collection of nonnegative Poisson rates, a real-valued statistic of the count vector that is square-integrable and whose coordinate add-one increments are square-integrable has variance bounded by the sum of rate-weighted expected squared coordinate increments.","s":"∀ {iota : Type u_1} [inst : Fintype iota] [inst_1 : DecidableEq iota] (lambda : iota → NNReal) (F : (iota → ℕ) → ℝ), MeasureTheory.MemLp F 2 (Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi lambda) → (∀ (i : iota), MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.coordinateAddOne i F) 2 (Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi lambda)) → ProbabilityTheory.variance F (Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi lambda) ≤ ∑ i, ↑(lambda i) * ∫ (x : iota → ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.coordinateAddOne i F x ^ 2 ∂Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonPi lambda","l":559,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A nonnegative Poisson rate and count value determine the real Poisson probability mass at that count, given by the standard exponential--factorial mass formula.","s":"NNReal → ℕ → ℝ","l":25,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight_eq_measureReal_singleton","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, the singleton probability of a count equals its explicit real Poisson mass.","s":"∀ (lambda : NNReal) (n : ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda n = (ProbabilityTheory.poissonMeasure lambda).real {n}","l":42,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight_hasSum_one","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, the Poisson probability masses sum to one.","s":"∀ (lambda : NNReal), HasSum (Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda) 1","l":35,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, the mass at any count is nonnegative.","s":"∀ (lambda : NNReal) (n : ℕ), 0 ≤ Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda n","l":29,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight_shift","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, a positive count satisfies the predecessor form of the Poisson size-bias shift identity.","s":"∀ (lambda : NNReal) {n : ℕ}, 0 < n → ↑n * Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda n = ↑lambda * Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda (n - 1)","l":58,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight_succ_shift","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate and a count, successive Poisson masses obey the size-bias shift identity.","s":"∀ (lambda : NNReal) (n : ℕ), ↑(n + 1) * Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda (n + 1) = ↑lambda * Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda n","l":48,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poisson_addOne_poincare","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Scalar","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"At a nonnegative Poisson rate, a real sequence whose values are square-integrable and whose add-one increments are square-integrable has variance no larger than the rate times the expected squared add-one increment.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure lambda) → MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f) 2 (ProbabilityTheory.poissonMeasure lambda) → ProbabilityTheory.variance f (ProbabilityTheory.poissonMeasure lambda) ≤ ↑lambda * ∫ (n : ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f n ^ 2 ∂ProbabilityTheory.poissonMeasure lambda","l":782,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Scalar.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poisson_addOne_poincare_finiteSupport","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Scalar","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, a real sequence with finite support has variance no larger than the rate times the expected squared add-one increment.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), (Function.support f).Finite → ProbabilityTheory.variance f (ProbabilityTheory.poissonMeasure lambda) ≤ ↑lambda * ∫ (n : ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f n ^ 2 ∂ProbabilityTheory.poissonMeasure lambda","l":695,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Scalar.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.poisson_addOne_poincare_tsum","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Scalar","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, a real sequence whose values are square-integrable and whose add-one increments are square-integrable has the Poisson Poincaré variance bound in explicit countable-series form.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure lambda) → MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f) 2 (ProbabilityTheory.poissonMeasure lambda) → ProbabilityTheory.variance f (ProbabilityTheory.poissonMeasure lambda) ≤ ↑lambda * ∑' (n : ℕ), Causalean.Mathlib.Probability.PoissonAddOnePoincare.poissonWeight lambda n * Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f n ^ 2","l":816,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Scalar.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.sq_sub_le_nat_sub_mul_sum_sq_step","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Series","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"For a real sequence and ordered endpoints, the squared endpoint difference is bounded by path length times the sum of squared adjacent increments.","s":"∀ (f : ℕ → ℝ) {n m : ℕ}, n ≤ m → (f m - f n) ^ 2 ≤ ↑(m - n) * ∑ k ∈ Finset.Ico n m, (f (k + 1) - f k) ^ 2","l":124,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Series.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation","k":"def","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"A cutoff index and real sequence determine its support truncation, given by retaining exactly the values below the cutoff and setting all later values to zero.","s":"ℕ → (ℕ → ℝ) → ℕ → ℝ","l":82,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation_addOne_integral_sq_tendsto_zero","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, support truncations of a real sequence whose values are square-integrable and whose add-one increments are square-integrable converge to those increments in squared L² distance.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure lambda) → MeasureTheory.MemLp (Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f) 2 (ProbabilityTheory.poissonMeasure lambda) → Filter.Tendsto (fun K => ∫ (n : ℕ), (Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne (Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation K f) n - Causalean.Mathlib.Probability.PoissonAddOnePoincare.addOne f n) ^ 2 ∂ProbabilityTheory.poissonMeasure lambda) Filter.atTop (nhds 0)","l":163,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation_finiteSupport","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"A cutoff index of a real sequence has finite support.","s":"∀ (K : ℕ) (f : ℕ → ℝ), (Function.support (Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation K f)).Finite","l":86,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation_integral_sq_tendsto_zero","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"At a nonnegative Poisson rate, support truncations of a real sequence that is square-integrable converge to it in squared L² distance.","s":"∀ (lambda : NNReal) (f : ℕ → ℝ), MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure lambda) → Filter.Tendsto (fun K => ∫ (n : ℕ), (Causalean.Mathlib.Probability.PoissonAddOnePoincare.supportTruncation K f n - f n) ^ 2 ∂ProbabilityTheory.poissonMeasure lambda) Filter.atTop (nhds 0)","l":127,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.variance_eq_half_integral_prod_sq_sub","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":1,"d":"Under a probability law, a square-integrable real statistic has variance equal to one half of the expected squared difference between two independent draws.","s":"∀ {alpha : Type u_1} [inst : MeasurableSpace alpha] (mu : MeasureTheory.Measure alpha) [MeasureTheory.IsProbabilityMeasure mu] (f : alpha → ℝ), MeasureTheory.MemLp f 2 mu → ProbabilityTheory.variance f mu = 1 / 2 * ∫ (z : alpha × alpha), (f z.1 - f z.2) ^ 2 ∂mu.prod mu","l":30,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.variance_pi_le_sum_integral_coordinateVariance","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Tensorization","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"For a finite family of probability laws, a real-valued statistic of their product observation that is square-integrable has variance bounded by the sum of expected conditional variances from resampling one coordinate at a time.","s":"∀ {iota : Type u_1} [inst : Fintype iota] [inst_1 : DecidableEq iota] {X : iota → Type u_2} [inst_2 : (i : iota) → MeasurableSpace (X i)] (mu : (i : iota) → MeasureTheory.Measure (X i)) [∀ (i : iota), MeasureTheory.IsProbabilityMeasure (mu i)] (F : ((i : iota) → X i) → ℝ), MeasureTheory.MemLp F 2 (MeasureTheory.Measure.pi mu) → ProbabilityTheory.variance F (MeasureTheory.Measure.pi mu) ≤ ∑ i, ∫ (x : (i : iota) → X i), ProbabilityTheory.variance (fun y => F (Causalean.Mathlib.Probability.PoissonAddOnePoincare.coordinateReplace i x y)) (mu i) ∂MeasureTheory.Measure.pi mu","l":373,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Tensorization.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.variance_poissonMeasure_zero","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.Scalar","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"A real sequence that is square-integrable under the zero-rate Poisson law has zero variance.","s":"∀ (f : ℕ → ℝ), MeasureTheory.MemLp f 2 (ProbabilityTheory.poissonMeasure 0) → ProbabilityTheory.variance f (ProbabilityTheory.poissonMeasure 0) = 0","l":830,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/Scalar.lean"},{"n":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.variance_tendsto_of_integral_sq_sub_tendsto_zero","k":"theorem","m":"Causalean.Mathlib.Probability.PoissonAddOnePoincare.L2Closure","a":"Mathlib/Probability/PoissonAddOnePoincare","t":2,"d":"Under a probability law, a sequence of real statistics and a limiting statistic, with each sequence member square-integrable, the limit square-integrable, and squared L² distance tending to zero, have variances converging to the limiting variance.","s":"∀ {alpha : Type u_1} [inst : MeasurableSpace alpha] (mu : MeasureTheory.Measure alpha) [MeasureTheory.IsProbabilityMeasure mu] (f : ℕ → alpha → ℝ) (g : alpha → ℝ), (∀ (K : ℕ), MeasureTheory.MemLp (f K) 2 mu) → MeasureTheory.MemLp g 2 mu → Filter.Tendsto (fun K => ∫ (x : alpha), (f K x - g x) ^ 2 ∂mu) Filter.atTop (nhds 0) → Filter.Tendsto (fun K => ProbabilityTheory.variance (f K) mu) Filter.atTop (nhds (ProbabilityTheory.variance g mu))","l":207,"f":"Causalean/Mathlib/Probability/PoissonAddOnePoincare/L2Closure.lean"},{"n":"Causalean.Mathlib.Probability.ae_normalizedRestrict_iff","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"On a positive-mass cell, an almost-sure assertion holds under the normalized cell law exactly when it holds under the unnormalized restricted measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {C : Set Ω}, 0 < P C → ∀ {p : Ω → Prop}, (∀ᵐ (ω : Ω) ∂Causalean.Mathlib.Probability.normalizedRestrict P C, p ω) ↔ ∀ᵐ (ω : Ω) ∂P.restrict C, p ω","l":86,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"For any real number interpreted as a success parameter, the Boolean-valued Bernoulli measure assigns mass $\\max(p,0)$ to true and mass $\\max(1-p,0)$ to false.","s":"ℝ → MeasureTheory.Measure Bool","l":272,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_bind","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Drawing a Bool-valued Bernoulli variable and then selecting a distribution according to its value produces the corresponding success-probability mixture of the two distributions.","s":"∀ {β : Type u_1} [inst : MeasurableSpace β] (p : ℝ) (K : Bool → MeasureTheory.Measure β), (Causalean.Mathlib.Probability.bernoulliBool p).bind K = ENNReal.ofReal p • K true + ENNReal.ofReal (1 - p) • K false","l":320,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_integral","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The expectation of a real-valued function of a Bool-valued Bernoulli draw is its value at success times the success probability plus its value at failure times the failure probability.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → ∀ (f : Bool → ℝ), ∫ (z : Bool), f z ∂Causalean.Mathlib.Probability.bernoulliBool p = p * f true + (1 - p) * f false","l":302,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bool-valued Bernoulli distribution is a probability distribution whenever its success probability lies between zero and one.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.bernoulliBool p)","l":288,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliBool_map","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Transforming a Bool-valued Bernoulli draw produces a two-point distribution concentrated on the transformed success and failure values with their original probabilities.","s":"∀ {β : Type u_1} [inst : MeasurableSpace β] (p : ℝ) (f : Bool → β), MeasureTheory.Measure.map f (Causalean.Mathlib.Probability.bernoulliBool p) = ENNReal.ofReal p • MeasureTheory.Measure.dirac (f true) + ENNReal.ofReal (1 - p) • MeasureTheory.Measure.dirac (f false)","l":332,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"For any real number interpreted as a success parameter, the real-valued Bernoulli measure assigns mass $\\max(p,0)$ to $1$ and mass $\\max(1-p,0)$ to $0$.","s":"ℝ → MeasureTheory.Measure ℝ","l":34,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ac_of_reference_interior","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bernoulli law is absolutely continuous with respect to any Bernoulli law whose success probability is strictly between zero and one.","s":"∀ {p q : ℝ}, 0 < q → q < 1 → (Causalean.Mathlib.Probability.bernoulliLaw p).AbsolutelyContinuous (Causalean.Mathlib.Probability.bernoulliLaw q)","l":107,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_le_one","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bernoulli law on the real line is almost surely at most one, since all of its mass is placed at 0 and 1.","s":"∀ {p : ℝ}, (fun y => y) ≤ᵐ[Causalean.Mathlib.Probability.bernoulliLaw p] fun x => 1","l":85,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bernoulli law on the real line is almost surely nonnegative, since all of its mass is placed at 0 and 1.","s":"∀ {p : ℝ}, 0 ≤ᵐ[Causalean.Mathlib.Probability.bernoulliLaw p] fun y => y","l":72,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_ae_zero_or_one","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A draw from the real-valued Bernoulli law is almost surely either 0 or 1.","s":"∀ {p : ℝ}, ∀ᵐ (y : ℝ) ∂Causalean.Mathlib.Probability.bernoulliLaw p, y = 0 ∨ y = 1","l":98,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_integral","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"Two-point integral for the custom bernoulliLaw: since bernoulliLaw p = ENNReal.ofReal p • dirac 1 + ENNReal.ofReal (1-p) • dirac 0, its integral splits via integral_add_measure / integral_smul_measure / integral_dirac into the two-point weighted sum, with 0 ≤ p ≤ 1 collapsing the ENNReal → ℝ coercions to p and 1 - p.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → ∀ (f : ℝ → ℝ), ∫ (y : ℝ), f y ∂Causalean.Mathlib.Probability.bernoulliLaw p = p * f 1 + (1 - p) * f 0","l":40,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The custom bernoulliLaw is a probability measure for 0 ≤ p ≤ 1: its total mass is p + (1 - p) = 1.","s":"∀ {p : ℝ}, 0 ≤ p → p ≤ 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.bernoulliLaw p)","l":59,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_klDiv_le_four_sq_sub","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"If both success probabilities $p$ and $q$ lie in the middle half of the unit interval, $1/4 \\le p \\le 3/4$ and $1/4 \\le q \\le 3/4$, then the Kullback–Leibler divergence from the Bernoulli($p$) law to the Bernoulli($q$) law is at most four times the squared difference of the probabilities, $4(p-q)^2$.","s":"∀ {p q : ℝ}, 1 / 4 ≤ p → p ≤ 3 / 4 → 1 / 4 ≤ q → q ≤ 3 / 4 → InformationTheory.klDiv (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q) ≤ ENNReal.ofReal (4 * (p - q) ^ 2)","l":237,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_klDiv_toReal","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"Let $p$ be a Bernoulli success probability with $0 \\le p \\le 1$, and let $q$ be a reference success probability with $0 < q < 1$. Then the Kullback–Leibler divergence from the Bernoulli($p$) law to the Bernoulli($q$) law equals $p \\log(p/q) + (1-p)\\log((1-p)/(1-q))$, the usual two-point KL formula: a success contribution plus a failure contribution.","s":"∀ {p q : ℝ}, 0 ≤ p → p ≤ 1 → 0 < q → q < 1 → (InformationTheory.klDiv (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q)).toReal = p * Real.log (p / q) + (1 - p) * Real.log ((1 - p) / (1 - q))","l":155,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_lintegral_ofReal","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"ℝ≥0∞/lintegral analogue of bernoulliLaw_integral: a Bernoulli law integrates an ℝ≥0∞-valued function as the two-point weighted sum.","s":"∀ {p : ℝ} (f : ℝ → ENNReal), ∫⁻ (y : ℝ), f y ∂Causalean.Mathlib.Probability.bernoulliLaw p = ENNReal.ofReal p * f 1 + ENNReal.ofReal (1 - p) * f 0","l":263,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulliLaw_llr_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The log-likelihood ratio between any two real-parameter Bernoulli laws is integrable under the first law.","s":"∀ {p q : ℝ}, MeasureTheory.Integrable (MeasureTheory.llr (Causalean.Mathlib.Probability.bernoulliLaw p) (Causalean.Mathlib.Probability.bernoulliLaw q)) (Causalean.Mathlib.Probability.bernoulliLaw p)","l":134,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.bernoulli_mean_channel_kl","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":1,"d":"KL band for the signed two-point mean channel. For a strictly positive spread parameter B and two channel means u and v each confined to the interval [-B/2, B/2], the Kullback–Leibler divergence between the two-point channels with means u and v is bounded by the quadratic (u − v)²/B². This is the affine transport of the {0,1} Bernoulli KL band onto the {−B, B} mean parametrization.","s":"∀ (B u v : ℝ), 0 < B → |u| ≤ B / 2 → |v| ≤ B / 2 → InformationTheory.klDiv (Causalean.Mathlib.Probability.twoPointMean B u) (Causalean.Mathlib.Probability.twoPointMean B v) ≤ ENNReal.ofReal ((u - v) ^ 2 / B ^ 2)","l":170,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.bernstein","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":1,"d":"Bernstein's theorem (finite-variance Darmois–Skitovich, two variables). Let X and Y be real random variables that are independent and each have finite second moment. If their sum X + Y and difference X − Y are independent, then both X and Y have Gaussian laws.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → MeasureTheory.MemLp X 2 P → MeasureTheory.MemLp Y 2 P → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ProbabilityTheory.IsGaussian (MeasureTheory.Measure.map X P) ∧ ProbabilityTheory.IsGaussian (MeasureTheory.Measure.map Y P)","l":621,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.bernstein_charFun_funeq","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"Functional equation for Bernstein's theorem. If X and Y are independent and the pair (X + Y, X − Y) is independent, then the characteristic functions f = charFun (P.map X) and g = charFun (P.map Y) satisfy f (u+v) · g (u−v) = f u · g u · f v · g (−v) for all u, v.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ∀ (u v : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map X P) (u + v) * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) (u - v) = MeasureTheory.charFun (MeasureTheory.Measure.map X P) u * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) u * (MeasureTheory.charFun (MeasureTheory.Measure.map X P) v * MeasureTheory.charFun (MeasureTheory.Measure.map Y P) (-v))","l":58,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.bernstein_charFun_gaussian_nhds_zero","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"Analytic core of Bernstein's theorem. Under the Bernstein hypotheses with finite second moments, the characteristic functions of X and Y coincide, on a neighbourhood of 0, with Gaussian characteristic functions that share one variance σ² ≥ 0 (with means mf, mg). The shared σ² — a consequence of the single constant c from gaussianForm_of_funeq — is exactly what the doubling bootstrap in bernstein needs to extend the Gaussian form to all of ℝ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable X → Measurable Y → MeasureTheory.MemLp X 2 P → MeasureTheory.MemLp Y 2 P → ProbabilityTheory.IndepFun X Y P → ProbabilityTheory.IndepFun (X + Y) (X - Y) P → ∃ mf mg σ2, 0 ≤ σ2 ∧ (∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun (MeasureTheory.Measure.map X P) t = Complex.exp (↑mf * ↑t * Complex.I - ↑σ2 * ↑t ^ 2 / 2)) ∧ ∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun (MeasureTheory.Measure.map Y P) t = Complex.exp (↑mg * ↑t * Complex.I - ↑σ2 * ↑t ^ 2 / 2)","l":486,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.binomialWeight","k":"def","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":1,"d":"For a nonnegative number of trials, a real success parameter, and a nonnegative count, the binomial weight is $\\binom{m}{j}p^j(1-p)^{m-j}$.","s":"ℕ → ℝ → ℕ → ℝ","l":349,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.binomial_inverse_two_arms_interior_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"If the overlap margin is positive and the success probability lies between that margin and one minus the margin, the binomial expectation of the two inverse arm counts on the interior event is at most four divided by the trial count plus one times the margin.","s":"∀ (m : ℕ) (p epsilon : ℝ), 0 < epsilon → epsilon ≤ p → p ≤ 1 - epsilon → (∑ j ∈ Finset.range (m + 1), Causalean.Mathlib.Probability.binomialWeight m p j * if 0 < j ∧ j < m then (↑j)⁻¹ + (↑(m - j))⁻¹ else 0) ≤ 4 / (↑(m + 1) * epsilon)","l":526,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.binomial_totalized_inverse_count_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"When the success probability is positive and at most one, the binomial expectation of the zero-safe inverse success count is at most twice the reciprocal of the trial count plus one times that probability.","s":"∀ (m : ℕ) (p : ℝ), 0 < p → p ≤ 1 → (∑ j ∈ Finset.range (m + 1), Causalean.Mathlib.Probability.binomialWeight m p j * if 0 < j then (↑j)⁻¹ else 0) ≤ 2 / (↑(m + 1) * p)","l":443,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.charFun_contDiff_two","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"charFun of a pushforward with a finite second moment is C². This is a specialisation of contDiff_charFun (the characteristic function is Cⁿ whenever the n-th moment is finite), transferring MemLp Z 2 P to MemLp id 2 (P.map Z) along the pushforward map.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {Z : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], MeasureTheory.MemLp Z 2 P → ContDiff ℝ 2 (MeasureTheory.charFun (MeasureTheory.Measure.map Z P))","l":147,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.charFun_eventually_ne_zero","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":2,"d":"The characteristic function of a probability measure on ℝ is nonzero on a neighbourhood of 0: it is continuous and equals 1 at 0. This is what makes the logarithmic derivative (charFun μ)' / charFun μ well defined near 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], ∀ᶠ (t : ℝ) in nhds 0, MeasureTheory.charFun μ t ≠ 0","l":138,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.crossMomentMatrix","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, two finite vector dimensions, a measure, and two finite-dimensional real random vectors, the cross-moment matrix has at each ordered pair of coordinates the integral of the product of the corresponding coordinates of the two random vectors.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {m n : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin m → ℝ) → (Ω → Fin n → ℝ) → Matrix (Fin m) (Fin n) ℝ","l":189,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.designVector","k":"def","m":"Causalean.Mathlib.Probability.WeightedProduct","a":"Mathlib/Probability","t":1,"d":"For a map assigning each observation a design value and a finite sample of observations, the design vector assigns to every sample coordinate the design value of the observation at that coordinate.","s":"{N : ℕ} → {Omega : Type u_1} → {D : Type u_2} → (Omega → D) → (Fin N → Omega) → Fin N → D","l":24,"f":"Causalean/Mathlib/Probability/WeightedProduct.lean"},{"n":"Causalean.Mathlib.Probability.firstMomentVector","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, a finite vector dimension, a measure, and a finite-dimensional real random vector, the first-moment vector has at each coordinate the integral of the corresponding coordinate of the random vector.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {n : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin n → ℝ) → Fin n → ℝ","l":177,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.gaussianForm_of_funeq","k":"theorem","m":"Causalean.Mathlib.Probability.DarmoisSkitovich","a":"Mathlib/Probability","t":1,"d":"Pure analytic core: the Bernstein functional equation forces a Gaussian form. Let f, g : ℝ → ℂ be functions that are twice continuously differentiable and equal to 1 at the origin, and suppose they satisfy the Bernstein functional equation f (u+v) · g (u−v) = f u · g u · (f v · g (−v)) for all real u, v. Then there is a single constant c such that, on a neighbourhood of 0, f t = exp (f'(0)·t + c·t²/2) and g t = exp (g'(0)·t + c·t²/2), with the same c in both formulas.","s":"∀ {f g : ℝ → ℂ}, ContDiff ℝ 2 f → ContDiff ℝ 2 g → f 0 = 1 → g 0 = 1 → (∀ (u v : ℝ), f (u + v) * g (u - v) = f u * g u * (f v * g (-v))) → ∃ c, (∀ᶠ (t : ℝ) in nhds 0, f t = Complex.exp (deriv f 0 * ↑t + c * ↑t ^ 2 / 2)) ∧ ∀ᶠ (t : ℝ) in nhds 0, g t = Complex.exp (deriv g 0 * ↑t + c * ↑t ^ 2 / 2)","l":157,"f":"Causalean/Mathlib/Probability/DarmoisSkitovich.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_integral","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"An average of n independent, identically distributed observations has the same expectation as the population statistic being averaged.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (m : ℕ), 0 < m → ∀ (F : Ω → ℝ), MeasureTheory.Integrable F μ → (∫ (sample : Fin m → Ω), (↑m)⁻¹ * ∑ i, F (sample i) ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), F o ∂μ","l":362,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_integral_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"An average of independent, identically distributed observations has the same expectation as the population statistic being averaged.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ), MeasureTheory.Integrable F μ → (∫ (sample : ι → Ω), (↑(Fintype.card ι))⁻¹ * ∑ i, F (sample i) ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), F o ∂μ","l":227,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_variance","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The variance of an average of independent, identically distributed observations is the population variance divided by the sample size (including the zero-length case).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (m : ℕ) (F : Ω → ℝ), MeasureTheory.MemLp F 2 μ → ProbabilityTheory.variance (fun sample => (↑m)⁻¹ * ∑ i, F (sample i)) (MeasureTheory.Measure.pi fun x => μ) = (↑m)⁻¹ * ProbabilityTheory.variance F μ","l":375,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_average_variance_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The variance of an average of independent, identically distributed observations is the population variance divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ), MeasureTheory.MemLp F 2 μ → ProbabilityTheory.variance (fun sample => (↑(Fintype.card ι))⁻¹ * ∑ i, F (sample i)) (MeasureTheory.Measure.pi fun x => μ) = (↑(Fintype.card ι))⁻¹ * ProbabilityTheory.variance F μ","l":250,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean absolute error of a square-integrable scalar sample average of n independent, identically distributed observations is at most the square root of the population second moment divided by n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : Fin n → Ω), |(↑n)⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ| ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑n)","l":301,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_abs_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean absolute error of a square-integrable scalar sample average from independent, identically distributed observations is at most the square root of the population second moment divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : ι → Ω), |(↑(Fintype.card ι))⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ| ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑(Fintype.card ι))","l":140,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_euclidean_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":1,"d":"For a strictly positive sample size n and finitely many square-integrable real-valued statistics indexed by k, each observed on the same n-point independent, identically distributed sample, the expected Euclidean norm of the vector of centered sample averages — one coordinate per statistic — is at most the square root of the sum of the population second moments divided by n.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.MemLp (ξ k) 2 μ) → (∫ (s : Fin n → Ω), √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2) ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∑ k, ∫ (ω : Ω), ξ k ω ^ 2 ∂μ) / ↑n)","l":313,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_euclidean_abs_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The expected Euclidean error of finitely many square-integrable sample averages from the same independent, identically distributed sample is controlled by their summed population second moments and the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {κ : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] [inst_3 : Fintype κ] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : κ → Ω → ℝ), (∀ (k : κ), MeasureTheory.MemLp (ξ k) 2 μ) → (∫ (s : ι → Ω), √(∑ k, ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2) ∂MeasureTheory.Measure.pi fun x => μ) ≤ √((∑ k, ∫ (ω : Ω), ξ k ω ^ 2 ∂μ) / ↑(Fintype.card ι))","l":182,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_sq_le","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean squared error of a square-integrable scalar sample average of n independent, identically distributed observations is at most the population second moment divided by n.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ}, 0 < n → ∀ (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : Fin n → Ω), ((↑n)⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ) ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) ≤ (∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑n","l":290,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.iid_mean_sq_le_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The mean squared error of a square-integrable scalar sample average from independent, identically distributed observations is at most the population second moment divided by the sample size.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [Nonempty ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : Ω → ℝ), MeasureTheory.MemLp ξ 2 μ → (∫ (s : ι → Ω), ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ (s i) - ∫ (ω : Ω), ξ ω ∂μ) ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) ≤ (∫ (ω : Ω), ξ ω ^ 2 ∂μ) / ↑(Fintype.card ι)","l":77,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.indepFun_of_boundedTestFactorization","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"Under a probability law, measurable random elements whose all bounded measurable real-valued tests factor are independent.","s":"∀ {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace S] [inst_2 : MeasurableSpace T] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {X : Ω → S} {Y : Ω → T}, Measurable X → Measurable Y → Causalean.Mathlib.Probability.BoundedTestFactorization μ X Y → ProbabilityTheory.IndepFun X Y μ","l":131,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.integrable_euclidean_of_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite family of integrable real functions is itself integrable.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {μ : MeasureTheory.Measure Ω} (v : Ω → ι → ℝ), (∀ (i : ι), MeasureTheory.Integrable (fun ω => v ω i) μ) → MeasureTheory.Integrable (fun ω => √(∑ i, v ω i ^ 2)) μ","l":44,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.integrable_iid_mean_euclidean","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many integrable statistics over an n-point independent, identically distributed sample have integrable Euclidean length.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.Integrable (ξ k) μ) → MeasureTheory.Integrable (fun s => √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) (MeasureTheory.Measure.pi fun x => μ)","l":343,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.klDiv_map_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"KL-divergence is invariant under a measurable equivalence. Pushing both finite measures μ, ν forward through a measurable equivalence e leaves their Kullback–Leibler divergence unchanged: KL(e_* μ, e_* ν) = KL(μ, ν).","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (e : α ≃ᵐ β) (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν) = InformationTheory.klDiv μ ν","l":124,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.measurable_bernoulliBool","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The Bool-valued Bernoulli distribution varies measurably with its success probability, so a measurable probability parameter can be used to construct a measurable kernel.","s":"Measurable Causalean.Mathlib.Probability.bernoulliBool","l":281,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.measurable_designVector","k":"theorem","m":"Causalean.Mathlib.Probability.WeightedProduct","a":"Mathlib/Probability","t":2,"d":"For a design map that is measurable, the corresponding finite-sample vector of design values is measurable.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (design : Omega → D), Measurable design → Measurable (Causalean.Mathlib.Probability.designVector design)","l":31,"f":"Causalean/Mathlib/Probability/WeightedProduct.lean"},{"n":"Causalean.Mathlib.Probability.measurable_twoPointMean","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"For a fixed scale B, the signed two-point mean channel is measurable as a function of the target mean u.","s":"∀ (B : ℝ), Measurable fun u => Causalean.Mathlib.Probability.twoPointMean B u","l":44,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.measure_eq_sum_mul_cond_of_finite_partition","k":"theorem","m":"Causalean.Mathlib.Probability.FinitePartitionConditional","a":"Mathlib/Probability","t":2,"d":"Given a probability measure, a finite collection of cells, measurability of every cell, pairwise disjointness of distinct cells, coverage of the whole sample space, an event, and measurability of that event, its probability equals the finite sum of cell mass times conditional event probability.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (C : ι → Set Ω), (∀ (i : ι), MeasurableSet (C i)) → (∀ (i j : ι), i ≠ j → Disjoint (C i) (C j)) → ⋃ i, C i = Set.univ → ∀ (E : Set Ω), MeasurableSet E → μ E = ∑ i, μ (C i) * μ[E | C i]","l":19,"f":"Causalean/Mathlib/Probability/FinitePartitionConditional.lean"},{"n":"Causalean.Mathlib.Probability.measure_le_of_cond_le_on_finite_partition","k":"theorem","m":"Causalean.Mathlib.Probability.FinitePartitionConditional","a":"Mathlib/Probability","t":1,"d":"Given a probability measure, a finite collection of cells, measurability of every cell, pairwise disjointness of distinct cells, coverage of the whole sample space, an event, measurability of that event, a probability bound, and that bound conditional on each positive-mass cell, the unconditional event probability is at most the same bound.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (C : ι → Set Ω), (∀ (i : ι), MeasurableSet (C i)) → (∀ (i j : ι), i ≠ j → Disjoint (C i) (C j)) → ⋃ i, C i = Set.univ → ∀ (E : Set Ω), MeasurableSet E → ∀ (β : ENNReal), (∀ (i : ι), μ (C i) ≠ 0 → μ[E | C i] ≤ β) → μ E ≤ β","l":47,"f":"Causalean/Mathlib/Probability/FinitePartitionConditional.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_iid_mean_euclidean","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many square-integrable statistics over an n-point independent, identically distributed sample form a square-integrable Euclidean length.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : ι → Ω → ℝ), (∀ (k : ι), MeasureTheory.MemLp (ξ k) 2 μ) → MeasureTheory.MemLp (fun s => √(∑ k, ((↑n)⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) 2 (MeasureTheory.Measure.pi fun x => μ)","l":329,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_iid_mean_euclidean_fintype","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The centred coordinatewise sample averages of finitely many square-integrable statistics form a square-integrable Euclidean length on the product sample space.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {κ : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [inst_2 : Fintype κ] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (ξ : κ → Ω → ℝ), (∀ (k : κ), MeasureTheory.MemLp (ξ k) 2 μ) → MeasureTheory.MemLp (fun s => √(∑ k, ((↑(Fintype.card ι))⁻¹ * ∑ i, ξ k (s i) - ∫ (ω : Ω), ξ k ω ∂μ) ^ 2)) 2 (MeasureTheory.Measure.pi fun x => μ)","l":164,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.memLp_two_sqrt_sum_sq","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite family of square-integrable real functions is itself square-integrable.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {μ : MeasureTheory.Measure Ω} {Y : ι → Ω → ℝ}, (∀ (k : ι), MeasureTheory.MemLp (Y k) 2 μ) → MeasureTheory.MemLp (fun ω => √(∑ k, Y k ω ^ 2)) 2 μ","l":60,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestrict","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, a measure on it, and a cell in that sample space, the normalized restricted measure is the measure restricted to the cell and scaled by the reciprocal of the measure of that cell.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Set Ω → MeasureTheory.Measure Ω","l":26,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestrict_apply","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"For a positive-mass cell and a measurable event, the normalized cell law of that event is its intersection mass divided by the cell mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {C A : Set Ω}, 0 < P C → MeasurableSet A → (Causalean.Mathlib.Probability.normalizedRestrict P C) A = (P C)⁻¹ * P (A ∩ C)","l":64,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestrict_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"For a finite sampling measure, a measurable cell with strictly positive mass has a normalized restricted law that is a probability measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {C : Set Ω}, MeasurableSet C → 0 < P C → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.normalizedRestrict P C)","l":51,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestrictedIntegral","k":"def","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"Given a measurable sample space, a real normed vector-valued outcome space, a measure, a cell, and a function on the sample space, the normalized restricted integral is the integral of the function with respect to the normalized restricted measure of that cell.","s":"{Ω : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : NormedAddCommGroup E] → [NormedSpace ℝ E] → MeasureTheory.Measure Ω → Set Ω → (Ω → E) → E","l":38,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestrictedIntegral_eq","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"On a positive-mass cell, integrating a function under the normalized cell law is the restricted integral rescaled by the reciprocal cell mass.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P] {C : Set Ω}, 0 < P C → ∀ (f : Ω → E), Causalean.Mathlib.Probability.normalizedRestrictedIntegral P C f = (P C).toReal⁻¹ • ∫ (ω : Ω) in C, f ω ∂P","l":73,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestricted_coordinate_factorization","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":2,"d":"For a measurable positive-mass cell, measurable finite-coordinate random vectors, integrable individual coordinates, and factorization of every bounded measurable test under the normalized cell law, each coordinate product is integrable and its cell cross moment factors into the two cell first moments.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {C : Set Ω}, MeasurableSet C → 0 < P C → ∀ {m n : ℕ} {X : Ω → Fin m → ℝ} {Y : Ω → Fin n → ℝ}, Measurable X → Measurable Y → (∀ (i : Fin m), MeasureTheory.Integrable (fun ω => X ω i) (Causalean.Mathlib.Probability.normalizedRestrict P C)) → (∀ (j : Fin n), MeasureTheory.Integrable (fun ω => Y ω j) (Causalean.Mathlib.Probability.normalizedRestrict P C)) → Causalean.Mathlib.Probability.NormalizedRestrictedBoundedTestFactorization P C X Y → ∀ (i : Fin m) (j : Fin n), MeasureTheory.Integrable (fun ω => X ω i * Y ω j) (Causalean.Mathlib.Probability.normalizedRestrict P C) ∧ (Causalean.Mathlib.Probability.normalizedRestrictedIntegral P C fun ω => X ω i * Y ω j) = (Causalean.Mathlib.Probability.normalizedRestrictedIntegral P C fun ω => X ω i) * Causalean.Mathlib.Probability.normalizedRestrictedIntegral P C fun ω => Y ω j","l":203,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.normalizedRestricted_crossMomentMatrix_eq_outer","k":"theorem","m":"Causalean.Mathlib.Probability.FiniteCellConditionalMomentBridge","a":"Mathlib/Probability","t":1,"d":"For a measurable positive-mass cell, measurable finite-coordinate random vectors, integrable individual coordinates, and factorization of every bounded measurable test under the normalized cell law, the complete normalized cross-moment matrix is the outer product of the two normalized first-moment vectors.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {C : Set Ω}, MeasurableSet C → 0 < P C → ∀ {m n : ℕ} {X : Ω → Fin m → ℝ} {Y : Ω → Fin n → ℝ}, Measurable X → Measurable Y → (∀ (i : Fin m), MeasureTheory.Integrable (fun ω => X ω i) (Causalean.Mathlib.Probability.normalizedRestrict P C)) → (∀ (j : Fin n), MeasureTheory.Integrable (fun ω => Y ω j) (Causalean.Mathlib.Probability.normalizedRestrict P C)) → Causalean.Mathlib.Probability.NormalizedRestrictedBoundedTestFactorization P C X Y → Causalean.Mathlib.Probability.crossMomentMatrix (Causalean.Mathlib.Probability.normalizedRestrict P C) X Y = Matrix.vecMulVec (Causalean.Mathlib.Probability.firstMomentVector (Causalean.Mathlib.Probability.normalizedRestrict P C) X) (Causalean.Mathlib.Probability.firstMomentVector (Causalean.Mathlib.Probability.normalizedRestrict P C) Y)","l":238,"f":"Causalean/Mathlib/Probability/FiniteCellConditionalMomentBridge.lean"},{"n":"Causalean.Mathlib.Probability.product_weighted_centered_l1_le","k":"theorem","m":"Causalean.Mathlib.Probability.WeightedProduct","a":"Mathlib/Probability","t":1,"d":"For a one-observation probability law, a measurable design map, a measurable outcome confined to the unit interval, a measurable design regression that is its conditional expectation, a measurable array of weights depending on the full design vector, and the two natural square-integrability conditions, the expected absolute weighted centered sum is at most one half the square root of the expected sum of squared weights.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → (∀ (omega : Omega), 0 ≤ Y omega) → (∀ (omega : Omega), Y omega ≤ 1) → ∀ (mD : D → ℝ), Measurable mD → P[Y | MeasurableSpace.comap design inferInstance] =ᵐ[P] mD ∘ design → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → MeasureTheory.Integrable (fun z => ∑ i, w (Causalean.Mathlib.Probability.designVector design z) i ^ 2) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (fun z => Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∫ (z : Fin N → Omega), |Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w z| ∂MeasureTheory.Measure.pi fun x => P) ≤ 1 / 2 * √(∫ (z : Fin N → Omega), ∑ i, w (Causalean.Mathlib.Probability.designVector design z) i ^ 2 ∂MeasureTheory.Measure.pi fun x => P)","l":813,"f":"Causalean/Mathlib/Probability/WeightedProduct.lean"},{"n":"Causalean.Mathlib.Probability.product_weighted_centered_sq_integral_le","k":"theorem","m":"Causalean.Mathlib.Probability.WeightedProduct","a":"Mathlib/Probability","t":2,"d":"For a one-observation probability law, a measurable design map, a measurable outcome confined to the unit interval, a measurable design regression that is its conditional expectation, a measurable array of weights depending on the full design vector, and the two natural square-integrability conditions, the second moment of the weighted centered sum is at most one quarter of the expected sum of squared weights.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → (∀ (omega : Omega), 0 ≤ Y omega) → (∀ (omega : Omega), Y omega ≤ 1) → ∀ (mD : D → ℝ), Measurable mD → P[Y | MeasurableSpace.comap design inferInstance] =ᵐ[P] mD ∘ design → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → MeasureTheory.Integrable (fun z => ∑ i, w (Causalean.Mathlib.Probability.designVector design z) i ^ 2) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (fun z => Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∫ (z : Fin N → Omega), Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w z ^ 2 ∂MeasureTheory.Measure.pi fun x => P) ≤ 1 / 4 * ∫ (z : Fin N → Omega), ∑ i, w (Causalean.Mathlib.Probability.designVector design z) i ^ 2 ∂MeasureTheory.Measure.pi fun x => P","l":653,"f":"Causalean/Mathlib/Probability/WeightedProduct.lean"},{"n":"Causalean.Mathlib.Probability.sqrt_sum_sq_le_sum_abs","k":"theorem","m":"Causalean.Mathlib.Probability.IidMeanVariance","a":"Mathlib/Probability","t":2,"d":"The Euclidean length of a finite real-valued vector is no greater than the sum of the absolute values of its components.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (v : ι → ℝ), √(∑ i, v i ^ 2) ≤ ∑ i, |v i|","l":33,"f":"Causalean/Mathlib/Probability/IidMeanVariance.lean"},{"n":"Causalean.Mathlib.Probability.sum_bernoulli_eq_binomial","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"A Bernoulli-weighted sum of any function of the success count equals the corresponding sum against the binomial mass function.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (p : ℝ) (F : ℕ → ℝ), ∑ b, (∏ i, if b i = true then p else 1 - p) * F {i | b i = true}.card = ∑ j ∈ Finset.range (Fintype.card ι + 1), Causalean.Mathlib.Probability.binomialWeight (Fintype.card ι) p j * F j","l":398,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.totalized_inverse_count_le","k":"theorem","m":"Causalean.Mathlib.Probability.BernoulliMeasure","a":"Mathlib/Probability","t":2,"d":"The reciprocal of a positive integer, totalized to zero at the origin, is at most twice the reciprocal of its successor.","s":"∀ (j : ℕ), (if 0 < j then (↑j)⁻¹ else 0) ≤ 2 * (↑j + 1)⁻¹","l":423,"f":"Causalean/Mathlib/Probability/BernoulliMeasure.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean","k":"def","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":1,"d":"For a real scale and a real target mean, the symmetric two-point mean measure is the sum of a point mass at $B$ weighted by $\\max((1+u/B)/2,0)$ and a point mass at $-B$ weighted by $\\max((1-u/B)/2,0)$.","s":"ℝ → ℝ → MeasureTheory.Measure ℝ","l":37,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_bad_support_zero","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The channel is supported in [−M, M] whenever |B| ≤ M. The mass that any nonnegative mixture of point masses at B and −B places outside the interval [−M, M] is 0.","s":"∀ {B M : ℝ} (wplus wminus : ENNReal), |B| ≤ M → (wplus • MeasureTheory.Measure.dirac B + wminus • MeasureTheory.Measure.dirac (-B)) {y | y ∉ Set.Icc (-M) M} = 0","l":98,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_coef_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"When the scale is positive and the target mean lies within that scale, both weights in the signed two-point distribution are nonnegative.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → 0 ≤ (1 + u / B) / 2 ∧ 0 ≤ (1 - u / B) / 2","l":51,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_eq_map_bernoulli","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"Affine-image representation. The {−B, B} mean channel is the pushforward of the {0,1} Bernoulli law bernoulliLaw ((1 + u/B)/2) under the affine map x ↦ 2Bx − B (for B ≠ 0).","s":"∀ (B u : ℝ) (hB : B ≠ 0), Causalean.Mathlib.Probability.twoPointMean B u = MeasureTheory.Measure.map (⇑(affineHomeomorph (2 * B) (-B) (mul_ne_zero (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false)) hB)).toMeasurableEquiv) (Causalean.Mathlib.Probability.bernoulliLaw ((1 + u / B) / 2))","l":150,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_integral","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"Two-point integral. For 0 < B and |u| ≤ B, integrating f against twoPointMean B u returns the two-point weighted average ((1 + u/B)/2)·f(B) + ((1 − u/B)/2)·f(−B).","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → ∀ (f : ℝ → ℝ), ∫ (y : ℝ), f y ∂Causalean.Mathlib.Probability.twoPointMean B u = (1 + u / B) / 2 * f B + (1 - u / B) / 2 * f (-B)","l":72,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_isProbabilityMeasure","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The signed two-point channel is a probability measure. For 0 < B and |u| ≤ B the total mass of twoPointMean B u is 1.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.Probability.twoPointMean B u)","l":58,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.twoPointMean_mean","k":"theorem","m":"Causalean.Mathlib.Probability.SignedTwoPoint","a":"Mathlib/Probability","t":2,"d":"The mean is u. For 0 < B and |u| ≤ B, the expectation of the identity under twoPointMean B u is exactly u.","s":"∀ {B u : ℝ}, 0 < B → |u| ≤ B → ∫ (y : ℝ), y ∂Causalean.Mathlib.Probability.twoPointMean B u = u","l":90,"f":"Causalean/Mathlib/Probability/SignedTwoPoint.lean"},{"n":"Causalean.Mathlib.Probability.variance_prod_eq_integral_variance_add","k":"theorem","m":"Causalean.Mathlib.Probability.VarianceProd","a":"Mathlib/Probability","t":1,"d":"For independent coordinates with laws μ and ν, and a real-valued statistic F on the product that is square-integrable under the product law, suppose that almost every slice of F along the first coordinate, at a fixed value of the second coordinate, is square-integrable, that the map m of slice means is square-integrable under ν, and that m records the correct slice means: the integral of F over the first coordinate at almost every value of the second coordinate equals m there. Then the variance of F under the product measure equals the ν-average of the within-slice variances of F plus the variance, across the second coordinate, of the slice-mean map m.","s":"∀ {Ω : Type u_1} {T : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace T] (μ : MeasureTheory.Measure Ω) (ν : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (F : Ω × T → ℝ) (m : T → ℝ), MeasureTheory.MemLp F 2 (μ.prod ν) → (∀ᵐ (t : T) ∂ν, MeasureTheory.MemLp (fun s => F (s, t)) 2 μ) → MeasureTheory.MemLp m 2 ν → (∀ᵐ (t : T) ∂ν, ∫ (s : Ω), F (s, t) ∂μ = m t) → ProbabilityTheory.variance F (μ.prod ν) = ∫ (t : T), ProbabilityTheory.variance (fun s => F (s, t)) μ ∂ν + ProbabilityTheory.variance m ν","l":23,"f":"Causalean/Mathlib/Probability/VarianceProd.lean"},{"n":"Causalean.Mathlib.Probability.weightedCenteredSum","k":"def","m":"Causalean.Mathlib.Probability.WeightedProduct","a":"Mathlib/Probability","t":1,"d":"For a map assigning each observation a design value, a real-valued outcome function, a real-valued function of design values, weights that may depend on the complete design vector, and a finite sample of observations, the weighted centered sum is the sum over sample coordinates of each weight times the outcome minus the given function evaluated at that coordinate's design value.","s":"{N : ℕ} → {Omega : Type u_1} → {D : Type u_2} → (Omega → D) → (Omega → ℝ) → (D → ℝ) → ((Fin N → D) → Fin N → ℝ) → (Fin N → Omega) → ℝ","l":41,"f":"Causalean/Mathlib/Probability/WeightedProduct.lean"},{"n":"Causalean.Mathlib.ProbabilityTheory.ProductCondDistrib.condDistrib_fst_of_compProd_prod","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ProductCondDistrib","a":"Mathlib/Probability/Kernel","t":1,"d":"For the joint law formed by combining a finite base measure ρ with the product of two Markov kernels κ and η, the conditional distribution of the first product-kernel coordinate given the base coordinate equals κ, for ρ-almost every base point.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace β] [inst_4 : Nonempty β] {ρ : MeasureTheory.Measure α} [inst_5 : MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel α γ) [inst_6 : ProbabilityTheory.IsMarkovKernel κ] [inst_7 : ProbabilityTheory.IsMarkovKernel η], ⇑(ProbabilityTheory.condDistrib (fun z => z.2.1) (fun z => z.1) (ρ.compProd (κ.prod η))) =ᵐ[ρ] ⇑κ","l":31,"f":"Causalean/Mathlib/Probability/Kernel/ProductCondDistrib.lean"},{"n":"Causalean.Mathlib.ProbabilityTheory.ProductCondDistrib.condDistrib_snd_of_compProd_prod","k":"theorem","m":"Causalean.Mathlib.Probability.Kernel.ProductCondDistrib","a":"Mathlib/Probability/Kernel","t":1,"d":"For the joint law formed by combining a finite base measure ρ with the product of two Markov kernels κ and η, the conditional distribution of the second product-kernel coordinate given the base coordinate equals η, for ρ-almost every base point.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] {ρ : MeasureTheory.Measure α} [inst_5 : MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel α γ) [inst_6 : ProbabilityTheory.IsMarkovKernel κ] [inst_7 : ProbabilityTheory.IsMarkovKernel η], ⇑(ProbabilityTheory.condDistrib (fun z => z.2.2) (fun z => z.1) (ρ.compProd (κ.prod η))) =ᵐ[ρ] ⇑η","l":73,"f":"Causalean/Mathlib/Probability/Kernel/ProductCondDistrib.lean"},{"n":"Causalean.Mathlib.RpowArith.div_rpow_of_nonneg_of_pos","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Factoring a real power of a quotient. For a nonnegative numerator A and a strictly positive denominator n, the real power (A/n)^p equals A^p times n raised to the power −p, for any real exponent p.","s":"∀ (A p n : ℝ), 0 ≤ A → 0 < n → (A / n) ^ p = A ^ p * n ^ (-p)","l":29,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.inv_eq_rpow_neg_one","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"For any real number x, its reciprocal equals its real power raised to the exponent −1.","s":"∀ (x : ℝ), x⁻¹ = x ^ (-1)","l":24,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.inv_mul_sqrt_eq_rpow_neg_half","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Reciprocal times square root as a negative-half power. For a nonnegative real number q, the reciprocal of q times the square root of q equals q raised to the power −1/2.","s":"∀ (q : ℝ), 0 ≤ q → q⁻¹ * √q = q ^ (-(1 / 2))","l":66,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.RpowArith.rpow_natCast_nonpos_le_one","k":"theorem","m":"Causalean.Mathlib.Analysis.RpowArith","a":"Mathlib/Analysis","t":1,"d":"Nonpositive real power of a natural number is at most one. For a nonpositive real exponent e, the real power of any natural-number cast raised to e is at most 1.","s":"∀ (n : ℕ), ∀ e ≤ 0, ↑n ^ e ≤ 1","l":51,"f":"Causalean/Mathlib/Analysis/RpowArith.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.PositiveExclusionRadius","k":"structure","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":1,"d":"A positive exclusion radius is a positive lower bound for a residual on the part of K outside U, together with a candidate where that lower bound is attained.","s":"{X : Type u_1} → Set X → Set X → (X → ℝ) → Type","l":38,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius","k":"structure","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":1,"d":"A uniform positive exclusion radius records a far feasible pair where the residual reaches a strictly positive value that lower-bounds its value at every far feasible pair.","s":"{P : Type u_1} → {X : Type u_2} → [TopologicalSpace P] → [PseudoMetricSpace X] → Set (P × X) → (P → X) → (P → ℝ) → (P × X → ℝ) → Type (max u_1 u_2)","l":100,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius.dist_lt_of_residual_le_half","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a uniform positive exclusion radius, a feasible pair, and a residual at most half that radius, then the candidate lies strictly inside its parameter-dependent reference neighborhood.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [inst_1 : PseudoMetricSpace X] {K : Set (P × X)} {x₀ : P → X} {ρ : P → ℝ} {r : P × X → ℝ} (h : Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius K x₀ ρ r) {p : P} {x : X}, (p, x) ∈ K → r (p, x) ≤ h.radius / 2 → dist x (x₀ p) < ρ p","l":241,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius.dist_lt_of_residual_lt","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a uniform positive exclusion radius, a feasible pair, and a residual strictly below that radius, then the candidate lies strictly inside its parameter-dependent reference neighborhood.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [inst_1 : PseudoMetricSpace X] {K : Set (P × X)} {x₀ : P → X} {ρ : P → ℝ} {r : P × X → ℝ} (h : Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius K x₀ ρ r) {p : P} {x : X}, (p, x) ∈ K → r (p, x) < h.radius → dist x (x₀ p) < ρ p","l":228,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.exists_positiveExclusionRadius","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"For a compact candidate set, an open local neighborhood, a continuous residual, and a reference point, if the candidate set is compact, the neighborhood is open, the reference belongs to the candidate set, the reference belongs to the neighborhood, the far set is nonempty, the residual is continuous, the residual is nonnegative on candidates, and its only zero among candidates is the reference, then the far set has a strictly positive attained residual minimum.","s":"∀ {X : Type u_1} [inst : TopologicalSpace X] (K U : Set X) (r : X → ℝ) (x₀ : X), IsCompact K → IsOpen U → x₀ ∈ K → x₀ ∈ U → (K \\ U).Nonempty → Continuous r → (∀ x ∈ K, 0 ≤ r x) → (∀ x ∈ K, r x = 0 → x = x₀) → Nonempty (Causalean.Mathlib.Topology.CompactExclusion.PositiveExclusionRadius K U r)","l":50,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.exists_uniformExclusionTolerance_le","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, a residual, and a radius function, if the correspondence is compact, the reference section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference candidate has zero residual among feasible pairs, then one strictly positive tolerance sends every feasible pair with residual at most that tolerance inside its own reference neighborhood.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [T2Space P] [inst_2 : MetricSpace X] (K : Set (P × X)) (x₀ : P → X) (r : P × X → ℝ) (ρ : P → ℝ), IsCompact K → Continuous x₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) = 0 → x = x₀ p) → ∃ ε₀, 0 < ε₀ ∧ ∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) ≤ ε₀ → dist x (x₀ p) < ρ p","l":282,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.exists_uniformExclusionTolerance_lt","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, a residual, and a radius function, if the correspondence is compact, the reference section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference candidate has zero residual among feasible pairs, then one strictly positive tolerance sends every feasible pair with smaller residual inside its own reference neighborhood.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [T2Space P] [inst_2 : MetricSpace X] (K : Set (P × X)) (x₀ : P → X) (r : P × X → ℝ) (ρ : P → ℝ), IsCompact K → Continuous x₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) = 0 → x = x₀ p) → ∃ ε₀, 0 < ε₀ ∧ ∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) < ε₀ → dist x (x₀ p) < ρ p","l":253,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.exists_uniformPositiveExclusionRadius_of_isClosed","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, a residual, and a positive-radius function, if the correspondence is compact, the far feasible set is closed, the far feasible set is nonempty, the residual is continuous, the residual is nonnegative on feasible pairs, only the reference candidate has zero residual among feasible pairs, and every radius is strictly positive, then some far pair attains a strictly positive residual minimum on the far feasible set.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [inst_1 : PseudoMetricSpace X] (K : Set (P × X)) (x₀ : P → X) (r : P × X → ℝ) (ρ : P → ℝ), IsCompact K → IsClosed (Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ) → (Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ).Nonempty → Continuous r → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) = 0 → x = x₀ p) → (∀ (p : P), 0 < ρ p) → Nonempty (Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius K x₀ ρ r)","l":141,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet","k":"def","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":1,"d":"For a parameter space, a candidate pseudometric space, a set of feasible parameter--candidate pairs, a reference candidate assigned to each parameter, and a real-valued radius assigned to each parameter, the far feasible set consists exactly of the feasible pairs whose candidate is at least that parameter's assigned radius from its reference candidate.","s":"{P : Type u_1} → {X : Type u_2} → [TopologicalSpace P] → [PseudoMetricSpace X] → Set (P × X) → (P → X) → (P → ℝ) → Set (P × X)","l":95,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.isClosed_farFeasibleSet","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, and a radius function, if the correspondence is compact, the reference section is continuous, and the radius function is continuous, then the parameter-dependent far feasible set is closed.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [T2Space P] [inst_2 : MetricSpace X] (K : Set (P × X)) (x₀ : P → X) (ρ : P → ℝ), IsCompact K → Continuous x₀ → Continuous ρ → IsClosed (Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ)","l":120,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.isCompact_farFeasibleSet_of_isClosed","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, and a radius function, if the correspondence is compact and its far feasible set is closed, then the far feasible set is compact.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [inst_1 : PseudoMetricSpace X] (K : Set (P × X)) (x₀ : P → X) (ρ : P → ℝ), IsCompact K → IsClosed (Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ) → IsCompact (Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ)","l":132,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.uniformCompactCorrespondence_dichotomy","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":1,"d":"Given a feasible correspondence, a reference section, a residual, and a radius function, if the correspondence is compact, the reference section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference candidate has zero residual among feasible pairs, then either no far feasible pair exists or one attains a strictly positive uniform residual minimum.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [T2Space P] [inst_2 : MetricSpace X] (K : Set (P × X)) (x₀ : P → X) (r : P × X → ℝ) (ρ : P → ℝ), IsCompact K → Continuous x₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) = 0 → x = x₀ p) → Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ = ∅ ∨ Nonempty (Causalean.Mathlib.Topology.CompactExclusion.UniformPositiveExclusionRadius K x₀ ρ r)","l":178,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.CompactExclusion.uniformCompactCorrespondence_dichotomy_explicit","k":"theorem","m":"Causalean.Mathlib.Topology.CompactExclusion","a":"Mathlib/Topology","t":2,"d":"Given a feasible correspondence, a reference section, a residual, and a radius function, if the correspondence is compact, the reference section is continuous, the residual is continuous, the radius function is continuous, every radius is strictly positive, the residual is nonnegative on feasible pairs, and only the reference candidate has zero residual among feasible pairs, then either the far feasible set is empty or a far pair and its strictly positive attained residual minimum can be exhibited explicitly.","s":"∀ {P : Type u_1} {X : Type u_2} [inst : TopologicalSpace P] [T2Space P] [inst_2 : MetricSpace X] (K : Set (P × X)) (x₀ : P → X) (r : P × X → ℝ) (ρ : P → ℝ), IsCompact K → Continuous x₀ → Continuous r → Continuous ρ → (∀ (p : P), 0 < ρ p) → (∀ z ∈ K, 0 ≤ r z) → (∀ (p : P) (x : X), (p, x) ∈ K → r (p, x) = 0 → x = x₀ p) → Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ = ∅ ∨ ∃ pStar xStar ε₀, (pStar, xStar) ∈ Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ ∧ 0 < ε₀ ∧ r (pStar, xStar) = ε₀ ∧ ∀ z ∈ Causalean.Mathlib.Topology.CompactExclusion.farFeasibleSet K x₀ ρ, ε₀ ≤ r z","l":203,"f":"Causalean/Mathlib/Topology/CompactExclusion.lean"},{"n":"Causalean.Mathlib.Topology.continuous_uniformOnFun_affine_of_compact","k":"theorem","m":"Causalean.Mathlib.Topology.UniformConvergence.Affine","a":"Mathlib/Topology/UniformConvergence","t":1,"d":"For a compact set that is nonempty, the affine interpolation between two endpoint functions continuous on that set is continuous in the topology of uniform convergence on the set.","s":"∀ {α : Type u_1} {E : Type u_2} [inst : TopologicalSpace α] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] {K : Set α}, IsCompact K → K.Nonempty → ∀ {f g : α → E}, ContinuousOn f K → ContinuousOn g K → Continuous fun t => (UniformOnFun.ofFun {K}) fun x => (1 - t) • f x + t • g x","l":21,"f":"Causalean/Mathlib/Topology/UniformConvergence/Affine.lean"},{"n":"Causalean.Mathlib.Topology.continuous_uniformOnFun_of_eq_affine_on_compact","k":"theorem","m":"Causalean.Mathlib.Topology.UniformConvergence.Affine","a":"Mathlib/Topology/UniformConvergence","t":2,"d":"For a compact set that is nonempty, a path that agrees there with the affine interpolation of two endpoint functions continuous on the set is continuous in the topology of uniform convergence on the set.","s":"∀ {α : Type u_1} {E : Type u_2} [inst : TopologicalSpace α] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] {K : Set α}, IsCompact K → K.Nonempty → ∀ {f g : α → E}, ContinuousOn f K → ContinuousOn g K → ∀ (path : ℝ → α → E), (∀ (t : ℝ), ∀ x ∈ K, path t x = (1 - t) • f x + t • g x) → Continuous fun t => (UniformOnFun.ofFun {K}) (path t)","l":49,"f":"Causalean/Mathlib/Topology/UniformConvergence/Affine.lean"},{"n":"Causalean.Mathlib.Topology.isSubsequentialLimit_iff_mapClusterPt","k":"theorem","m":"Causalean.Mathlib.Topology.SubsequentialLimits","a":"Mathlib/Topology","t":2,"d":"For a sequence in a first-countable space, a point is the limit along a strictly increasing subsequence exactly when it is a mapped cluster point at infinity.","s":"∀ {X : Type u_1} [inst : TopologicalSpace X] [FirstCountableTopology X] (x : ℕ → X) (a : X), (∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)) ↔ MapClusterPt a Filter.atTop x","l":23,"f":"Causalean/Mathlib/Topology/SubsequentialLimits.lean"},{"n":"Causalean.Mathlib.Topology.subsequentialLimitSet_nonempty_compact_of_eventually_mem_compact","k":"theorem","m":"Causalean.Mathlib.Topology.SubsequentialLimits","a":"Mathlib/Topology","t":1,"d":"If a sequence is eventually contained in a compact set, then its set of limits along strictly increasing subsequences is nonempty and compact.","s":"∀ {X : Type u_1} [inst : TopologicalSpace X] [FirstCountableTopology X] [T2Space X] (x : ℕ → X) {s : Set X}, IsCompact s → (∀ᶠ (n : ℕ) in Filter.atTop, x n ∈ s) → (∃ y φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds y)) ∧ IsCompact {y | ∃ φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds y)}","l":36,"f":"Causalean/Mathlib/Topology/SubsequentialLimits.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.one_minus_a_b_abs_lt_one","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":2,"d":"Pointwise spectral gap: |1 - a - b| < 1 when a + b lies strictly between zero and two.","s":"∀ {K : Type u_1} [inst : CommRing K] [inst_1 : LinearOrder K] [IsStrictOrderedRing K] {a b : K}, 0 < a + b → a + b < 2 → |1 - a - b| < 1","l":123,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.one_minus_a_b_uniform_gap_on_compact","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":2,"d":"Uniform spectral gap on compact subsets of the open strip: the continuous function (a,b) ↦ |1 - a - b| attains its supremum when 0 < a + b < 2 at some point of K, and that supremum is strictly less than 1 by the pointwise bound.","s":"∀ (K : Set (ℝ × ℝ)), IsCompact K → K ⊆ {p | 0 < p.1 + p.2 ∧ p.1 + p.2 < 2} → ∃ ρ < 1, ∀ p ∈ K, |1 - p.1 - p.2| ≤ ρ","l":137,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.stationaryProjection","k":"def","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For two transition parameters in an arbitrary field, the stationary projection matrix is the $2\\times2$ matrix whose two rows both equal $(b/(a+b),a/(a+b))$.","s":"{K : Type u_1} → [Field K] → K → K → Matrix (Fin 2) (Fin 2) K","l":57,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix","k":"def","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For two transition parameters in an arbitrary field, the two-state transition matrix is the $2\\times2$ matrix whose first row is $(1-a,a)$ and whose second row is $(b,1-b)$.","s":"{K : Type u_1} → [Field K] → K → K → Matrix (Fin 2) (Fin 2) K","l":46,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix_pow_eq_spectral","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For parameters a, b in a field with a + b nonzero, the k-th power of the two-state transition matrix transitionMatrix a b decomposes as the stationary projection stationaryProjection a b plus (1 - a - b)^k times its complement, for every k.","s":"∀ {K : Type u_1} [inst : Field K] (a b : K), a + b ≠ 0 → ∀ (k : ℕ), Causalean.Mathlib.TwoStateMarkov.transitionMatrix a b ^ k = Causalean.Mathlib.TwoStateMarkov.stationaryProjection a b + (1 - a - b) ^ k • (1 - Causalean.Mathlib.TwoStateMarkov.stationaryProjection a b)","l":64,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.TwoStateMarkov.transitionMatrix_pow_tendsto_stationary_uniform","k":"theorem","m":"Causalean.Mathlib.TwoStateMarkov","a":"Mathlib","t":1,"d":"For a set K of transition-parameter pairs (a,b) that is compact and contained in the open unit square, the k-th power of the transition matrix converges to the stationary projection entrywise, uniformly over K: for every ε > 0 there is a threshold N such that every entry of (transitionMatrix a b)^k - stationaryProjection a b has absolute value at most ε once k ≥ N, for every (a,b) in K.","s":"∀ (K : Set (ℝ × ℝ)), IsCompact K → K ⊆ {p | 0 < p.1 ∧ p.1 < 1 ∧ 0 < p.2 ∧ p.2 < 1} → ∀ ε > 0, ∃ N, ∀ (k : ℕ), N ≤ k → ∀ p ∈ K, ∀ (i j : Fin 2), |(Causalean.Mathlib.TwoStateMarkov.transitionMatrix p.1 p.2 ^ k - Causalean.Mathlib.TwoStateMarkov.stationaryProjection p.1 p.2) i j| ≤ ε","l":165,"f":"Causalean/Mathlib/TwoStateMarkov.lean"},{"n":"Causalean.Mathlib.abs_covariance_le_sqrt_mul","k":"theorem","m":"Causalean.Mathlib.Probability.CovarianceCauchySchwarz","a":"Mathlib/Probability","t":1,"d":"Cauchy–Schwarz for covariance. Under a finite measure, if $X$ is square-integrable and $Y$ is square-integrable, then the absolute value of their covariance is at most the product of their standard deviations, $|\\mathrm{Cov}(X,Y)| \\le \\sqrt{\\mathrm{Var}(X)} \\cdot \\sqrt{\\mathrm{Var}(Y)}$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {X Y : Ω → ℝ}, MeasureTheory.MemLp X 2 μ → MeasureTheory.MemLp Y 2 μ → |ProbabilityTheory.covariance X Y μ| ≤ √(ProbabilityTheory.variance X μ) * √(ProbabilityTheory.variance Y μ)","l":69,"f":"Causalean/Mathlib/Probability/CovarianceCauchySchwarz.lean"},{"n":"Causalean.Mathlib.affine_preimage_Ioi","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The inverse image of a right-hand tail under a positive affine transformation is a right-hand tail whose threshold is transformed by the inverse affine formula.","s":"∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {s c m : α}, 0 < s → (fun z => s * z + m) ⁻¹' Set.Ioi c = Set.Ioi ((c - m) / s)","l":53,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.bddBelow_condQuantileSet","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a strictly positive conditional-quantile level, the response values whose conditional cumulative distribution function reaches that level are bounded below.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, 0 < τ a → BddBelow {x | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x}","l":49,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.cdf_continuous_of_noAtoms","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The CDF of an atomless real probability measure is continuous.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.NullSingletonClass μ], Continuous ↑(ProbabilityTheory.cdf μ)","l":97,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.cdf_neg_of_map_neg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The CDF of an atomless probability measure invariant under reflection is symmetric.","s":"∀ {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.NullSingletonClass μ], MeasureTheory.Measure.map (fun x => -x) μ = μ → ∀ (t : ℝ), ↑(ProbabilityTheory.cdf μ) (-t) = 1 - ↑(ProbabilityTheory.cdf μ) t","l":67,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.centered_sq_lintegral_le_eLpNorm_two_sq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Centering a square-integrable real variable cannot increase its second moment beyond its uncentered L2 energy under a probability distribution.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, MeasureTheory.MemLp f 2 P → ∫⁻ (x : X), ENNReal.ofReal ((f x - ∫ (y : X), f y ∂P) ^ 2) ∂P ≤ ENNReal.ofReal ((MeasureTheory.eLpNorm f 2 P).toReal ^ 2)","l":153,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.condQuantile","k":"def","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":1,"d":"For any measurable parameter space, a measure on the product of that space and the real line, a real-valued target-level function on the parameter space, and a parameter value, the conditional quantile is the infimum of the real response values at which the conditional cumulative distribution function reaches the target level at that parameter value.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure (α × ℝ) → (α → ℝ) → α → ℝ","l":31,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantileSet_up_closed","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"If a response value's conditional cumulative distribution function reaches a target level, then every larger response value also reaches that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x x' : ℝ}, x ∈ {y | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) y} → x ≤ x' → x' ∈ {y | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) y}","l":42,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantile_le_iff","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At an interior target level, a point is above the conditional generalized inverse exactly when its conditional cumulative distribution function has reached that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, 0 < τ a → τ a < 1 → (Causalean.Mathlib.condQuantile ρ τ a ≤ x ↔ τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x)","l":99,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.condQuantile_le_of_le_condCDF","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"When the conditional cumulative distribution function has reached a positive target level at a point, the conditional generalized inverse is no larger than that point.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, 0 < τ a → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x → Causalean.Mathlib.condQuantile ρ τ a ≤ x","l":92,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.covarianceBilin_stdGaussian","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The covariance bilinear form of the standard Gaussian measure is the ambient inner product.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] (u v : E), ((ProbabilityTheory.covarianceBilin (Causalean.Mathlib.stdGaussian E)) u) v = inner ℝ u v","l":245,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.covariance_sq_le_variance_mul","k":"theorem","m":"Causalean.Mathlib.Probability.CovarianceCauchySchwarz","a":"Mathlib/Probability","t":2,"d":"Cauchy–Schwarz for covariance (squared form). For square-integrable statistics under a finite measure, the squared covariance is at most the product of the two variances.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {X Y : Ω → ℝ}, MeasureTheory.MemLp X 2 μ → MeasureTheory.MemLp Y 2 μ → ProbabilityTheory.covariance X Y μ ^ 2 ≤ ProbabilityTheory.variance X μ * ProbabilityTheory.variance Y μ","l":31,"f":"Causalean/Mathlib/Probability/CovarianceCauchySchwarz.lean"},{"n":"Causalean.Mathlib.eLpNorm_two_sq_toReal_eq_integral_sq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"The square of a square-integrable variable's L2 norm equals the integral of its squared norm, for any measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] {f : X → E}, MeasureTheory.MemLp f 2 P → ENNReal.ofReal ((MeasureTheory.eLpNorm f 2 P).toReal ^ 2) = ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P)","l":123,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.exists_orthogonalProjection_of_posSemidef","k":"theorem","m":"Causalean.Mathlib.SemiInnerProjection","a":"Mathlib","t":1,"d":"In a vector space over a linearly ordered field equipped with a bilinear form B that is symmetric and positive-semidefinite, every finite-dimensional linear subspace H admits a linear self-map P of the ambient space, valued in H, whose residual X - P X is B-orthogonal to every vector of H, for every X.","s":"∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : LinearOrder K] [IsStrictOrderedRing K] [inst_3 : AddCommGroup V] [inst_4 : Module K V] (B : LinearMap.BilinForm K V), (∀ (x y : V), (B x) y = (B y) x) → (∀ (x : V), 0 ≤ (B x) x) → ∀ (H : Submodule K V) [FiniteDimensional K ↥H], ∃ P, (∀ (X : V), P X ∈ H) ∧ ∀ (X h : V), h ∈ H → (B (X - P X)) h = 0","l":42,"f":"Causalean/Mathlib/SemiInnerProjection.lean"},{"n":"Causalean.Mathlib.gaussianReal_Ioi_eq","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":1,"d":"Gaussian survival. For a normal law with mean m and a nonnegative variance parameter v that is nonzero, the probability mass above a threshold c equals one minus the standard-normal CDF evaluated at the standardized threshold (c − m)/√v.","s":"∀ (m : ℝ) (v : NNReal), v ≠ 0 → ∀ (c : ℝ), ((ProbabilityTheory.gaussianReal m v) (Set.Ioi c)).toReal = 1 - Causalean.Mathlib.stdNormalCDF ((c - m) / √↑v)","l":144,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.gaussianReal_eq_map_std","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"A real Gaussian distribution with nonnegative variance is the standard normal distribution after scaling by the standard deviation and shifting by the mean.","s":"∀ (m : ℝ) (v : NNReal), ProbabilityTheory.gaussianReal m v = MeasureTheory.Measure.map (fun z => √↑v * z + m) (ProbabilityTheory.gaussianReal 0 1)","l":31,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.gradient_coord_eq_deriv","k":"theorem","m":"Causalean.Mathlib.Analysis.GradientCoord","a":"Mathlib/Analysis","t":1,"d":"Gradient coordinate as a directional derivative. For a real-valued function f on a finite-dimensional Euclidean space that is differentiable at the point x, the a-th coordinate of its gradient at x equals the derivative at 0 of the one-variable restriction t ↦ f(x + t·eₐ) along the a-th standard basis direction.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (f : EuclideanSpace ℝ ι → ℝ) (x : EuclideanSpace ℝ ι) (a : ι), DifferentiableAt ℝ f x → (gradient f x).ofLp a = deriv (fun t => f (x + t • EuclideanSpace.single a 1)) 0","l":26,"f":"Causalean/Mathlib/Analysis/GradientCoord.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":1,"d":"Conditional second-moment bound for a centered i.i.d. sum. For a nonempty finite index set s, a sub-σ-algebra m_A with m_A coarser than the ambient σ-algebra, and a finite family of measurable sample coordinates W i that is jointly independent of m_A and jointly distributed as the i.i.d. product law P^s, if the parametric integrand g is jointly measurable with respect to m_A and the σ-algebra on the sample space and g ω lies in L²(P) for every ω, then the second moment of the normalized centered sum (1/√|s|)·Σᵢ(g ω (W i ω) − ∫ g ω dP) is at most the integrated squared L²(P)-norm of g ω, stated in lintegral form via ENNReal.ofReal.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι), 0 < s.card → ∀ (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ → (MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal (((√↑s.card)⁻¹ * ∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":465,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_unscaled_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"A finite conditionally independent identically distributed family has integrated squared centered sum bounded by its size times the integrated squared L² norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι) (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ m_A ≤ mΩ, ProbabilityTheory.Indep m_A (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ → (MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":406,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.iid_centered_sum_sq_lintegral_unscaled_le_of_joined_law","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"When a finite family jointly has the product law of a base population measure and independent identically distributed draws, the integrated squared centered sum is bounded by the number of draws times the integrated squared L² norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} (s : Finset ι) (W : ι → Ω → X), (∀ i ∈ s, Measurable (W i)) → ∀ (m_A : MeasurableSpace Ω) (hm_A_le : m_A ≤ mΩ), MeasureTheory.Measure.map (fun ω => (ω, fun i => W (↑i) ω)) μ = (μ.trim hm_A_le).prod (MeasureTheory.Measure.pi fun x => P) → ∀ (g : Ω → X → ℝ), Measurable (Function.uncurry g) → (∀ (ω : Ω), MeasureTheory.MemLp (g ω) 2 P) → ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, (g ω (W i ω) - ∫ (x : X), g ω x ∂P)) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g ω) 2 P).toReal ^ 2) ∂μ","l":284,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.indep_trim_prod_map_eq","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Convert independence of a sub-σ-algebra and a random element into the product law of the joined map, with the first marginal trimmed to the sub-σ-algebra.","s":"∀ {Ω : Type u_1} {β : Type u_2} [mΩ : MeasurableSpace Ω] [mβ : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (m_A : MeasurableSpace Ω) (hm_A_le : m_A ≤ mΩ) {Z : Ω → β}, AEMeasurable Z μ → ProbabilityTheory.Indep m_A (MeasurableSpace.comap Z mβ) μ → MeasureTheory.Measure.map (fun ω => (ω, Z ω)) μ = (μ.trim hm_A_le).prod (MeasureTheory.Measure.map Z μ)","l":259,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.integrable_id_mul_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The standard normal density has a finite first absolute moment.","s":"MeasureTheory.Integrable (fun x => x * Causalean.Mathlib.stdNormalPDF x) MeasureTheory.volume","l":81,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_affine_gaussianReal_eq_density","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"Above a threshold, the integral of an affine function under the standard-normal law equals the integral of that function against its density.","s":"∀ (s m t : ℝ), ∫ (z : ℝ) in Set.Ioi t, s * z + m ∂ProbabilityTheory.gaussianReal 0 1 = ∫ (z : ℝ) in Set.Ioi t, ProbabilityTheory.gaussianPDFReal 0 1 z * (s * z + m)","l":129,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_affine_stdNormal","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"Above a threshold, integrating an affine function against the standard-normal density equals its slope times the truncated first moment plus its intercept times the tail mass.","s":"∀ (s m t : ℝ), ∫ (z : ℝ) in Set.Ioi t, ProbabilityTheory.gaussianPDFReal 0 1 z * (s * z + m) = (s * ∫ (z : ℝ) in Set.Ioi t, z * Causalean.Mathlib.stdNormalPDF z) + m * ∫ (z : ℝ) in Set.Ioi t, Causalean.Mathlib.stdNormalPDF z","l":100,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_id_gaussianReal","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":1,"d":"Gaussian truncated first moment. For a normal law with mean m and a nonnegative variance parameter v that is nonzero, the first moment integrated over the tail above a threshold c equals m·(1 − Φ(t)) + √v·φ(t), where t = (c − m)/√v — the affine image of the standard-normal truncated moment ∫_{t}^∞ z φ(z) dz = φ(t).","s":"∀ (m : ℝ) (v : NNReal), v ≠ 0 → ∀ (c : ℝ), ∫ (y : ℝ) in Set.Ioi c, y ∂ProbabilityTheory.gaussianReal m v = m * (1 - Causalean.Mathlib.stdNormalCDF ((c - m) / √↑v)) + √↑v * Causalean.Mathlib.stdNormalPDF ((c - m) / √↑v)","l":174,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_id_mul_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalMoments","a":"Mathlib/Probability","t":1,"d":"The standard-normal truncated first moment. For any real cutoff c, the tail integral of t times the standard normal density over (c, ∞) equals the density value at c.","s":"∀ (c : ℝ), ∫ (t : ℝ) in Set.Ioi c, t * Causalean.Mathlib.stdNormalPDF t = Causalean.Mathlib.stdNormalPDF c","l":53,"f":"Causalean/Mathlib/Probability/StdNormalMoments.lean"},{"n":"Causalean.Mathlib.integral_Ioi_stdNormalPDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalMoments","a":"Mathlib/Probability","t":2,"d":"The standard-normal survival integral: ∫_{c}^∞ φ(t) dt = 1 − Φ(c).","s":"∀ (c : ℝ), ∫ (t : ℝ) in Set.Ioi c, Causalean.Mathlib.stdNormalPDF t = 1 - Causalean.Mathlib.stdNormalCDF c","l":27,"f":"Causalean/Mathlib/Probability/StdNormalMoments.lean"},{"n":"Causalean.Mathlib.integral_id_map_equiv","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"Transporting a measure through a continuous linear equivalence transports its vector integral through the same equivalence.","s":"∀ {F : Type u_2} {G : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] [inst_2 : MeasurableSpace F] [BorelSpace F] [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] [inst_6 : MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {μ : MeasureTheory.Measure F} (L : F ≃L[ℝ] G), ∫ (x : G), x ∂MeasureTheory.Measure.map (⇑L) μ = L (∫ (x : F), x ∂μ)","l":144,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_euclideanStdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every real normed inner-product space, the transported coordinate-product standard Gaussian law on the associated Euclidean space is a Gaussian probability law.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.euclideanStdGaussian✝ E)","l":134,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_piGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every real normed inner-product space, the coordinate product of independent one-dimensional standard Gaussian laws is a Gaussian probability law.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.piGaussian E)","l":101,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isGaussian_stdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every finite-dimensional real normed inner-product space equipped with a measurable structure equal to its Borel structure, the standard Gaussian measure on that space is a Gaussian probability law.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], ProbabilityTheory.IsGaussian (Causalean.Mathlib.stdGaussian E)","l":139,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_euclideanStdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every real normed inner-product space, the transported coordinate-product standard Gaussian law on the associated Euclidean space is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.euclideanStdGaussian✝ E)","l":123,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_piGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every real normed inner-product space, the coordinate product of independent one-dimensional standard Gaussian laws is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.piGaussian E)","l":90,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.isProbabilityMeasure_stdGaussian","k":"instance","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"For every finite-dimensional real normed inner-product space equipped with a measurable structure equal to its Borel structure, the standard Gaussian measure on that space is a probability measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.IsProbabilityMeasure (Causalean.Mathlib.stdGaussian E)","l":129,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.le_condCDF_condQuantile","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a target level below one, the conditional generalized inverse reaches at least that level in the conditional cumulative distribution function.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, τ a < 1 → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) (Causalean.Mathlib.condQuantile ρ τ a)","l":68,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.le_condCDF_of_condQuantile_le","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"A conditional quantile below a value makes the conditional CDF reach any level below one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α} {x : ℝ}, τ a < 1 → Causalean.Mathlib.condQuantile ρ τ a ≤ x → τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x","l":85,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.lintegral_ofReal_inv_sqrt_smul_sum_sq_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"A finite sum whose squared integral is at most its number of terms times a nonnegative benchmark remains bounded by that benchmark after division by the square root of the number of terms.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (s : Finset ι), 0 < s.card → ∀ (Y : ι → Ω → ℝ) (B : Ω → ENNReal), ∫⁻ (ω : Ω), ENNReal.ofReal ((∑ i ∈ s, Y i ω) ^ 2) ∂μ ≤ ↑s.card * ∫⁻ (ω : Ω), B ω ∂μ → ∫⁻ (ω : Ω), ENNReal.ofReal (((√↑s.card)⁻¹ * ∑ i ∈ s, Y i ω) ^ 2) ∂μ ≤ ∫⁻ (ω : Ω), B ω ∂μ","l":65,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.measurable_condQuantile_and_attains","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":1,"d":"Measurable conditional quantile (selection). For a measure ρ on the product of a parameter space and the reals, and a measurable target level function τ that is everywhere strictly positive and everywhere strictly below one, if the conditional cumulative distribution function of ρ is continuous at the selected conditional quantile, for every parameter value, then the conditional quantile map condQuantile ρ τ is measurable and attains the target level — the conditional CDF at the selected quantile equals τ a for every a.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (ρ : MeasureTheory.Measure (α × ℝ)) (τ : α → ℝ), Measurable τ → (∀ (a : α), 0 < τ a) → (∀ (a : α), τ a < 1) → (∀ (a : α), ContinuousAt (↑(ProbabilityTheory.condCDF ρ a)) (Causalean.Mathlib.condQuantile ρ τ a)) → Measurable (Causalean.Mathlib.condQuantile ρ τ) ∧ ∀ (a : α), ↑(ProbabilityTheory.condCDF ρ a) (Causalean.Mathlib.condQuantile ρ τ a) = τ a","l":107,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_toReal_of_uncurry","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":1,"d":"Lp-norm measurable from joint measurability (top σ-algebra). For a σ-finite base measure P and an exponent p that is neither zero nor infinite, if the map (ω, x) ↦ g ω x is jointly measurable on the product of Ω and X, then the real-valued Lp norm ω ↦ ‖g ω‖_{Lp(P)} is measurable as a function of ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Ω] {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_3 : MeasurableSpace E] [inst_4 : TopologicalSpace E] [inst_5 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":50,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_toReal_of_uncurry_of_factor","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":1,"d":"Lp-norm measurable with respect to a sub-σ-algebra. For a σ-finite base measure P and an exponent p that is neither zero nor infinite, if the map (ω, x) ↦ g ω x is jointly measurable with respect to the product of a sub-σ-algebra mΩ on Ω and the σ-algebra on X, then the real-valued Lp norm ω ↦ ‖g ω‖_{Lp(P)} is measurable with respect to mΩ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_2 : MeasurableSpace E] [inst_3 : TopologicalSpace E] [inst_4 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":90,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_two_toReal_of_uncurry","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":2,"d":"Vestigial _two_ spelling of measurable_eLpNorm_toReal_of_uncurry; the exponent p was never fixed to 2.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Ω] {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_3 : MeasurableSpace E] [inst_4 : TopologicalSpace E] [inst_5 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":78,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.measurable_eLpNorm_two_toReal_of_uncurry_of_factor","k":"theorem","m":"Causalean.Mathlib.ELpNormMeasurable","a":"Mathlib","t":2,"d":"Vestigial _two_ spelling of measurable_eLpNorm_toReal_of_uncurry_of_factor; the exponent p was never fixed to 2.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure X} [MeasureTheory.SFinite P] {E : Type u_3} [inst_2 : MeasurableSpace E] [inst_3 : TopologicalSpace E] [inst_4 : ContinuousENorm E] [OpensMeasurableSpace E] {g : Ω → X → E} {p : ENNReal}, p ≠ 0 → p ≠ ⊤ → Measurable (Function.uncurry g) → Measurable fun ω => (MeasureTheory.eLpNorm (g ω) p P).toReal","l":120,"f":"Causalean/Mathlib/ELpNormMeasurable.lean"},{"n":"Causalean.Mathlib.memLp_eval","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"Each coordinate is MemLp of order 2 under the product measure.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (i : Fin (Module.finrank ℝ E)), MeasureTheory.MemLp (fun ω => ω i) 2 (Causalean.Mathlib.piGaussian E)","l":95,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.neg_log_coord_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"-log (p k) is convex on a convex set s whose k-th coordinate is positive throughout.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (k : ι), (∀ p ∈ s, 0 < p k) → ConvexOn ℝ s fun p => -Real.log (p k)","l":30,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_one_sub_coord_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"-log (1 - p k) is convex on a convex set s where the k-th coordinate stays below 1.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (k : ι), (∀ p ∈ s, p k < 1) → ConvexOn ℝ s fun p => -Real.log (1 - p k)","l":103,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_one_sub_sum_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"∑ k ∈ S, -log (1 - p k) is convex on a convex set s where the coordinates indexed by S stay below 1.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, p k < 1) → ConvexOn ℝ s fun p => ∑ k ∈ S, -Real.log (1 - p k)","l":123,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.neg_log_sum_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"∑ k ∈ S, -log (p k) is convex on a convex set s whose coordinates indexed by S are positive throughout.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, 0 < p k) → ConvexOn ℝ s fun p => ∑ k ∈ S, -Real.log (p k)","l":45,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.nonempty_condQuantileSet","k":"theorem","m":"Causalean.Mathlib.Probability.MeasurableCondQuantile","a":"Mathlib/Probability","t":2,"d":"At a conditional-quantile level strictly below one, some response value has a conditional cumulative distribution function that reaches that level.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {ρ : MeasureTheory.Measure (α × ℝ)} {τ : α → ℝ} {a : α}, τ a < 1 → {x | τ a ≤ ↑(ProbabilityTheory.condCDF ρ a) x}.Nonempty","l":60,"f":"Causalean/Mathlib/Probability/MeasurableCondQuantile.lean"},{"n":"Causalean.Mathlib.piGaussian","k":"def","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"For every real normed inner-product space, the coordinate product Gaussian measure is the product of independent standard normal laws, with one real-valued coordinate for every element of the finite index set whose size is the space's real rank.","s":"(E : Type u_1) → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → MeasureTheory.Measure (Fin (Module.finrank ℝ E) → ℝ)","l":59,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.pi_centered_sum_sq_lintegral_le","k":"theorem","m":"Causalean.Mathlib.IIDCenteredSum","a":"Mathlib","t":2,"d":"Finitely many independent coordinates, each with its own square-integrable real-valued law, have a centered sum whose second moment is bounded by the sum of their individual squared L² norms.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {X : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (X i)] {P : (i : ι) → MeasureTheory.Measure (X i)} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {f : (i : ι) → X i → ℝ}, (∀ (i : ι), MeasureTheory.MemLp (f i) 2 (P i)) → ∫⁻ (v : (i : ι) → X i), ENNReal.ofReal ((∑ i, (f i (v i) - ∫ (x : X i), f i x ∂P i)) ^ 2) ∂MeasureTheory.Measure.pi P ≤ ∑ i, ENNReal.ofReal ((MeasureTheory.eLpNorm (f i) 2 (P i)).toReal ^ 2)","l":179,"f":"Causalean/Mathlib/IIDCenteredSum.lean"},{"n":"Causalean.Mathlib.probit","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"For each real number $p$, the probit at $p$ is the infimum of the real numbers $x$ for which the standard-normal cumulative distribution function at $x$ is at least $p$.","s":"ℝ → ℝ","l":159,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.probit_stdNormalCDF","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"For any real score x, applying the probit transform to Φ(x) recovers x: the probit function Φ⁻¹ is a left inverse of the standard normal CDF Φ.","s":"∀ (x : ℝ), Causalean.Mathlib.probit (Causalean.Mathlib.stdNormalCDF x) = x","l":192,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.prod_inv_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":1,"d":"Convexity of a reciprocal coordinate product. On a convex subset of the coordinate space, if every coordinate indexed by a fixed finite index set stays strictly positive throughout the set, then the map sending a point to the reciprocal of the product of its coordinates over that index set is convex on the set.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, 0 < p k) → ConvexOn ℝ s fun p => (∏ k ∈ S, p k)⁻¹","l":68,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.prod_one_sub_inv_convexOn","k":"theorem","m":"Causalean.Mathlib.Analysis.Convex.ReciprocalProduct","a":"Mathlib/Analysis/Convex","t":2,"d":"Convexity of a reciprocal product of complements. On a convex set s whose coordinates indexed by S stay below 1, p ↦ (∏ k ∈ S, (1 - p k))⁻¹ is convex.","s":"∀ {ι : Type u_1} {s : Set (ι → ℝ)}, Convex ℝ s → ∀ (S : Finset ι), (∀ p ∈ s, ∀ k ∈ S, p k < 1) → ConvexOn ℝ s fun p => (∏ k ∈ S, (1 - p k))⁻¹","l":146,"f":"Causalean/Mathlib/Analysis/Convex/ReciprocalProduct.lean"},{"n":"Causalean.Mathlib.recipC","k":"def","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":1,"d":"Given a real cutoff scale $\\varepsilon$ and a real argument $x$, the floored reciprocal is $x^{-1}$ multiplied by the smooth transition evaluated at $(x-\\varepsilon/4)/(\\varepsilon/4)$. This definition applies for every real cutoff scale, including zero and negative values.","s":"ℝ → ℝ → ℝ","l":20,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":1,"d":"Global continuous differentiability of the floored reciprocal. For a strictly positive threshold ε, the floored reciprocal recipC ε is continuously differentiable on all of the reals.","s":"∀ (ε : ℝ), 0 < ε → ContDiff ℝ 1 (Causalean.Mathlib.recipC ε)","l":80,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff_of","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"The floored reciprocal is globally Cⁿ on all of ℝ, for EVERY smoothness order n: near the singularity the smooth cutoff vanishes to infinite order, absorbing the blow-up of x⁻¹, and away from the singularity it is a product of Cⁿ functions.","s":"∀ (n : ℕ∞) (ε : ℝ), 0 < ε → ContDiff ℝ (↑n) (Causalean.Mathlib.recipC ε)","l":43,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_contDiff_two","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"recipC ε is globally C². Specialization of recipC_contDiff_of; this is the order the reciprocal-product envelope's directional curvature modulus needs.","s":"∀ (ε : ℝ), 0 < ε → ContDiff ℝ 2 (Causalean.Mathlib.recipC ε)","l":89,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.recipC_eq_inv","k":"theorem","m":"Causalean.Mathlib.Analysis.SmoothReciprocal","a":"Mathlib/Analysis","t":2,"d":"On the region x ≥ ε/2 (with ε > 0) the floored reciprocal is exactly x⁻¹, because the smooth cutoff has already saturated to 1 there.","s":"∀ {ε x : ℝ}, 0 < ε → ε / 2 ≤ x → Causalean.Mathlib.recipC ε x = x⁻¹","l":31,"f":"Causalean/Mathlib/Analysis/SmoothReciprocal.lean"},{"n":"Causalean.Mathlib.stdGaussian","k":"def","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"For every finite-dimensional real inner-product space equipped with a measurable structure, the standard Gaussian measure on that space is obtained by transporting the coordinate product of independent standard normal laws through a chosen orthonormal-coordinate identification. It is therefore the centered Gaussian law whose covariance form is the inner product.","s":"(E : Type u_1) → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst : MeasurableSpace E] → MeasureTheory.Measure E","l":74,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdGaussian_map_normSq_eq_pi","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":1,"d":"Product-of-1-D-Gaussians model for the squared norm. The law of the squared norm under the standard Gaussian measure on E equals the law of the sum of squared coordinates under a product of finrank ℝ E independent standard real Gaussians.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.Measure.map (fun x => ‖x‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = MeasureTheory.Measure.map (fun w => ∑ i, w i ^ 2) (MeasureTheory.Measure.pi fun x => ProbabilityTheory.gaussianReal 0 1)","l":263,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdGaussian_mean","k":"theorem","m":"Causalean.Mathlib.StandardGaussian","a":"Mathlib","t":2,"d":"The standard Gaussian measure on a finite-dimensional real inner-product space has mean zero.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], ∫ (x : E), x ∂Causalean.Mathlib.stdGaussian E = 0","l":185,"f":"Causalean/Mathlib/StandardGaussian.lean"},{"n":"Causalean.Mathlib.stdNormalCDF","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"For each real number $x$, the standard-normal cumulative distribution function at $x$ is the probability that a mean-zero, variance-one normal random variable is at most $x$.","s":"ℝ → ℝ","l":42,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"The standard normal CDF Φ is continuous: the standard normal has no atoms.","s":"Continuous Causalean.Mathlib.stdNormalCDF","l":115,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_def","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The named standard-normal CDF is Mathlib's CDF for the standard real Gaussian law.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x = ↑(ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)) x","l":53,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_le_one","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The standard-normal CDF is at most one at every real point.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x ≤ 1","l":64,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_lt_one","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(x) < 1 for every real x.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalCDF x < 1","l":153,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_monotone","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ is monotone (inherited from the StieltjesFunction structure of cdf).","s":"Monotone Causalean.Mathlib.stdNormalCDF","l":56,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_neg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Symmetry of the standard normal CDF: Φ(−t) = 1 − Φ(t), from the reflection symmetry of the Gaussian law and its atomlessness.","s":"∀ (t : ℝ), Causalean.Mathlib.stdNormalCDF (-t) = 1 - Causalean.Mathlib.stdNormalCDF t","l":82,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_nonneg","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(x) ∈ [0,1].","s":"∀ (x : ℝ), 0 ≤ Causalean.Mathlib.stdNormalCDF x","l":61,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_pos","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"0 < Φ(x) for every real x (full support of the Gaussian).","s":"∀ (x : ℝ), 0 < Causalean.Mathlib.stdNormalCDF x","l":147,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_probit","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ(Φ⁻¹(p)) = p for p ∈ (0,1) (exact inversion, using continuity + strict monotonicity).","s":"∀ {p : ℝ}, 0 < p → p < 1 → Causalean.Mathlib.stdNormalCDF (Causalean.Mathlib.probit p) = p","l":166,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_strictMono","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ is strictly monotone: the standard normal has full support.","s":"StrictMono Causalean.Mathlib.stdNormalCDF","l":122,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_tendsto_atBot","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ → 0 at -∞.","s":"Filter.Tendsto Causalean.Mathlib.stdNormalCDF Filter.atBot (nhds 0)","l":91,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalCDF_tendsto_atTop","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"Φ → 1 at +∞.","s":"Filter.Tendsto Causalean.Mathlib.stdNormalCDF Filter.atTop (nhds 1)","l":94,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalMeasure_Ioi_toReal_eq_integral","k":"theorem","m":"Causalean.Mathlib.Probability.GaussianMoments","a":"Mathlib/Probability","t":2,"d":"The standard normal probability of values above t equals the integral of the standard-normal density over that upper-tail region.","s":"∀ (t : ℝ), ((ProbabilityTheory.gaussianReal 0 1) (Set.Ioi t)).toReal = ∫ (x : ℝ) in Set.Ioi t, Causalean.Mathlib.stdNormalPDF x","l":69,"f":"Causalean/Mathlib/Probability/GaussianMoments.lean"},{"n":"Causalean.Mathlib.stdNormalPDF","k":"def","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":1,"d":"For each real number $x$, the standard-normal density at $x$ is $\\exp(-x^2/2)/\\sqrt{2\\pi}$.","s":"ℝ → ℝ","l":36,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.Mathlib.stdNormalPDF_def","k":"theorem","m":"Causalean.Mathlib.Probability.StdNormalCDF","a":"Mathlib/Probability","t":2,"d":"The named standard-normal density is Mathlib's real Gaussian density with mean zero and variance one.","s":"∀ (x : ℝ), Causalean.Mathlib.stdNormalPDF x = ProbabilityTheory.gaussianPDFReal 0 1 x","l":49,"f":"Causalean/Mathlib/Probability/StdNormalCDF.lean"},{"n":"Causalean.MonotonicityKind","k":"inductive","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":1,"d":"A monotonicity classification is one of nondecreasing, nonincreasing, strictly increasing, or strictly decreasing.","s":"Type","l":30,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite ordered support with $K$ points, the adjacent-threshold set consists of every strictly positive support position, each representing the boundary between its immediately preceding position and itself.","s":"ℕ → Type","l":44,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj.lower","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Lower endpoint j-1 of an adjacent threshold.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → Fin K","l":51,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj.upper","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Upper endpoint j of an adjacent threshold.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → Fin K","l":55,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"A finite ordered first-stage index for the saturated multiple-instrument characterization: instrument support masses that are nonnegative and sum to one, together with a first-stage index that is weakly increasing in the displayed support order.","s":"ℕ → Type","l":61,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.centeredIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and a support point, the centered first-stage index at that point is its first-stage score minus the support-mass-weighted mean score.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Fin K → ℝ","l":141,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.centered_weight_sum_zero","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Centering identity Σ_k ρ_k (dhat_k - dbar) = 0.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), ∑ k, I.rho k * I.centeredIndex k = 0","l":158,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.fromMeasureScore","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a measurable sample space, a finite support of size $K$, a probability measure, a measurable instrument with that support, a real first-stage score at each support point, and the condition that this score is weakly increasing in support order, the ordered finite first-stage index has support masses equal to the instrument probabilities and the supplied score.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (Z : Ω → Fin K) → Measurable Z → (dhat : Fin K → ℝ) → (∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":115,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.meanIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index, the mean first-stage index is the support-mass-weighted average $\\sum_k \\rho_k\\widehat d_k$.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":136,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and an adjacent threshold, the MTW tail coefficient is the sum, over the corresponding upper tail, of each support mass times its centered first-stage index.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ","l":152,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff_eq_cov","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Covariance identity. For an adjacent threshold j, the MTW tail coefficient B_j equals the finite-support covariance between the first-stage index dhat(Z) and the upper-tail indicator 1_{Z ∈ T_j} under the instrument distribution ρ.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (j : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K), I.tailCoeff j = ∑ k, I.rho k * (I.dhat k - I.meanIndex) * ((if k ∈ Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail j then 1 else 0) - ∑ l, I.rho l * if l ∈ Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail j then 1 else 0)","l":278,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tailCoeff_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an adjacent threshold j with the support ordered by the first-stage index, the MTW tail coefficient B_j is nonnegative (def:po-estimand-mtw-tail-coefficients).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (j : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K), 0 ≤ I.tailCoeff j","l":175,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.tail_sum_interchange","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite upper-tail interchange identity used by the signed decomposition. It is the algebraic form of moving from Σ_k ρ_k a_k Σ_{j≤k} x_j to Σ_j B_j x_j.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (x : Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ), (∑ k, I.rho k * I.centeredIndex k * ∑ j, if ↑↑j ≤ ↑k then x j else 0) = ∑ j, I.tailCoeff j * x j","l":244,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex.upperTail","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite ordered support of size $K$ and an adjacent threshold, the upper-tail set contains exactly the support points at or above that threshold.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → Finset (Fin K)","l":147,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite-support matrix first stage for the source population-2SLS definition. The score vector q, support masses rho, reduced-form first-stage moments firstStageMoment, and invertible Gram matrix determine the projection coefficient by the displayed matrix inverse. The resulting fitted values are required to be in the displayed weakly increasing order so they can feed the MTW tail-coefficient algebra.","s":"ℕ → ℕ → Type","l":370,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.fittedValue","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite matrix first-stage specification and an instrument support point, the fitted first-stage value is the inner product of that point's score vector and the population first-stage projection coefficients.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Fin K → ℝ","l":428,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.gram","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite matrix first-stage specification, the Gram matrix is its mass-weighted second-moment matrix of instrument score vectors.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Matrix (Fin L) (Fin L) ℝ","l":408,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.instInvertibleGram","k":"instance","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"For a finite matrix first-stage specification, the assertion that its mass-weighted second-moment (Gram) matrix is invertible holds.","s":"{K L : ℕ} → (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L) → Invertible S.gram","l":413,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.projectionCoeff","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite matrix first-stage specification and a score-coordinate index, the population first-stage projection coefficient at that coordinate is the corresponding entry of the inverse Gram matrix times the vector of first-stage moments.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Fin L → ℝ","l":422,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage.toFiniteIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite matrix first-stage specification, the ordered finite first-stage index retains its support masses and uses its fitted first-stage values as the ordered score.","s":"{K L : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.MatrixFirstStage K L → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":434,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Multiple-instrument IV potential-outcome subsystem. Inside a potential-outcome system, this bundles a node serving as the instrument, taking values in a finite support of size K, a node serving as the binary treatment, and a node serving as the real-valued outcome, subject to the instrument, treatment, and outcome being pairwise distinct nodes. This is the Fin K generalization of PO.POIVSystem.","s":"Causalean.PO.POSystem → ℕ → Type (max u_1 u_2)","l":61,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Classical multi-instrument IV assumption bundle (ass:po-estimand-mtw-iv-validity). For a multiple-instrument potential-outcome subsystem, this packages the observed treatment and outcome equaling the potential treatment and outcome realized under the actual instrument value (SUTVA), and at each instrument support point, the instrument being independent of the counterfactual triple consisting of the potential treatment under that instrument value together with the two treatment-arm potential outcomes (IV exogeneity).","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Prop","l":197,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and an instrument support point, the potential treatment maps each unit to its binary treatment were the instrument fixed at that point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → P.Ω → Bool","l":100,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"On {Z = zᵏ}, the counterfactual treatment D(zᵏ) equals the factual D.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), S.Assumptions → ∀ (k : Fin K) {ω : P.Ω}, ω ∈ S.zEvent k → S.DofZ k ω = S.factualD ω","l":222,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.DofZ_eq_on_gEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"On {G = g}, the potential treatment D(zᵏ) is the constant g k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K) (k : Fin K) {ω : P.Ω}, ω ∈ S.gEvent g → S.DofZ k ω = g k","l":233,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and a binary treatment value, the potential outcome maps each unit to its real outcome were treatment fixed at that value.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Bool → P.Ω → ℝ","l":103,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and an instrument support point, the potential outcome under the induced treatment gives each unit's treated potential outcome if that instrument point induces treatment and its untreated potential outcome otherwise.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → P.Ω → ℝ","l":124,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.YofDofZ_def","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The potential outcome under the treatment that an instrument value induces sends a unit to that unit's treated potential outcome when the induced treatment is one, and to its untreated potential outcome otherwise.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), S.YofDofZ k = fun ω => if S.DofZ k ω = true then S.YofD true ω else S.YofD false ω","l":128,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.baseOutcome","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support size $K$, a multiple-instrument IV subsystem, the condition $K>0$, and a response type, the baseline response-type outcome is the conditional expectation, within that type, of the potential outcome under treatment induced by the first instrument support point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":384,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.baseTreatment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support size $K$, a multiple-instrument IV subsystem, and the condition $K>0$, the baseline treatment mean is the expectation of treatment induced by the first instrument support point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → ℝ","l":388,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.cfCell","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and an instrument support point, the counterfactual cell bundles the potential treatment at that point with the treated and untreated potential outcomes.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Causalean.PO.POCFBundle P","l":144,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and an instrument support point, the regimed treatment variable represents treatment under an intervention fixing the instrument at that point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Causalean.PO.RegimedVar P Bool","l":136,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the treatment variable is that subsystem's binary treatment represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P Bool","l":94,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.effect","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and a response type, the within-type causal effect is the conditional expectation of the treated-minus-untreated potential outcome among units of that type.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":377,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the factual treatment maps each unit to its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Bool","l":109,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the factual outcome maps each unit to its observed real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → ℝ","l":112,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Factual Y equals Y(factualD) pointwise.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":228,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the factual instrument maps each unit to its observed instrument value.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Fin K","l":106,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and a response type, the response-type event is the set of units whose vector of potential treatments equals that response type.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Set P.Ω","l":118,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent_iUnion","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type events cover the sample space.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), ⋃ g, S.gEvent g = Set.univ","l":361,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.gEvent_pairwise_disjoint","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type events are pairwise disjoint.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Pairwise (Function.onFun Disjoint S.gEvent)","l":353,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.integral_partition","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Total law over response types for an integrable integrand.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] {f : P.Ω → ℝ}, MeasureTheory.Integrable f P.μ → ∫ (ω : P.Ω), f ω ∂P.μ = ∑ g, (P.μ (S.gEvent g)).toReal * Causalean.PO.eventCondExp P.μ (S.gEvent g) f","l":365,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and a response type, the response-type mass is the probability that a unit has that response type.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":374,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type masses are nonnegative because they are real-valued probabilities.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ S.mass g","l":392,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.mass_sum_one","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type masses sum to one.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), ∑ g, S.mass g = 1","l":395,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurableSet_gEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Each response-type cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), MeasurableSet (S.gEvent g)","l":180,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Each instrument support cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), MeasurableSet (S.zEvent k)","l":184,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The potential treatment under any instrument support point is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), Measurable (S.DofZ k)","l":153,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The potential outcome under either treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (d : Bool), Measurable (S.YofD d)","l":158,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The outcome under the treatment induced by an instrument support point is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (k : Fin K), Measurable (S.YofDofZ k)","l":188,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualD","l":167,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual outcome value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualY","l":171,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The factual instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.factualZ","l":163,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.measurable_responseType","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The response-type map collecting all treatment responses is measurable.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K), Measurable S.responseType","l":175,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.observedBeta2SLS_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Potential-outcome multiple-IV 2SLS characterization (prop:po-estimand-mtw-response-type-form, PO level). Consider a finite instrument support of positive size K together with weights dhat that are weakly increasing in the support order, under the potential-outcome identifying assumption bundle. If every instrument-support cell has positive probability, the potential outcome under treatment and under control are integrable, the centered-instrument-weighted outcome and treatment are integrable, and the observed first-stage moment is nonzero, then the observed population 2SLS ratio E[h(Z)Y] / E[h(Z)D] equals the response-type weighted sum Σ_g ω_g Δ_g of within-type conditional causal effects Δ_g = E[Y(1) - Y(0) | G = g].","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (dhat : Fin K → ℝ) (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l), (∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable (fun ω => (S.toFiniteIndex dhat fun {k l} => hmono).centeredIndex (S.factualZ ω) * S.factualY ω) P.μ → MeasureTheory.Integrable (fun ω => (S.toFiniteIndex dhat fun {k l} => hmono).centeredIndex (S.factualZ ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) P.μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment P.μ S.factualZ S.factualD (S.toFiniteIndex dhat fun {k l} => hmono) ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS P.μ S.factualZ S.factualD S.factualY (S.toFiniteIndex dhat fun {k l} => hmono) = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand (S.toFiniteIndex dhat fun {k l} => hmono) (S.toPopulationBridge hK).stats","l":654,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.outcomeDrop","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"E[Y | Z = zᵏ] = E[Y(D(zᵏ))].","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.eventCondExp P.μ (S.zEvent k) S.factualY = ∫ (ω : P.Ω), S.YofDofZ k ω ∂P.μ","l":284,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.outcome_cell_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Outcome cell identity: E[Y | Z = zᵏ] = outcomeAtSupport k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.eventCondExp P.μ (S.zEvent k) S.factualY = (S.toPopulationBridge hK).outcomeAtSupport k","l":555,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.responseType","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the response-type map assigns each unit the binary vector of its potential treatments at every instrument support point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → P.Ω → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K","l":115,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.rho_eq_zMass","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The instrument masses ρ_k from toFiniteIndex are the probabilities of the observed instrument cells.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (dhat : Fin K → ℝ) (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) (k : Fin K), (S.toFiniteIndex dhat fun {k l} => hmono).rho k = (P.μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent S.factualZ k)).toReal","l":628,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.telescoped_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The telescoped adjacent step at support point k collapses to the difference of endpoint treatments: Σ_{j≤k}(d_j - d_{j-1}) = g k - g 0.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) (hK : 0 < K) (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K) (k : Fin K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.telescopedTypeStep g k = Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (g k) - Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (g (S.z0 hK))","l":434,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toFiniteIndex","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, a real first-stage score at each support point, and the condition that this score is weakly increasing in support order, the ordered finite first-stage index has the subsystem's instrument probabilities and the supplied score.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → (dhat : Fin K → ℝ) → (∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K","l":426,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toObservedBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support size $K$, a multiple-instrument IV subsystem, the condition $K>0$, its IV assumptions, a real first-stage score, weak monotonicity of that score in support order, positive probability for every instrument cell, and integrable treated and untreated potential outcomes, the observed population bridge links factual variables to the finite response-type representation.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) → (hK : 0 < K) → S.Assumptions → (dhat : Fin K → ℝ) → (hmono : ∀ {k l : Fin K}, ↑k ≤ ↑l → dhat k ≤ dhat l) → (∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge P.μ S.factualZ S.factualD S.factualY (S.toFiniteIndex dhat fun {k l} => hmono) (S.toPopulationBridge hK)","l":636,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toPopulationBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support size $K$, a multiple-instrument IV subsystem, and the condition $K>0$, the saturated finite-support population bridge is assembled from its response-type statistics and baseline quantities.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K","l":421,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.toStats","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the finite response-type statistics consist of the subsystem's response-type masses and within-type causal effects.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K","l":414,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.treatmentDrop","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"E[D | Z = zᵏ] = E[D(zᵏ)] (0/1-coded).","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → (Causalean.PO.eventCondExp P.μ (S.zEvent k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.DofZ k ω) ∂P.μ","l":248,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.treatment_cell_eq","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Treatment cell identity: E[D | Z = zᵏ] = baseTreatment + treatmentAtSupport k.","s":"∀ {P : Causalean.PO.POSystem} {K : ℕ} (S : Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K) [MeasureTheory.IsFiniteMeasure P.μ] (hK : 0 < K), S.Assumptions → ∀ (k : Fin K), P.μ (S.zEvent k) ≠ 0 → (Causalean.PO.eventCondExp P.μ (S.zEvent k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (S.factualD ω)) = S.baseTreatment hK + (S.toPopulationBridge hK).treatmentAtSupport k","l":495,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and a binary treatment value, the regimed outcome variable represents outcome under an intervention fixing treatment at that value.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Bool → Causalean.PO.RegimedVar P ℝ","l":140,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the outcome variable is that subsystem's real-valued outcome represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P ℝ","l":97,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.z0","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support size $K$, a multiple-instrument IV subsystem, and the condition $K>0$, the reference support point is the first point of the support.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → 0 < K → Fin K","l":381,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, a multiple-instrument IV subsystem, and an instrument support point, the instrument cell is the set of units whose factual instrument equals that point.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Fin K → Set P.Ω","l":121,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.POBridge","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a potential-outcome system, a finite instrument support of size $K$, and a multiple-instrument IV subsystem, the instrument variable is that subsystem's instrument represented as a potential-outcome variable with the stated finite support.","s":"{P : Causalean.PO.POSystem} → {K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.POMultipleIVSystem P K → Causalean.PO.POVar P (Fin K)","l":91,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/POBridge.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a response type is a binary treatment response specified for every instrument support point.","s":"ℕ → Type","l":57,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite response-type statistics. For a finite family of latent response types, this records the probability mass of each type and the type-specific average causal effect Δ_g, subject to every mass being nonnegative and the masses summing to one, so together they form a probability vector over response types.","s":"ℕ → Type","l":68,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.ComponentwiseMonotoneRestriction","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Opaque response-type restriction interface.","s":"{K : ℕ} → Type","l":293,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Saturated finite-support population bridge for the MTW identification step. Bundles a finite response-type statistics record supplying the type masses and type-specific effects together with a response-type-specific baseline outcome mean, the term the centered first-stage index cancels, leaving the telescoped adjacent treatment increments used by the finite algebra.","s":"ℕ → Type","l":111,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge","k":"structure","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Bridge from an observed finite-instrument population to the response-type algebra. Given an instrument, a binary treatment, an outcome, a finite index, and a saturated response-type population P, this bundles the underlying law being a probability measure, the finite-index masses reproducing the probabilities of the observed instrument cells, the observed conditional mean of the outcome on each instrument cell equaling the response-type expansion P.outcomeAtSupport, a baseline treatment mean common across cells after exogeneity, and the observed conditional mean of the treatment on each instrument cell equaling that common baseline plus the response-type expansion P.treatmentAtSupport.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Type","l":132,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_beta2SLSFiniteAlgebra","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"End-to-end bridge to the finite response-type algebra. Given a bridge B linking the observed data to the response-type population, provided the instrument is measurable, the centered-instrument-weighted outcome is integrable, and the centered-instrument-weighted treatment is integrable, the observed population 2SLS ratio E[h(Z)Y] / E[h(Z)D] equals the finite response-type algebra estimand beta2SLSFiniteAlgebra.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I P.stats","l":251,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_beta2SLSPopulationBridge","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Observed population 2SLS, written as E[h(Z)Y]/E[h(Z)D], equals the saturated finite-support population bridge ratio.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = P.beta2SLSPopulationBridge I","l":240,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedBeta2SLS_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"End-to-end corollary (prop:po-estimand-mtw-response-type-form, observed level). Given a bridge B linking the data-generating instrument, treatment, and outcome to the response-type population, provided the instrument is measurable, the centered-instrument-weighted outcome is integrable, the centered-instrument-weighted treatment is integrable, and the observed first-stage moment observedFirstStageMoment μ Z D I is nonzero, the observed population 2SLS ratio equals the response-type weighted sum of within-type causal effects.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS μ Z D Y I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats","l":129,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedFirstStageMoment_eq_firstStageMoment","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The observed first-stage moment equals the finite response-type first-stage moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I = P.firstStageMoment I","l":201,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge.observedReducedFormMoment_eq_reducedFormMoment","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The observed reduced-form moment equals the finite response-type reduced-form moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} {μ : MeasureTheory.Measure Ω} {Z : Ω → Fin K} {D : Ω → Bool} {Y : Ω → ℝ} {I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K} {P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K} (B : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.ObservedBridge μ Z D Y I P) [MeasureTheory.IsFiniteMeasure μ], Measurable Z → MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment μ Z Y I = P.reducedFormMoment I","l":189,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.baselineMoment_eq_zero","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The baseline outcome component vanishes because the first-stage index is centered.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), ∑ k, I.rho k * I.centeredIndex k * ∑ g, P.stats.mass g * P.baseOutcome g = 0","l":153,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a saturated finite-support population bridge, and an ordered finite first-stage index, the population multiple-IV 2SLS ratio is its reduced-form moment divided by its first-stage moment.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":149,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_beta2SLSFiniteAlgebra","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Saturated finite-support MTW identification. The population 2SLS ratio E[h(Z)Y] / E[h(Z)D], after consistency, exogeneity, exclusion, and telescoping, is exactly the finite response-type ratio.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I P.stats","l":277,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_positiveResponseTypeAverage","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Positive response-type average (population bridge). When the response types are sign-aligned with the instrument order and the first-stage type-weight denominator is strictly positive, the population-bridge 2SLS estimand beta2SLSPopulationBridge equals the response-type estimand, its normalized response-type weights are all nonnegative, and those weights sum to one.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I P.stats → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I P.stats g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I P.stats g = 1","l":103,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type weighted-sum form (population bridge). Provided the first-stage type-weight denominator is nonzero, the population- bridge 2SLS estimand equals the response-type-weighted sum Σ_g ω_g Δ_g of within-type causal effects.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats ≠ 0 → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I P.stats","l":93,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.beta2SLSPopulationBridge_eq_signedAdjacentRatio","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Signed adjacent ratio form (population bridge). Provided the signed adjacent first-stage denominator is nonzero, the population- bridge 2SLS estimand equals the signed adjacent numerator divided by the signed adjacent denominator.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I P.stats ≠ 0 → P.beta2SLSPopulationBridge I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator I P.stats / Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I P.stats","l":82,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.firstStageMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a saturated finite-support population bridge, and an ordered finite first-stage index, the population first-stage moment is the support-mass-weighted sum of centered first-stage indices times response-type treatment expansions.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":145,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.firstStageMoment_eq_typeWeightDenom","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The finite-support first-stage moment is the response-type denominator. This is the denominator half of the MTW identification step.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.firstStageMoment I = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I P.stats","l":170,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedBeta2SLS","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a measurable sample space, a finite instrument support of size $K$, a measure, an instrument, a binary treatment, a real outcome, and an ordered finite first-stage index, the observed population 2SLS ratio is the observed reduced-form moment divided by the observed first-stage moment.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":57,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a measurable sample space, a finite instrument support of size $K$, a measure, an instrument, a binary treatment, and an ordered finite first-stage index, the observed first-stage moment is the integral of the centered first-stage index evaluated at the instrument times the zero-one treatment indicator.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → Bool) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":52,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment_eq_sum_eventCondExp","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite-support total-law rewrite of E[h(Z)D] into instrument-cell conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Z : Ω → Fin K), Measurable Z → ∀ (D : Ω → Bool) (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedFirstStageMoment μ Z D I = ∑ k, (μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k)).toReal * I.centeredIndex k * Causalean.PO.eventCondExp μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k) fun ω => Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal (D ω)","l":119,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a measurable sample space, a finite instrument support of size $K$, a measure, an instrument, a real outcome, and an ordered finite first-stage index, the observed reduced-form moment is the integral of the centered first-stage index evaluated at the instrument times the outcome.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → (Ω → ℝ) → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":47,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment_eq_sum_eventCondExp","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Finite-support total-law rewrite of E[h(Z)Y] into instrument-cell conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Z : Ω → Fin K), Measurable Z → ∀ (Y : Ω → ℝ) (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K), MeasureTheory.Integrable (fun ω => I.centeredIndex (Z ω) * Y ω) μ → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.observedReducedFormMoment μ Z Y I = ∑ k, (μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k)).toReal * I.centeredIndex k * Causalean.PO.eventCondExp μ (Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent Z k) Y","l":97,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.outcomeAtSupport","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a saturated finite-support population bridge, and a support point, the response-type outcome expansion is the baseline outcome plus the response-type-mass-weighted sum of within-type effects times telescoped treatment responses.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Fin K → ℝ","l":131,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.reducedFormMoment","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a saturated finite-support population bridge, and an ordered finite first-stage index, the population reduced-form moment is the support-mass-weighted sum of centered first-stage indices times response-type outcome expansions.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → ℝ","l":141,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.reducedFormMoment_eq_typeWeightNumerator","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"The finite-support reduced-form moment is the response-type numerator. This is the numerator half of the MTW identification step.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (P : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K), P.reducedFormMoment I = ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight I P.stats g * P.stats.effect g","l":212,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.telescopedTypeStep","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a response type, and a support point, the telescoped adjacent treatment response is the sum of that type's adjacent treatment-response increments from the first support point through that point.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Fin K → ℝ","l":127,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.treatmentAtSupport","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a saturated finite-support population bridge, and a support point, the response-type treatment expansion is the response-type-mass-weighted sum of telescoped treatment responses at that point.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge K → Fin K → ℝ","l":137,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.PopulationBridge.zEvent","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Population","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a sample space, a finite instrument support of size $K$, an instrument on that sample space, and a support point, the instrument cell is the set of units whose instrument equals that point.","s":"{Ω : Type u_1} → {K : ℕ} → (Ω → Fin K) → Fin K → Set Ω","l":43,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Population.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, sign alignment means that every response type with strictly positive mass has a nonnegative tail-coefficient-weighted sum of adjacent treatment-response increments.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Prop","l":288,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, the finite-algebra 2SLS estimand is the unnormalized response-type-weighted sum of causal effects divided by the sum of unnormalized response-type weights.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":104,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_positiveResponseTypeAverage","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Positive-weight response-type characterization. When the response types are sign-aligned with the instrument order and the first-stage type-weight denominator is strictly positive, the finite-algebra 2SLS estimand equals the response-type estimand, every normalized response-type weight is nonnegative, and the weights sum to one.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":377,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_positiveResponseTypeAverage'","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Positive response-type average under MTW sign alignment and a positive finite first-stage denominator (prop:po-estimand-mtw-positive-weights, finite algebra layer).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R ∧ (∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g) ∧ ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":65,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_responseTypeWeightedSum","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Response-type form of the finite MTW ratio (prop:po-estimand-mtw-response-type-form). Provided the first-stage type-weight denominator is nonzero, the finite-algebra 2SLS estimand equals the response-type-weighted sum of within-type causal effects.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R","l":358,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_responseTypeWeightedSum'","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Response-type weighted-sum form of the finite MTW algebra (prop:po-estimand-mtw-response-type-form, finite algebra layer).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand I R","l":58,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra_eq_signedAdjacentRatio","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Finite response-type ratio as signed adjacent ratio. Provided the signed adjacent first-stage denominator is nonzero, the finite-algebra MTW estimand beta2SLSFiniteAlgebra equals the signed adjacent numerator Σ_g λ_g Δ_g divided by the signed adjacent denominator Σ_g λ_g.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I R ≠ 0 → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.beta2SLSFiniteAlgebra I R = Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator I R / Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator I R","l":44,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index, finite response-type statistics, and a response type, the normalized response-type weight is that type's unnormalized weight divided by the sum of all unnormalized weights.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":96,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight_nonneg_of_signAligned","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Normalized response-type weights are nonnegative when sign alignment holds and the denominator is positive (prop:po-estimand-mtw-positive-weights).","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → ∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g","l":341,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight_sum_eq_one_of_pos","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Normalized response-type weights sum to one when the first-stage denominator is positive.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), 0 < Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom I R → ∑ g, Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g = 1","l":350,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.responseTypeEstimand","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, the response-type estimand is the sum of each within-type causal effect weighted by its normalized response-type weight.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":100,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentDenominator","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, the signed adjacent first-stage denominator is the sum of all unnormalized response-type weights.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":40,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.signedAdjacentNumerator","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.Main","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, the signed adjacent reduced-form numerator is the sum of each within-type causal effect multiplied by its unnormalized response-type weight.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":36,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/Main.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.typeWeightDenom","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index and finite response-type statistics, the response-type weight denominator is the sum of the unnormalized weights over all response types.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → ℝ","l":92,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For an ordered finite first-stage index, finite response-type statistics, and a response type, the unnormalized MTW response-type weight is that type's mass times the sum of each tail coefficient times its adjacent treatment-response increment.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → ℝ","l":88,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight_nonneg_of_signAligned","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Sign alignment and nonnegative type masses imply nonnegative unnormalized response-type weights.","s":"∀ {K : ℕ} (I : Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex K) (R : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats K), Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.SignAligned I R → ∀ (g : Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.unnormTypeWeight I R g","l":329,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.boolToReal","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a binary treatment indicator, the real-valued treatment indicator equals one for treatment and zero otherwise.","s":"Bool → ℝ","l":60,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.exists_negativeNormalizedTypeWeight","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Negative-weights theorem. There exists a finite-support instrument index, a response-type population, and a response type such that, with two support points and a 3/4-defier population, that type has positive mass yet a negative normalized response-type weight (equal to −1/2).","s":"∃ I R g, 0 < R.mass g ∧ Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypeStats.normalizedTypeWeight I R g < 0","l":582,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.firstStageGram","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a finite instrument support of size $K$, a score-vector dimension $L$, a real mass assigned to each support point, and a real score vector at each support point, the first-stage Gram matrix has entry $(a,b)$ equal to the mass-weighted sum of the products of score coordinates $a$ and $b$.","s":"{K L : ℕ} → (Fin K → ℝ) → (Fin K → Fin L → ℝ) → Matrix (Fin L) (Fin L) ℝ","l":361,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"Given a sample space equipped with a σ-algebra, a finite instrument support of size $K$, a measure on that sample space, an instrument taking values in that support, and a support point, the support mass is the real-valued measure of the event that the instrument equals that point.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {K : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin K) → Fin K → ℝ","l":77,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"Support masses obtained from a measure are nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) (Z : Ω → Fin K) (k : Fin K), 0 ≤ Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass μ Z k","l":85,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass_sum_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.FiniteIndex","a":"PO/ID/Exact/MultipleInstrumentIV","t":2,"d":"For a probability measure, the finite support masses induced by Z sum to one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {K : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Z : Ω → Fin K), Measurable Z → ∑ k, Causalean.PO.ID.Exact.MultipleInstrumentIV.supportMass μ Z k = 1","l":91,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/FiniteIndex.lean"},{"n":"Causalean.PO.ID.Exact.MultipleInstrumentIV.typeStep","k":"def","m":"Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseTypes","a":"PO/ID/Exact/MultipleInstrumentIV","t":1,"d":"For a finite instrument support of size $K$, a response type, and an adjacent threshold, the adjacent treatment-response increment is the real-valued treatment response at the upper support point minus that at the preceding point.","s":"{K : ℕ} → Causalean.PO.ID.Exact.MultipleInstrumentIV.ResponseType K → Causalean.PO.ID.Exact.MultipleInstrumentIV.Adj K → ℝ","l":64,"f":"Causalean/PO/ID/Exact/MultipleInstrumentIV/ResponseTypes.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.Crossing","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins, an initial treatment level, a final treatment level, and a margin, the crossing condition holds exactly when the movement starts below that margin's upper endpoint and ends at or above it.","s":"{J : ℕ} → Fin (J + 1) → Fin (J + 1) → Fin J → Prop","l":51,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins, an initial treatment level, a final treatment level, and a margin, the crossing indicator equals one when the movement crosses that margin and zero otherwise.","s":"{J : ℕ} → Fin (J + 1) → Fin (J + 1) → Fin J → ℝ","l":55,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins and an ordered treatment level, the numeric intensity value is that level's position in the ordered list, expressed as a real number.","s":"{J : ℕ} → Fin (J + 1) → ℝ","l":43,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.lowerLevel","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins and a margin, the lower treatment level is the lower endpoint of that margin in the ordered list of $J+1$ levels.","s":"{J : ℕ} → Fin J → Fin (J + 1)","l":35,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.marginIncrement","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins, a real-valued function on the $J+1$ ordered treatment levels, and a margin, the margin increment is the function value at that margin's upper endpoint minus its value at the lower endpoint.","s":"{J : ℕ} → (Fin (J + 1) → ℝ) → Fin J → ℝ","l":47,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a finite collection of indices, a real weight assigned to each index, and one index, the normalized finite weight is that index's weight divided by the sum of all weights.","s":"{ι : Type u_1} → [Fintype ι] → (ι → ℝ) → ι → ℝ","l":117,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Nonnegativity of normalized weights from nonnegative raw weights and a positive normalizing sum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), (∀ (i : ι), 0 ≤ a i) → 0 < ∑ i, a i → ∀ (i : ι), 0 ≤ Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight a i","l":121,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.ordered_telescope_identity","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Ordered telescoping for the identity intensity map. For an ordered treatment level a no larger than b among J + 1 ordered intensity levels, the numeric gap b − a equals the number of unit margins j → j+1 that the movement from a to b crosses.","s":"∀ {J : ℕ} {a b : Fin (J + 1)}, a ≤ b → Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue b - Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue a = ∑ j, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator a b j","l":108,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.ordered_telescope_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Ordered telescoping across crossed margins for an arbitrary real-valued function on finite ordered levels.","s":"∀ {J : ℕ} (f : Fin (J + 1) → ℝ) {a b : Fin (J + 1)}, a ≤ b → f b - f a = ∑ j, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.marginIncrement f j * Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.crossingIndicator a b j","l":78,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.sum_normalizedWeight_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Normalized finite weights sum to one when the normalizing sum is positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), 0 < ∑ i, a i → ∑ i, Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.normalizedWeight a i = 1","l":128,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.upperLevel","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"For a sequence of $J$ adjacent margins and a margin, the upper treatment level is the upper endpoint of that margin in the ordered list of $J+1$ levels.","s":"{J : ℕ} → Fin J → Fin (J + 1)","l":39,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/OrderedTreatment.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"A variable-intensity IV system records an instrument, an ordered treatment intensity, and an outcome inside a potential-outcome system, with the treatment taking at least one margin.","s":"Causalean.PO.POSystem → (𝒵 : Type u_1) → [inst : MeasurableSpace 𝒵] → [Fintype 𝒵] → [MeasurableSingletonClass 𝒵] → ℕ → Type (max (max u_1 u_2) u_3)","l":44,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the potential treatment intensity assigns every unit the ordered treatment level that would be observed if the instrument were fixed to that value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → P.Ω → Fin (J + 1)","l":91,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"On zEvent z, the counterfactual intensity D(z) equals the factual D.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), P.Consistency → ∀ (z : 𝒵) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":414,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Centered finite instrument score used to define an interface-only population 2SLS estimand for multivalued instruments.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → ℕ → Type u_1","l":918,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore.beta2SLS","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a population 2SLS score, the population 2SLS estimand is the ratio of the population mean fitted-treatment--outcome product to the population mean fitted-treatment--treatment product.","s":"{k : ℕ} → {P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → {S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J} → S.PopulationTwoSLSScore k → ℝ","l":943,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.PopulationTwoSLSScore.fittedTreatment","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a population 2SLS score, the first-stage fitted treatment assigns each unit the linear projection of its factual treatment intensity onto that score.","s":"{k : ℕ} → {P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → {S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J} → S.PopulationTwoSLSScore k → P.Ω → ℝ","l":936,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given the condition that there is exactly one treatment margin, the unique binary-treatment margin is the sole margin of the ordered treatment scale.","s":"{J : ℕ} → J = 1 → Fin J","l":818,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.crossingEvent_eq_complianceEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Under J = 1, the unique crossing event coincides with the complier event {D(z1) = Fin.last J ∧ D(z0) = 0}, i.e. the unit jump from level 0 to the maximum level. When J = 1, Fin.last 1 = 1 : Fin 2, so this recovers exactly the binary-treatment LATE complier event {D(z1) = 1 ∧ D(z0) = 0} of Imbens-Angrist (1994): β_Wald(z0,z1) = E[Y(1) − Y(0) | D(z1) = Fin.last J ∧ D(z0) = 0].","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z0 z1 : 𝒵) (hJ : J = 1), S.crossingEvent z0 z1 (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hJ) = {ω | S.DofZ z1 ω = Fin.last J ∧ S.DofZ z0 ω = 0}","l":842,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_constantResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Constant marginal response specialization.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵} {τ : ℝ}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → (∀ (j : Fin J), S.marginResponse j =ᵐ[P.μ] fun x => τ) → S.wald z0 z1 = τ","l":874,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_late_of_binaryIntensity","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Binary-intensity specialization: Wald recovers LATE. Under the variable-intensity IV validity assumptions, with positive probability of the instrument cell Z = z0, positive probability of the instrument cell Z = z1, and a single treatment margin, J = 1, the directed Wald estimand equals the conditional mean unit causal response given the unique crossing event — the classical binary-treatment local average treatment effect.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → ∀ (hBinaryIntensity : J = 1), S.wald z0 z1 = Causalean.PO.eventCondExp P.μ (S.crossingEvent z0 z1 (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hBinaryIntensity)) (S.marginResponse (Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.binaryMargin hBinaryIntensity))","l":822,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.SpecialCases.wald_eq_marginResponseAverage","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Margin-specific response average specialization. Under the variable-intensity IV validity assumptions, with positive probability of the instrument cell Z = z0, positive probability of the instrument cell Z = z1, and a candidate margin-response function m that agrees, on each treatment-intensity margin, with the conditional mean causal response given that margin's crossing event, the directed Wald estimand equals the crossing-probability-weighted average of m across margins.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵} (m : Fin J → ℝ), S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → (∀ (j : Fin J), m j = S.conditionalMarginResponse z0 z1 j) → S.wald z0 z1 = ∑ j, S.crossingWeight z0 z1 j * m j","l":900,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.TwoSLSContrastDecomposition","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Deferred interface for expanding a multivalued-instrument 2SLS estimand into finite directed contrasts. A later theorem can add sign-alignment assumptions to turn this signed decomposition into a convex ACR average.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) → {k : ℕ} → S.PopulationTwoSLSScore k → Type u_1","l":955,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.ValidContrastAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"IV-validity assumptions for a fixed directed contrast (z0,z1).","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Prop","l":356,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and a treatment level, the treatment-indexed potential outcome assigns every unit the outcome that would be observed if treatment intensity were fixed to that level.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin (J + 1) → P.Ω → ℝ","l":96,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the instrument-induced potential outcome assigns each unit the potential outcome at the treatment level induced by that instrument value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → P.Ω → ℝ","l":288,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.YofDofZ_def","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential outcome under the treatment intensity that an instrument value induces sends a unit to that unit's potential outcome at the induced intensity.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), S.YofDofZ z = fun ω => S.YofD (S.DofZ z ω) ω","l":296,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.averageCausalResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the Angrist--Imbens average causal response is the crossing-probability-weighted sum of conditional margin responses.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":284,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.cfContrastBundle","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the contrast counterfactual bundle contains the potential treatments under those values and all treatment-indexed potential outcomes.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Causalean.PO.POCFBundle P","l":162,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the observed first-stage conditional mean is the conditional mean of factual treatment intensity in the instrument cell at that value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → ℝ","l":338,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ_left_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage bridge (left cell z0): the observed first-stage conditional mean equals the unconditional expectation of the potential intensity D(z0).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → S.condExpDZ z0 = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z0 ω) ∂P.μ","l":425,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpDZ_right_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage bridge (right cell z1).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z1)).toReal → S.condExpDZ z1 = ∫ (ω : P.Ω), Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z1 ω) ∂P.μ","l":468,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the observed reduced-form conditional mean is the conditional mean of the factual outcome in the instrument cell at that value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → ℝ","l":345,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ_left_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Reduced-form bridge (left cell z0): the observed reduced-form conditional mean equals the unconditional expectation of Y(D(z0)).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → S.condExpYZ z0 = ∫ (ω : P.Ω), S.YofDofZ z0 ω ∂P.μ","l":512,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.condExpYZ_right_eq_integral","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Reduced-form bridge (right cell z1).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z1)).toReal → S.condExpYZ z1 = ∫ (ω : P.Ω), S.YofDofZ z1 ω ∂P.μ","l":591,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.conditionalMarginResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, two instrument values, and a treatment margin, the conditional margin response is the average unit-level causal response among units whose potential treatment crosses that margin when the instrument changes from the first value to the second.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":278,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingEvent","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, two instrument values, and a treatment margin, the crossing event is the set of units whose potential treatment moves across that margin when the instrument changes from the first value to the second.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → Set P.Ω","l":174,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingProb","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, two instrument values, and a treatment margin, the crossing probability is the probability of the corresponding crossing event.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":178,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingWeight","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, two instrument values, and a treatment margin, the normalized margin-crossing weight is that margin's crossing probability divided by total crossing probability.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":256,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.crossingWeight_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Crossing weights are nonnegative for a valid directed contrast.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∀ (j : Fin J), 0 ≤ S.crossingWeight z0 z1 j","l":701,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the regimed treatment variable is the treatment variable under the regime that fixes the instrument to that value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → Causalean.PO.RegimedVar P (Fin (J + 1))","l":147,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the treatment-intensity potential-outcome variable is its ordered treatment, whose admissible levels run from zero through $J$.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P (Fin (J + 1))","l":85,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the factual treatment intensity assigns each unit its observed ordered treatment level.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → Fin (J + 1)","l":105,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the factual outcome assigns each unit its observed real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → ℝ","l":108,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Factual Y equals the counterfactual Y(factualD ω).","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), P.Consistency → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":420,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the factual instrument assigns each unit its observed instrument value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → P.Ω → 𝒵","l":102,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.firstStageContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the potential first-stage contrast is the population mean of the difference between the real-valued potential treatment intensities under the second and first values.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":326,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.firstStage_eq_sum_crossingProb","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"First-stage denominator equals the sum of crossing probabilities.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → S.firstStageContrast z0 z1 = ∑ j, S.crossingProb z0 z1 j","l":669,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedACR","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the indicator-weighted average causal response is the unnormalized contrast divided by total crossing probability.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":271,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedACR_eq_averageCausalResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Indicator-weighted and conditional-mean ACR forms agree. For any pair of instrument values z0 and z1, the indicator-weighted average causal response — the ratio of the summed crossing-indicator-weighted outcome contrasts to the summed crossing probabilities — equals the crossing-probability- weighted average of the conditional-mean margin responses.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.indicatorWeightedACR z0 z1 = S.averageCausalResponse z0 z1","l":763,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.indicatorWeightedEffect","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, two instrument values, and a treatment margin, the indicator-weighted margin effect is the population mean of the unit-level margin response restricted to the corresponding crossing event.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → Fin J → ℝ","l":260,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.integrable_intensityValue_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The real intensity of the potential treatment at a fixed instrument value is integrable, being a bounded measurable function under a finite measure.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), MeasureTheory.Integrable (fun ω => Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z ω)) P.μ","l":197,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.integrable_marginResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Integrability of a margin response, derived from H7.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∀ (j : Fin J), MeasureTheory.Integrable (S.marginResponse j) P.μ","l":380,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.marginResponse","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and a treatment margin, the unit-level margin response assigns each unit the difference between its potential outcomes at the upper and lower levels of that margin.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin J → P.Ω → ℝ","l":169,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual instrument cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), MeasurableSet (S.zEvent z)","l":136,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential treatment under a fixed instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), Measurable (S.DofZ z)","l":114,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (d : Fin (J + 1)), Measurable (S.YofD d)","l":119,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The potential outcome under the instrument-induced treatment level is a measurable function of the unit.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), Measurable (S.YofDofZ z)","l":303,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualD","l":128,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualY","l":132,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J), Measurable S.factualZ","l":124,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_intensityValue","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Reading an ordered treatment level as a real intensity is a measurable map on the finite level set.","s":"∀ {J : ℕ}, Measurable fun d => Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue d","l":182,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.measurable_intensityValue_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"The real intensity of the potential treatment at a fixed instrument value is a measurable function of the unit.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) (z : 𝒵), Measurable fun ω => Causalean.PO.ID.Exact.VariableIntensityIV.OrderedTreatment.intensityValue (S.DofZ z ω)","l":190,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.outcomeBundle","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the outcome counterfactual bundle contains the potential outcomes at every one of the $J+1$ ordered treatment levels.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POCFBundle P","l":155,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.reducedFormContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the potential reduced-form contrast is the population mean of the difference between instrument-induced potential outcomes under the second and first values.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":334,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.reducedForm_eq_sum_crossingEffects","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Reduced-form decomposition across crossed margins. Fix a directed instrument contrast (z0, z1). Under the variable-intensity IV validity assumptions — SUTVA consistency of treatment and outcome, instrument independence from the potential treatments and treatment-indexed potential outcomes, almost-sure directed monotonicity of the potential treatment intensity in the instrument, a positive first stage, and integrability of every treatment-indexed potential outcome, the potential reduced-form contrast E[Y(D(z1)) − Y(D(z0))] equals the sum, over treatment-intensity margins, of the expected unit causal response on each margin restricted to the event that the instrument move from z0 to z1 crosses that margin.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → S.reducedFormContrast z0 z1 = ∑ j, S.indicatorWeightedEffect z0 z1 j","l":722,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.sum_crossingWeight_eq_one","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":2,"d":"Crossing weights sum to one for a valid directed contrast.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → ∑ j, S.crossingWeight z0 z1 j = 1","l":713,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.totalCrossingProb","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the total crossing probability is the sum of the crossing probabilities over all treatment margins.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":249,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.unnormalizedACRContrast","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the unnormalized average-causal-response contrast is the sum of indicator-weighted margin effects over all treatment margins.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":267,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.wald","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and two instrument values, the directed Wald estimand is the difference in observed conditional outcome means divided by the corresponding difference in observed conditional treatment-intensity means.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → 𝒵 → ℝ","l":351,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.wald_eq_averageCausalResponse","k":"theorem","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Angrist-Imbens average-causal-response characterization. Under the variable-intensity IV validity assumptions, with the instrument cell Z = z0 having positive probability and the instrument cell Z = z1 having positive probability, the directed Wald estimand — the ratio of the reduced-form to first-stage conditional-mean contrasts across the two instrument cells — equals the Angrist-Imbens average causal response: the crossing-probability-weighted average, over treatment-intensity margins, of the conditional mean causal response given that the instrument move from z0 to z1 crosses that margin.","s":"∀ {P : Causalean.PO.POSystem} {𝒵 : Type u_1} [inst : MeasurableSpace 𝒵] [inst_1 : Fintype 𝒵] [inst_2 : MeasurableSingletonClass 𝒵] {J : ℕ} (S : Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J) {z0 z1 : 𝒵}, S.ValidContrastAssumptions z0 z1 → 0 < (P.μ (S.zEvent z0)).toReal → 0 < (P.μ (S.zEvent z1)).toReal → S.wald z0 z1 = S.averageCausalResponse z0 z1","l":785,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and a treatment level, the regimed outcome variable is the outcome variable under the regime that fixes treatment to that level.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Fin (J + 1) → Causalean.PO.RegimedVar P ℝ","l":151,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the outcome potential-outcome variable is its real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P ℝ","l":88,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system and an instrument value, the instrument cell is the set of units whose factual instrument equals that value.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → 𝒵 → Set P.Ω","l":111,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensity","a":"PO/ID/Exact/VariableIntensityIV","t":1,"d":"Given a variable-intensity IV system, the instrument potential-outcome variable is that system's instrument, represented with its finite instrument-value space.","s":"{P : Causalean.PO.POSystem} → {𝒵 : Type u_1} → [inst : MeasurableSpace 𝒵] → [inst_1 : Fintype 𝒵] → [inst_2 : MeasurableSingletonClass 𝒵] → {J : ℕ} → Causalean.PO.ID.Exact.VariableIntensityIV.VariableIntensityIVSystem P 𝒵 J → Causalean.PO.POVar P 𝒵","l":82,"f":"Causalean/PO/ID/Exact/VariableIntensityIV/VariableIntensity.lean"},{"n":"Causalean.PO.IsRegressionFunction","k":"structure","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":1,"d":"A regression-function representative for the conditional mean of a response g given a conditioning variable X: a candidate function f on the real line that is measurable, for which the response g is integrable and the composite f ∘ X is integrable, and such that the integral of g over every measurable event determined by X equals the integral of f ∘ X over that same event.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → (ℝ → ℝ) → Prop","l":40,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.aeEq_of_aeEq_response","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":1,"d":"Fix a conditioning variable X that is μ-almost-everywhere measurable. If two response variables g₁ and g₂ are μ-almost- everywhere equal, and f₁ is a regression-function representative of the conditional mean of g₁ given X while f₂ is a regression-function representative of the conditional mean of g₂ given X, then f₁ and f₂ agree (μ.map X)-almost everywhere.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g₁ g₂ : Ω → ℝ} {f₁ f₂ : ℝ → ℝ}, AEMeasurable X μ → g₁ =ᵐ[μ] g₂ → Causalean.PO.IsRegressionFunction μ X g₁ f₁ → Causalean.PO.IsRegressionFunction μ X g₂ f₂ → f₁ =ᵐ[MeasureTheory.Measure.map X μ] f₂","l":98,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.integrable_pushforward","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":2,"d":"Pushforward integrability: f is integrable under μ.map X whenever X is AEMeasurable and f represents E[g|X=·].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g : Ω → ℝ} {f : ℝ → ℝ}, Causalean.PO.IsRegressionFunction μ X g f → AEMeasurable X μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map X μ)","l":64,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.IsRegressionFunction.sub","k":"theorem","m":"Causalean.PO.Analysis.Regression","a":"PO/Analysis","t":2,"d":"Linearity (subtraction) of IsRegressionFunction: difference of representatives is the representative of the difference.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X g₁ g₂ : Ω → ℝ} {f₁ f₂ : ℝ → ℝ}, Causalean.PO.IsRegressionFunction μ X g₁ f₁ → Causalean.PO.IsRegressionFunction μ X g₂ f₂ → Causalean.PO.IsRegressionFunction μ X (fun ω => g₁ ω - g₂ ω) fun x => f₁ x - f₂ x","l":74,"f":"Causalean/PO/Analysis/Regression.lean"},{"n":"Causalean.PO.OR","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a first real number and a second real number, the odds ratio is the first number's odds divided by the second number's odds, $(p₁/(1-p₁))/(p₂/(1-p₂))$.","s":"ℝ → ℝ → ℝ","l":38,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.ObsIdx","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model, the observed-node index set consists of all random observed nodes of that model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max 0 uN)","l":65,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POBackdoorSystem","k":"structure","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"A backdoor system consists of a binary treatment variable, a real-valued outcome variable, and an adjustment covariate with an arbitrary measurable value space, where the three variables are pairwise distinct.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max u_1 u_2) u_3)","l":54,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the average treatment effect is the population mean of the treated potential outcome minus the untreated potential outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":198,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the average treatment effect on the treated is the population mean treatment effect weighted by the treated-arm indicator and divided by the marginal treatment probability. It is ATT = E[A · (Y(1) − Y(0))] / π_T.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":49,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Backdoor assumptions for ATT identification. These are the standard ATT conditions, with overlap required on only one side — every covariate stratum keeps a positive chance of the control arm (e(X) < 1) — which is strictly weaker than the two-sided overlap the ATE needs, because on the treated the outcome is observed directly.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":66,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions.integrable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"Under the ATT assumption bundle the potential outcome of either arm is integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → ∀ (d : Bool), MeasureTheory.Integrable (S.YofD d) P.μ","l":104,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"Under the ATT assumption bundle the observed outcome is integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → MeasureTheory.Integrable S.factualY P.μ","l":113,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATTAssumptions.propScore_false_ne","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"Under one-sided overlap (e(X) < 1), the control propensity score P[D=0 | σ(X)] = 1 - e(X) is a.s. nonzero.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore false ω ≠ 0","l":146,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.ATT_eq_adjustedATT","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"ATT identification (one-sided overlap). Under consistency, unconfoundedness, one-sided overlap (e(X) < 1, the control arm), and positivity of the marginal treatment probability, the potential-outcome-level average treatment effect on the treated equals the observable adjusted-ATT functional:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → S.ATT = S.adjustedATT","l":382,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"The backdoor (Rosenbaum–Rubin / Firpo) identifying assumptions for the ATE, at the potential-outcome level, for binary treatment D and real outcome Y adjusted for covariates X: the observed outcome equals the potential outcome of the realized treatment arm, treatment is conditionally independent of the treated and untreated potential outcomes given the covariates, every covariate stratum has a positive chance of either treatment arm, and the treated and untreated potential outcomes are integrable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":226,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions.integrable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Under the backdoor assumption bundle the potential outcome of either treatment arm is integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (d : Bool), MeasureTheory.Integrable (S.YofD d) P.μ","l":262,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome is integrable under the backdoor assumption bundle.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":313,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Assumptions.toDistributional","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"The ATE backdoor bundle projects to the weaker distributional bundle.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → S.DistributionalAssumptions","l":95,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.CATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system and a treatment arm, the conditional mean potential outcome assigns each unit the conditional mean of that arm's potential outcome given the covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":134,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.CATE_eq","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The conditional mean potential outcome at a treatment arm is the conditional expectation of that arm's potential outcome given the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), S.CATE d = P.μ[S.YofD d | S.sigmaX]","l":148,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.Calibrated","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a candidate complete-propensity function, the calibration condition holds exactly when the conditional mean, given the covariate σ-algebra, of the treated-arm indicator divided by that function equals one almost surely.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → Prop","l":52,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.Calibrated0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a candidate complete control propensity, the control-arm calibration condition holds when the conditional expectation, given the factual covariate, of the untreated indicator divided by that candidate propensity equals one almost surely.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → Prop","l":39,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.DistributionalAssumptions","k":"structure","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Distributional backdoor assumptions. Firpo's distributional identification of the potential-outcome law under arm d uses consistency, conditional ignorability of treatment given covariates, and common support. It does not require the outcome-integrability assumptions bundled in the ATE backdoor theorem, because laws and quantiles are defined without first moments.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":75,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.GaussianOutcomeModel","k":"structure","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional-Gaussian outcome model. Strengthens GaussianTreatedModel (conditional CDF Φ((·−m)/σ)) with the two conditional moments it implies: the treated conditional mean E[A·Y|σX] = e(X)·m(X) and the truncated mean above any σ(X)-measurable cutoff. This is the faithful \"treated outcomes are conditionally Gaussian N(m(X), σ(X)²)\" premise of the Dorn–Guo corollary.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → Prop","l":173,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.GaussianTreatedModel","k":"structure","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The conditional-Gaussian treated-outcome model. The treated conditional law of Y given X is N(m(X), σ(X)²): a measurable mean m and positive standard deviation σ with condCDF treatedXYLaw a t = Φ((t − m a)/σ a) for all a, t.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → Prop","l":89,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the MSM ambiguity set contains exactly the candidate complete-propensity functions that are almost surely strictly between zero and one and whose odds ratio relative to the observed propensity is almost surely between the reciprocal of the sensitivity level and the sensitivity level.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":94,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, at a sensitivity level, the control marginal- sensitivity-model ambiguity set consists of candidate complete control propensities that are strictly between zero and one almost surely and whose odds ratio relative to the observed control propensity is between $1/Λ$ and $Λ$ almost surely.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":75,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Monotonicity of the control ambiguity set in the sensitivity parameter.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → S.MSMSet0 Λ ⊆ S.MSMSet0 Λ'","l":203,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet0_one_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Point identification at Λ = 1 (control arm, no-unmeasured-confounding collapse). Assume the observed control propensity score P[D=0∣σ(X)] is almost surely strictly between 0 and 1. Then for any candidate complete control propensity ẽ belonging to the (uncalibrated) control ambiguity set at sensitivity level Λ = 1, ẽ agrees almost everywhere with the observed control propensity score: at Λ = 1 the odds-ratio ambiguity set collapses to the single point given by no unmeasured confounding.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (etilde : P.Ω → ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → etilde ∈ S.MSMSet0 1 → ∀ᵐ (ω : P.Ω) ∂P.μ, etilde ω = S.propScore false ω","l":237,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the calibrated MSM ambiguity set contains exactly the candidate complete-propensity functions that belong to the MSM ambiguity set at that level and satisfy the calibration condition.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":61,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the calibrated control marginal-sensitivity-model ambiguity set consists of candidate complete control propensities that belong to the control ambiguity set and satisfy the control-arm calibration condition.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → Set (P.Ω → ℝ)","l":47,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib0_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibrated control set is a subset of the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.MSMSetCalib0 Λ ⊆ S.MSMSet0 Λ","l":147,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSetCalib_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibrated set is a subset of the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.MSMSetCalib Λ ⊆ S.MSMSet Λ","l":161,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Monotonicity of the ambiguity set in the sensitivity parameter. A larger budget Λ ≤ Λ' (with 1 ≤ Λ) admits every candidate that the smaller budget admits: MSMSet Λ ⊆ MSMSet Λ'. The odds-ratio window [1/Λ, Λ] widens with Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → S.MSMSet Λ ⊆ S.MSMSet Λ'","l":248,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.MSMSet_one_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Point identification at Λ = 1 (no-unmeasured-confounding collapse). Assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If a candidate complete propensity ẽ belongs to the marginal-sensitivity ambiguity set at Λ = 1, then it equals the observed propensity score almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (etilde : P.Ω → ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → etilde ∈ S.MSMSet 1 → ∀ᵐ (ω : P.Ω) ∂P.μ, etilde ω = S.propScore true ω","l":288,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the control-mean estimand is the population mean of the potential outcome under control.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":70,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control MSM bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the control MSM lower and upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSet0 Λ → S.candMean0 S.completeProp0 = S.Y0mean → BddBelow (S.candMean0 '' S.MSMSet0 Λ) → BddAbove (S.candMean0 '' S.MSMSet0 Λ) → S.Y0mean ∈ Set.Icc (S.msmLower0 Λ) (S.msmUpper0 Λ)","l":184,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y0mean_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the calibrated control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the calibrated ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the sharp control lower and upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSetCalib0 Λ → S.candMean0 S.completeProp0 = S.Y0mean → BddBelow (S.candMean0 '' S.MSMSetCalib0 Λ) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ)","l":127,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system, the treated potential-outcome mean is the population mean of the potential outcome under treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":91,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The MSM bound: E[Y(1)] lies in the MSM interval. Assuming the true complete propensity belongs to the marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the ambiguity set, the estimand E[Y(1)] is sandwiched between the infimum and supremum of the candidate mean over the ambiguity set:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSet Λ → S.candMean S.completeProp = S.Y1mean → BddBelow (S.candMean '' S.MSMSet Λ) → BddAbove (S.candMean '' S.MSMSet Λ) → S.Y1mean ∈ Set.Icc (S.msmLower Λ) (S.msmUpper Λ)","l":226,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.Y1mean_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp bound is valid. Assuming the true complete propensity belongs to the calibrated (sharp) marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the calibrated ambiguity set, then the estimand E[Y(1)] lies in the calibrated (sharp) interval [msmLowerCalib Λ, msmUpperCalib Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSetCalib Λ → S.candMean S.completeProp = S.Y1mean → BddBelow (S.candMean '' S.MSMSetCalib Λ) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ)","l":143,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system and a treatment arm, the potential outcome under that arm assigns every unit its real-valued outcome if treatment were set to that arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":82,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the adjusted average treatment effect is the population mean of the adjusted conditional functional for treatment minus that for control.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":219,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATT","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the adjusted average treatment effect on the treated is the treated-indicator-weighted population mean of factual outcome minus the adjusted control conditional functional, divided by the marginal treatment probability. It is the observable, control-regression form: E[A · (Y − μ₀(X))] / π_T. Only the CONTROL regression μ₀(X) = adjustedCE false appears — the treated potential outcome is observed directly on {D = 1} via consistency (A · Y = A · Y(1)), so no treated regression μ₁(X) and hence no 0 < e(X) is needed. This is the standard ATT estimand and requires only one-sided overlap e(X) < 1. The AIPW form is recovered as a corollary (adjustedATT_eq_aipwForm).","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":55,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedATT_eq_aipwForm","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"AIPW corollary. Under the ATT identification assumptions — consistency, unconfoundedness, one-sided control-arm overlap, and a positive marginal treatment probability, provided the observed inverse-propensity-weighted correction term is integrable, the adjustedATT functional equals its augmented inverse-propensity-weighted (AIPW) form:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → MeasureTheory.Integrable (fun ω => (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω)) P.μ → S.adjustedATT = (∫ (ω : P.Ω), S.dVar.indicator true ω * (S.factualY ω - S.adjustedCE false ω) - (1 - S.dVar.indicator true ω) * (S.propScore true ω / (1 - S.propScore true ω)) * (S.factualY ω - S.adjustedCE false ω) ∂P.μ) / S.propTreated","l":443,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedCE","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system and a treatment arm, the adjusted conditional functional is the covariate-conditional mean of factual outcome times the indicator for that arm, divided by that arm's propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":201,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.adjustedCE_def","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The adjusted conditional functional at a treatment arm sends a unit to the conditional expectation, given the covariate σ-algebra, of the factual outcome times that arm's treatment indicator, divided by the propensity score of that arm.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), S.adjustedCE d = fun ω => P.μ[fun ω' => S.factualY ω' * S.dVar.indicator d ω' | S.sigmaX] ω / S.propScore d ω","l":208,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the average treatment effect is the population mean of the potential outcome under treatment minus the population mean of the potential outcome under control.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateCalib_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval is contained in the ZSB ATE interval. If the treated arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, the treated arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, the control arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, and the control arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, then the sharp ATE interval [ateLowerCalib Λ, ateUpperCalib Λ] is contained in the ZSB ATE interval [ateLower Λ, ateUpper Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.msmUpperCalib Λ ≤ S.msmUpper Λ → S.msmLower Λ ≤ S.msmLowerCalib Λ → S.msmUpperCalib0 Λ ≤ S.msmUpper0 Λ → S.msmLower0 Λ ≤ S.msmLowerCalib0 Λ → Set.Icc (S.ateLowerCalib Λ) (S.ateUpperCalib Λ) ⊆ Set.Icc (S.ateLower Λ) (S.ateUpper Λ)","l":111,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the uncalibrated lower bound for the average treatment effect is the uncalibrated treated-arm lower bound minus the uncalibrated control-arm upper bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":72,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateLowerCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the sharp lower bound for the average treatment effect is the sharp treated-arm lower bound minus the sharp control- arm upper bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":60,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateUpper","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the uncalibrated upper bound for the average treatment effect is the uncalibrated treated-arm upper bound minus the uncalibrated control-arm lower bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":66,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ateUpperCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the sharp upper bound for the average treatment effect is the sharp treated-arm upper bound minus the sharp control- arm lower bound.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":54,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Integrated backdoor ATE identification. Under the backdoor identification assumptions — consistency, unconfoundedness, two-sided overlap, and integrable potential outcomes, the average treatment effect equals the adjusted-outcome functional obtained by averaging the CATE-level backdoor identity over the covariate distribution.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → S.ATE = S.adjustedATE","l":523,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_endpoints_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATEClosedForm","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval endpoints in closed form. Fix a sensitivity parameter Λ greater than 1. Given, for the treated arm, two-sided propensity overlap, an atomless conditional outcome distribution, that the upper and lower calibration levels each lie strictly between 0 and 1 almost everywhere, that the calibrated candidate mean is bounded above over its ambiguity set, that every candidate propensity in that ambiguity set is measurable up to null sets, and integrability regularity, for every σ(X)-measurable cutoff candidate, feeding both the upper and lower calibration constructions — together with the symmetric conditions for the control arm (overlap, atomlessness, calibration-level regularity, boundedness, measurability, and cutoff integrability regularity) — then there exist σ(X)-measurable conditional-quantile cutoffs cTU, cTL, cCU, cCL such that the sharp calibrated ATE upper endpoint equals the treated upper-cutoff candidate mean minus the control lower-cutoff candidate mean, and the sharp calibrated ATE lower endpoint equals the treated lower-cutoff candidate mean minus the control upper-cutoff candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ cTU cTL cCU cCL, (Measurable cTU ∧ Measurable cTL ∧ Measurable cCU ∧ Measurable cCL) ∧ S.ateUpperCalib Λ = S.candMean (S.cutoffProp Λ cTU) - S.candMean0 (S.lowerCutoffProp0 Λ cCL) ∧ S.ateLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ cTL) - S.candMean0 (S.cutoffProp0 Λ cCU)","l":47,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATEClosedForm.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The ZSB ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the uncalibrated ZSB interval [msmLower Λ, msmUpper Λ] and the control arm's mean potential outcome E[Y(0)] lies in the uncalibrated ZSB interval [msmLower0 Λ, msmUpper0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the ZSB interval [ateLower Λ, ateUpper Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.Y1mean ∈ Set.Icc (S.msmLower Λ) (S.msmUpper Λ) → S.Y0mean ∈ Set.Icc (S.msmLower0 Λ) (S.msmUpper0 Λ) → S.ate ∈ Set.Icc (S.ateLower Λ) (S.ateUpper Λ)","l":99,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc_calib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the calibrated sharp interval [msmLowerCalib Λ, msmUpperCalib Λ] and the control arm's mean potential outcome E[Y(0)] lies in the calibrated sharp interval [msmLowerCalib0 Λ, msmUpperCalib0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the sharp interval [ateLowerCalib Λ, ateUpperCalib Λ].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ) → S.ate ∈ Set.Icc (S.ateLowerCalib Λ) (S.ateUpperCalib Λ)","l":87,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.ate_mem_Icc_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATEClosedForm","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The true ATE lies in the sharp closed-form interval. Under the same treated-arm and control-arm regularity conditions as ate_endpoints_eq_cutoff — a sensitivity parameter Λ greater than 1; for the treated arm, propensity overlap, an atomless conditional outcome distribution, calibration-level regularity, boundedness of the calibrated candidate mean, measurability of every candidate propensity, and cutoff integrability regularity; and symmetrically for the control arm (overlap, atomlessness, calibration-level regularity, boundedness, measurability, and cutoff integrability regularity) — together with validity of the treated arm's calibrated sharp interval for E[Y(1)] and validity of the control arm's calibrated sharp interval for E[Y(0)], there exist σ(X)-measurable conditional-quantile cutoffs cTU, cTL, cCU, cCL such that the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies between the closed-form lower endpoint (treated lower-cutoff candidate mean minus control upper-cutoff candidate mean) and the closed-form upper endpoint (treated upper-cutoff candidate mean minus control lower-cutoff candidate mean).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → S.Y1mean ∈ Set.Icc (S.msmLowerCalib Λ) (S.msmUpperCalib Λ) → S.Y0mean ∈ Set.Icc (S.msmLowerCalib0 Λ) (S.msmUpperCalib0 Λ) → ∃ cTU cTL cCU cCL, (Measurable cTU ∧ Measurable cTL ∧ Measurable cCU ∧ Measurable cCL) ∧ S.ate ∈ Set.Icc (S.candMean (S.lowerCutoffProp Λ cTL) - S.candMean0 (S.cutoffProp0 Λ cCU)) (S.candMean (S.cutoffProp Λ cTU) - S.candMean0 (S.lowerCutoffProp0 Λ cCL))","l":140,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATEClosedForm.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a sensitivity level, and a unit, the calibration quantile level is one minus the target treated-survival probability divided by that unit's propensity for treatment. It is the treated conditional-distribution-function level whose quantile supplies the calibrating cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":322,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a sensitivity level, and a unit, the control calibration quantile level is one minus the target control-survival probability divided by that unit's propensity for control. It is the control conditional-distribution-function level whose quantile supplies the calibrating cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":300,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevelLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a sensitivity level, and a unit, the lower calibration quantile level is one minus the lower target survival probability divided by that unit's propensity for treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":66,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevelLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, and a sample point, the lower calibration quantile level is one minus the lower target control-survival probability divided by the control propensity score at that point.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":46,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.calibLevel_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The calibration level is exactly Λ/(Λ+1). Under overlap and 1 < Λ, the sharp-upper-bound quantile level calibLevel = 1 − survTarget/e equals Λ/(Λ+1) a.e., independent of the propensity e(X) — because survTarget = e/(Λ+1). Pure algebra from wMin, wMax.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.calibLevel Λ ω = Λ / (Λ + 1)","l":75,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a candidate complete-propensity function, the candidate inverse-probability-weighted mean is the population mean of the treated-arm indicator times factual outcome divided by that candidate function.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → ℝ","l":73,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a candidate complete control propensity, the candidate inverse-probability-weighted control mean is the population expectation of the factual outcome times the indicator of being untreated, divided by that candidate propensity.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → ℝ","l":61,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean0_completeProp0_eq_Y0mean","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"IPW / tower bridge (control arm). Evaluated at the true complete control propensity e₀ = P[D=0 | σ(X, Y(0))], the candidate IPW mean recovers E[Y(0)]. The D=0 reflection of candMean_completeProp_eq_Y1mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω) → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.factualY ω / S.completeProp0 ω) P.μ → S.candMean0 S.completeProp0 = S.Y0mean","l":99,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean_completeProp_eq_Y1mean","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"IPW / tower bridge. Assuming potential and observed outcomes agree under the realized treatment (consistency), the true complete propensity e₀ = P[D=1 | σ(X, Y(1))] is positive almost surely, the treated potential outcome Y(1) is integrable, and the inverse-propensity-weighted candidate A·Y/e₀ is integrable, then evaluated at the true complete propensity, the candidate IPW mean recovers the estimand: candMean e₀ = E[A · Y / e₀] = E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω) → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.factualY ω / S.completeProp ω) P.μ → S.candMean S.completeProp = S.Y1mean","l":119,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.candMean_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The candidate inverse-probability-weighted mean for a candidate complete propensity is the population mean of the treatment indicator times the factual outcome, divided by that candidate propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (etilde : P.Ω → ℝ), S.candMean etilde = ∫ (ω : P.Ω), S.dVar.indicator true ω * S.factualY ω / etilde ω ∂P.μ","l":82,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.cate_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"CATE-level backdoor identification. Under the backdoor identification assumptions — consistency, unconfoundedness, and two-sided overlap of the propensity score, together with integrability of the potential outcomes, for a treatment arm d, the conditional average treatment effect given the covariates equals the adjusted-outcome functional:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (d : Bool), S.CATE d =ᵐ[P.μ] S.adjustedCE d","l":505,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cate_backdoor_of_propScore_ne","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"CATE-level backdoor identification (per-arm overlap form). The conditional ATE equals the adjusted-outcome functional under consistency, unconfoundedness, arm-wise outcome integrability, and the single one-sided requirement that the d-arm propensity score is a.s. nonzero (P[D=d | σ(X)] ≠ 0):","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → P.CondIndepCF (Causalean.PO.RegimedVar.ofFactual S.dVar) S.cfBundle (Causalean.PO.RegimedVar.ofFactual S.xVar) P.μ → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → ∀ (d : Bool), (∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0) → S.CATE d =ᵐ[P.μ] S.adjustedCE d","l":361,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the counterfactual bundle contains the treated and untreated potential outcomes for unconfoundedness.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POCFBundle P","l":123,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.cfUnderLaw_eq_ipwLaw","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Distributional backdoor identification. Under the distributional backdoor assumption bundle — consistency, unconfoundedness, and common support, for each treatment arm d, the law of the potential outcome Y(d) equals the observable inverse-probability-weighted outcome law ipwLaw d.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), S.yVar.cfUnderLaw S.dVar d P.μ = S.ipwLaw d P.μ","l":442,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system, the complete propensity assigns each unit the conditional probability of treatment given the factual covariate and treated potential outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":64,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the complete control propensity assigns to each unit the conditional probability of being untreated given its factual covariate and its control potential outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":51,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_calibrated","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete control propensity is calibrated. E[(1−Z) / e₀ | σ(X)] = 1 a.e., where e₀ = P[D=0 | σ(X, Y(0))]. The D=0 reflection of completeProp_calibrated.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω) → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.completeProp0 ω) P.μ → S.Calibrated0 S.completeProp0","l":67,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM assumption: the truth lies in the ambiguity set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), ((∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp0 ω ∧ S.completeProp0 ω < 1) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ, 1 / Λ ≤ Causalean.PO.OR (S.completeProp0 ω) (S.propScore false ω) ∧ Causalean.PO.OR (S.completeProp0 ω) (S.propScore false ω) ≤ Λ) → S.completeProp0 ∈ S.MSMSet0 Λ","l":173,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp0_mem_MSMSetCalib0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete control propensity lies in the calibrated set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp0 ∈ S.MSMSet0 Λ → S.Calibrated0 S.completeProp0 → S.completeProp0 ∈ S.MSMSetCalib0 Λ","l":121,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_calibrated","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete propensity is calibrated. E[Z / e₀ | σ(X)] = 1 a.e., where e₀ = P[D=1 | σ(X, Y(1))]. By the tower property (σ(X) ⊆ σ(X, Y(1))): E[Z/e₀ | σX] = E[ E[Z/e₀ | σ(X,Y(1))] | σX] = E[ (1/e₀)·E[Z|σ(X,Y(1))] | σX] = E[ e₀/e₀ | σX] = 1. Uses the same condExp pull-out + cancellation as the IPW bridge in Setup.lean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.completeProp ω) P.μ → S.Calibrated S.completeProp","l":74,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM assumption: the truth lies in the ambiguity set. The true complete propensity e₀ belongs to MSMSet Λ, i.e. it is a.e. interior and its odds ratio against the observed propensity e(X) is a.e. within [1/Λ, Λ]. This is precisely the marginal sensitivity assumption; the membership facts are supplied as hMSM.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), ((∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.completeProp ω ∧ S.completeProp ω < 1) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ, 1 / Λ ≤ Causalean.PO.OR (S.completeProp ω) (S.propScore true ω) ∧ Causalean.PO.OR (S.completeProp ω) (S.propScore true ω) ≤ Λ) → S.completeProp ∈ S.MSMSet Λ","l":212,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.completeProp_mem_MSMSetCalib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The true complete propensity lies in the calibrated set when it satisfies MSM membership and calibration.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.completeProp ∈ S.MSMSet Λ → S.Calibrated S.completeProp → S.completeProp ∈ S.MSMSetCalib Λ","l":136,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_control_wMax0_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"All-upper-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the upper calibration weight wMax0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the upper weight times the control propensity P[D=0∣X].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax0 Λ ω * S.propScore false ω","l":145,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_control_wMin0_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"All-lower-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the lower calibration weight wMin0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the lower weight times the control propensity P[D=0∣X].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin0 Λ ω * S.propScore false ω","l":110,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_treat_wMax_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The all-wMax conditional calibration value is wMax·e. Assuming the product of the treatment indicator and the upper marginal-sensitivity-model weight wMax is integrable, the conditional expectation, given the σ-algebra generated by the covariates, of the treatment indicator times wMax equals wMax times the propensity score, almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator true ω * S.wMax Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax Λ ω * S.propScore true ω","l":133,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.condExp_treat_wMin_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The all-wMin conditional calibration value is wMin·e. Assuming the product of the treatment indicator and the lower marginal-sensitivity-model weight wMin is integrable, the conditional expectation, given the σ-algebra generated by the covariates, of the treatment indicator times wMin equals wMin times the propensity score, almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → P.μ[fun ω => S.dVar.indicator true ω * S.wMin Λ ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin Λ ω * S.propScore true ω","l":95,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlCondCDF","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a unit, and a real threshold, the control conditional distribution-function value is the probability that a control unit's factual outcome does not exceed the threshold, conditional on having that unit's factual covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ → ℝ","l":62,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlCutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"A cutoff solving the target control-survival equation is calibrated. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c is calibrated: the conditional expectation of the control indicator divided by it, given the covariates, equals 1 almost everywhere.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ → S.Calibrated0 (S.cutoffProp0 Λ c)","l":273,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the control set is the set of all units whose factual treatment status is control.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Set P.Ω","l":41,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system and a real-valued cutoff function on the sample space, the conditional control-survival function assigns to each sample point the conditional expectation, given the covariates, of the control-arm indicator times the indicator that the factual outcome exceeds the cutoff at that point.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → P.Ω → ℝ","l":52,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv_const_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (constant cutoff). The weighted conditional survival equals the control conditional survival scaled by the propensity: E[(1-Z)·1{Y>t} | σ(X)] = e₀(X)·(1 − F₀(t | X)) a.e. The genuine measure-theoretic content relates a conditional expectation under μ to the conditional CDF of the control push-forward law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), P.μ[fun ω => S.dVar.indicator false ω * if t < S.factualY ω then 1 else 0 | S.sigmaX] =ᵐ[P.μ] fun ω => S.propScore false ω * (1 - S.controlCondCDF ω t)","l":308,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlSurv_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (functional cutoff). The version of controlSurv_const_eq evaluated at a σ(X)-measurable cutoff c: E[(1-Z)·1{Y>c(X)} | σ(X)] = e₀(X)·(1 − F₀(c(X) | X)) a.e. Since c is σ(X)-measurable it is frozen inside the conditional expectation, reducing to the constant-cutoff bridge fibrewise. This is the form consumed by exists_calibrating_cutoff0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → S.controlSurv c =ᵐ[P.μ] fun ω => S.propScore false ω * (1 - S.controlCondCDF ω (c ω))","l":616,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlXYLaw","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the control covariate--outcome law is the image law of the factual covariate and factual outcome under the population measure restricted to control units.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure (γ × ℝ)","l":46,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.controlXYLaw_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control push-forward law of the covariate and outcome is the image, under the map recording the factual covariate and factual outcome, of the population measure restricted to the control set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.controlXYLaw = MeasureTheory.Measure.map (fun ω => (S.factualX ω, S.factualY ω)) (P.μ.restrict S.controlSet)","l":53,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.control_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Cutoff control calibration value decomposes into a lower bracket plus a survival term. Fix a sensitivity parameter Λ and a cutoff function c measurable with respect to the covariate σ-algebra. If the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, and the product of the weight spread wMax0 Λ - wMin0 Λ with the control-indicator survival term is integrable, then the conditional expectation, given the covariate σ-algebra, of the control indicator divided by the cutoff-calibration weight equals, almost everywhere, the lower weight times the control propensity plus the weight spread times the conditional control-survival functional controlSurv c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin0 Λ ω * S.propScore false ω + (S.wMax0 Λ ω - S.wMin0 Λ ω) * S.controlSurv c ω","l":180,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, a real-valued cutoff function on the sample space, and a sample point, the treated-arm quantile-cutoff complete propensity is the reciprocal of the largest admissible treated inverse-propensity weight when the factual outcome exceeds the cutoff and of the smallest such weight otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":53,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, a real-valued cutoff function on the sample space, and a sample point, the control quantile-cutoff complete propensity is the reciprocal of the largest admissible control inverse-propensity weight when the factual outcome exceeds the cutoff and of the smallest such weight otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":46,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The control cutoff propensity always lies inside the control odds-ratio box. Fix a sensitivity parameter Λ at least 1 and a cutoff function c. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then the cutoff-calibration candidate propensity cutoffProp0 Λ c always belongs to the control marginal-sensitivity-model ambiguity set MSMSet0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), S.cutoffProp0 Λ c ∈ S.MSMSet0 Λ","l":315,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp0_mem_MSMSetCalib0_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The cutoff belongs to the calibrated control MSM set whenever it solves the target control-survival equation. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c belongs to the calibrated control MSM set MSMSetCalib0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ","l":405,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Calibration from the survival equation. If the conditional treated-survival of the cutoff equals the target, the cutoff is calibrated.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ → S.Calibrated (S.cutoffProp Λ c)","l":153,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The cutoff propensity is always in the odds-ratio box. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Then for any measurable cutoff function c, the induced cutoff propensity always lies in the marginal-sensitivity-model odds-ratio ambiguity set, since at each point it equals either the lower or the upper marginal-sensitivity-model weight.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), S.cutoffProp Λ c ∈ S.MSMSet Λ","l":185,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoffProp_mem_MSMSetCalib_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Calibrated feasibility reduces to the survival equation. If the cutoff c solves the conditional treated-survival equation G(c) = survTarget, then cutoffProp Λ c ∈ MSMSetCalib Λ, discharging the hcut_mem hypothesis of msmUpperCalib_eq_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ","l":276,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Decomposition of the cutoff calibration value. For a σ(X)-measurable cutoff function c, assume the treatment indicator divided by the cutoff propensity is integrable, the treated-survival indicator, weighted by the treatment indicator, is integrable, the product of the treatment indicator and the lower marginal-sensitivity-model weight wMin is integrable, and the gap between the upper and lower weights, weighted by the treatment-weighted treated-survival indicator, is integrable. Then pulling the σ(X)-measurable weights wMin, wMax out of the conditional expectation decomposes the cutoff calibration value: E[Z/cutoffProp Λ c | σ(X)] = wMin·e + (wMax − wMin)·G(c), where G(c) is the conditional treated-survival at cutoff c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMin Λ ω * S.propScore true ω + (S.wMax Λ ω - S.wMin Λ ω) * S.treatedSurv c ω","l":61,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the quantile-cutoff weight (Neyman–Pearson exchange). Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, ẽ's candidate mean is at most the cutoff candidate mean — the cutoff weight maximizes the candidate mean among calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib Λ → AEMeasurable etilde P.μ → S.candMean etilde ≤ S.candMean (S.cutoffProp Λ c)","l":59,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlQuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the control quantile-cutoff weight. Fix a sensitivity parameter Λ at least one. Assume the control propensity score is almost surely strictly between 0 and 1 (two-sided overlap), and let c be a σ(X)-measurable, integrable cutoff function such that the quantile-cutoff propensity it induces lies in the calibrated control ambiguity set. If the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands hold, then for any almost-everywhere measurable candidate complete control propensity ẽ in the calibrated ambiguity set, the candidate mean at ẽ is no greater than the candidate mean at the quantile-cutoff weight: the control cutoff attains the maximum over all calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib0 Λ → AEMeasurable etilde P.μ → S.candMean0 etilde ≤ S.candMean0 (S.cutoffProp0 Λ c)","l":30,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlQuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal0_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least one. Assume the control propensity score is almost surely strictly between 0 and 1 (two-sided overlap), and let c be a σ(X)-measurable, integrable cutoff function such that the lower quantile-cutoff propensity it induces lies in the calibrated control ambiguity set. If the envelope 1_{D=0}·|Y|·wMax0(Λ) is integrable, the weighted control indicator 1_{D=0}·wMax0(Λ) is integrable, and the cutoff-weighted envelope |c|·1_{D=0}·wMax0(Λ) is integrable, then for any almost-everywhere measurable candidate complete control propensity ẽ in the calibrated ambiguity set, the candidate mean at the lower quantile-cutoff weight is no greater than the candidate mean at ẽ — the lower cutoff attains the minimum over all calibrated candidates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib0 Λ → AEMeasurable etilde P.μ → S.candMean0 (S.lowerCutoffProp0 Λ c) ≤ S.candMean0 etilde","l":52,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.cutoff_optimal_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced lower-cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, the lower-cutoff candidate mean is at most ẽ's candidate mean — the lower cutoff minimizes the candidate mean among calibrated candidates. The ≥-mirror of cutoff_optimal.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → ∀ {etilde : P.Ω → ℝ}, etilde ∈ S.MSMSetCalib Λ → AEMeasurable etilde P.μ → S.candMean (S.lowerCutoffProp Λ c) ≤ S.candMean etilde","l":94,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the treatment potential-outcome variable is the system's treatment represented with the two treatment arms.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P Bool","l":75,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.eLift","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the value-space propensity representative maps each covariate value to the treated-arm propensity score at that value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → γ → ℝ","l":655,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Existence of a calibrating cutoff (treated arm). Fix a sensitivity parameter Λ strictly greater than one. Assume the treated propensity score is almost surely strictly between 0 and 1 (two-sided overlap), that the treated-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the calibration quantile level lies strictly between 0 and 1 almost everywhere. Then there exists a σ(X)-measurable cutoff function c such that the treatment-weighted conditional survival function at c agrees almost everywhere with the target survival function survTarget Λ; the cutoff is realized as the conditional quantile of the treated outcome law at the calibration level.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → ∃ c, Measurable c ∧ S.treatedSurv c =ᵐ[P.μ] S.survTarget Λ","l":777,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Existence of a calibrating cutoff. Under overlap, 1 < Λ, an atomless control conditional outcome law (condCDF of the control push-forward continuous), and a strictly-interior calibration level, there is a σ(X)-measurable cutoff c solving the survival equation controlSurv c =ᵐ survTarget0 Λ. The cutoff is the conditional quantile Q_{calibLevel0}(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → ∃ c, Measurable c ∧ S.controlSurv c =ᵐ[P.μ] S.survTarget0 Λ","l":657,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff0_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Existence of a calibrating lower cutoff. The survTargetLower0 analogue of exists_calibrating_cutoff: a σ(X)-measurable c with controlSurv c =ᵐ survTargetLower0 Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → ∃ c, Measurable c ∧ S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ","l":621,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_calibrating_cutoff_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Existence of a calibrating lower cutoff. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If the treated outcome's conditional distribution given each covariate value has a continuous cumulative distribution function and the lower calibration quantile level lies strictly between 0 and 1 almost surely, then there exists a σ(X)-measurable cutoff function whose conditional treated-survival equals the lower target survival almost everywhere. The survTargetLower analogue of exists_calibrating_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → ∃ c, Measurable c ∧ S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ","l":630,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_factor_through_factualX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"σ(X)-measurable functions factor through X. If f is σ(X)-measurable then f = g ∘ X for a measurable g : γ → ℝ. This is standard comap factorization: the conditioning is on the value of X.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {f : P.Ω → ℝ}, Measurable f → ∃ g, Measurable g ∧ f = fun ω => g (S.factualX ω)","l":768,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_propScoreLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The σ(X)-measurable propensity propScore true factors through factualX: a measurable value-space propensity e : γ → ℝ with propScore true = e ∘ factualX (Doob–Dynkin). This is the value-space propensity an estimator learns.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), ∃ e, Measurable e ∧ S.propScore true = fun ω => e (S.factualX ω)","l":645,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.exists_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The σ(D, X)-measurable outcomeReg factors through (D, X): there is a measurable value-space regression function r : Bool × γ → ℝ with outcomeReg = r ∘ factualDX. This is Doob–Dynkin (exists_eq_measurable_comp); r is the function an ML regressor of Y on (D, X) targets.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), ∃ r, Measurable r ∧ S.outcomeReg = fun ω => r (S.factualDX ω)","l":618,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the factual treatment assigns every unit its observed treatment arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → Bool","l":84,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualDX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the factual treatment--covariate pair assigns each unit its observed treatment arm and covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → Bool × γ","l":549,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the factual covariate assigns every unit its observed adjustment-covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → γ","l":88,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the factual outcome assigns every unit its observed real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":86,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianCutoff","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a covariate-indexed mean function and standard-deviation function, a sensitivity parameter, and a sample point, the explicit Gaussian cutoff is $m(X(\\omega))+\\sigma(X(\\omega))\\Phi^{-1}(\\Lambda/(\\Lambda+1))$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (γ → ℝ) → (γ → ℝ) → ℝ → P.Ω → ℝ","l":99,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianCutoff_calibrates","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The explicit Gaussian cutoff calibrates. Under the conditional-Gaussian model, the cutoff m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)) solves the survival equation treatedSurv c =ᵐ survTarget Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianTreatedModel m σ → S.treatedSurv (S.gaussianCutoff m σ Λ) =ᵐ[P.μ] S.survTarget Λ","l":116,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.gaussianMSMHalfWidthFactor","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a real sensitivity parameter, the Dorn–Guo Gaussian half-width factor is $(\\Lambda^2-1)/\\Lambda\\,\\phi(\\Phi^{-1}(\\Lambda/(\\Lambda+1)))$, where $\\Phi$ and $\\phi$ are the standard normal distribution and density functions, respectively.","s":"ℝ → ℝ","l":47,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_CATE","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The conditional mean potential outcome is integrable: it is a conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), MeasureTheory.Integrable (S.CATE d) P.μ","l":192,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_adjustedCE_control","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"Under the ATT assumption bundle the control-arm backdoor adjustment functional is integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.ATTAssumptions → MeasureTheory.Integrable (S.adjustedCE false) P.μ","l":184,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The conditional CDF evaluated at a measurable cutoff is integrable under the first marginal.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {q : γ → ℝ}, Measurable q → MeasureTheory.Integrable (fun a => ↑(ProbabilityTheory.condCDF ρ a) (q a)) ρ.fst","l":405,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_factualY_of_consistency","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Factual Y is integrable whenever both treatment-arm potential outcomes are integrable and consistency links the observed outcome to the realized arm.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), P.Consistency → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → MeasureTheory.Integrable S.factualY P.μ","l":271,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.integrable_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The propensity score is integrable: it is a conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), MeasureTheory.Integrable (S.propScore d) P.μ","l":173,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.integral_comp_YofD_eq","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"Core distributional backdoor identity. Under the distributional backdoor assumption bundle, for every treatment arm d and every measurable real function g that is bounded by a constant C, the mean of g applied to the potential outcome Y(d) equals the mean of g applied to the factual outcome, weighted by the inverse-probability-weighting density ipwDensity d.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool) {g : ℝ → ℝ}, Measurable g → ∀ {C : ℝ}, (∀ (y : ℝ), |g y| ≤ C) → ∫ (ω : P.Ω), g (S.YofD d ω) ∂P.μ = ∫ (ω : P.Ω), g (S.factualY ω) * S.ipwDensity d ω ∂P.μ","l":308,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.integral_mul_indicator_eq_integral_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Conditional-expectation pull-out. For a σ(X)-strongly-measurable h, ∫ h·1_{D=d} dμ = ∫ h·e_d dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool) {h : P.Ω → ℝ}, MeasureTheory.StronglyMeasurable h → MeasureTheory.Integrable (fun ω => h ω * S.dVar.indicator d ω) P.μ → ∫ (ω : P.Ω), h ω * S.dVar.indicator d ω ∂P.μ = ∫ (ω : P.Ω), h ω * S.propScore d ω ∂P.μ","l":150,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwDensity","k":"def","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"For a potential-outcome backdoor system and a treatment arm, the observable inverse-probability-weighting density assigns to each sample point the indicator that its factual treatment equals that arm divided by the conditional probability of that arm given the covariates.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":63,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwDensity_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Integrability of the IPW weight under bare common support.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), MeasureTheory.Integrable (S.ipwDensity d) P.μ","l":178,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.ipwLaw","k":"def","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":1,"d":"For a potential-outcome backdoor system, a treatment arm, and a measure on the sample space, the observable inverse-probability-weighted outcome law is the distribution of the factual outcome under the measure obtained by weighting each sample point by the nonnegative version of its inverse-probability-weighting density for that arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":69,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerControl_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Decomposition of the lower-cutoff calibration value. E[(1-Z)/lowerCutoffProp0 Λ c | σ(X)] = wMax0·e₀ − (wMax0 − wMin0)·G(c), with G(c) = controlSurv c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax0 Λ ω * S.propScore false ω - (S.wMax0 Λ ω - S.wMin0 Λ ω) * S.controlSurv c ω","l":400,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a sensitivity level, a cutoff function, and a unit, the lower quantile-cutoff complete propensity is the reciprocal of the lower inverse-probability weight when that unit's factual outcome exceeds its cutoff and of the upper inverse-probability weight otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":46,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, a real-valued cutoff function on the sample space, and a sample point, the lower quantile-cutoff complete propensity is the reciprocal of the lower admissible control inverse-propensity weight when the factual outcome exceeds the cutoff and of the upper admissible weight otherwise: $[w_{\\min,0}(\\Lambda,\\omega)]^{-1}$ if $c(\\omega)<Y(\\omega)$, and $[w_{\\max,0}(\\Lambda,\\omega)]^{-1}$ otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → (P.Ω → ℝ) → P.Ω → ℝ","l":34,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Lower-cutoff calibration from the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ → S.Calibrated0 (S.lowerCutoffProp0 Λ c)","l":487,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_mem_MSMSet0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff propensity is always in the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), S.lowerCutoffProp0 Λ c ∈ S.MSMSet0 Λ","l":518,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp0_mem_MSMSetCalib0_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff is calibrated-feasible given the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.controlSurv c =ᵐ[P.μ] S.survTargetLower0 Λ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ","l":604,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_calibrated_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Lower-cutoff calibration from the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ → S.Calibrated (S.lowerCutoffProp Λ c)","l":496,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_mem_MSMSet","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff propensity is always in the odds-ratio box.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), S.lowerCutoffProp Λ c ∈ S.MSMSet Λ","l":527,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoffProp_mem_MSMSetCalib_of_survival","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower cutoff is calibrated-feasible given the survival equation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → S.treatedSurv c =ᵐ[P.μ] S.survTargetLower Λ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ","l":613,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.lowerCutoff_calibValue_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Decomposition of the lower-cutoff calibration value. E[Z/lowerCutoffProp Λ c | σ(X)] = wMax·e − (wMax − wMin)·G(c) (with G(c) = treatedSurv c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ → P.μ[fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω | S.sigmaX] =ᵐ[P.μ] fun ω => S.wMax Λ ω * S.propScore true ω - (S.wMax Λ ω - S.wMin Λ ω) * S.treatedSurv c ω","l":421,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_CATE","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The conditional mean potential outcome is measurable for the ambient sigma-algebra on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.CATE d)","l":185,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Potential outcomes under fixed treatment arms are measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.YofD d)","l":91,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_adjustedCE","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"The backdoor adjustment functional is measurable for the ambient sigma-algebra on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.adjustedCE d)","l":168,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_calibLevelLower_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower calibration quantile level is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.calibLevelLower Λ)","l":84,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The conditional CDF remains measurable when evaluated at a measurable cutoff.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {q : γ → ℝ}, Measurable q → Measurable fun a => ↑(ProbabilityTheory.condCDF ρ a) (q a)","l":378,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_controlCondCDF_const","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"At a fixed outcome level, the control conditional CDF depends on the unit only through the observed covariate, so it is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), Measurable fun ω => S.controlCondCDF ω t","l":105,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_controlCondCDF_const_ambient","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control conditional CDF at a fixed outcome level is measurable for the ambient σ-algebra as well as for the covariate one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), Measurable fun ω => S.controlCondCDF ω t","l":122,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_controlCondCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Evaluated at a cutoff that itself depends on the unit only through the observed covariate, the control conditional CDF is still measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → Measurable fun ω => S.controlCondCDF ω (c ω)","l":384,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_controlCondCDF_variable_ambient","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control conditional CDF at a covariate-measurable cutoff is measurable for the ambient σ-algebra as well as for the covariate one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → Measurable fun ω => S.controlCondCDF ω (c ω)","l":407,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_eLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The value-space propensity representative is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.eLift","l":658,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualD","l":95,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment-covariate pair is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualDX","l":556,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualDX_sigmaDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment-covariate pair is measurable for the joint sigma-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualDX","l":571,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualD_sigmaDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable for the joint treatment-covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualD","l":577,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualX","l":101,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualX_sigmaDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed covariate is measurable for the joint treatment-covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualX","l":583,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualX_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed covariate is measurable for the covariate sigma-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualX","l":113,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.factualY","l":98,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_gaussianCutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The explicit Gaussian cutoff is σ(X)-measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {m σ : γ → ℝ}, Measurable m → Measurable σ → ∀ (Λ : ℝ), Measurable (S.gaussianCutoff m σ Λ)","l":105,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_ipwDensity","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"The observable inverse-probability weight for a treatment arm is a measurable function of the unit.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.ipwDensity d)","l":109,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_outcomeReg","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome regression is measurable for the ambient sigma-algebra on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.outcomeReg","l":612,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The propensity score is measurable for the ambient sigma-algebra on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.propScore d)","l":167,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_propScore_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The propensity score is measurable for the covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), Measurable (S.propScore d)","l":162,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The value-space outcome regression is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), Measurable S.regFn","l":635,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_survTargetLower_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower target survival is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.survTargetLower Λ)","l":73,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_treatedCondCDF_const","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"At a fixed outcome level, the treated conditional CDF depends on the unit only through the observed covariate, so it is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), Measurable fun ω => S.treatedCondCDF ω t","l":127,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_treatedCondCDF_const_ambient","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The treated conditional CDF at a fixed outcome level is measurable for the ambient σ-algebra as well as for the covariate one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), Measurable fun ω => S.treatedCondCDF ω t","l":144,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_treatedCondCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Evaluated at a cutoff that itself depends on the unit only through the observed covariate, the treated conditional CDF is still measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → Measurable fun ω => S.treatedCondCDF ω (c ω)","l":495,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_treatedCondCDF_variable_ambient","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The treated conditional CDF at a covariate-measurable cutoff is measurable for the ambient σ-algebra as well as for the covariate one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → Measurable fun ω => S.treatedCondCDF ω (c ω)","l":518,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_wMax","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The upper sensitivity weight is measurable for the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.wMax Λ)","l":123,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_wMax_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The upper marginal-sensitivity weight is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.wMax Λ)","l":97,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_wMin","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower sensitivity weight is measurable for the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.wMin Λ)","l":118,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.measurable_wMin_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The lower marginal-sensitivity weight is measurable for the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), Measurable (S.wMin Λ)","l":88,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the MSM lower bound is the infimum of candidate inverse-probability-weighted means over the MSM ambiguity set at that level.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":113,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the control marginal- sensitivity-model lower bound is the infimum of the candidate inverse-probability- weighted control means over the control ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":93,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0_anti","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM interval widens with Λ (lower endpoint).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean0 '' S.MSMSet0 Λ).Nonempty → BddBelow (S.candMean0 '' S.MSMSet0 Λ') → S.msmLower0 Λ' ≤ S.msmLower0 Λ","l":227,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower0_le_msmLowerCalib0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp control lower bound is tighter than the ZSB bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean0 '' S.MSMSetCalib0 Λ).Nonempty → BddBelow (S.candMean0 '' S.MSMSet0 Λ) → S.msmLower0 Λ ≤ S.msmLowerCalib0 Λ","l":161,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the sharp MSM lower bound is the infimum of candidate inverse-probability-weighted means over the calibrated MSM ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":71,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the sharp control lower bound is the infimum of the candidate inverse-probability-weighted control means over the calibrated control ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":61,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least one. Under two-sided overlap of the control propensity score, given a σ(X)-measurable, integrable cutoff c whose induced lower quantile-cutoff propensity is itself calibrated-feasible, and assuming the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands together with almost-everywhere measurability of every calibrated candidate propensity, the sharp (infimum) lower bound for E[Y(0)] over the calibrated ambiguity set equals the candidate mean evaluated at the lower quantile-cutoff propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → S.msmLowerCalib0 Λ = S.candMean0 (S.lowerCutoffProp0 Λ c)","l":365,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib0_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the control propensity score, that the control-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the lower calibration quantile level lies strictly between 0 and 1 almost everywhere. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced lower quantile-cutoff propensity is calibrated-feasible, at which the sharp (infimum) lower bound for E[Y(0)] equals the candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower0 Λ ω ∧ S.calibLevelLower0 Λ ω < 1) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.lowerCutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ c, Measurable c ∧ S.lowerCutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ ∧ S.msmLowerCalib0 Λ = S.candMean0 (S.lowerCutoffProp0 Λ c)","l":699,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp lower bound has the quantile-balancing closed form. Given a calibrating lower cutoff, msmLowerCalib Λ = candMean (lowerCutoffProp Λ c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → S.msmLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ c)","l":393,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerCalib_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp lower bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If the treated outcome's conditional distribution given each covariate value has a continuous cumulative distribution function, the lower calibration quantile level lies strictly between 0 and 1 almost surely, every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, and the regularity conditions needed for the lower-cutoff candidate mean and calibration to be well defined hold for every σ(X)-measurable cutoff, then there exists a σ(X)-measurable, calibrated, box-feasible cutoff function at which the sharp lower bound on E[Y(1)] equals the candidate mean of the induced lower-cutoff propensity. The sInf-mirror of msmUpperCalib_eq_cutoff_unconditional.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevelLower Λ ω ∧ S.calibLevelLower Λ ω < 1) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.lowerCutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → ∃ c, Measurable c ∧ S.lowerCutoffProp Λ c ∈ S.MSMSetCalib Λ ∧ S.msmLowerCalib Λ = S.candMean (S.lowerCutoffProp Λ c)","l":703,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLowerForm","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system and a real sensitivity level, the lower marginal-sensitivity functional is the expectation of the factual outcome among treated units, weighted by the lower endpoint weight when that outcome is nonnegative and by the upper endpoint weight otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":133,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_anti","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM interval widens with Λ (lower endpoint). Since MSMSet Λ ⊆ MSMSet Λ', the infimum of the candidate mean can only decrease. Boundedness-below of the larger image is required for the sInf comparison.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean '' S.MSMSet Λ).Nonempty → BddBelow (S.candMean '' S.MSMSet Λ') → S.msmLower Λ' ≤ S.msmLower Λ","l":276,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Closed form of the MSM lower bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the infimum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM lower bound equals E[A·Y·(wMin if Y≥0 else wMax)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ etilde ∈ S.MSMSet Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → S.msmLower Λ = S.msmLowerForm Λ","l":365,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmLower_le_msmLowerCalib","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp lower bound is tighter than the ZSB bound: msmLower Λ ≤ msmLowerCalib Λ.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean '' S.MSMSetCalib Λ).Nonempty → BddBelow (S.candMean '' S.MSMSet Λ) → S.msmLower Λ ≤ S.msmLowerCalib Λ","l":179,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the MSM upper bound is the supremum of candidate inverse-probability-weighted means over the MSM ambiguity set at that level.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":107,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the control marginal- sensitivity-model upper bound is the supremum of the candidate inverse-probability- weighted control means over the control ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":87,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper0_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The control MSM interval widens with Λ (upper endpoint).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean0 '' S.MSMSet0 Λ).Nonempty → BddAbove (S.candMean0 '' S.MSMSet0 Λ') → S.msmUpper0 Λ ≤ S.msmUpper0 Λ'","l":217,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system and a sensitivity level, the sharp MSM upper bound is the supremum of candidate inverse-probability-weighted means over the calibrated MSM ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":68,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, and a sensitivity level, the sharp control upper bound is the supremum of the candidate inverse-probability-weighted control means over the calibrated control ambiguity set.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":55,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlQuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp calibrated control upper bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least one. Under two-sided overlap of the control propensity score, given a σ(X)-measurable, integrable cutoff c whose induced quantile-cutoff propensity is itself calibrated-feasible, and assuming the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands together with almost-everywhere measurability of every calibrated candidate propensity, the sharp (supremum) upper bound for E[Y(0)] over the calibrated ambiguity set equals the candidate mean evaluated at the quantile-cutoff propensity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → S.msmUpperCalib0 Λ = S.candMean0 (S.cutoffProp0 Λ c)","l":340,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlQuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp control upper bound has a quantile-balancing closed form, unconditionally. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the control outcome's conditional law given the covariates is atomless (its conditional CDF is continuous), the calibration level lies strictly between 0 and 1 almost everywhere, every candidate propensity in the calibrated control ambiguity set is almost-everywhere measurable, and every covariate-measurable cutoff function satisfies the integrability conditions needed to evaluate the calibration and candidate-mean functionals at it, then there exists a covariate-measurable cutoff c such that the cutoff-calibration propensity cutoffProp0 Λ c lies in the calibrated control MSM set and the sharp control upper bound equals the candidate mean at that cutoff, msmUpperCalib0 Λ = candMean0 (cutoffProp0 Λ c).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.controlXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel0 Λ ω ∧ S.calibLevel0 Λ ω < 1) → BddAbove (S.candMean0 '' S.MSMSetCalib0 Λ) → (∀ etilde ∈ S.MSMSetCalib0 Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω / S.cutoffProp0 Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMin0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax0 Λ ω - S.wMin0 Λ ω) * (S.dVar.indicator false ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * |S.factualY ω| * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator false ω * S.wMax0 Λ ω) P.μ) → ∃ c, Measurable c ∧ S.cutoffProp0 Λ c ∈ S.MSMSetCalib0 Λ ∧ S.msmUpperCalib0 Λ = S.candMean0 (S.cutoffProp0 Λ c)","l":736,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib0_le_msmUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSharp","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The sharp control upper bound is tighter than the ZSB bound.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean0 '' S.MSMSetCalib0 Λ).Nonempty → BddAbove (S.candMean0 '' S.MSMSet0 Λ) → S.msmUpperCalib0 Λ ≤ S.msmUpper0 Λ","l":151,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_eq_cutoff","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.QuantileBalance","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, and assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, then the sharp upper bound on E[Y(1)] equals the candidate mean of the cutoff propensity built from c.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ → S.cutoffProp Λ c ∈ S.MSMSetCalib Λ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ c)","l":372,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/QuantileBalance.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_eq_cutoff_unconditional","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp treated upper bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the treated propensity score, that the treated-arm conditional outcome law given covariates is atomless, i.e. its conditional CDF is continuous, that the calibration quantile level lies strictly between 0 and 1 almost everywhere, and that the candidate means over the calibrated ambiguity set are bounded above. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced quantile-cutoff propensity is calibrated-feasible, at which the sharp (supremum) upper bound for E[Y(1)] equals the candidate mean.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ (a : γ), Continuous ↑(ProbabilityTheory.condCDF S.treatedXYLaw a)) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.calibLevel Λ ω ∧ S.calibLevel Λ ω < 1) → BddAbove (S.candMean '' S.MSMSetCalib Λ) → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → (∀ (c : P.Ω → ℝ), Measurable c → MeasureTheory.Integrable c P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ c ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if c ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |c ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ) → ∃ c, Measurable c ∧ S.cutoffProp Λ c ∈ S.MSMSetCalib Λ ∧ S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ c)","l":861,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_gaussian","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Gaussian","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound at the explicit Gaussian cutoff. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Under the conditional-Gaussian treated-outcome model, i.e. the treated conditional law of the outcome given the covariates is Gaussian with mean m and standard deviation σ, assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable and the integrability conditions needed to make the candidate means, the survival decomposition, and the cutoff propensity well defined, the sharp (Dorn–Guo) upper bound on E[Y(1)] equals the candidate IPW mean evaluated at the cutoff propensity built from the explicit Gaussian quantile cutoff m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianTreatedModel m σ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (S.gaussianCutoff m σ Λ) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ (S.gaussianCutoff m σ Λ) ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |S.gaussianCutoff m σ Λ ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → S.msmUpperCalib Λ = S.candMean (S.cutoffProp Λ (S.gaussianCutoff m σ Λ))","l":150,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Gaussian.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_gaussian_halfWidth","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.GaussianHalfWidth","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The Dorn–Guo Gaussian sharp upper bound, closed form. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Under the conditional-Gaussian treated-outcome model with conditional mean m and conditional standard deviation σ, strengthened with the implied treated conditional mean and truncated-mean identities, assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, the conditional mean m(X) is integrable, (1 − e(X))·σ(X) is integrable, the regularity conditions making the Gaussian-cutoff candidate mean and survival decomposition well defined, and the candidate-mean integrability at the lower weight and at the truncated difference, then the sharp upper bound on E[Y(1)] equals the point-identified mean ∫ m(X) plus the half-width (Λ²−1)/Λ · φ(Φ⁻¹(Λ/(Λ+1))) · ∫ (1 − e(X))·σ(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 < Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ {m σ : γ → ℝ}, S.GaussianOutcomeModel m σ → (∀ etilde ∈ S.MSMSetCalib Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => m (S.factualX ω)) P.μ → MeasureTheory.Integrable (fun ω => (1 - S.propScore true ω) * σ (S.factualX ω)) P.μ → MeasureTheory.Integrable (S.gaussianCutoff m σ Λ) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω / S.cutoffProp Λ (S.gaussianCutoff m σ Λ) ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMin Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * S.wMax Λ ω) P.μ ∧ MeasureTheory.Integrable (fun ω => |S.gaussianCutoff m σ Λ ω| * S.dVar.indicator true ω * S.wMax Λ ω) P.μ → MeasureTheory.Integrable (fun ω => S.wMin Λ ω * (S.dVar.indicator true ω * S.factualY ω)) P.μ → MeasureTheory.Integrable (fun ω => (S.wMax Λ ω - S.wMin Λ ω) * (S.dVar.indicator true ω * S.factualY ω * if S.gaussianCutoff m σ Λ ω < S.factualY ω then 1 else 0)) P.μ → S.msmUpperCalib Λ = ∫ (ω : P.Ω), m (S.factualX ω) ∂P.μ + Causalean.PO.POBackdoorSystem.gaussianMSMHalfWidthFactor Λ * ∫ (ω : P.Ω), (1 - S.propScore true ω) * σ (S.factualX ω) ∂P.μ","l":191,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/GaussianHalfWidth.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperCalib_le_msmUpper","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Sharp","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"The sharp upper bound is tighter than the ZSB bound. Assuming the calibrated candidate-mean image is nonempty and the candidate mean is bounded above over the uncalibrated odds-ratio-box ambiguity set, the sharp (calibrated) upper bound is at most the ZSB (uncalibrated) upper bound: msmUpperCalib Λ ≤ msmUpper Λ. The calibrated set is smaller, so its candidate-mean supremum can only decrease.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), (S.candMean '' S.MSMSetCalib Λ).Nonempty → BddAbove (S.candMean '' S.MSMSet Λ) → S.msmUpperCalib Λ ≤ S.msmUpper Λ","l":165,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Sharp.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpperForm","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system and a real sensitivity level, the upper marginal-sensitivity functional is the expectation of the factual outcome among treated units, weighted by the upper endpoint weight when that outcome is nonnegative and by the lower endpoint weight otherwise.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → ℝ","l":128,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Closed form of the MSM upper bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the supremum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM upper bound equals E[A·Y·(wMax if Y≥0 else wMin)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → (∀ etilde ∈ S.MSMSet Λ, AEMeasurable etilde P.μ) → MeasureTheory.Integrable (fun ω => S.dVar.indicator true ω * |S.factualY ω| * S.wMax Λ ω) P.μ → S.msmUpper Λ = S.msmUpperForm Λ","l":193,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.msmUpper_mono","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The MSM interval widens with Λ (upper endpoint). Since MSMSet Λ ⊆ MSMSet Λ', the supremum of the candidate mean can only increase. Boundedness of the larger image is required for the sSup comparison.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) {Λ Λ' : ℝ}, 1 ≤ Λ → Λ ≤ Λ' → (S.candMean '' S.MSMSet Λ).Nonempty → BddAbove (S.candMean '' S.MSMSet Λ') → S.msmUpper Λ ≤ S.msmUpper Λ'","l":264,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.one_le_wMax0_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Upper control calibration bracket is at least one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the upper calibration weight wMax0 Λ times the control propensity is at least 1.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, 1 ≤ S.wMax0 Λ ω * S.propScore false ω","l":88,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.one_le_wMax_mul_propScore","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Upper calibration bracket (algebraic). Fix a sensitivity parameter Λ at least 1. Assuming the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap), almost surely the upper marginal-sensitivity-model weight satisfies 1 ≤ wMax(X)·e(X) = e(X) + Λ·(1 − e(X)).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, 1 ≤ S.wMax Λ ω * S.propScore true ω","l":73,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.outcomeReg","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the outcome regression assigns each unit the conditional mean of the factual outcome given its factual treatment and covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ","l":597,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.outcomeReg_eq_regFn","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome regression equals the value-space regression evaluated at each unit's observed treatment and covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.outcomeReg = fun ω => S.regFn (S.factualDX ω)","l":639,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system and a treatment arm, the propensity score assigns each unit the conditional probability of receiving that arm given the covariate σ-algebra.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → P.Ω → ℝ","l":130,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_eq","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The propensity score at a treatment arm is the conditional expectation of that arm's treatment indicator given the covariate σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), S.propScore d = P.μ[S.dVar.indicator d | S.sigmaX]","l":141,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_ne_of_overlap","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"From two-sided overlap, the per-arm propensity score P[D=d | σ(X)] is a.s. nonzero: for d = true directly from 0 < e(X), for d = false from e(X) < 1 via propScore false = 1 - e(X).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ], (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0","l":321,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_nonneg","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"propScore d ≥ 0 a.e. (conditional expectation of a nonnegative function).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), 0 ≤ᵐ[P.μ] S.propScore d","l":116,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_pos","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"Under bare overlap, both arms have a.e.-positive propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.DistributionalAssumptions → ∀ (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore d ω","l":122,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.propScore_true_eq_eLift","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The treated-arm propensity score equals the value-space propensity evaluated at each unit's observed covariate.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.propScore true = fun ω => S.eLift (S.factualX ω)","l":662,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.propTreated","k":"def","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the marginal treatment probability is the population mean of the indicator for the treated arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ","l":45,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.qtdQuantile","k":"def","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"For a potential-outcomes backdoor system, a treatment arm, a measure on its sample space, and a real quantile level, the arm-specific potential-outcome quantile is the generalized $\\tau$-quantile of the potential outcome under that treatment arm, evaluated under that measure.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":72,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qtdQuantile_eq_quantile_cfUnderLaw","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":2,"d":"The quantile of Y(d) is the quantile of the law of Y(d).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ), S.qtdQuantile d μ τ = Causalean.Stat.quantile (S.yVar.cfUnderLaw S.dVar d μ) τ","l":82,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte","k":"def","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"For a potential-outcomes backdoor system, a measure on its sample space, and a real quantile level, the totalized quantile treatment effect is the generalized $\\tau$-quantile of the treated potential outcome minus that of the control potential outcome. It is defined for every real $\\tau$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":76,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"Backdoor QTE identification. Under the ATE backdoor identifying assumption bundle, the totalized generalized-quantile treatment effect at any real level τ equals the difference of the quantiles of the two observable inverse-probability-weighted outcome laws:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (τ : ℝ), S.qte P.μ τ = Causalean.Stat.quantile (S.ipwLaw true P.μ) τ - Causalean.Stat.quantile (S.ipwLaw false P.μ) τ","l":110,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.qte_eq_of_law_eq","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.QuantileEffect","a":"PO/ID/Exact/QTE","t":1,"d":"QTE as a functional of the identified potential-outcome laws. If the law of the potential outcome Y(1) equals a given observable measure ν₁ and the law of the potential outcome Y(0) equals a given observable measure ν₀, then for every real quantile level τ, the totalized quantile treatment effect equals the difference of the τ-quantiles of ν₁ and ν₀. This is the payoff of the quantile layer: identification of the QTE reduces to distributional identification of the two potential-outcome laws.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ) {ν₁ ν₀ : MeasureTheory.Measure ℝ}, S.yVar.cfUnderLaw S.dVar true μ = ν₁ → S.yVar.cfUnderLaw S.dVar false μ = ν₀ → S.qte μ τ = Causalean.Stat.quantile ν₁ τ - Causalean.Stat.quantile ν₀ τ","l":87,"f":"Causalean/PO/ID/Exact/QTE/QuantileEffect.lean"},{"n":"Causalean.PO.POBackdoorSystem.regFn","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the value-space outcome regression maps a treatment--covariate value to the corresponding conditional mean factual outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool × γ → ℝ","l":632,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.regression_adjustment","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Regression-adjustment identification (observable). For a treatment arm d, if the observed outcome is integrable and the covariate- conditional probability of receiving arm d is almost surely nonzero, then the backdoor adjustment functional for arm d equals the value-space outcome regression evaluated at the arm d:","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), MeasureTheory.Integrable S.factualY P.μ → (∀ᵐ (ω : P.Ω) ∂P.μ, S.propScore d ω ≠ 0) → S.adjustedCE d =ᵐ[P.μ] fun ω => S.regFn (d, S.factualX ω)","l":668,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.regression_adjustment_of_assumptions","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"Regression-adjustment, packaged for identified systems. The same observable identity adjustedCE d =ᵐ E[Y|D=d,X], with the two regularity hypotheses of regression_adjustment discharged from the back-door Assumptions: integrability of the observed outcome from Assumptions.integrable_factualY, and overlap from propScore_ne_of_overlap. Lets callers in an identified context apply it with just hA.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [inst_1 : StandardBorelSpace P.Ω] [inst_2 : MeasureTheory.IsFiniteMeasure P.μ], S.Assumptions → ∀ (d : Bool), S.adjustedCE d =ᵐ[P.μ] fun ω => S.regFn (d, S.factualX ω)","l":781,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.setLIntegral_condCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Set-lintegral form of the conditional CDF identity at a measurable variable cutoff.","s":"∀ {γ : Type u_1} [inst : MeasurableSpace γ] (ρ : MeasureTheory.Measure (γ × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {B : Set γ}, MeasurableSet B → ∀ {q : γ → ℝ}, Measurable q → ∫⁻ (a : γ) in B, ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) (q a)) ∂ρ.fst = ρ {p | p.1 ∈ B ∧ p.2 ≤ q p.1}","l":417,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaDX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the joint treatment--covariate σ-algebra is the σ-algebra on the sample space generated by the factual treatment--covariate pair.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":561,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaDX_le","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The joint treatment-covariate sigma-algebra is a sub-sigma-algebra of the ambient one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaDX ≤ inferInstance","l":567,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the covariate σ-algebra is the σ-algebra on the sample space generated by the factual covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":105,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the joint σ-algebra generated by the factual covariate and the control potential outcome is the least σ-algebra making both quantities measurable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":35,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY0_le","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlSetup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The joint σ-algebra σ(X, Y(0)) is coarser than the ambient σ-algebra on Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaXY0 ≤ inferInstance","l":45,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlSetup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY1","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Given a binary-treatment backdoor system, the complete-information σ-algebra is the σ-algebra on the sample space generated jointly by the factual covariate and the potential outcome under treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasurableSpace P.Ω","l":49,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaXY1_le","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Setup","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The joint σ-algebra σ(X, Y(1)) is coarser than the ambient σ-algebra on Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaXY1 ≤ inferInstance","l":58,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX_le","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The covariate sigma-algebra is a sub-sigma-algebra of the ambient one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaX ≤ inferInstance","l":109,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.sigmaX_le_sigmaDX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"σ(X) ≤ σ(D, X): the covariate σ-algebra is coarser than the joint one.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.sigmaX ≤ S.sigmaDX","l":589,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_CATE_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The conditional mean potential outcome is strongly measurable for the covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), MeasureTheory.StronglyMeasurable (S.CATE d)","l":178,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_adjustedCE_comap","k":"theorem","m":"Causalean.PO.ID.Exact.ATT","a":"PO/ID/Exact","t":2,"d":"The backdoor adjustment functional is strongly measurable with respect to the covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), MeasureTheory.StronglyMeasurable (S.adjustedCE d)","l":155,"f":"Causalean/PO/ID/Exact/ATT.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_controlCondCDF_const","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the control conditional CDF at a fixed outcome level.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), MeasureTheory.StronglyMeasurable fun ω => S.controlCondCDF ω t","l":115,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_controlCondCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the control conditional CDF at a covariate-measurable cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.StronglyMeasurable fun ω => S.controlCondCDF ω (c ω)","l":399,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_outcomeReg","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The observed outcome regression is strongly measurable with respect to the joint treatment-covariate sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), MeasureTheory.StronglyMeasurable S.outcomeReg","l":605,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_propScore","k":"theorem","m":"Causalean.PO.ID.Exact.QTE.DistributionalBackdoor","a":"PO/ID/Exact/QTE","t":2,"d":"propScore d is σ(X)-strongly-measurable (it is a conditional expectation).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) [StandardBorelSpace P.Ω] [MeasureTheory.IsFiniteMeasure P.μ] (d : Bool), MeasureTheory.StronglyMeasurable (S.propScore d)","l":102,"f":"Causalean/PO/ID/Exact/QTE/DistributionalBackdoor.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_propScore_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":2,"d":"The propensity score is strongly measurable for the covariate sigma-algebra: it is a conditional expectation given that sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (d : Bool), MeasureTheory.StronglyMeasurable (S.propScore d)","l":155,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_treatedCondCDF_const","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the treated conditional CDF at a fixed outcome level.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), MeasureTheory.StronglyMeasurable fun ω => S.treatedCondCDF ω t","l":137,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_treatedCondCDF_variable","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the treated conditional CDF at a covariate-measurable cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → MeasureTheory.StronglyMeasurable fun ω => S.treatedCondCDF ω (c ω)","l":510,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_wMax_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the upper sensitivity weight.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.StronglyMeasurable (S.wMax Λ)","l":112,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.stronglyMeasurable_wMin_sigmaX","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The strongly-measurable form, for the covariate σ-algebra, of the lower sensitivity weight.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), MeasureTheory.StronglyMeasurable (S.wMin Λ)","l":106,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.sub_mem_Icc_of_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ATE","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"Interval subtraction. If a ∈ [aₗ, aᵤ] and b ∈ [bₗ, bᵤ], then a − b ∈ [aₗ − bᵤ, aᵤ − bₗ]. The arithmetic core of the ATE-interval theorems.","s":"∀ {a aₗ aᵤ b bₗ bᵤ : ℝ}, a ∈ Set.Icc aₗ aᵤ → b ∈ Set.Icc bₗ bᵤ → a - b ∈ Set.Icc (aₗ - bᵤ) (aᵤ - bₗ)","l":78,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTarget","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system, a real sensitivity level, and a unit in its sample space, the target survival probability for cutoff calibration is $(1-w_{\\min}e)/(w_{\\max}-w_{\\min})$, where $e$ is the unit's treated propensity score and $w_{\\min},w_{\\max}$ are its endpoint weights.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":57,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTarget0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, and a sample point, the target control-survival probability is $(1-w_{\\min,0}(\\Lambda,\\omega)e_0(\\omega))/(w_{\\max,0}(\\Lambda,\\omega)-w_{\\min,0}(\\Lambda,\\omega))$, where $e_0(\\omega)$ is the control propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":58,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTargetLower","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.LowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a sensitivity level, and a unit, the lower target survival probability is the upper inverse-probability weight times the treatment propensity minus one, divided by the difference between the upper and lower inverse-probability weights.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":56,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/LowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.survTargetLower0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlLowerBound","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, and a sample point, the lower target control-survival probability is $(w_{\\max,0}(\\Lambda,\\omega)e_0(\\omega)-1)/(w_{\\max,0}(\\Lambda,\\omega)-w_{\\min,0}(\\Lambda,\\omega))$, where $e_0(\\omega)$ is the control propensity score.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":40,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlLowerBound.lean"},{"n":"Causalean.PO.POBackdoorSystem.toBackdoorEstimationSystem","k":"def","m":"Causalean.Estimation.ATE.Setup","a":"Estimation/ATE","t":1,"d":"Given a potential-outcome system with a standard-Borel sample space and finite probability measure, a measurable covariate space, a back-door potential-outcome system, the assumption that its treated propensity is strictly between zero and one almost surely, and an integrable factual outcome, the associated back-door estimation system uses the value-space outcome regression and a propensity score that equals the lifted propensity on its strict-overlap support and equals one half elsewhere.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → (S : Causalean.PO.POBackdoorSystem P γ) → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → MeasureTheory.Integrable S.factualY P.μ → Causalean.Estimation.ATE.BackdoorEstimationSystem P γ","l":217,"f":"Causalean/Estimation/ATE/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.toTreatedEstimationSystem","k":"def","m":"Causalean.Estimation.ATT.Setup","a":"Estimation/ATT","t":1,"d":"Given a potential-outcome system with a measurable covariate space, a potential-outcome back-door system, the condition that its conditional probability of treatment lies strictly between zero and one almost surely, and an integrable factual outcome, the constructed treated estimation system extends that back-door system with a control-arm outcome regression and a propensity-score representative bounded strictly below one.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → (S : Causalean.PO.POBackdoorSystem P γ) → [inst_1 : StandardBorelSpace P.Ω] → [inst_2 : MeasureTheory.IsFiniteMeasure P.μ] → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → MeasureTheory.Integrable S.factualY P.μ → Causalean.Estimation.ATT.TreatedEstimationSystem P γ","l":252,"f":"Causalean/Estimation/ATT/Setup.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedCondCDF","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, a back-door system on them, a unit, and a real threshold, the treated conditional distribution-function value is the probability that a treated unit's factual outcome does not exceed the threshold, conditional on having that unit's factual covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → P.Ω → ℝ → ℝ","l":84,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSet","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the treated set is the set of all units whose factual treatment status is treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Set P.Ω","l":63,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffSelection","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system and a real-valued cutoff function on its sample space, the conditional treated-survival function assigns each unit the conditional expectation, given its covariates, of the treatment indicator times the indicator that its factual outcome exceeds its cutoff.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → (P.Ω → ℝ) → P.Ω → ℝ","l":53,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffSelection.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv_const_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (constant cutoff). The weighted conditional survival equals the treated conditional survival scaled by the propensity: E[Z·1{Y>t} | σ(X)] = e(X)·(1 − F(t | X)) a.e. The genuine measure-theoretic content relates a conditional expectation under μ (weighted by the treatment indicator) to the conditional CDF of the treated push-forward law.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (t : ℝ), P.μ[fun ω => S.dVar.indicator true ω * if t < S.factualY ω then 1 else 0 | S.sigmaX] =ᵐ[P.μ] fun ω => S.propScore true ω * (1 - S.treatedCondCDF ω t)","l":330,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedSurv_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The survival bridge (functional cutoff). The version of treatedSurv_const_eq evaluated at a σ(X)-measurable cutoff c: E[Z·1{Y>c(X)} | σ(X)] = e(X)·(1 − F(c(X) | X)) a.e. Since c is σ(X)-measurable, it is \"frozen\" inside the conditional expectation, reducing to the constant-cutoff bridge fibrewise. This is the form consumed by exists_calibrating_cutoff.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (c : P.Ω → ℝ), Measurable c → S.treatedSurv c =ᵐ[P.μ] fun ω => S.propScore true ω * (1 - S.treatedCondCDF ω (c ω))","l":727,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedXYLaw","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes system, a measurable covariate space, and a back-door system on them, the treated covariate--outcome law is the image law of the factual covariate and factual outcome under the population measure restricted to treated units.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → MeasureTheory.Measure (γ × ℝ)","l":68,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.treatedXYLaw_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffConstruct","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The treated push-forward law of the covariate and outcome is the image, under the map recording the factual covariate and factual outcome, of the population measure restricted to the treated set.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ), S.treatedXYLaw = MeasureTheory.Measure.map (fun ω => (S.factualX ω, S.factualY ω)) (P.μ.restrict S.treatedSet)","l":75,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffConstruct.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMax","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system, a real sensitivity level, and a unit in its sample space, the upper endpoint inverse-propensity weight is $1+\\Lambda(1-e)/e$, where $e$ is that unit's treated propensity score. It is the endpoint corresponding to an odds ratio of $\\Lambda$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":64,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMax0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, and a sample point, the largest admissible control inverse-propensity weight is $1+\\Lambda(1-e_0(\\omega))/e_0(\\omega)$, where $e_0(\\omega)$ is the probability of the control arm conditional on the covariates.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":40,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMax_def","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The largest admissible inverse-propensity weight at a sensitivity level sends a unit to one plus the sensitivity level times the complementary propensity score, divided by the propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.wMax Λ = fun ω => 1 + Λ * (1 - S.propScore true ω) / S.propScore true ω","l":79,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcomes backdoor system, a real sensitivity level, and a unit in its sample space, the lower endpoint inverse-propensity weight is $1+(1-e)/(\\Lambda e)$, where $e$ is that unit's treated propensity score. It is the endpoint corresponding to an odds ratio of $1/\\Lambda$.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":58,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin0","k":"def","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"For a potential-outcome backdoor system, a sensitivity parameter, and a sample point, the smallest admissible control inverse-propensity weight is $1+(1-e_0(\\omega))/(\\Lambda e_0(\\omega))$, where $e_0(\\omega)$ is the probability of the control arm conditional on the covariates.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → ℝ → P.Ω → ℝ","l":34,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin0_mul_propScore_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.ControlCutoff","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Lower control calibration bracket is at most one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the lower calibration weight wMin0 Λ times the control propensity is at most 1.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore false ω ∧ S.propScore false ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.wMin0 Λ ω * S.propScore false ω ≤ 1","l":64,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/ControlCutoff.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin_def","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.Bounds","a":"PO/ID/Partial/Sensitivity/MSM","t":2,"d":"The smallest admissible inverse-propensity weight at a sensitivity level sends a unit to one plus the ratio of the complementary propensity score to the sensitivity level times the propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), S.wMin Λ = fun ω => 1 + (1 - S.propScore true ω) / (Λ * S.propScore true ω)","l":70,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/Bounds.lean"},{"n":"Causalean.PO.POBackdoorSystem.wMin_mul_propScore_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.Sensitivity.MSM.CutoffExists","a":"PO/ID/Partial/Sensitivity/MSM","t":1,"d":"Lower calibration bracket (algebraic). Fix a sensitivity parameter Λ at least 1. Assuming the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap), almost surely the lower marginal-sensitivity-model weight satisfies wMin(X)·e(X) = e(X) + (1 − e(X))/Λ ≤ 1, where e(X) denotes the propensity score.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POBackdoorSystem P γ) (Λ : ℝ), 1 ≤ Λ → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < S.propScore true ω ∧ S.propScore true ω < 1) → ∀ᵐ (ω : P.Ω) ∂P.μ, S.wMin Λ ω * S.propScore true ω ≤ 1","l":48,"f":"Causalean/PO/ID/Partial/Sensitivity/MSM/CutoffExists.lean"},{"n":"Causalean.PO.POBackdoorSystem.xVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the adjustment-covariate potential-outcome variable is its specified covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P γ","l":79,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system and a treatment arm, the regimed outcome variable is the outcome variable under the regime fixing treatment to that arm.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Bool → Causalean.PO.RegimedVar P ℝ","l":119,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBackdoorSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.ATE","a":"PO/ID/Exact","t":1,"d":"Given a binary-treatment backdoor system, the outcome potential-outcome variable is the system's real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POBackdoorSystem P γ → Causalean.PO.POVar P ℝ","l":77,"f":"Causalean/PO/ID/Exact/ATE.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SOmega","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical sample space consists of one observed binary instrument value together with four binary latent response values: treatment under each instrument value and outcome under each treatment value.","s":"Type","l":49,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SV","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical variable-index set has exactly three elements, representing respectively the instrument, treatment, and outcome.","s":"Type","l":52,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.SX","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The canonical measurement-scale assignment assigns the binary scale to each of the three canonical variables.","s":"Causalean.PO.POBalkePearlSharp.SV → Type","l":55,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalBP","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, a table of real weights indexed by the four binary latent response values, the condition that every table entry is nonnegative, and the condition that all table entries sum to one, the canonical Balke--Pearl system designates the first, second, and third canonical variables as instrument, treatment, and outcome, respectively.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → (π : Bool → Bool → Bool → Bool → ℝ) → (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) → Causalean.PO.POBalkePearlSystem (Causalean.PO.POBalkePearlSharp.canonicalPOSystem S π hπ_nn hπ_sum)","l":299,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and a table of real weights indexed by the four binary latent response values, the canonical product measure is the product of the system's instrument marginal measure and the latent-table measure.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → (Bool → Bool → Bool → Bool → ℝ) → MeasureTheory.Measure Causalean.PO.POBalkePearlSharp.SOmega","l":173,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonicalPOSystem","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, a table of real weights indexed by the four binary latent response values, the condition that every table entry is nonnegative, and the condition that all table entries sum to one, the canonical potential-outcomes system has the canonical variables, binary measurement scales, canonical sample space, canonical product probability measure, and canonical evaluator. Its probability-measure property follows from the two conditions on the latent table.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → (π : Bool → Bool → Bool → Bool → ℝ) → (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → Causalean.PO.POSystem","l":282,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_ATE_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.ATE = BPObjective π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).ATE = Causalean.PO.POBalkePearlSystem.BPObjective π","l":834,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_DofZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.DofZ z ω = (POBalkePearlSystem.dArm) z ω.2.1 ω.2.2.1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).DofZ z ω = Causalean.PO.POBalkePearlSystem.dArm z ω.2.1 ω.2.2.1","l":501,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.YofD d ω = (POBalkePearlSystem.yArm) d ω.2.2.2.1 ω.2.2.2.2.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofD d ω = Causalean.PO.POBalkePearlSystem.yArm d ω.2.2.2.1 ω.2.2.2.2","l":518,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_YofZD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.YofZD z d ω = (POBalkePearlSystem.yArm) d ω.2.2.2.1 ω.2.2.2.2.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z d : Bool) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofZD z d ω = Causalean.PO.POBalkePearlSystem.yArm d ω.2.2.2.1 ω.2.2.2.2","l":535,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_baseAssumptions","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Bundle the canonical model's BaseAssumptions.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).BaseAssumptions","l":700,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_cellProb_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.cellProb y d z = S.cellProb y d z for any feasible π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (hA : S.BaseAssumptions), S.BPFeasible hA π → ∀ (y d z : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cellProb y d z = S.cellProb y d z","l":822,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_cfBundle_factors_through_snd","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The cfBundle's jointValue factors through Prod.snd: it depends only on the latent factor.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), ∃ g, Measurable g ∧ (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cfBundle.jointValue = fun ω => g ω.2","l":629,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_consistency","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Consistency for the canonical PO system.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).Consistency","l":344,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_exclusion","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Exclusion: Y(z,d) = Y(d) pointwise (and so a.e.).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z d : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofZD z d = (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).YofD d","l":574,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_exogeneity","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Exogeneity: Z ⊥ cfBundle under the canonical product measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).IndepCF (Causalean.PO.RegimedVar.ofFactual (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zVar) (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).cfBundle","l":676,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualD ω = D(ω.1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualD ω = Causalean.PO.POBalkePearlSharp.dArmω ω.1 ω","l":475,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualY ω = Y(D(ω.1)).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualY ω = Causalean.PO.POBalkePearlSharp.yArmω (Causalean.PO.POBalkePearlSharp.dArmω ω.1 ω) ω","l":487,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.factualZ ω = ω.1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (ω : Causalean.PO.POBalkePearlSharp.SOmega), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualZ ω = ω.1","l":466,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_factualZ_eq_fst","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factualZ on the canonical model factors through Prod.fst.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).factualZ = fun ω => ω.1","l":624,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentProb_eq","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"S'.latentProb = π.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentProb d0 d1 y0 y1 = π d0 d1 y0 y1","l":815,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentSet","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The canonical latent set is Set.univ ×ˢ {(d0, d1, y0, y1)}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentSet d0 d1 y0 y1 = Set.univ ×ˢ {(d0, d1, y0, y1)}","l":775,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_latentSet_measure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"μ'(latentSet) = ENNReal.ofReal (π d0 d1 y0 y1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).latentSet d0 d1 y0 y1) = ENNReal.ofReal (π d0 d1 y0 y1)","l":804,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_posZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Positive Z probability.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1), S.BaseAssumptions → ∀ (z : Bool), 0 < (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zVar.event z)","l":615,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Z-event in the canonical model is {ω | ω.1 = z}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool), (Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zEvent z = {ω | ω.1 = z}","l":582,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.canonical_zEvent_measure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"μ' (S'.zEvent z) = P.μ (S.zEvent z).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (π : Bool → Bool → Bool → Bool → ℝ) (hπ_nn : ∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) (hπ_sum : ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1) (z : Bool), (Causalean.PO.POBalkePearlSharp.P'✝ S π hπ_nn hπ_sum).μ ((Causalean.PO.POBalkePearlSharp.S'✝ S π hπ_nn hπ_sum).zEvent z) = P.μ (S.zEvent z)","l":590,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.dArmω","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a binary instrument value and a canonical sample-space point, the latent treatment response is that point's treatment response under the specified instrument value.","s":"Bool → Causalean.PO.POBalkePearlSharp.SOmega → Bool","l":60,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For an intervention regime on the canonical instrument, treatment, and outcome variables, a canonical sample-space point, and one of those variables, the canonical evaluator returns the variable's assigned value when the regime intervenes on it; otherwise it returns the point's instrument value, its treatment response to the resulting instrument value, or its outcome response to the resulting treatment value, respectively. For an intervened variable, the assigned value is used; for the instrument, the observed coordinate is used; for the treatment, the treatment response is used; and for the outcome, the outcome response is used.","s":"Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX → Causalean.PO.POBalkePearlSharp.SOmega → (v : Causalean.PO.POBalkePearlSharp.SV) → Causalean.PO.POBalkePearlSharp.SX v","l":68,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The D-coordinate folded via eval_zero.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.dArmω (Causalean.PO.POBalkePearlSharp.eval r ω ⟨0, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":118,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_one_raw","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The D-coordinate (raw).","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.dArmω (if h0 : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h0 else ω.1) ω","l":101,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_two","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Y-coordinate folded via eval_one.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.yArmω (Causalean.PO.POBalkePearlSharp.eval r ω ⟨1, of_decide_eq_true (id (Eq.refl true))⟩) ω","l":125,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_two_raw","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Y-coordinate (raw).","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨2, of_decide_eq_true (id (Eq.refl true))⟩ h else Causalean.PO.POBalkePearlSharp.yArmω (if h1 : ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨1, of_decide_eq_true (id (Eq.refl true))⟩ h1 else Causalean.PO.POBalkePearlSharp.dArmω (if h0 : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h0 else ω.1) ω) ω","l":108,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.eval_zero","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Z-coordinate eval r ω 0: assignment if intervened on, else ω.1.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX) (ω : Causalean.PO.POBalkePearlSharp.SOmega), Causalean.PO.POBalkePearlSharp.eval r ω ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ = if h : ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ ∈ r.target then r.assign ⟨0, of_decide_eq_true (id (Eq.refl true))⟩ h else ω.1","l":96,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbCanonicalMeasure","k":"instance","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"For a potential-outcomes system, a binary Balke--Pearl system on that system, and a table of real weights indexed by the four binary latent response values, if the probability measure induced by that latent table is a probability measure, then the canonical product measure combining the instrument marginal and latent-table measure is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) {π : Bool → Bool → Bool → Bool → ℝ} [MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.piMeasure π)], MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.canonicalMeasure S π)","l":268,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbPiMeasure","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"A nonnegative latent table that sums to one induces a probability measure.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.piMeasure π)","l":258,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.instIsProbZMeasure","k":"instance","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"For a potential-outcomes system and a binary Balke--Pearl system on that system, the canonical marginal distribution of the binary instrument is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), MeasureTheory.IsProbabilityMeasure (Causalean.PO.POBalkePearlSharp.zMeasure S)","l":203,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.measurable_eval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The canonical evaluator is measurable under every intervention regime.","s":"∀ (r : Causalean.PO.Regime Causalean.PO.POBalkePearlSharp.SV Causalean.PO.POBalkePearlSharp.SX), Measurable (Causalean.PO.POBalkePearlSharp.eval r)","l":91,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a table of real weights indexed by the four binary latent response values, the latent-table measure is the discrete measure that places at each response type the nonnegative extended-real part of its table weight.","s":"(Bool → Bool → Bool → Bool → ℝ) → MeasureTheory.Measure (Bool × Bool × Bool × Bool)","l":167,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"piMeasure π {(d0, d1, y0, y1)} = ENNReal.ofReal (π d0 d1 y0 y1).","s":"∀ (π : Bool → Bool → Bool → Bool → ℝ) (d0 d1 y0 y1 : Bool), (Causalean.PO.POBalkePearlSharp.piMeasure π) {(d0, d1, y0, y1)} = ENNReal.ofReal (π d0 d1 y0 y1)","l":731,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.piMeasure_univ_of_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"piMeasure π has total mass 1 when π has nonneg entries summing to 1.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → (Causalean.PO.POBalkePearlSharp.piMeasure π) Set.univ = 1","l":211,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.yArmω","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a binary treatment value and a canonical sample-space point, the latent outcome response is that point's outcome response under the specified treatment value.","s":"Bool → Causalean.PO.POBalkePearlSharp.SOmega → Bool","l":64,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.zMeasure","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the canonical instrument marginal measure assigns to each binary instrument value the probability of that system's corresponding factual instrument event.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → MeasureTheory.Measure Bool","l":163,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSharp.zMeasure_univ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":2,"d":"zMeasure S has total mass 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), (Causalean.PO.POBalkePearlSharp.zMeasure S) Set.univ = 1","l":180,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Binary-IV system for Balke–Pearl ATE bounds (def:po-iv-balke-pearl-system). Inside a potential-outcome system, this bundles a binary instrument node Z, a binary treatment node D, and a binary outcome node Y, subject to the instrument, treatment, and outcome being pairwise distinct system variables.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":36,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the average treatment effect is the expectation, under the system's probability measure, of the real-valued potential outcome under treatment minus that under control.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":173,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_eq_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":2,"d":"ATE = BPObjective applied to the realized latent table.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → S.ATE = Causalean.PO.POBalkePearlSystem.BPObjective S.latentProb","l":90,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_eq_sum_latent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The average treatment effect equals the probability-weighted sum, over the sixteen latent response types, of each type's treatment effect.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.ATE = ∑ d0, ∑ d1, ∑ y0, ∑ y1, (Causalean.PO.POBalkePearlSystem.boolToReal y1 - Causalean.PO.POBalkePearlSystem.boolToReal y0) * S.latentProb d0 d1 y0 y1","l":155,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Necessity. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.ATE ∈ S.BPIdentifiedInterval hA","l":98,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_Icc_bpLower_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl bound in closed form. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies between the largest of eight closed-form lower expressions and the smallest of eight closed-form upper expressions, both computable directly from the observed cell probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → S.ATE ∈ Set.Icc S.bpLower S.bpUpper","l":213,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.ATE_mem_Icc_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Necessity, closed-interval form. Under the Balke-Pearl IV base assumptions, the true average treatment effect lies between the infimum and supremum of the Balke-Pearl identified interval — the LP minimum and maximum. This is ATE_mem_BPIdentifiedInterval rephrased through the engine bridge Causalean.PartialID.mem_Icc_csInf_csSup.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.ATE ∈ Set.Icc (sInf (S.BPIdentifiedInterval hA)) (sSup (S.BPIdentifiedInterval hA))","l":109,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPFeasible","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Balke–Pearl latent-table feasibility. A latent response-type table π — the joint distribution over the instrument's and treatment's potential values together with the outcome's potential values — is feasible for a Balke–Pearl IV system under a given assumption bundle when every table entry is nonnegative, the entries sum to one, and aggregating the table over the response types compatible with each observed instrument-treatment-outcome cell reproduces the observed conditional cell probability.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → S.BaseAssumptions → (Bool → Bool → Bool → Bool → ℝ) → Prop","l":47,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPIdentifiedInterval","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and proof that the system satisfies the Balke--Pearl base assumptions, the sharp identified interval for the average treatment effect is the set of values of the Balke--Pearl linear-program objective over all feasible latent response-type tables.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POBalkePearlSystem P) → S.BaseAssumptions → Set ℝ","l":74,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPIdentifiedInterval_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The identified interval is contained in the closed-form interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.BPIdentifiedInterval hA ⊆ Set.Icc S.bpLower S.bpUpper","l":206,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a table of real weights indexed by the four binary latent response values, the Balke--Pearl linear-program objective is the sum, over all latent response types, of that type's weight times its binary outcome response under treatment minus its binary outcome response under control.","s":"(Bool → Bool → Bool → Bool → ℝ) → ℝ","l":67,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The closed-form upper endpoint bounds the objective on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → Causalean.PO.POBalkePearlSystem.BPObjective π ≤ S.bpUpper","l":200,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_bpUpperTerm","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Every upper expression bounds the objective from above on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → ∀ (i : Fin 8), Causalean.PO.POBalkePearlSystem.BPObjective π ≤ S.bpUpperTerm i","l":163,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BPObjective_le_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Upper bound on the objective. For a feasible (nonneg, sum-one) table π, the LP objective ∑ (y1 - y0) · π is at most 1, since every contrast is ≤ 1 and π is a probability vector.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → Causalean.PO.POBalkePearlSystem.BPObjective π ≤ 1","l":60,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Balke-Pearl IV assumptions — def:po-iv-balke-pearl-assumptions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Prop","l":60,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_ne_top","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Instrument-cell probabilities are finite.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), P.μ (S.zVar.event z) ≠ ⊤","l":86,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_ne_zero","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Positive instrument probabilities are nonzero.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), P.μ (S.zVar.event z) ≠ 0","l":82,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.BaseAssumptions.posZ_toReal_pos","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Under the Balke-Pearl IV base assumptions — IV-specific consistency of treatment and outcome on their respective cells, the exclusion restriction, instrument exogeneity, and positive instrument probability, the real-valued probability of each instrument cell z is strictly positive.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POBalkePearlSystem P}, S.BaseAssumptions → ∀ (z : Bool), 0 < (P.μ (S.zVar.event z)).toReal","l":91,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.DofZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and an instrument value, the potential treatment function maps each unit to the treatment it would receive were the instrument set to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → Bool","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and a treatment value, the potential outcome function maps each unit to the binary outcome it would have under that treatment value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → Bool","l":72,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofD_real","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and a treatment value, the real-valued potential outcome function maps each unit's binary potential outcome under that treatment to its zero--one real encoding.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → P.Ω → ℝ","l":107,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.YofZD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, an instrument value, and a treatment value, the joint-intervention potential outcome function maps each unit to its outcome when the instrument and treatment are set jointly to those values.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → P.Ω → Bool","l":86,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.balkePearl_sharp","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Sharpness. Under the Balke-Pearl IV base assumptions, for a latent treatment-response table π that is feasible for the linear program — nonnegative, summing to one, and reproducing the observed cell probabilities as its marginals, there exists another potential-outcome system, satisfying the same base assumptions, whose observed cell probabilities agree with the original system's and whose average treatment effect equals the LP objective value of π: every feasible latent table is realised by some potential-outcome model.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) (π : Bool → Bool → Bool → Bool → ℝ), S.BPFeasible hA π → ∃ P' S', ∃ (_ : S'.BaseAssumptions), (∀ (y d z : Bool), S'.cellProb y d z = S.cellProb y d z) ∧ S'.ATE = Causalean.PO.POBalkePearlSystem.BPObjective π","l":856,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem.balkePearl_sharp_of_mem","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Sharp","a":"PO/ID/Partial/BalkePearl","t":1,"d":"Corollary of sharpness. Under the Balke-Pearl IV base assumptions, for a real number τ lying in the Balke-Pearl identified interval, there exists another potential-outcome system, satisfying the same base assumptions, whose observed cell probabilities agree with the original system's and whose average treatment effect equals τ: every value in the identified interval is the ATE of some BP-feasible model with matching observed cell probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), ∀ τ ∈ S.BPIdentifiedInterval hA, ∃ P' S', ∃ (_ : S'.BaseAssumptions), (∀ (y d z : Bool), S'.cellProb y d z = S.cellProb y d z) ∧ S'.ATE = τ","l":881,"f":"Causalean/PO/ID/Partial/BalkePearl/Sharp.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bddAbove_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Balke-Pearl identified interval is bounded above (by 1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), BddAbove (S.BPIdentifiedInterval hA)","l":100,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bddBelow_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Balke-Pearl identified interval is bounded below (by -1).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), BddBelow (S.BPIdentifiedInterval hA)","l":93,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.boolToReal","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The binary-to-real encoding maps every binary value to a real number: for true, its value is one, and for false, its value is zero.","s":"Bool → ℝ","l":102,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux0t","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass is $\\max\\{0,p_{001\\mid 0}+p_{011\\mid 0}-p_{000\\mid 0}-p_{100\\mid 0}\\}$, where $p_{yd\\mid z}$ denotes the observed probability of outcome $y$ and treatment $d$ at instrument value $z$.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":45,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux1v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass is $\\max\\{0,p_{000\\mid0}-p_{001\\mid0}+p_{010\\mid0}-p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability at instrument value $z$.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":144,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux2u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass for the third lower-bound witness is $\\max\\{0,p_{011\\mid0}-p_{010\\mid0}-p_{100\\mid0} +p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":240,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux2v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass for the third lower-bound witness is $\\max\\{0,p_{011\\mid0}-p_{100\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":247,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux3u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass for the fourth lower-bound witness is $\\max\\{0,p_{010\\mid0}-p_{011\\mid0}+p_{100\\mid0} -p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":348,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpAux3v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass for the fourth lower-bound witness is $\\max\\{0,p_{010\\mid0}-p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":355,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the closed-form lower endpoint is the largest of its eight affine lower-bound expressions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":112,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerTerm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, and one of the eight indices, the corresponding affine lower-bound expression is calculated from the observed outcome--treatment cell probabilities by naming those probabilities and applying the appropriate first, second, third, fourth, fifth, sixth, seventh, or eighth displayed formula.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Fin 8 → ℝ","l":64,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerTerm_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Every lower expression bounds the objective from below on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → ∀ (i : Fin 8), S.bpLowerTerm i ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":134,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the first lower-bound expression assigns its displayed mass to each of the eight listed latent profiles in the first case, second case, third case, fourth case, fifth case, sixth case, seventh case, and eighth case, and assigns zero in the remaining case.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":53,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 0 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 0) → S.BPFeasible hA S.bpLowerWitness0","l":77,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness0_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 0 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness0 = S.bpLowerTerm 0","l":135,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the second lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, and eighth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":151,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 1 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 1) → S.BPFeasible hA S.bpLowerWitness1","l":173,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness1_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 1 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness1 = S.bpLowerTerm 1","l":231,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the third lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":253,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 2 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 2) → S.BPFeasible hA S.bpLowerWitness2","l":278,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness2_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 2 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness2 = S.bpLowerTerm 2","l":338,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the fourth lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":361,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 3 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 3) → S.BPFeasible hA S.bpLowerWitness3","l":386,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness3_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 3 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness3 = S.bpLowerTerm 3","l":446,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the fifth lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":456,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 4 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 4) → S.BPFeasible hA S.bpLowerWitness4","l":476,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness4_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 4 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness4 = S.bpLowerTerm 4","l":530,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the sixth lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":539,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 5 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 5) → S.BPFeasible hA S.bpLowerWitness5","l":557,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness5_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 5 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness5 = S.bpLowerTerm 5","l":611,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the seventh lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":620,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 6 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 6) → S.BPFeasible hA S.bpLowerWitness6","l":638,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness6_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 6 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness6 = S.bpLowerTerm 6","l":692,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the eighth lower-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":701,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 7 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpLowerTerm j ≤ S.bpLowerTerm 7) → S.BPFeasible hA S.bpLowerWitness7","l":721,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLowerWitness7_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpLowerTerm 7 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpLowerWitness7 = S.bpLowerTerm 7","l":775,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_bpUpper_eq_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The Balke-Pearl closed form is sharp. Under the Balke-Pearl IV base assumptions, the interval [bpLower, bpUpper], computed from the observed cell probabilities alone, has both endpoints attained by observationally-equivalent latent tables — equivalently, bpLower is the infimum and bpUpper is the supremum of the Balke-Pearl identified interval. So no smaller interval contains every average treatment effect compatible with the data: the bound cannot be improved without further assumptions.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpLower = sInf (S.BPIdentifiedInterval hA) ∧ S.bpUpper = sSup (S.BPIdentifiedInterval hA)","l":69,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The closed-form lower endpoint bounds the objective on feasible tables.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π → S.bpLower ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":194,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpLower_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Lower","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, the closed-form Balke-Pearl lower bound, computed from the observed cell probabilities, is itself attained as the average treatment effect of some feasible latent treatment-response table — it lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpLower ∈ S.BPIdentifiedInterval hA","l":786,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Lower.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux0u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass is $\\max\\{0,1-p_{000\\mid0}-p_{001\\mid0}-p_{011\\mid0}-p_{100\\mid0}\\}$, where $p_{yd\\mid z}$ denotes the observed probability of outcome $y$ and treatment $d$ at instrument value $z$.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":48,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux1u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass is $\\max\\{0,1-p_{000\\mid0}-p_{001\\mid0}-p_{010\\mid0}-p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":145,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux2u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass for the third upper-bound witness is $\\max\\{0,p_{010\\mid0}-p_{011\\mid0}+p_{100\\mid0} -p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":242,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux2v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass for the third upper-bound witness is $\\max\\{0,1-p_{000\\mid0}-p_{001\\mid0}-p_{010\\mid0} -p_{100\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":249,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux3u","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the first auxiliary free mass for the fourth upper-bound witness is $\\max\\{0,-p_{010\\mid0}+p_{011\\mid0}-p_{100\\mid0} +p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":355,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUAux3v","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the second auxiliary free mass for the fourth upper-bound witness is $\\max\\{0,1-p_{000\\mid0}-p_{001\\mid0}-p_{011\\mid0} -p_{101\\mid0}\\}$, where $p_{yd\\mid z}$ is the observed outcome--treatment cell probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":362,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpper","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the closed-form upper endpoint is the smallest of its eight affine upper-bound expressions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → ℝ","l":120,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperTerm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedForm","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, and one of the eight indices, the corresponding affine upper-bound expression is calculated from the observed outcome--treatment cell probabilities by naming those probabilities and applying the appropriate first, second, third, fourth, fifth, sixth, seventh, or eighth displayed formula.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Fin 8 → ℝ","l":88,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the first upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, and eighth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":55,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 0 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 0 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness0","l":77,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness0_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 0 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness0 = S.bpUpperTerm 0","l":135,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the second upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, and eighth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":152,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 1 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 1 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness1","l":174,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness1_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 1 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness1 = S.bpUpperTerm 1","l":232,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the third upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":256,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 2 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 2 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness2","l":283,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness2_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 2 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness2 = S.bpUpperTerm 2","l":345,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the fourth upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":369,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 3 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 3 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness3","l":396,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness3_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 3 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness3 = S.bpUpperTerm 3","l":458,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the fifth upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":468,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 4 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 4 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness4","l":490,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness4_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 4 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness4 = S.bpUpperTerm 4","l":544,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the sixth upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":553,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 5 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 5 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness5","l":574,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness5_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 5 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness5 = S.bpUpperTerm 5","l":628,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the seventh upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":637,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 6 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 6 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness6","l":658,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness6_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 6 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness6 = S.bpUpperTerm 6","l":712,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the latent-type witness table for the eighth upper-bound expression assigns its displayed mass in the first, second, third, fourth, fifth, sixth, and seventh listed latent-profile cases, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":721,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 7 is feasible on its optimality region.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions) {π₀ : Bool → Bool → Bool → Bool → ℝ}, S.BPFeasible hA π₀ → (∀ (j : Fin 8), S.bpUpperTerm 7 ≤ S.bpUpperTerm j) → S.BPFeasible hA S.bpUpperWitness7","l":743,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpperWitness7_objective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"The witness for bpUpperTerm 7 attains it.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → Causalean.PO.POBalkePearlSystem.BPObjective S.bpUpperWitness7 = S.bpUpperTerm 7","l":797,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.bpUpper_mem_BPIdentifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Upper","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, the closed-form Balke-Pearl upper bound, computed from the observed cell probabilities, is itself attained as the average treatment effect of some feasible latent treatment-response table — it lies in the Balke-Pearl identified interval.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.bpUpper ∈ S.BPIdentifiedInterval hA","l":808,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Upper.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, an outcome value, a treatment value, and an instrument value, the conditional cell probability is the probability that the factual outcome and treatment equal the specified values conditional on the factual instrument equaling the specified instrument value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → ℝ","l":177,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb_eq_sum_latent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Each observed conditional cell probability equals the sum of compatible latent type probabilities.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → ∀ (y d z : Bool), S.cellProb y d z = ∑ d0, ∑ d1, ∑ y0, ∑ y1, (if Causalean.PO.POBalkePearlSystem.dArm z d0 d1 = d ∧ Causalean.PO.POBalkePearlSystem.yArm d y0 y1 = y then 1 else 0) * S.latentProb d0 d1 y0 y1","l":284,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cellProb_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Basic","a":"PO/ID/Partial/BalkePearl/Attainment","t":2,"d":"Observed cell probabilities are nonnegative.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (y d z : Bool), 0 ≤ S.cellProb y d z","l":28,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Basic.lean"},{"n":"Causalean.PO.POBalkePearlSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, the counterfactual bundle collects, in order, treatment under instrument values zero and one and outcome under treatment values zero and one.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POCFBundle P","l":48,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.csInf_BPIdentifiedInterval_eq_bpLower","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The closed-form lower endpoint is exactly the LP minimum. Under the Balke-Pearl IV base assumptions, the infimum of the Balke-Pearl identified interval equals the closed-form lower bound bpLower.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), sInf (S.BPIdentifiedInterval hA) = S.bpLower","l":43,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.csSup_BPIdentifiedInterval_eq_bpUpper","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.ClosedFormAttainment","a":"PO/ID/Partial/BalkePearl","t":1,"d":"The closed-form upper endpoint is exactly the LP maximum. Under the Balke-Pearl IV base assumptions, the supremum of the Balke-Pearl identified interval equals the closed-form upper bound bpUpper.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), sSup (S.BPIdentifiedInterval hA) = S.bpUpper","l":56,"f":"Causalean/PO/ID/Partial/BalkePearl/ClosedFormAttainment.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dArm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a binary instrument value and two binary treatment potential outcomes, respectively under instrument values zero and one, the selected treatment potential outcome is the first when the instrument value is zero and the second when it is one.","s":"Bool → Bool → Bool → Bool","l":38,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and a treatment value, the treatment event is the set of units whose factual treatment equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":115,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, and a binary instrument value, the treatment potential-outcome variable under that instrument value is the factual treatment variable evaluated in the intervention that fixes the instrument at that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":34,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.dVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the treatment variable is the system's treatment, packaged together with its binary measurement scale.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":61,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the factual treatment function maps each unit to its observed binary treatment value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":94,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the factual outcome function maps each unit to its observed binary outcome value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":97,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.factualZ","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the factual instrument function maps each unit to its observed binary instrument value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → P.Ω → Bool","l":91,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system and binary values for treatment under instrument zero and one and outcome under treatment zero and one, the latent-type probability is the real-valued probability of the corresponding latent-type event.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → ℝ","l":59,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_feasible","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Main","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The actual latent type distribution latentProb is feasible.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (hA : S.BaseAssumptions), S.BPFeasible hA S.latentProb","l":81,"f":"Causalean/PO/ID/Partial/BalkePearl/Main.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Latent type probabilities are nonnegative.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d0 d1 y0 y1 : Bool), 0 ≤ S.latentProb d0 d1 y0 y1","l":81,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentProb_sum_eq_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The 16 latent sets partition Ω, so their probabilities sum to 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), ∑ d0, ∑ d1, ∑ y0, ∑ y1, S.latentProb d0 d1 y0 y1 = 1","l":106,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.latentSet","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system and binary values for treatment under instrument zero and one and outcome under treatment zero and one, the latent-type event is the set of sample points at which all four corresponding potential outcomes equal those values.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Bool → Bool → Set P.Ω","l":51,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":153,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_latentSet","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Each latent type event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d0 d1 y0 y1 : Bool), MeasurableSet (S.latentSet d0 d1 y0 y1)","l":68,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_yEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual outcome event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (y : Bool), MeasurableSet (S.yEvent y)","l":157,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":149,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a fixed instrument value z, the potential treatment D(z) is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z : Bool), Measurable (S.DofZ z)","l":123,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), Measurable (S.YofD d)","l":129,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofD_real","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The real-valued potential outcome under a fixed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (d : Bool), Measurable (S.YofD_real d)","l":166,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_YofZD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The outcome under fixed instrument and treatment values is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P) (z d : Bool), Measurable (S.YofZD z d)","l":134,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_boolToReal","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The Boolean-to-real embedding is measurable.","s":"Measurable Causalean.PO.POBalkePearlSystem.boolToReal","l":161,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualD","l":142,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualY","l":145,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), Measurable S.factualZ","l":139,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.neg_one_le_BPObjective","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.IntervalForm","a":"PO/ID/Partial/BalkePearl","t":2,"d":"Lower bound on the objective. Symmetrically, -1 ≤ BPObjective π.","s":"∀ {π : Bool → Bool → Bool → Bool → ℝ}, (∀ (d0 d1 y0 y1 : Bool), 0 ≤ π d0 d1 y0 y1) → ∑ d0, ∑ d1, ∑ y0, ∑ y1, π d0 d1 y0 y1 = 1 → -1 ≤ Causalean.PO.POBalkePearlSystem.BPObjective π","l":74,"f":"Causalean/PO/ID/Partial/BalkePearl/IntervalForm.lean"},{"n":"Causalean.PO.POBalkePearlSystem.regimeZD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, an instrument value, and a treatment value, the joint intervention regime sets the instrument to the specified instrument value and the treatment to the specified treatment value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Bool → Causalean.PO.Regime P.V P.X","l":77,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.sum_cellProb_eq_one","k":"theorem","m":"Causalean.PO.ID.Partial.BalkePearl.Attainment.Basic","a":"PO/ID/Partial/BalkePearl/Attainment","t":1,"d":"Under the Balke-Pearl IV base assumptions, for every instrument value z, the four observed outcome-treatment cell probabilities sum to one.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POBalkePearlSystem P), S.BaseAssumptions → ∀ (z : Bool), ∑ y, ∑ d, S.cellProb y d z = 1","l":32,"f":"Causalean/PO/ID/Partial/BalkePearl/Attainment/Basic.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yArm","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.LatentTable","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a binary treatment value and two binary outcome potential outcomes, respectively under treatment values zero and one, the selected outcome potential outcome is the first when treatment is zero and the second when it is one.","s":"Bool → Bool → Bool → Bool","l":44,"f":"Causalean/PO/ID/Partial/BalkePearl/LatentTable.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and an outcome value, the outcome event is the set of units whose factual outcome equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":118,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Assumptions","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcome system and its Balke--Pearl observational system, and a binary treatment value, the outcome potential-outcome variable under that treatment value is the factual outcome variable evaluated in the intervention that fixes treatment at that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":40,"f":"Causalean/PO/ID/Partial/BalkePearl/Assumptions.lean"},{"n":"Causalean.PO.POBalkePearlSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the outcome variable is the system's outcome, packaged together with its binary measurement scale.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":64,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.zEvent","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system, a binary Balke--Pearl system on it, and an instrument value, the instrument event is the set of units whose factual instrument equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Bool → Set P.Ω","l":112,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POBalkePearlSystem.zVar","k":"def","m":"Causalean.PO.ID.Partial.BalkePearl.Setup","a":"PO/ID/Partial/BalkePearl","t":1,"d":"For a potential-outcomes system and a binary Balke--Pearl system on it, the instrument variable is the system's instrument, packaged together with its binary measurement scale.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POBalkePearlSystem P → Causalean.PO.POVar P Bool","l":58,"f":"Causalean/PO/ID/Partial/BalkePearl/Setup.lean"},{"n":"Causalean.PO.POCFBundle","k":"structure","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"A finite heterogeneous bundle of n regimed potential-outcome variables indexed by Fin n: each coordinate carries its own value type equipped with a measurable-space structure, and each coordinate is itself a potential-outcome variable paired with the intervention regime under which it is evaluated.","s":"Causalean.PO.POSystem → Type (max (max (u_1 + 1) u_2) u_3)","l":39,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system, a counterfactual bundle, a real-valued quantity on the sample space, and a measure on that sample space, the bundle conditional expectation is the conditional expectation of the quantity given the σ-algebra generated by jointly observing the bundle.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":67,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_add","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Linearity of the bundle conditional expectation: additivity.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → B.condExpGiven (f + g) μ =ᵐ[μ] B.condExpGiven f μ + B.condExpGiven g μ","l":118,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_add'","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Additivity of the bundle conditional expectation with the integrability side conditions discharged automatically.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (MeasureTheory.Integrable f μ) Causalean.PO.POCFBundle.condExpGiven_add'._auto_1 → autoParam (MeasureTheory.Integrable g μ) Causalean.PO.POCFBundle.condExpGiven_add'._auto_3 → B.condExpGiven (f + g) μ =ᵐ[μ] B.condExpGiven f μ + B.condExpGiven g μ","l":130,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_congr_ae","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"a.e. congruence for condExpGiven: if f =ᵐ g, then their bundle conditional expectations agree a.e.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, f =ᵐ[μ] g → B.condExpGiven f μ =ᵐ[μ] B.condExpGiven g μ","l":228,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_const","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"A constant is its own bundle conditional expectation: on a finite measure, averaging a constant given a counterfactual bundle returns that constant, everywhere (not merely almost everywhere).","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (k : ℝ) {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ], B.condExpGiven (fun x => k) μ = fun x => k","l":161,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_eq","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Averaging a real-valued sample-space quantity given a counterfactual bundle is exactly the standard conditional expectation given the σ-algebra that the bundle generates.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) (μ : MeasureTheory.Measure P.Ω), B.condExpGiven g μ = μ[g | B.sigma]","l":76,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_finsetSum'","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The bundle conditional expectation commutes with a finite sum, with the family-wide integrability side condition discharged automatically by fun_prop.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {ι : Type u_1} {s : Finset ι} {F : ι → P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (∀ i ∈ s, MeasureTheory.Integrable (F i) μ) Causalean.PO.POCFBundle.condExpGiven_finsetSum'._auto_1 → B.condExpGiven (∑ i ∈ s, F i) μ =ᵐ[μ] ∑ i ∈ s, B.condExpGiven (F i) μ","l":168,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_indicator_mul","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Indicator pull-out for condExpGiven: if s is B.sigma-measurable, the indicator 1_s factors out of the bundle conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {s : Set P.Ω} {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSet s → MeasureTheory.Integrable g μ → B.condExpGiven ((s.indicator fun x => 1) * g) μ =ᵐ[μ] (s.indicator fun x => 1) * B.condExpGiven g μ","l":209,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_consistency_CondIndepCFBundle","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExpBundle","a":"PO/Conditioning","t":1,"d":"Bundle product-form workhorse (analogue of POSystem.eventCondExp_of_consistency_IndepCF). Suppose a factual variable a is conditionally independent, given the σ-algebra of a bundle C, of a counterfactual bundle B, where h is a measurable function whose composite with B's joint value is integrable, and {x} is a measurable singleton in the range of a. If the factual integrand factualF times the indicator of {a = x} agrees almost everywhere with h composed with B's joint value, times the same indicator, then the bundle conditional expectation, given C, of factualF times the indicator of {a = x} factorises almost everywhere as the bundle conditional expectation of h composed with B's joint value, times the bundle conditional expectation of the indicator of {a = x}:","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (C : Causalean.PO.POCFBundle P) [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.POVar P α}, P.CondIndepCFBundle (Causalean.PO.RegimedVar.ofFactual a) B C P.μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (B.jointValue ω)) P.μ → ∀ {x : α}, MeasurableSet {x} → ((fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => h (B.jointValue ω) * a.indicator x ω) → (C.condExpGiven fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => C.condExpGiven (fun ω' => h (B.jointValue ω')) P.μ ω * C.condExpGiven (a.indicator x) P.μ ω","l":70,"f":"Causalean/PO/Conditioning/EventCondExpBundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_stronglyMeasurable_left","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Pull-out-left for condExpGiven: if f is strongly measurable w.r.t. B.sigma, then μ[f·g | B.sigma] =ᵐ f · μ[g | B.sigma].","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable g μ → B.condExpGiven (f * g) μ =ᵐ[μ] f * B.condExpGiven g μ","l":188,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_mul_of_stronglyMeasurable_right","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Pull-out-right for condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable g → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable f μ → B.condExpGiven (f * g) μ =ᵐ[μ] B.condExpGiven f μ * g","l":199,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_neg","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Negation passes through the bundle conditional expectation. No integrability is required.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, B.condExpGiven (-g) μ =ᵐ[μ] -B.condExpGiven g μ","l":155,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_smul","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Scalar homogeneity of the bundle conditional expectation. No integrability is required.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (k : ℝ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, B.condExpGiven (k • g) μ =ᵐ[μ] k • B.condExpGiven g μ","l":149,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_sub","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Linearity of the bundle conditional expectation: subtraction.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → B.condExpGiven (f - g) μ =ᵐ[μ] B.condExpGiven f μ - B.condExpGiven g μ","l":124,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_sub'","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Subtractivity of the bundle conditional expectation with the integrability side conditions discharged automatically; see POCFBundle.condExpGiven_add'.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (MeasureTheory.Integrable f μ) Causalean.PO.POCFBundle.condExpGiven_sub'._auto_1 → autoParam (MeasureTheory.Integrable g μ) Causalean.PO.POCFBundle.condExpGiven_sub'._auto_3 → B.condExpGiven (f - g) μ =ᵐ[μ] B.condExpGiven f μ - B.condExpGiven g μ","l":142,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpGiven_tower_of_le","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Tower property over an arbitrary smaller sub-σ-algebra m ≤ B.sigma. Requires the usual SigmaFinite hypothesis on the trim of μ to B.sigma.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} {m : MeasurableSpace P.Ω}, m ≤ B.sigma → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POCFBundle.sigma_le B))], μ[B.condExpGiven g μ | m] =ᵐ[μ] μ[g | m]","l":178,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system, a counterfactual bundle, two real-valued quantities on the sample space, and a measure on that sample space, the bundle conditional-expectation ratio maps each sample point to the conditional expectation of the first quantity given the bundle divided by that of the second at that sample point.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → (P.Ω → ℝ) → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":237,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio_def","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The bundle conditional-expectation ratio is, pointwise, the quotient of the two bundle conditional expectations.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g h : P.Ω → ℝ) (μ : MeasureTheory.Measure P.Ω), B.condExpRatio g h μ = fun ω => B.condExpGiven g μ ω / B.condExpGiven h μ ω","l":248,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio_eq_of_mul","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Characterisation of bundle condExpRatio via a product identity.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) {g h target : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, B.condExpGiven g μ =ᵐ[μ] B.condExpGiven h μ * target → (∀ᵐ (ω : P.Ω) ∂μ, B.condExpGiven h μ ω ≠ 0) → B.condExpRatio g h μ =ᵐ[μ] target","l":276,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.condExpRatio_of_consistency_CondIndepCFBundle","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExpBundle","a":"PO/Conditioning","t":1,"d":"Bundle ratio-form workhorse: ratio version of condExpGiven_mul_of_consistency_CondIndepCFBundle. Under the same hypotheses as that theorem — bundle-conditional independence of a from B given C, a measurable and integrable composite h ∘ B.jointValue, a measurable singleton {x}, and the consistency-on-event product identity — plus an almost-surely nonzero bundle-conditional probability of {a = x}, the conditional ratio condExpRatio (factualF · 1_{a=x}) (1_{a=x}) collapses almost everywhere to the bundle conditional mean of h ∘ B.jointValue.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (C : Causalean.PO.POCFBundle P) [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.POVar P α}, P.CondIndepCFBundle (Causalean.PO.RegimedVar.ofFactual a) B C P.μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (B.jointValue ω)) P.μ → ∀ {x : α}, MeasurableSet {x} → ((fun ω => factualF ω * a.indicator x ω) =ᵐ[P.μ] fun ω => h (B.jointValue ω) * a.indicator x ω) → (∀ᵐ (ω : P.Ω) ∂P.μ, C.condExpGiven (a.indicator x) P.μ ω ≠ 0) → C.condExpRatio (fun ω => factualF ω * a.indicator x ω) (a.indicator x) =ᵐ[P.μ] C.condExpGiven fun ω' => h (B.jointValue ω')","l":160,"f":"Causalean/PO/Conditioning/EventCondExpBundle.lean"},{"n":"Causalean.PO.POCFBundle.cons","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"For a regimed potential-outcome variable and a finite counterfactual bundle in the same potential-outcome system, the extended counterfactual bundle places that variable first and retains every original coordinate.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.POCFBundle P","l":88,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.instMeasurableSpaceType","k":"instance","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"For a potential-outcomes system, a finite bundle of regimed potential-outcome variables in that system, and a coordinate of that bundle, the measurable-space structure on the value space at that coordinate is the structure declared by the bundle.","s":"{P : Causalean.PO.POSystem} → (B : Causalean.PO.POCFBundle P) → (i : Fin B.n) → MeasurableSpace (B.type i)","l":59,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.integrable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Mathlib's totalized conditional expectation given the bundle's σ-algebra is integrable, even when the original sample-space function is not assumed integrable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable (B.condExpGiven g μ) μ","l":103,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.jointValue","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"For a finite bundle of regimed potential-outcome variables, the joint counterfactual-value map assigns to every sample point the tuple whose coordinate is that variable's potential outcome under its associated regime.","s":"{P : Causalean.PO.POSystem} → (B : Causalean.PO.POCFBundle P) → P.Ω → (i : Fin B.n) → B.type i","l":69,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.measurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditional-ratio function of two real integrands given a counterfactual bundle is measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, Measurable (B.condExpRatio g h μ)","l":260,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.measurable_jointValue","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"The joint counterfactual-value map of a bundle is measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P), Measurable B.jointValue","l":75,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.measurable_jointValue_sigma","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The bundle's joint counterfactual-value map is measurable for the very σ-algebra it generates: conditioning information always determines the values that were conditioned on. This is the generator lemma that lets automation discharge measurability of bundle-value compositions relative to B.sigma.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P), Measurable B.jointValue","l":59,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.nil","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, the empty counterfactual bundle has no coordinates.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.POCFBundle P","l":80,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POCFBundle.sigma","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system and a finite bundle of counterfactual variables, the conditioning σ-algebra of the bundle is the σ-algebra on the sample space generated by jointly observing every counterfactual value in that bundle.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCFBundle P → MeasurableSpace P.Ω","l":45,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.sigma_le","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditioning σ-algebra B.sigma is a sub-σ-algebra of the ambient σ-algebra on P.Ω.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P), B.sigma ≤ inferInstance","l":54,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a strongly measurable function w.r.t. the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpGiven g μ)","l":96,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpGiven_comap","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a function strongly measurable w.r.t. B.sigma.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpGiven g μ)","l":89,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCFBundle.stronglyMeasurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"The conditional-ratio function of two real integrands given a counterfactual bundle is strongly measurable.","s":"∀ {P : Causalean.PO.POSystem} (B : Causalean.PO.POCFBundle P) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (B.condExpRatio g h μ)","l":269,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POCSDIDSystem","k":"structure","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"A staggered-adoption DID system has a binary treatment variable at each of T periods, where at least two periods are observed, together with a real outcome variable at each period, such that the treatment nodes are pairwise distinct across periods and likewise for the outcome nodes, and no treatment node coincides with any outcome node.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":47,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, a cohort period, an outcome period, and proof that the cohort is not the first period, the group-time average treatment effect on the treated is the cohort-event conditional mean of the period-$t$ difference between the cohort and never-treated potential outcomes.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → Fin S.T → 1 ≤ ↑g → ℝ","l":276,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"The Callaway-Sant'Anna assumptions combine consistency of the underlying potential-outcome system: no unit is treated at period zero and once treated, a unit remains treated in every later period (irreversible adoption); pre-treatment outcomes do not anticipate future treatment; each cohort's mean untreated trend matches the never-treated group's mean untreated trend; each cohort and the never-treated group occur with positive, finite probability; and the cohort and never-treated potential outcomes are integrable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Prop","l":286,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Y_notin_dTargetUpTo","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"An outcome node is not among the treatment targets before any cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), ∀ k ≤ S.T, S.Y t ∉ S.dTargetUpTo k","l":204,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.Y_notin_regimeBy","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"An outcome node is not targeted by any full-horizon treatment-path regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T) (b : Fin S.T → Bool), S.Y t ∉ (S.regimeBy b).target","l":211,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.YofCohort","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, an outcome period, and a cohort period, the cohort potential-outcome function assigns each unit its outcome at that period under the treatment path that begins in the cohort period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Fin S.T → P.Ω → ℝ","l":229,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.YofNT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and an outcome period, the never-treated potential-outcome function assigns each unit its outcome at that period under the regime that never treats.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → ℝ","l":235,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.att_csdid","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"Callaway--Sant'Anna group-time DID identification of ATT(g, t). Under the group-time assumptions — consistency, no-anticipation, and group-time parallel trends, for a treatment cohort g that starts treatment no earlier than period 1 and a calendar period t no earlier than g, the group-time average treatment effect on the treated equals the difference between the cohort-g mean outcome change from the period before g to period t and the corresponding mean outcome change for the never-treated group:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), S.Assumptions → ∀ (g t : Fin S.T) (hg : 1 ≤ ↑g), ↑g ≤ ↑t → S.ATT g t hg = (Causalean.PO.eventCondExp P.μ (S.cohortEvent g hg) fun ω => S.factualY t ω - S.factualY (S.predFin g hg) ω) - Causalean.PO.eventCondExp P.μ S.neverTreatedEvent fun ω => S.factualY t ω - S.factualY (S.predFin g hg) ω","l":463,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.cohortEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, a cohort period, and proof that it is not the first period, the cohort event is the set of units untreated immediately before that period and treated in that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → 1 ≤ ↑g → Set P.Ω","l":257,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, a period, and a binary treatment value, the period-specific treatment event is the set of units whose observed treatment at that period equals that value.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Bool → Set P.Ω","l":87,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dTargetUpTo","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, the treatment-target-set function maps each cutoff to the empty set at cutoff zero and the treatment nodes from periods strictly before the cutoff at every positive cutoff.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → ℕ → Finset P.V","l":114,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dTargetUpTo_mem_iff","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Membership in the target set is exactly being a treatment node before the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), ∀ k ≤ S.T, ∀ (v : P.V), v ∈ S.dTargetUpTo k ↔ ∃ i, ↑i < k ∧ v = S.D i","l":123,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a period, the binary treatment potential-outcome variable is the treatment node for that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.POVar P Bool","l":71,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a period, the factual treatment function assigns each unit its observed treatment at that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → Bool","l":79,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a period, the factual outcome function assigns each unit its observed outcome at that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → P.Ω → ℝ","l":83,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_cohortEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Each cohort event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (g : Fin S.T) (hg : 1 ≤ ↑g), MeasurableSet (S.cohortEvent g hg)","l":263,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Each period-specific treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T) (b : Bool), MeasurableSet (S.dEvent s b)","l":102,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurableSet_neverTreatedEvent","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The never-treated event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P), MeasurableSet S.neverTreatedEvent","l":272,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_YofCohort","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Cohort potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t g : Fin S.T), Measurable (S.YofCohort t g)","l":241,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_YofNT","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"Never-treated potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), Measurable (S.YofNT t)","l":246,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The observed treatment at each period is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T), Measurable (S.factualD s)","l":92,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The observed outcome at each period is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (s : Fin S.T), Measurable (S.factualY s)","l":97,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.neverTreatedEvent","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, the never-treated event is the set of units whose observed treatment is false in every period.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Set P.Ω","l":268,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.predFin","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, a cohort period, and proof that the cohort is not the first period, the predecessor period is the period immediately before that cohort.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (g : Fin S.T) → 1 ≤ ↑g → Fin S.T","l":251,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regNT","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, the never-treated regime fixes treatment to false in every period.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POCSDIDSystem P → Causalean.PO.Regime P.V P.X","l":197,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regOf","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a cohort period, the cohort treatment regime leaves all earlier periods untreated and fixes treatment to true in that cohort period and every later period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.Regime P.V P.X","l":191,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regUpToAux","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system, a binary treatment path over all periods, a cutoff, and proof that the cutoff does not exceed the horizon, the partial treatment-path regime together with its target-set identity fixes treatments before the cutoff according to that path and has exactly the treatment nodes before the cutoff as its target.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (Fin S.T → Bool) → (k : ℕ) → k ≤ S.T → { r // r.target = S.dTargetUpTo k }","l":145,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regimeBy","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a binary treatment path over all periods, the full-horizon treatment regime fixes every period's treatment to the corresponding value on that path.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → (Fin S.T → Bool) → Causalean.PO.Regime P.V P.X","l":179,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.regimeBy_target_eq","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The target of a full-horizon treatment-path regime is the full set of treatment nodes.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (b : Fin S.T → Bool), (S.regimeBy b).target = S.dTargetUpTo S.T","l":186,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":1,"d":"For a staggered-adoption DID system and a period, the real-valued outcome potential-outcome variable is the outcome node for that period.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POCSDIDSystem P) → Fin S.T → Causalean.PO.POVar P ℝ","l":75,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar_v_notin_regNT_target","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The outcome variable at a period is not targeted by the never-treated regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t : Fin S.T), (S.yVar t).v ∉ S.regNT.target","l":222,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.POCSDIDSystem.yVar_v_notin_regOf_target","k":"theorem","m":"Causalean.PO.ID.Exact.CSDID","a":"PO/ID/Exact","t":2,"d":"The outcome variable at a period is not targeted by the cohort regime.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POCSDIDSystem P) (t g : Fin S.T), (S.yVar t).v ∉ (S.regOf g).target","l":217,"f":"Causalean/PO/ID/Exact/CSDID.lean"},{"n":"Causalean.PO.PODIDSystem","k":"structure","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"A two-period DID system packages a treatment node whose value space is identified with the booleans, together with a pre-period outcome node and a post-period outcome node each of whose value spaces is identified with the real line; the treatment node is required to be distinct from the pre-period outcome node and distinct from the post-period outcome node.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":34,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.ATT","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the average treatment effect on the treated is the mean, conditional on observed treatment, of the difference between each treated unit's post-period potential outcomes under treatment and no treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → ℝ","l":118,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"Assumptions for two-period difference-in-differences identification of the ATT (def:po-did-assumptions). In words: the observed outcomes coincide with the realized-arm potential outcomes; in the pre-period the treated and control groups have the same potential outcome regardless of treatment; and absent treatment the two groups would have changed in parallel between the two periods. The treated group and the control group each occur with positive probability, and the control pre-period outcome, the control post-period outcome, and the treated post-period outcome are integrable, so the group-conditional means are well-defined and finite.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Prop","l":124,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Y0ofD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system and a binary treatment arm, the pre-period potential-outcome function gives each unit's pre-period outcome when treatment is fixed to that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → P.Ω → ℝ","l":66,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.Y1ofD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system and a binary treatment arm, the post-period potential-outcome function gives each unit's post-period outcome when treatment is fixed to that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → P.Ω → ℝ","l":71,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.att_did","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"Under the two-period DID assumptions — consistency, no-anticipation, parallel trends, and positive-probability, integrable treatment and control groups, the average treatment effect on the treated equals the difference between the treated group's mean pre-to-post outcome change and the control group's mean pre-to-post outcome change.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), S.Assumptions → S.ATT = (Causalean.PO.eventCondExp P.μ (S.dEvent true) fun ω => S.factualY₁ ω - S.factualY₀ ω) - Causalean.PO.eventCondExp P.μ (S.dEvent false) fun ω => S.factualY₁ ω - S.factualY₀ ω","l":173,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system and a binary treatment arm, the treatment event is the set of units whose observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Bool → Set P.Ω","l":88,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the binary treatment potential-outcome variable is its treatment node with values represented as false or true.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P Bool","l":54,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the factual treatment function assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → Bool","l":76,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualY₀","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the factual pre-period outcome function assigns each unit its observed pre-period outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → ℝ","l":80,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.factualY₁","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the factual post-period outcome function assigns each unit its observed post-period outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → P.Ω → ℝ","l":84,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"Each observed treatment-arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":114,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_Y0ofD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The pre-period potential outcome under a fixed treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), Measurable (S.Y0ofD d)","l":92,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_Y1ofD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The post-period potential outcome under a fixed treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P) (d : Bool), Measurable (S.Y1ofD d)","l":97,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualD","l":102,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualY₀","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed pre-period outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualY₀","l":106,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.measurable_factualY₁","k":"theorem","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":2,"d":"The observed post-period outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.PODIDSystem P), Measurable S.factualY₁","l":110,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.y0Var","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the real-valued pre-period outcome potential-outcome variable is its pre-period outcome node.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P ℝ","l":58,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODIDSystem.y1Var","k":"def","m":"Causalean.PO.ID.Exact.DID","a":"PO/ID/Exact","t":1,"d":"For a two-period DID system, the real-valued post-period outcome potential-outcome variable is its post-period outcome node.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.PODIDSystem P → Causalean.PO.POVar P ℝ","l":62,"f":"Causalean/PO/ID/Exact/DID.lean"},{"n":"Causalean.PO.PODTRSystem","k":"structure","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"A finite-horizon dynamic treatment-regime system packages the variables for sequential potential-outcome identification: a stage-indexed state history observed before each treatment, the treatment chosen at each stage whose value space is identified with a common treatment alphabet across stages, and a terminal outcome whose value space is identified with the real line, subject to the state nodes being pairwise distinct across stages, the treatment nodes being pairwise distinct across stages, and no state, treatment, or outcome node coinciding with another.","s":"Causalean.PO.POSystem → (n : ℕ) → (δ : Type) → (γ : Fin n → Type) → [inst : MeasurableSpace δ] → [MeasurableSingletonClass δ] → [(k : Fin n) → MeasurableSpace (γ k)] → Type (max u_1 u_2)","l":53,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"Sequential backdoor assumptions for dynamic-treatment-regime identification at a general horizon n: potential-outcome consistency for the ambient system; per-sequence sequential exchangeability, i.e. at each stage the treatment is conditionally independent of the counterfactual terminal outcome under the treatment sequence given the history observed up to that stage; pointwise positivity of the stagewise propensity given the same history, almost surely; and integrability of the counterfactual terminal outcome under every treatment sequence together with integrability of the factual terminal outcome.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":336,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.S_of","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, a treatment sequence, and a stage, the stage counterfactual-state function assigns each unit its state at that stage under interventions fixing the preceding treatments to the sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → (k : Fin n) → P.Ω → γ k","l":225,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.Y_of","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the terminal counterfactual-outcome function assigns each unit its terminal outcome under the full regime fixing treatments to that sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → P.Ω → ℝ","l":219,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.adjustedDtr","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the observable adjusted dynamic-regime functional is the probability-measure integral of the outermost backward regression when at least one stage exists, and is zero at a zero horizon.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℝ","l":424,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":1,"d":"General-n CDTR backdoor identification. Under the dynamic-treatment-regime identification assumptions, for a positive horizon n, the σ(historyBundle 0)-conditional mean of Y(dbar) equals the outermost iterated-CE functional innerReg dbar (n - 1).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), (S.historyBundle 0 hn).condExpGiven (S.Y_of dbar) =ᵐ[P.μ] S.innerReg dbar (n - 1)","l":96,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_base","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Induction","a":"PO/ID/Exact/DTR","t":1,"d":"Base case of the backward induction for a dynamic treatment regime. Under the dynamic-treatment-regime identification assumptions — consistency and stage-wise sequential exchangeability/overlap, provided the horizon n is positive, the depth-zero adjusted-regression functional, multiplied by the indicator that the observed treatment matches the target regime dbar through stage n-1, agrees almost everywhere with that same indicator multiplied by the conditional mean of the regime outcome given the treatment-and-covariate history through stage n-1.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), (fun ω => S.innerReg dbar 0 ω * S.indD dbar (n - 1) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - 1) ω * (S.historyBundle (n - 1) (Nat.sub_lt hn Nat.one_pos)).condExpGiven (S.Y_of dbar) P.μ ω","l":64,"f":"Causalean/PO/ID/Exact/DTR/Induction.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_iter","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":2,"d":"Backward-induction iterate combining cdtr_base (j = 0) with cdtr_step (j → j + 1). At depth j, asserts that innerReg dbar j multiplied by the partial indicator product indD dbar (n - 1 - j) agrees a.e. with the conditional expectation (historyBundle (n - 1 - j)).condExpGiven (Y_of dbar) multiplied by the same partial indicator.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), ∀ j < n, (fun ω => S.innerReg dbar j ω * S.indD dbar (n - 1 - j) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - 1 - j) ω * (S.historyBundle (n - 1 - j) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_iter._proof_1 hn j a)).condExpGiven (S.Y_of dbar) P.μ ω","l":45,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_step","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Induction","a":"PO/ID/Exact/DTR","t":2,"d":"The induction step moves the dynamic-regime cancellation identity one stage outward.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (j : ℕ) (hj : j + 1 < n) (hk : n - j - 2 < n), MeasureTheory.Integrable (S.innerReg dbar j) P.μ → ((fun ω => S.innerReg dbar j ω * S.indD dbar (n - j - 1) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - j - 1) ω * (S.historyBundle (n - j - 1) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_step._proof_1 j hj a)).condExpGiven (S.Y_of dbar) P.μ ω) → (fun ω => S.innerReg dbar (j + 1) ω * S.indD dbar (n - j - 2) ω) =ᵐ[P.μ] fun ω => S.indD dbar (n - j - 2) ω * (S.historyBundle (n - j - 2) hk).condExpGiven (S.Y_of dbar) P.μ ω","l":404,"f":"Causalean/PO/ID/Exact/DTR/Induction.lean"},{"n":"Causalean.PO.PODTRSystem.cdtr_strong","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.StrongCancellation","a":"PO/ID/Exact/DTR","t":1,"d":"Joint inductive invariant for the strengthened cancellation. Consider a dynamic-treatment-regime system for which the identifying assumption bundle holds, fix a positive number of decision stages, and fix a treatment history dbar. Then for every stage index j below the horizon, the partial regression term innerReg dbar j agrees almost surely with the product of the treatment-regime indicator at the mirrored stage n - 1 - j and the conditional expectation of the outcome given the history up to that stage, and this term is integrable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ) (hn : 0 < n), ∀ j < n, (S.innerReg dbar j =ᵐ[P.μ] fun ω => S.indD dbar (n - 1 - j) ω * (S.historyBundle (n - 1 - j) (Decidable.byContradiction fun a => Causalean.PO.PODTRSystem.cdtr_strong._proof_1 hn j a)).condExpGiven (S.Y_of dbar) P.μ ω) ∧ MeasureTheory.Integrable (S.innerReg dbar j) P.μ","l":90,"f":"Causalean/PO/ID/Exact/DTR/StrongCancellation.lean"},{"n":"Causalean.PO.PODTRSystem.cfYBundle","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the counterfactual-outcome bundle is the singleton bundle containing the terminal potential outcome under that sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → Causalean.PO.POCFBundle P","l":327,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a stage, the treatment potential-outcome variable is that stage's treatment node with the common treatment value space.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Fin n → Causalean.PO.POVar P δ","l":109,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dtrEffect","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the dynamic-regime mean potential outcome is the probability-measure expectation of the terminal potential outcome under that sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℝ","l":418,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.dtr_backdoor","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Main","a":"PO/ID/Exact/DTR","t":1,"d":"General-n integrated DTR backdoor identification. Under the dynamic-treatment-regime identification assumptions, for a positive horizon n, the mean potential outcome E[Y(dbar)] equals the integral of the outermost adjusted functional innerReg dbar (n - 1).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ), 0 < n → S.dtrEffect dbar = S.adjustedDtr dbar","l":120,"f":"Causalean/PO/ID/Exact/DTR/Main.lean"},{"n":"Causalean.PO.PODTRSystem.factualAgrees_regime","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"General multi-target consistency: every ω in the full agreement event factually agrees with S.regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), ∀ ω ∈ {ω | ∀ (i : Fin n), S.factualD i ω = dbar i}, P.FactualAgrees (S.regime dbar) ω","l":387,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.factualAgrees_regimeUpTo","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Helper: for each k ≤ n, FactualAgrees for regimeUpTo dbar k holds on the event \"factualD i = dbar i for all i.val < k\".","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (h : k ≤ n) (ω : P.Ω), (∀ (i : Fin n), ↑i < k → S.factualD i ω = dbar i) → P.FactualAgrees (S.regimeUpTo dbar k h) ω","l":361,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a stage, the factual treatment function assigns each unit its observed treatment at that stage.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Fin n → P.Ω → δ","l":232,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.factualS","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a stage, the factual state function assigns each unit its observed state at that stage.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (k : Fin n) → P.Ω → γ k","l":242,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, the factual terminal-outcome function assigns each unit its observed terminal outcome.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → P.Ω → ℝ","l":237,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.hb_jointValue_step_eq","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Key algebraic identity: the stage-k joint value factors through the stage-(k+1) joint value via hb_step_proj.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), (S.historyBundle k (Nat.lt_of_succ_lt h)).jointValue = S.hb_step_proj k h ∘ (S.historyBundle (k + 1) h).jointValue","l":66,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.hb_step_proj","k":"def","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, a stage index, and proof that the next stage exists, the projection from the next-stage history vector to the current- stage history vector drops the newly added next state and current treatment coordinates.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → (S : Causalean.PO.PODTRSystem P n δ γ) → (k : ℕ) → (h : k + 1 < n) → ((i : Fin (S.historyBundle (k + 1) h).n) → (S.historyBundle (k + 1) h).type i) → (j : Fin (S.historyBundle k (Nat.lt_of_succ_lt h)).n) → (S.historyBundle k (Nat.lt_of_succ_lt h)).type j","l":44,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, a stage cutoff, and proof that the cutoff is below the horizon, the history bundle collects the factual states and treatments observed before that stage together with the factual state at the stage.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (k : ℕ) → k < n → Causalean.PO.POCFBundle P","l":305,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle_sigma_mono","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Iterated σ-algebra monotonicity: σ(history j) ≤ σ(history k) for j ≤ k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (j k : ℕ) (hjk : j ≤ k) (hk : k < n), (S.historyBundle j (lt_of_le_of_lt hjk hk)).sigma ≤ (S.historyBundle k hk).sigma","l":85,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.historyBundle_sigma_mono_step","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"One-step σ-algebra inclusion: σ(history k) ≤ σ(history (k+1)).","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), (S.historyBundle k (Nat.lt_of_succ_lt h)).sigma ≤ (S.historyBundle (k + 1) h).sigma","l":74,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the joint treatment-agreement indicator function maps each cutoff to one at cutoff zero and the preceding indicator times the next treatment-agreement indicator at a positive cutoff, provided that stage exists, otherwise the preceding indicator.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℕ → P.Ω → ℝ","l":273,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.indD_eq_indicator_event","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"Event expression for indD: as a set-indicator of the agreement event.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), ∀ k ≤ n, S.indD dbar k = {ω | ∀ (i : Fin n), ↑i < k → S.factualD i ω = dbar i}.indicator fun x => 1","l":131,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_eq_zero_or_one","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k ω ∈ {0, 1} for every ω and every k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (ω : P.Ω), S.indD dbar k ω = 0 ∨ S.indD dbar k ω = 1","l":105,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_factor_split","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":1,"d":"For any stage k within the horizon n, the indicator that a unit's observed treatment path matches the regime dbar through stage k+1 factors as the indicator through stage k times the indicator that the observed treatment at stage k equals dbar's value at stage k: indD dbar (k+1) = indD dbar k · 1_{D k = dbar k}.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (hk : k < n), S.indD dbar (k + 1) = fun ω => S.indD dbar k ω * (S.dVar ⟨k, hk⟩).indicator (dbar ⟨k, hk⟩) ω","l":188,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k is bounded by 1, hence integrable for finite μ.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) [MeasureTheory.IsFiniteMeasure P.μ], MeasureTheory.Integrable (S.indD dbar k) P.μ","l":121,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.indD_mul_Y_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"indD dbar k · Y(dbar) is integrable, bounded by |Y(dbar)|.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ), MeasureTheory.Integrable (S.Y_of dbar) P.μ → MeasureTheory.Integrable (fun ω => S.indD dbar k ω * S.Y_of dbar ω) P.μ","l":397,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.innerReg","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the backward adjusted-regression function maps its recursion index to the final-history conditional-expectation ratio when the horizon is positive, and zero otherwise, then the analogous recursively defined earlier-history ratio when the indicated stage exists, and the preceding value otherwise.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → ℕ → P.Ω → ℝ","l":371,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.innerReg_integrable","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.StrongCancellation","a":"PO/ID/Exact/DTR","t":2,"d":"innerReg dbar j is integrable for every j < n.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] [inst_3 : StandardBorelSpace P.Ω] [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] (S : Causalean.PO.PODTRSystem P n δ γ), S.Assumptions → ∀ (dbar : Fin n → δ), ∀ j < n, MeasureTheory.Integrable (S.innerReg dbar j) P.μ","l":495,"f":"Causalean/PO/ID/Exact/DTR/StrongCancellation.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_S_of","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each stage counterfactual state under the earlier treatment interventions is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : Fin n), Measurable (S.S_of dbar k)","l":252,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_Y_of","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The terminal counterfactual outcome under a treatment sequence is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), Measurable (S.Y_of dbar)","l":247,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each observed treatment process is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : Fin n), Measurable (S.factualD k)","l":258,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualD_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"factualD k' is (historyBundle m).sigma-measurable for k'.val < m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' < m → Measurable (S.factualD k')","l":246,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualD_sigma_history_last","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The last treatment recorded in a history bundle is measurable with respect to that history bundle's σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m + 1 < n), Measurable (S.factualD ⟨m, Nat.lt_of_succ_lt hm⟩)","l":314,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"Each observed state process is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : Fin n), Measurable (S.factualS k)","l":266,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"factualS k' is (historyBundle m).sigma-measurable for k'.val ≤ m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' ≤ m → Measurable (S.factualS k')","l":215,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS_sigma_history_last","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The state observed at the stage a history bundle ends on is measurable with respect to that history bundle's σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n), Measurable (S.factualS ⟨m, hm⟩)","l":296,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualS_sigma_history_pred","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The state observed one stage before the stage a history bundle ends on is measurable with respect to that history bundle's σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m + 1 < n), Measurable (S.factualS ⟨m, Nat.lt_of_succ_lt hm⟩)","l":304,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The observed terminal outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ), Measurable S.factualY","l":262,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_hb_step_proj","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The projection from a stage's extended history to the previous history is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (k : ℕ) (h : k + 1 < n), Measurable (S.hb_step_proj k h)","l":52,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_indD","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The joint treatment-agreement indicator up to any cutoff is measurable.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ), Measurable (S.indD dbar k)","l":289,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.measurable_innerReg","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"innerReg dbar j is measurable for every j.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (j : ℕ), Measurable (S.innerReg dbar j)","l":408,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.regime","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system and a treatment sequence, the full treatment regime fixes every stage's treatment to the corresponding sequence value.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → Causalean.PO.Regime P.V P.X","l":211,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeTarget","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, the regime-target-set function maps each nonnegative cutoff to the empty set at cutoff zero and the treatment nodes at stages strictly before the cutoff at every positive cutoff.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → ℕ → Finset P.V","l":121,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeTarget_mem_iff","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic-treatment-regime system S, a variable belongs to the regime target built up to stage k if and only if it is the treatment node of some earlier stage i < k.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ), ∀ k ≤ n, ∀ (v : P.V), v ∈ S.regimeTarget k ↔ ∃ i, ↑i < k ∧ v = S.D i","l":133,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpTo","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, a treatment sequence, a cutoff, and proof that the cutoff does not exceed the horizon, the partial treatment regime fixes exactly the treatments before the cutoff to that sequence.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → (Fin n → δ) → (k : ℕ) → k ≤ n → Causalean.PO.Regime P.V P.X","l":198,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpToAux","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, a treatment sequence, a cutoff, and proof that the cutoff does not exceed the horizon, the partial regime together with its target-set identity fixes the earlier treatments to that sequence and records that its target contains exactly those earlier treatment nodes.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → (S : Causalean.PO.PODTRSystem P n δ γ) → (Fin n → δ) → (k : ℕ) → k ≤ n → { r // r.target = S.regimeTarget k }","l":159,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.regimeUpTo_target_eq","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":2,"d":"The target of the partial treatment regime is the standalone target set for the cutoff.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ) (k : ℕ) (h : k ≤ n), (S.regimeUpTo dbar k h).target = S.regimeTarget k","l":205,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.stronglyMeasurable_indD_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The joint-agreement indicator indD dbar m' is (historyBundle m).sigma-strongly-measurable for m' ≤ m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (dbar : Fin n → δ), ∀ m' ≤ m, MeasureTheory.StronglyMeasurable (S.indD dbar m')","l":335,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.stronglyMeasurable_indicator_dVar_sigma_history","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The real-valued indicator dVar ⟨k', hk⟩.indicator (dbar ⟨k', hk⟩) is (historyBundle m).sigma-strongly-measurable for k' < m.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m < n) (k' : Fin n), ↑k' < m → ∀ (x : δ), MeasureTheory.StronglyMeasurable ((S.dVar k').indicator x)","l":271,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.stronglyMeasurable_indicator_dVar_sigma_history_last","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The indicator that the last treatment recorded in a history bundle equals a given value is strongly measurable with respect to that history bundle's σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (m : ℕ) (hm : m + 1 < n) (x : δ), MeasureTheory.StronglyMeasurable ((S.dVar ⟨m, Nat.lt_of_succ_lt hm⟩).indicator x)","l":324,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODTRSystem.toDTREstimationSystem","k":"def","m":"Causalean.Estimation.DTR.Constructor","a":"Estimation/DTR","t":1,"d":"For a two-stage potential-outcome system on a standard Borel sample space with a finite measure, measurable treatment values that have measurable singletons, and measurable state spaces at both stages, a two-stage dynamic-treatment-regime system, and a fixed two-stage target treatment regime, suppose the conditional probability of the target treatment at stage 0, given the stage-0 history, lies strictly between zero and one almost everywhere, and the analogous conditional probability at stage 1, given the stage-1 history, also lies strictly between zero and one almost everywhere. The dynamic-treatment-regime estimation system is then constructed from those data.","s":"{P : Causalean.PO.POSystem} → {δ : Type} → {γ : Fin 2 → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin 2) → MeasurableSpace (γ k)] → [inst_3 : StandardBorelSpace P.Ω] → [inst_4 : MeasureTheory.IsFiniteMeasure P.μ] → (S : Causalean.PO.PODTRSystem P 2 δ γ) → (dbar : Fin 2 → δ) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ∧ (S.historyBundle 0 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨0, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω < 1) → (∀ᵐ (ω : P.Ω) ∂P.μ, 0 < (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω ∧ (S.historyBundle 1 (of_decide_eq_true (id (Eq.refl true)))).condExpGiven ((S.dVar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩).indicator (dbar ⟨1, of_decide_eq_true (id (Eq.refl true))⟩)) P.μ ω < 1) → Causalean.Estimation.DTR.DTREstimationSystem P δ γ","l":123,"f":"Causalean/Estimation/DTR/Constructor.lean"},{"n":"Causalean.PO.PODTRSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.DTR.Setup","a":"PO/ID/Exact/DTR","t":1,"d":"For a dynamic treatment-regime system, the terminal-outcome potential-outcome variable is its terminal outcome node represented on the real line.","s":"{P : Causalean.PO.POSystem} → {n : ℕ} → {δ : Type} → {γ : Fin n → Type} → [inst : MeasurableSpace δ] → [inst_1 : MeasurableSingletonClass δ] → [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] → Causalean.PO.PODTRSystem P n δ γ → Causalean.PO.POVar P ℝ","l":115,"f":"Causalean/PO/ID/Exact/DTR/Setup.lean"},{"n":"Causalean.PO.PODTRSystem.yVar_notMem_regime","k":"theorem","m":"Causalean.PO.ID.Exact.DTR.Helpers","a":"PO/ID/Exact/DTR","t":2,"d":"The outcome node Y is not a target of S.regime dbar.","s":"∀ {P : Causalean.PO.POSystem} {n : ℕ} {δ : Type} {γ : Fin n → Type} [inst : MeasurableSpace δ] [inst_1 : MeasurableSingletonClass δ] [inst_2 : (k : Fin n) → MeasurableSpace (γ k)] (S : Causalean.PO.PODTRSystem P n δ γ) (dbar : Fin n → δ), S.yVar.v ∉ (S.regime dbar).target","l":451,"f":"Causalean/PO/ID/Exact/DTR/Helpers.lean"},{"n":"Causalean.PO.PODynLATESystem","k":"structure","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"A two-period dynamic instrumental-variable (LATE) model in the potential-outcome framework. A unit is observed over two periods: baseline covariates S₀, then in period 1 a binary encouragement / instrument Z₁ whose value space is identified with the booleans and the treatment D₁ it shifts, likewise identified with the booleans; an intermediate state S₁; then in period 2 a second encouragement Z₂ and treatment D₂, each identified with the booleans; and finally a real-valued outcome Y, identified with the real line. Sequential instrument variation identifies dynamic complier treatment effects (def:po-dynamic-late-system), and all seven nodes are required to be pairwise distinct.","s":"Causalean.PO.POSystem → (γ₀ γ₁ : Type) → [MeasurableSpace γ₀] → [MeasurableSpace γ₁] → Type (max u_1 u_2)","l":48,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"The dynamic instrumental-variable / LATE assumptions for the two-period system (def:po-dynamic-late-assumptions): potential-outcome consistency for the ambient system; conditional independence of the stage-1 encouragement from the counterfactual outcome and treatment path given the baseline state, and conditional independence of the stage-2 encouragement from its counterfactual outcome and treatment given the full stage-2 history; positive stage-1 and positive stage-2 propensities almost surely; a positive stage-1 and a positive stage-2 first-stage effect of encouragement on treatment, almost surely; one-sided noncompliance, whereby each stage's counterfactual treatment never exceeds its encouragement; an exclusion restriction under which the outcome depends on the encouragements only through the resulting treatments, the first-stage treatment does not depend on the second-period encouragement, and the first-period encouragement does not depend on the second-period encouragement; and integrability of the counterfactual outcome under every fixed treatment vector, under every fixed encouragement vector, and under every fixed second-period encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → [StandardBorelSpace P.Ω] → [MeasureTheory.IsFiniteMeasure P.μ] → Prop","l":641,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Compatibility projection for older call sites: factual outcome integrability is derived from consistency plus integrability of the two YofZ2 cells.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) [inst_2 : StandardBorelSpace P.Ω], S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":731,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The two treatment nodes are distinct.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D1 ≠ S.D2","l":146,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Y is distinct from each treatment.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D1 ≠ S.Y","l":151,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1ofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the first-treatment potential-outcome function under encouragement assigns each unit its first treatment when both encouragements are fixed to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Bool","l":257,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D1ofZ_eq_factualD1_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₁ = z 0}, D₁ under encouragementRegime z agrees with the factual D₁.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.D1ofZ z ω = S.factualD1 ω","l":202,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.D2_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment node is distinct from the outcome node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.D2 ≠ S.Y","l":156,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the second-treatment potential-outcome function under encouragement assigns each unit its second treatment when both encouragements are fixed to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Bool","l":267,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a binary second-stage encouragement, the second-treatment potential-outcome function under a stage-two intervention assigns each unit its second treatment when only the second encouragement is fixed.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → P.Ω → Bool","l":279,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ2_eq_factualD2_on_z2Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₂ = z₂}, D₂(Z₁, z₂) = factualD2 pointwise.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z₂ : Bool) {ω : P.Ω}, S.factualZ2 ω = z₂ → S.D2ofZ2 z₂ ω = S.factualD2 ω","l":67,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.D2ofZ_eq_D2ofZ2_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Second-stage treatment does not depend on first-stage regime detail once the actual first-period instrument matches the counterfactual one. At any outcome at which the factual first-period instrument equals the first coordinate of the instrument path z, the potential second-period treatment under the encouragement regime z coincides pointwise with the potential second-period treatment under the regime that fixes only the second-period instrument to z 1.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.D2ofZ z ω = S.D2ofZ2 (z 1) ω","l":188,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the joint treatment potential-outcome function assigns each unit the two-vector formed by its first and second treatment under that encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → Fin 2 → Bool","l":273,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZEq","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the dynamic-complier event is the set of units whose treatment vector under that encouragement equals that treatment vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → Set P.Ω","l":571,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ_in_two_values","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"For d ≼ z and one-sided noncompliance, D(z) ∈ {d, 0} a.s. when d is one of the two single-period treatment vectors (1,0) or (0,1), because the only d' ≼ z with d' ≠ d and d' ≠ 0 would have to flip a coordinate that one-sided rules out. Used to decompose Y(D(z)) on the encouragement event into the two-piece sum Y(d)·1_{D(z)=d} + Y(0)·1_{D(z)=0}.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ d ω = d ∨ S.DofZ d ω = ![false, false]","l":182,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.DofZ_zero_eq_zero","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, the zero-encouragement counterfactual treatment is identically zero a.s.: D(0) = 0 a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ ![false, false] ω = ![false, false]","l":153,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the dynamic local average treatment effect is the mean contrast between the outcome under that vector and under no treatment among its dynamic compliers, with a zero denominator totalized by real-valued division.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → ℝ","l":581,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Preceq","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a first binary vector and a second binary vector, the coordinatewise order relation holds exactly when each coordinate of the first is no greater than the corresponding coordinate of the second.","s":"(Fin 2 → Bool) → (Fin 2 → Bool) → Prop","l":634,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a treatment vector, the outcome potential-outcome function under treatment assigns each unit its outcome when both treatments are fixed to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":288,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the outcome potential-outcome function under encouragement assigns each unit its outcome when both encouragements are fixed to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":294,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_decomposition","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, the encouragement-regime outcome under encouragement d ∈ {(1,0), (0,1)} decomposes pointwise as Y(d)·1_{D(d)=d} + Y(0)·1_{D(d)=0} a.s.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.YofDofZ d =ᵐ[P.μ] fun ω => S.YofD d ω * (S.DofZEq d d).indicator (fun x => 1) ω + S.YofD ![false, false] ω * (S.DofZEq d ![false, false]).indicator (fun x => 1) ω","l":240,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_eq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a two-target encouragement is the counterfactual outcome variable evaluated at the encouragement regime induced by the encouragement values.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), S.YofDofZ z = S.yVar.cf (S.encouragementRegime z)","l":306,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_eq_YofD_on_DofZEq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On the event {D(z) = d}, the encouragement-regime outcome Y(D(z)) agrees pointwise with the treatment-regime outcome Y(d).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ z ω = d → S.YofDofZ z ω = S.YofD d ω","l":81,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_eq_YofZ2_on_z1Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₁ = z 0}, Y under encouragementRegime z agrees with Y under encZ2Regime (z 1).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool) {ω : P.Ω}, S.factualZ1 ω = z 0 → S.YofDofZ z ω = S.YofZ2 (z 1) ω","l":179,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.YofDofZ_zero_ae_eq_YofD_zero","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"A.s. Y(D(0)) = Y(0), the zero-encouragement composition collapse.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → S.YofDofZ ![false, false] =ᵐ[P.μ] S.YofD ![false, false]","l":174,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.YofZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a binary second-stage encouragement, the outcome potential-outcome function under a stage-two intervention assigns each unit its outcome when only the second encouragement is fixed.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → P.Ω → ℝ","l":313,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.YofZ2_eq_factualY_on_z2Event","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Consistency","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"On {Z₂ = z₂}, Y(D₁, D₂(Z₁, z₂)) = factualY pointwise.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [inst_2 : StandardBorelSpace P.Ω] [inst_3 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z₂ : Bool) {ω : P.Ω}, S.factualZ2 ω = z₂ → S.YofZ2 z₂ ω = S.factualY ω","l":58,"f":"Causalean/PO/ID/Exact/DynamicLATE/Consistency.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_D1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Encouragements are distinct from treatments.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.D1","l":171,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The first encouragement node is distinct from the second treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.D2","l":176,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Y is distinct from each encouragement.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.Y","l":161,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z1_ne_Z2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The most-used corollary: the two encouragement nodes are distinct.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z1 ≠ S.Z2","l":141,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_D1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the first treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.D1","l":181,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_D2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the second treatment node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.D2","l":186,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.Z2_ne_Y","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second encouragement node is distinct from the outcome node.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), S.Z2 ≠ S.Y","l":166,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cCompliance_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Compliance bridge (S₀-conditional). Under the dynamic LATE identifying assumptions, for an encouragement vector z and a treatment path d with d weakly dominated coordinatewise by z, the baseline-conditional probability of the dynamic complier event D(z) = d agrees almost surely with the inner-outer observable regression cObsProb z d.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → S.historyBundle1.condExpGiven ((S.DofZEq z d).indicator fun x => 1) =ᵐ[P.μ] S.cObsProb z d","l":548,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.cLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the baseline-conditional dynamic local average treatment-effect function is the baseline-state conditional version of the dynamic LATE contrast.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":592,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cMixtureLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the baseline-conditional mixture local average treatment-effect function is the baseline-state conditional version of the mixture LATE contrast.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":623,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cMixtureLATE_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Mixture dynamic LATE Wald identity (heterogeneous in S₀). Under the dynamic LATE identifying assumptions, for an encouragement vector z, the baseline-conditional mixture dynamic LATE cMixtureLATE z agrees almost surely with the ratio of the baseline-conditional mean contrast cObsMean z − cObsMean(0,0) to the baseline-conditional noncompliance probability 1 − cObsProb(z, 0,0).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), S.cMixtureLATE z =ᵐ[P.μ] fun ω => (S.cObsMean z ω - S.cObsMean ![false, false] ω) / (1 - S.cObsProb z ![false, false] ω)","l":520,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.cObsMean","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the conditional observable mean function is the baseline-state conditional outer regression of the inner outcome regression for that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":541,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cObsProb","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the conditional observable treatment-probability function is the baseline-state conditional outer regression of the corresponding inner probability.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":552,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cOutcome_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outcome bridge (S₀-conditional). Under the dynamic LATE identifying assumptions, the baseline-conditional expectation of the counterfactual outcome Y(D(z)) for an encouragement vector z agrees almost surely with the inner-outer observable regression cObsMean z.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → S.historyBundle1.condExpGiven (S.YofDofZ z) =ᵐ[P.μ] S.cObsMean z","l":411,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.cWhenToTreatLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a treatment vector, the baseline-conditional when-to-treat local average treatment-effect function is the baseline-state conditional dynamic LATE using that vector in both roles.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":610,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cWhenToTreat_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"When-to-treat dynamic LATE Wald identity (heterogeneous in S₀). For a treatment path d equal to (1,0) or (0,1), the baseline-covariate-conditional when-to-treat effect τ_d(S₀) agrees almost surely with the ratio of the conditional observable-mean contrast cObsMean(d;S₀) − cObsMean(0;S₀) to the conditional observable compliance probability cObsProb(d,d;S₀).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.cWhenToTreatLATE d =ᵐ[P.μ] fun ω => (S.cObsMean d ω - S.cObsMean ![false, false] ω) / S.cObsProb d d ω","l":443,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.cfBundle1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the first-stage counterfactual bundle contains the outcome, first treatment, and second treatment potential outcomes under that encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.POCFBundle P","l":470,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.cfBundle2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a second-stage encouragement, the second-stage counterfactual bundle contains the outcome and second-treatment potential outcomes under the corresponding stage-two intervention.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.POCFBundle P","l":482,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.compliance_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Compliance bridge (unconditional). Given a treatment path d that is coordinatewise no greater than the instrument path z, the probability of the counterfactual dynamic complier event that the treatment path realized under instrument assignment z equals d coincides with the observable nested regression obsProb z d.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), Causalean.PO.PODynLATESystem.Preceq d z → (P.μ (S.DofZEq z d)).toReal = S.obsProb z d","l":815,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.d1UnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the regimed first-treatment variable under encouragement packages the first treatment with the regime fixing both encouragements to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P Bool","l":420,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d1Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the first binary treatment potential-outcome variable is its first treatment node.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":96,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2UnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the regimed second-treatment variable under encouragement packages the second treatment with the regime fixing both encouragements to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P Bool","l":426,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2UnderZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a second-stage encouragement, the regimed second-treatment variable under the stage-two intervention packages the second treatment with the regime fixing that encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.RegimedVar P Bool","l":438,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.d2Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the second binary treatment potential-outcome variable is its second treatment node.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":99,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encTreatRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the joint encouragement-and-treatment regime fixes all four variables to those values.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":245,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encZ2Regime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a binary second-stage encouragement, the second-stage-only encouragement regime fixes only that encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.Regime P.V P.X","l":213,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encouragementRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a binary encouragement vector, the joint encouragement regime fixes both encouragements to its two coordinates.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":193,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.encouragementRegime_disjoint_treatmentRegime","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For an encouragement vector z and a treatment path d, the encouragement regime fixing Z₁, Z₂ to z and the treatment regime fixing D₁, D₂ to d target disjoint sets of variables.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z d : Fin 2 → Bool), (S.encouragementRegime z).Disjoint (S.treatmentRegime d)","l":221,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualD1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual first-treatment function assigns each unit its observed first-stage treatment.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":118,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualD2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual second-treatment function assigns each unit its observed second-stage treatment.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":121,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualS0","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual baseline-state function assigns each unit its observed baseline covariate.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → γ₀","l":106,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualS1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual intermediate-state function assigns each unit its observed intermediate covariate.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → γ₁","l":109,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual outcome function assigns each unit its observed outcome.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → ℝ","l":124,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualZ1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual first-encouragement function assigns each unit its observed first-stage instrument.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":112,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.factualZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the factual second-encouragement function assigns each unit its observed second-stage instrument.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → P.Ω → Bool","l":115,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.historyBundle1","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the first-stage history bundle is the singleton factual baseline-state bundle.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POCFBundle P","l":446,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.historyBundle2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the second-stage history bundle contains the factual baseline state, intermediate state, first encouragement, and first treatment.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POCFBundle P","l":454,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.indD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a treatment vector, the joint treatment indicator equals one exactly when both observed treatments equal that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":495,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.indZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the joint encouragement indicator equals one exactly when both observed encouragements equal that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":500,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondD","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the inner observable treatment-probability regression is the conditional probability of that treatment vector given the second-stage history and encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → P.Ω → ℝ","l":531,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondD_mul_z1_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Stage-2 compliance ratio bridge (multiplied by 1_{Z₁=z₁}). On the event {Z₁ = z₁},","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), (fun ω => S.innerCondD z d ω * S.z1Var.indicator (z 0) ω) =ᵐ[P.μ] fun ω => S.z1Var.indicator (z 0) ω * S.d1Var.indicator (d 0) ω * S.historyBundle2.condExpGiven (Causalean.PO.PODynLATESystem.d2ofZ2EqIndicator✝ S (z 1) (d 1)) P.μ ω","l":227,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondY","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the inner observable outcome regression is the conditional mean of the factual outcome given the second-stage history and that encouragement vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → P.Ω → ℝ","l":521,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.innerCondY_mul_z1_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The stage-2 outcome ratio bridge identifies the inner outcome regression on a first-encouragement cell.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), (fun ω => S.innerCondY z ω * S.z1Var.indicator (z 0) ω) =ᵐ[P.μ] fun ω => S.z1Var.indicator (z 0) ω * S.historyBundle2.condExpGiven (S.YofZ2 (z 1)) P.μ ω","l":95,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.int_outcome_difference_identity","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The numerator identity: the difference of the encouragement-regime outcome integrals collapses to the conditional contrast on {D(d) = d}.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → ∫ (ω : P.Ω), S.YofDofZ d ω ∂P.μ - ∫ (ω : P.Ω), S.YofDofZ ![false, false] ω ∂P.μ = ∫ (ω : P.Ω) in S.DofZEq d d, S.YofD d ω - S.YofD ![false, false] ω ∂P.μ","l":277,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.integrable_factualY_of_consistency_integrable_YofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Factual Y is integrable once the two stage-2 counterfactual outcomes are integrable. The proof partitions on the factual Boolean Z₂ cell and uses PO consistency to identify Y(Z₂ = z₂) with factual Y on that cell.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) [MeasureTheory.IsFiniteMeasure P.μ], P.Consistency → (∀ (z₂ : Bool), MeasureTheory.Integrable (S.YofZ2 z₂) P.μ) → MeasureTheory.Integrable S.factualY P.μ","l":375,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurableSet_DofZEq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The dynamic complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z d : Fin 2 → Bool), MeasurableSet (S.DofZEq z d)","l":576,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D1ofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The first treatment under a two-target encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.D1ofZ z)","l":321,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D2ofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment under a two-target encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.D2ofZ z)","l":325,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_D2ofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The second treatment under a stage-2-only encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z₂ : Bool), Measurable (S.D2ofZ2 z₂)","l":337,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint counterfactual treatment vector under encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.DofZ z)","l":329,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a fixed treatment vector is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (d : Fin 2 → Bool), Measurable (S.YofD d)","l":341,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a fixed encouragement vector is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.YofDofZ z)","l":345,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_YofZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The outcome under a stage-2-only encouragement regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z₂ : Bool), Measurable (S.YofZ2 z₂)","l":349,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualD1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed first treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualD1","l":365,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualD2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed second treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualD2","l":368,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualS0","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed baseline state is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualS0","l":353,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualS1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed intermediate state is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualS1","l":356,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualY","l":371,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualZ1","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed first encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualZ1","l":359,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_factualZ2","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The observed second encouragement is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁), Measurable S.factualZ2","l":362,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_indD","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint treatment indicator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (d : Fin 2 → Bool), Measurable (S.indD d)","l":506,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.measurable_indZ","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"The joint encouragement indicator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] (S : Causalean.PO.PODynLATESystem P γ₀ γ₁) (z : Fin 2 → Bool), Measurable (S.indZ z)","l":512,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.mixtureLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the mixture local average treatment effect is the mean contrast between the outcome induced by that encouragement and the no-treatment outcome among units induced to any nonzero treatment.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":615,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.mixtureLATE_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Mixture dynamic LATE Wald identity (unconditional). Under the dynamic LATE identifying assumptions, for an encouragement vector z, the mixture dynamic local average treatment effect mixtureLATE z equals the Wald ratio of the observable mean contrast obsMean z − obsMean(0,0) to the observable noncompliance probability 1 − obsProb(z, 0,0).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), S.mixtureLATE z = (S.obsMean z - S.obsMean ![false, false]) / (1 - S.obsProb z ![false, false])","l":366,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.obsMean","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the observable mean is the probability-measure expectation of its conditional observable mean function.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":560,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.obsProb","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, an encouragement vector, and a treatment vector, the observable treatment probability is the probability-measure expectation of its conditional observable treatment-probability function.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → (Fin 2 → Bool) → ℝ","l":564,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.outcome_bridge","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"Outcome bridge (unconditional). Under the dynamic LATE identifying assumptions, the counterfactual mean outcome E[Y(D(z))] for an encouragement vector z equals the observable nested regression obsMean z.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z : Fin 2 → Bool), ∫ (ω : P.Ω), S.YofDofZ z ω ∂P.μ = S.obsMean z","l":794,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.probDofZ_eq_zero_of_not_preceq","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.Bridges","a":"PO/ID/Exact/DynamicLATE","t":2,"d":"Under one-sided noncompliance, a treatment path not below the encouragement path has zero probability.","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (z d : Fin 2 → Bool), ¬Causalean.PO.PODynLATESystem.Preceq d z → P.μ (S.DofZEq z d) = 0","l":66,"f":"Causalean/PO/ID/Exact/DynamicLATE/Bridges.lean"},{"n":"Causalean.PO.PODynLATESystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a binary treatment vector, the joint treatment regime fixes both treatments to its two coordinates.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.Regime P.V P.X","l":203,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.whenToTreatLATE","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a treatment vector, the when-to-treat local average treatment effect is the dynamic LATE using that vector both as encouragement and as treatment.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → ℝ","l":605,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.whenToTreat_wald","k":"theorem","m":"Causalean.PO.ID.Exact.DynamicLATE.WhenToTreat","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"When-to-treat dynamic LATE Wald identity (unconditional). For a treatment path d equal to (1,0) or (0,1) — treatment started in period 1 and stopped, or delayed to period 2, the when-to-treat local average treatment effect τ_d equals the Wald ratio of the observable mean contrast obsMean(d) − obsMean(0,0) to the observable compliance probability obsProb(d,d).","s":"∀ {P : Causalean.PO.POSystem} {γ₀ γ₁ : Type} [inst : MeasurableSpace γ₀] [inst_1 : MeasurableSpace γ₁] [MeasurableSingletonClass γ₀] [MeasurableSingletonClass γ₁] [inst_4 : StandardBorelSpace P.Ω] [inst_5 : MeasureTheory.IsFiniteMeasure P.μ] {S : Causalean.PO.PODynLATESystem P γ₀ γ₁}, S.Assumptions → ∀ (d : Fin 2 → Bool), d = ![true, false] ∨ d = ![false, true] → S.whenToTreatLATE d = (S.obsMean d - S.obsMean ![false, false]) / S.obsProb d d","l":331,"f":"Causalean/PO/ID/Exact/DynamicLATE/WhenToTreat.lean"},{"n":"Causalean.PO.PODynLATESystem.yUnderZ","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and an encouragement vector, the regimed outcome variable under encouragement packages the outcome with the regime fixing both encouragements to that vector.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → (Fin 2 → Bool) → Causalean.PO.RegimedVar P ℝ","l":414,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.yUnderZ2","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system and a second-stage encouragement, the regimed outcome variable under the stage-two intervention packages the outcome with the regime fixing that encouragement.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Bool → Causalean.PO.RegimedVar P ℝ","l":432,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the real-valued outcome potential-outcome variable is its outcome node.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P ℝ","l":102,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.z1Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the first binary encouragement potential-outcome variable is its first encouragement node.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":90,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.PODynLATESystem.z2Var","k":"def","m":"Causalean.PO.ID.Exact.DynamicLATE.Setup","a":"PO/ID/Exact/DynamicLATE","t":1,"d":"For a two-period dynamic LATE system, the second binary encouragement potential-outcome variable is its second encouragement node.","s":"{P : Causalean.PO.POSystem} → {γ₀ γ₁ : Type} → [inst : MeasurableSpace γ₀] → [inst_1 : MeasurableSpace γ₁] → Causalean.PO.PODynLATESystem P γ₀ γ₁ → Causalean.PO.POVar P Bool","l":93,"f":"Causalean/PO/ID/Exact/DynamicLATE/Setup.lean"},{"n":"Causalean.PO.POFrontdoorSystem","k":"structure","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"A frontdoor system packages, within an ambient potential-outcome system, a binary treatment A, a finite discrete mediator M valued in a finite type, and a real-valued outcome Y, where the treatment, mediator, and outcome are pairwise distinct variables.","s":"Causalean.PO.POSystem → (β : Type u_1) → [inst : MeasurableSpace β] → [MeasurableSingletonClass β] → [Fintype β] → [DecidableEq β] → Type (max (max u_1 u_2) u_3)","l":49,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.ATE","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the average treatment effect is the population mean of the difference between each unit's potential outcome under treatment and under control.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → ℝ","l":185,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identifying assumptions. Bundles, for a frontdoor system, consistency of the underlying potential-outcome system, the full-mediation exclusion restriction that the two-treatment-and-mediator outcome does not depend on the treatment arm, treatment–mediator exchangeability, mediator–outcome exchangeability within treatment arms, positivity of each treatment arm, positivity of the mediator within the support of its counterfactual under a treatment arm, cross-world independence of the counterfactual mediator from the full-mediation outcome, and integrability of the treatment-arm and joint treatment–mediator potential outcomes.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Prop","l":193,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.Assumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Compatibility projection for older call sites: factual outcome integrability is derived from consistency plus integrability of the finite Y(a,m) cells.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, S.Assumptions → MeasureTheory.Integrable S.factualY P.μ","l":346,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.EYgivenAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, a treatment arm, and a mediator value, the conditional outcome mean is the event-conditional mean of the observed outcome among units with that observed treatment and mediator value.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → ℝ","l":173,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.EofY_eq_frontdoorTerm","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identification (individual regime). Under the frontdoor identifying assumption bundle, for each treatment arm a, the mean potential outcome under arm a equals the frontdoor functional frontdoorTerm(a), built from the mediator distribution given a and the outcome regression on mediator and treatment.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), S.Assumptions → ∀ (a : Bool), ∫ (ω : P.Ω), S.YofA a ω ∂P.μ = S.frontdoorTerm a","l":488,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.MofA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the potential mediator assigns to each unit the mediator value it would have under that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → P.Ω → β","l":86,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.MofA_eq_factualM_on_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"On the event {A = a}, M(a)(ω) = factualM ω. Single-target consistency.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) {ω : P.Ω}, ω ∈ S.aEvent a → S.MofA a ω = S.factualM ω","l":261,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the potential outcome assigns to each unit the real outcome it would have under that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → P.Ω → ℝ","l":83,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, a treatment arm, and a mediator value, the joint potential outcome assigns to each unit the outcome under simultaneously fixing treatment and mediator to those values.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → P.Ω → ℝ","l":98,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM_eq_YofA_on_MofA_event","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Composition lemma (def:po-consistency, composition clause): on the event {M(a) = m}, Y(a, m)(ω) = Y(a)(ω).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) (m : β) {ω : P.Ω}, S.MofA a ω = m → S.YofAM a m ω = S.YofA a ω","l":356,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.YofAM_eq_factualY_on_aMEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"On the event {A = a} ∩ {M = m}, Y(a,m)(ω) = factualY ω. Multi-target consistency via cf_eq_factual_of_factualAgrees.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → ∀ (a : Bool) (m : β) {ω : P.Ω}, ω ∈ S.aEvent a ∩ S.mEvent m → S.YofAM a m ω = S.factualY ω","l":267,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.aEvent","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the treatment event is the set of units whose observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → Set P.Ω","l":137,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.aVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the binary treatment potential-outcome variable is its treatment node equipped with its binary value representation.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P Bool","l":74,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.ate_frontdoor","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"Frontdoor identification of the ATE. Under the frontdoor identifying assumption bundle, the average treatment effect equals the frontdoor estimand frontdoorATE, the difference of the frontdoor functional evaluated at the two treatment arms.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), S.Assumptions → S.ATE = S.frontdoorATE","l":700,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the factual treatment assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → Bool","l":104,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the factual mediator assigns each unit its observed mediator value.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → β","l":106,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the factual outcome assigns each unit its observed real outcome.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → P.Ω → ℝ","l":108,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.frontdoorATE","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the observable frontdoor-adjusted average treatment effect is the frontdoor-adjusted functional under treatment minus that under control.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → ℝ","l":188,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.frontdoorTerm","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the frontdoor-adjusted functional is the finite sum over mediator values of the treatment-probability-weighted conditional outcome means, weighted by the mediator distribution conditional on that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → ℝ","l":177,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integrable_factualY_of_consistency_integrable_YofAM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed outcome is integrable when all joint treatment-mediator potential outcomes are integrable and consistency holds.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β}, P.Consistency → (∀ (a : Bool) (m : β), MeasureTheory.Integrable (S.YofAM a m) P.μ) → MeasureTheory.Integrable S.factualY P.μ","l":286,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integral_eq_sum_eventCondExp_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Bool-partition tower identity for eventCondExp.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β} (g : P.Ω → ℝ), MeasureTheory.Integrable g P.μ → ∫ (ω : P.Ω), g ω ∂P.μ = ∑ a', Causalean.PO.eventCondExp P.μ (S.aEvent a') g * (P.μ (S.aEvent a')).toReal","l":434,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.integral_eq_sum_integral_MofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Partition of ∫ f along fibers of MofA a (finite codomain β).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] {S : Causalean.PO.POFrontdoorSystem P β} (f : P.Ω → ℝ), MeasureTheory.Integrable f P.μ → ∀ (a : Bool), ∫ (ω : P.Ω), f ω ∂P.μ = ∑ m, ∫ (ω : P.Ω) in {ω' | S.MofA a ω' = m}, f ω ∂P.μ","l":409,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mBundle","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the counterfactual mediator bundle consists of the mediator potential outcomes under treatment and control.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POCFBundle P","l":158,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mEvent","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a mediator value, the mediator event is the set of units whose observed mediator equals that value.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → β → Set P.Ω","l":139,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mUnderA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the mediator under that treatment regime is the mediator potential outcome represented together with the intervention that fixes treatment to that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → Causalean.PO.RegimedVar P β","l":154,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.mVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the mediator potential-outcome variable is its mediator node equipped with its finite mediator-value representation.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P β","l":76,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurableSet_aEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The factual treatment event for a treatment arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), MeasurableSet (S.aEvent a)","l":142,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurableSet_mEvent","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The factual mediator event for a mediator value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (m : β), MeasurableSet (S.mEvent m)","l":145,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_MofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Treatment-arm potential mediators are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), Measurable (S.MofA a)","l":117,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Treatment-arm potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool), Measurable (S.YofA a)","l":113,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_YofAM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"Joint treatment-mediator potential outcomes are measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β) (a : Bool) (m : β), Measurable (S.YofAM a m)","l":121,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualA","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualA","l":125,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualM","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed mediator is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualM","l":128,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":2,"d":"The observed outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [inst_1 : MeasurableSingletonClass β] [inst_2 : Fintype β] [inst_3 : DecidableEq β] (S : Causalean.PO.POFrontdoorSystem P β), Measurable S.factualY","l":131,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.pA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system and a treatment arm, the treatment-arm probability is the probability that the observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → ℝ","l":166,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.pMgivenA","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, a mediator value, and a treatment arm, the conditional mediator probability is the event-conditional probability that the observed mediator equals that value given that observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → β → Bool → ℝ","l":169,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.regimeAM","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, a treatment arm, and a mediator value, the joint treatment--mediator intervention regime fixes treatment and mediator simultaneously.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Bool → β → Causalean.PO.Regime P.V P.X","l":89,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFrontdoorSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.Frontdoor","a":"PO/ID/Exact","t":1,"d":"For a frontdoor system, the real-valued outcome potential-outcome variable is its outcome node equipped with its real-valued representation.","s":"{P : Causalean.PO.POSystem} → {β : Type u_1} → [inst : MeasurableSpace β] → [inst_1 : MeasurableSingletonClass β] → [inst_2 : Fintype β] → [inst_3 : DecidableEq β] → Causalean.PO.POFrontdoorSystem P β → Causalean.PO.POVar P ℝ","l":78,"f":"Causalean/PO/ID/Exact/Frontdoor.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem","k":"structure","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy regression-discontinuity model in the potential-outcome framework. A unit has a continuous running variable X, an above-cutoff indicator Z recording whether X exceeds the cutoff c, a binary treatment D whose take-up probability jumps — but does not jump all the way from 0 to 1 — at the cutoff, and a real outcome Y, with the outcome, cutoff indicator, and treatment nodes pairwise distinct. The cutoff effect on the outcome divided by the cutoff jump in treatment identifies the cutoff-local LATE (def:po-fuzzy-rdd-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":55,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD assumption bundle. For a fuzzy regression-discontinuity system, this packages consistency (SUTVA), deterministic cutoff eligibility: the above-cutoff indicator agrees almost surely with whether the running variable has crossed the cutoff, and cutoff-neighborhood monotonicity: near the cutoff, a unit who would take treatment under the untreated instrument value would also take it under the treated instrument value. It supplies latent treatment and outcome regression representatives muD/muY and observable regression representatives nuD/nuY, each certified as a genuine regression function of the corresponding response on the running variable, with the latent representatives continuous at the cutoff; it also assumes the running variable has positive local probability mass on both sides of the cutoff, and that the observable treatment regression and outcome regression have well-defined one-sided limits at the cutoff. Finally, a complier outcome-difference representative mu_Ydiff_complier is likewise a regression function that is continuous at the cutoff, and the first-stage treatment-take-up jump at the cutoff is nonzero.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Type","l":169,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system and an instrument value, the potential treatment assigns each unit the binary treatment it would take under that instrument value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → Bool","l":93,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YdiffComplier","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system, the complier-weighted outcome difference assigns each unit the difference between its treated and untreated potential outcomes when it is a complier, and zero otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":113,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system and a treatment value, the potential outcome assigns each unit the real outcome it would have under that treatment value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → ℝ","l":97,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system and an instrument value, the instrument-induced potential outcome assigns each unit the potential outcome under the treatment that the instrument value would induce for that unit.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → P.Ω → ℝ","l":101,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.YofDofZ_def","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The potential outcome under the treatment that an instrument value induces sends a unit to that unit's treated potential outcome when the induced treatment is one, and to its untreated potential outcome otherwise.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), S.YofDofZ z = fun ω => if S.DofZ z ω = true then S.YofD true ω else S.YofD false ω","l":105,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.aemeasurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The observed running variable is almost-everywhere measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), AEMeasurable S.factualX P.μ","l":295,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system, the factual treatment assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → Bool","l":87,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system, the factual running variable assigns each unit its observed real-valued running variable.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":81,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system, the factual outcome assigns each unit its observed real outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → ℝ","l":90,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system, the factual cutoff-eligibility indicator assigns each unit its observed binary above-cutoff indicator.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → P.Ω → Bool","l":84,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.frd_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD identification at the cutoff, in regression-representative form. Under the fuzzy-RDD identifying assumption bundle, the Wald ratio of the latent right- and left-hand representative jumps at the cutoff equals the Wald ratio of the one-sided observable outcome- and treatment-regression limits at the cutoff:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), S.tau_FRD hA = (S.nuY_right_limit hA - S.nuY_left_limit hA) / (S.nuD_right_limit hA - S.nuD_left_limit hA)","l":606,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual eligibility event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":165,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The instrument-specific potential treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), Measurable (S.DofZ z)","l":134,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YdiffComplier","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The complier-weighted outcome difference is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.YdiffComplier","l":151,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (d : Bool), Measurable (S.YofD d)","l":139,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The outcome under the instrument-induced treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (z : Bool), Measurable (S.YofDofZ z)","l":144,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualD","l":127,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual running variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualX","l":121,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualY","l":130,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual cutoff-eligibility instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P), Measurable S.factualZ","l":124,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the left-hand treatment-regression limit is the selected limit of the observed treatment regression as the running variable approaches the cutoff from below.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":250,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-hand observable treatment-regression limit at the cutoff equals the untreated latent treatment regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.muD false S.c","l":571,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the right-hand treatment-regression limit is the selected limit of the observed treatment regression as the running variable approaches the cutoff from above.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":243,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuD_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-hand observable treatment-regression limit at the cutoff equals the treated latent treatment regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.muD true S.c","l":560,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the left-hand outcome-regression limit is the selected limit of the observed outcome regression as the running variable approaches the cutoff from below.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":264,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-hand observable outcome-regression limit at the cutoff equals the untreated latent outcome regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.muY false S.c","l":593,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the right-hand outcome-regression limit is the selected limit of the observed outcome regression as the running variable approaches the cutoff from above.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":257,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.nuY_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-hand observable outcome-regression limit at the cutoff equals the treated latent outcome regression there.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.muY true S.c","l":582,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_FRD","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the cutoff-local fuzzy-RDD Wald estimand is the difference between the treated- and untreated-instrument latent outcome regressions at the cutoff divided by the corresponding treatment-regression difference.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":229,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_LATE","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system satisfying the fuzzy-RDD assumptions, the cutoff-local complier-effect ratio is the complier outcome-difference regression at the cutoff divided by the latent first-stage treatment-regression jump.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POFuzzyRDDSystem P) → S.Assumptions → ℝ","l":234,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tau_late_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Fuzzy RDD identifies a cutoff representative complier ratio under an extra global monotonicity bridge. Under the local fuzzy-RDD identifying assumption bundle — consistency, deterministic cutoff eligibility, and local exclusion through potential outcomes, and additionally assuming treatment is almost-surely monotone in the instrument, i.e. whenever the potential treatment under the untreated instrument value is realized as true, the potential treatment under the treated instrument value is true as well (no defiers), the complier-weighted local average treatment effect at the cutoff tau_LATE equals the fuzzy-RDD Wald-ratio functional tau_FRD, i.e. the observable Wald ratio at the cutoff equals the complier-weighted outcome-difference representative divided by the first-stage jump.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), (∀ᵐ (ω : P.Ω) ∂P.μ, S.DofZ false ω = true → S.DofZ true ω = true) → S.tau_LATE hA = S.tau_FRD hA","l":662,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuD_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand treatment-regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nuD_left_limit hA))","l":277,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuD_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand treatment-regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuD (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nuD_right_limit hA))","l":271,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuY_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand outcome-regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nuY_left_limit hA))","l":289,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.tendsto_nuY_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand outcome-regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POFuzzyRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nuY (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nuY_right_limit hA))","l":283,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POFuzzyRDDSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.RDD.FuzzyRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a fuzzy regression-discontinuity system and an instrument value, the eligibility event is the set of units whose observed above-cutoff indicator equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POFuzzyRDDSystem P → Bool → Set P.Ω","l":162,"f":"Causalean/PO/ID/Exact/RDD/FuzzyRDD.lean"},{"n":"Causalean.PO.POHeckmanRoySystem","k":"structure","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"The Heckman–Vytlacil / generalized Roy instrumental-variable model in the potential-outcome framework, packaging an instrument, a binary treatment taken when a latent selection rank U falls below a propensity threshold p(Z) valued in [0,1], and a real outcome with potential outcomes Y(0) and Y(1), where the four nodes are pairwise distinct. This is the setup behind pairwise-Wald / LATE-type identification of treatment effects from instrument-induced variation in participation (def:po-iv-heckman-roy-system).","s":"Causalean.PO.POSystem → (α : Type u_1) → [inst : MeasurableSpace α] → [MeasurableSingletonClass α] → Type (max (max u_1 u_2) u_3)","l":43,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Heckman–Roy IV identifying assumptions (def:po-iv-heckman-roy-assumptions). Bundles consistency of the underlying potential-outcome system, exogeneity of the instrument, independent of the latent selection rank and the two potential outcomes, threshold-crossing selection: the potential treatment under instrument value z equals true exactly when the latent rank falls at or below the propensity p(z), and the latent rank being uniformly distributed on [0,1].","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Prop","l":224,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the potential treatment assigns each unit the treatment it would take under that instrument value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → P.Ω → Bool","l":105,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise D-consistency on {Z = z}: D(z)(ω) = D(ω). One-line specialisation of POVar.cf_eq_factual_on_event. Analogue of POIVSystem.DofZ_eq_factualD_on_zEvent in LATE.lean.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z : α) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":45,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and two instrument values, the latent interval average treatment effect is the mean difference between treated and untreated potential outcomes among units whose latent ranks lie strictly above the first propensity threshold and no greater than the second, with value zero when that event has zero probability.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → α → ℝ","l":252,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and a treatment value, the potential outcome assigns each unit the outcome it would have under that treatment value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → P.Ω → ℝ","l":108,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the outcome under the instrument-induced treatment assigns each unit its treated potential outcome if that instrument induces treatment and its untreated potential outcome otherwise.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → P.Ω → ℝ","l":169,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.YofDofZ_def","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The potential outcome under the treatment that an instrument value induces sends a unit to that unit's treated potential outcome when the induced treatment is one, and to its untreated potential outcome otherwise.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), S.YofDofZ z = fun ω => if S.DofZ z ω = true then S.YofD true ω else S.YofD false ω","l":173,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the counterfactual bundle contains the latent rank and the potential outcomes under treatment and control.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POCFBundle P","l":208,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.complier_measure","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Real-valued measure of the interval complier event: (μ {p z₀ < U ≤ p z₁}).toReal = p z₁ - p z₀, provided 0 ≤ p z₀ ≤ p z₁ ≤ 1 (which holds because p lands in [0,1] by S.hp_mem, plus the user-supplied ordering p z₀ < p z₁).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), S.p z₀ < S.p z₁ → (P.μ (S.intervalComplierEvent z₀ z₁)).toReal = S.p z₁ - S.p z₀","l":95,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the conditional treatment mean is the event-conditional mean of observed treatment among units with that instrument value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → ℝ","l":188,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the conditional outcome mean is the event-conditional mean of the observed outcome among units with that instrument value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → ℝ","l":195,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the binary treatment potential-outcome variable is its treatment node with its binary representation.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P Bool","l":86,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.event_conditioning_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Event-conditioning identity: ∫ (Y(1) - Y(0)) · 1_{C(z₀,z₁)} ∂μ = μ(C(z₀,z₁)).toReal · LATE z₀ z₁.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), ∫ (ω : P.Ω), (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω ∂P.μ = (P.μ (S.intervalComplierEvent z₀ z₁)).toReal * S.LATE z₀ z₁","l":433,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the factual treatment assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → Bool","l":114,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualU","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the factual latent selection rank assigns each unit its latent real-valued rank.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → ℝ","l":120,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the factual outcome assigns each unit its observed real outcome.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → ℝ","l":117,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise Y-consistency: Y(ω) = Y(D(ω))(ω) for every ω. One-line specialisation of POVar.factual_eq_cfUnder_self_selected.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":53,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the factual instrument assigns each unit its observed instrument value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → P.Ω → α","l":111,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.first_stage_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"First-stage identity: E[D | Z=z₁] - E[D | Z=z₀] = p z₁ - p z₀.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → S.p z₀ < S.p z₁ → S.condExpDZ z₁ - S.condExpDZ z₀ = S.p z₁ - S.p z₀","l":143,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.instrumentRegime","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the instrument intervention regime fixes the instrument to that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → Causalean.PO.Regime P.V P.X","l":97,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.intervalComplierEvent","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and two instrument values, the interval-complier event is the set of units whose latent rank is strictly above the first propensity threshold and no greater than the second.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → α → Set P.Ω","l":126,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.interval_indicator_sub","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise: for q₀ ≤ q₁, 1_{U ω ≤ q₁} - 1_{U ω ≤ q₀} = 1_{q₀ < U ω ≤ q₁}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), S.p z₀ ≤ S.p z₁ → ∀ (ω : P.Ω), ((if S.factualU ω ≤ S.p z₁ then 1 else 0) - if S.factualU ω ≤ S.p z₀ then 1 else 0) = (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω","l":61,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurableSet_intervalComplierEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The latent interval complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z₀ z₁ : α), MeasurableSet (S.intervalComplierEvent z₀ z₁)","l":160,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), MeasurableSet (S.zEvent z)","l":156,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a fixed instrument value z, the potential treatment D(z) is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), Measurable (S.DofZ z)","l":132,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (d : Bool), Measurable (S.YofD d)","l":138,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_YofDofZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The outcome composed with the instrument-induced treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α) (z : α), Measurable (S.YofDofZ z)","l":181,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualD","l":146,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualU","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual latent rank is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualU","l":152,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualY","l":149,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), Measurable S.factualZ","l":143,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.pointwise_threshold_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Pointwise threshold identity: Y(D(z₁)) - Y(D(z₀)) = (Y(1) - Y(0)) · 1_{p z₀ < U ≤ p z₁} a.s.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), S.p z₀ ≤ S.p z₁ → ∀ᵐ (ω : P.Ω) ∂P.μ, S.YofDofZ z₁ ω - S.YofDofZ z₀ ω = (S.YofD true ω - S.YofD false ω) * (S.intervalComplierEvent z₀ z₁).indicator (fun x => 1) ω","l":381,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.reduced_form_identity","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":2,"d":"Reduced-form identity: E[Y | Z=z₁] - E[Y | Z=z₀] = ∫ (Y(D(z₁)) - Y(D(z₀))) ∂μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → S.condExpYZ z₁ - S.condExpYZ z₀ = ∫ (ω : P.Ω), S.YofDofZ z₁ ω - S.YofDofZ z₀ ω ∂P.μ","l":251,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and a treatment value, the treatment intervention regime fixes treatment to that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → Causalean.PO.Regime P.V P.X","l":101,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.uVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the latent-rank potential-outcome variable is its latent selection-rank node with its real-valued representation.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P ℝ","l":92,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.wald_pairwise","k":"theorem","m":"Causalean.PO.ID.Exact.HeckmanRoy.Wald","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"Pairwise Wald identification of LATE (prop:po-iv-heckman-roy-wald). Under the Heckman–Roy identifying assumption bundle, for instrument values z₀, z₁ at which the event {Z = z₀} has positive probability, the event {Z = z₁} has positive probability, and the latent selection threshold at z₀ is strictly below the threshold at z₁, provided the potential outcome under treatment and the potential outcome under control are integrable, the Wald ratio of the conditional-mean outcome and treatment contrasts between Z = z₁ and Z = z₀ equals the pairwise local average treatment effect LATE z₀ z₁.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POHeckmanRoySystem P α), S.Assumptions → ∀ (z₀ z₁ : α), 0 < (P.μ (S.zEvent z₀)).toReal → 0 < (P.μ (S.zEvent z₁)).toReal → S.p z₀ < S.p z₁ → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → (S.condExpYZ z₁ - S.condExpYZ z₀) / (S.condExpDZ z₁ - S.condExpDZ z₀) = S.LATE z₀ z₁","l":468,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Wald.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and a treatment value, the outcome under the treatment regime is the potential outcome represented together with the intervention fixing treatment to that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Bool → Causalean.PO.RegimedVar P ℝ","l":204,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the real-valued outcome potential-outcome variable is its outcome node with its real-valued representation.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P ℝ","l":89,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system and an instrument value, the instrument event is the set of units whose observed instrument equals that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → α → Set P.Ω","l":123,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POHeckmanRoySystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.HeckmanRoy.Setup","a":"PO/ID/Exact/HeckmanRoy","t":1,"d":"For a Heckman--Roy system, the instrument potential-outcome variable is its instrument node with its instrument-value representation.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POHeckmanRoySystem P α → Causalean.PO.POVar P α","l":83,"f":"Causalean/PO/ID/Exact/HeckmanRoy/Setup.lean"},{"n":"Causalean.PO.POIVSystem","k":"structure","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"A binary instrumental-variables subsystem (def:po-iv-system) records, within an ambient potential-outcome system, an instrument, a treatment, and an outcome, where the three nodes are required to be pairwise distinct.","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":38,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Classical binary-instrument IV assumptions (def:po-iv-assumptions). Bundles consistency (SUTVA): the observed treatment and outcome equal the realized potential treatment and outcome, instrument exogeneity: the instrument is independent of the full counterfactual bundle of potential treatments and outcomes, monotonicity (no defiers): turning the instrument on never moves a unit out of treatment, and relevance: the complier event has positive probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Prop","l":181,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.DofZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the potential treatment assigns each unit the treatment it would take if the instrument were fixed to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → Bool","l":73,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.DofZ_eq_factualD_on_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"On zEvent z, the counterfactual treatment D(z) equals the factual D. Pointwise specialization of Consistency.factual with r = instrumentRegime z, Y = {D}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ (z : Bool) {ω : P.Ω}, ω ∈ S.zEvent z → S.DofZ z ω = S.factualD ω","l":226,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.LATE","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the local average treatment effect is the mean difference between treated and untreated potential outcomes among compliers, with value zero when the complier event has zero probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → ℝ","l":215,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and a treatment value, the potential outcome assigns each unit the outcome it would have if treatment were fixed to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → ℝ","l":82,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.YofDofZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the outcome under the instrument-induced treatment assigns each unit its treated potential outcome if the instrument induces treatment and its untreated potential outcome otherwise.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → P.Ω → ℝ","l":138,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.YofDofZ_def","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The potential outcome under the treatment that an instrument value induces sends a unit to that unit's treated potential outcome when the induced treatment is one, and to its untreated potential outcome otherwise.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (z : Bool), S.YofDofZ z = fun ω => if S.DofZ z ω = true then S.YofD true ω else S.YofD false ω","l":142,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.cfBundle","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the counterfactual bundle contains potential treatments under both instrument values and potential outcomes under both treatment values.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POCFBundle P","l":173,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.complierEvent","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the complier event is the set of units that would take treatment when the instrument is on and would not take treatment when it is off.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Set P.Ω","l":100,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.condExpDZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the conditional treatment mean is the event-conditional mean of observed treatment among units with that instrument value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → ℝ","l":150,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.condExpYZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the conditional outcome mean is the event-conditional mean of observed outcome among units with that instrument value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → ℝ","l":158,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.dUnderZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the treatment under the instrument regime is the potential treatment represented together with the intervention fixing the instrument to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.RegimedVar P Bool","l":165,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the binary treatment potential-outcome variable is its treatment node with its binary representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P Bool","l":59,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.event_conditioning_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 4 of rem:po-late: event-conditioning identity. E[(Y(1) - Y(0)) · 1_C] = P(C) · LATE.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), ∫ (ω : P.Ω), (S.YofD true ω - S.YofD false ω) * S.complierEvent.indicator (fun x => 1) ω ∂P.μ = (P.μ S.complierEvent).toReal * S.LATE","l":425,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the factual treatment assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → Bool","l":94,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the factual outcome assigns each unit its observed real outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → ℝ","l":97,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualY_eq_YofD_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Factual Y equals the counterfactual Y(factualD ω). Pointwise specialization of Consistency.factual with r = treatmentRegime (factualD ω), Y = {Y}.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ (ω : P.Ω), S.factualY ω = S.YofD (S.factualD ω) ω","l":234,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the factual instrument assigns each unit its observed binary instrument.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → P.Ω → Bool","l":91,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.first_stage_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 1 of rem:po-late: first-stage identity. E[D | Z=1] - E[D | Z=0] = P(C).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → S.condExpDZ true - S.condExpDZ false = (P.μ S.complierEvent).toReal","l":241,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.instrumentRegime","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the instrument intervention regime fixes the instrument to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.Regime P.V P.X","l":65,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.late_wald","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"Wald identification of LATE (prop:po-late). Under the binary- instrument LATE identifying assumption bundle, when the event {Z=1} has positive probability, the event {Z=0} has positive probability, and the potential outcomes under treatment and under control are integrable, the Wald ratio (E[Y|Z=1] − E[Y|Z=0]) / (E[D|Z=1] − E[D|Z=0]) equals the local average treatment effect LATE.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → (S.condExpYZ true - S.condExpYZ false) / (S.condExpDZ true - S.condExpDZ false) = S.LATE","l":452,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurableSet_complierEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The complier event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), MeasurableSet S.complierEvent","l":129,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual instrument event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (z : Bool), MeasurableSet (S.zEvent z)","l":134,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_DofZ","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The potential treatment under a fixed instrument value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (z : Bool), Measurable (S.DofZ z)","l":107,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The potential outcome under a fixed treatment value is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P) (d : Bool), Measurable (S.YofD d)","l":124,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualD","l":116,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualY","l":120,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"The factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), Measurable S.factualZ","l":112,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.pointwise_monotonicity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 3 of rem:po-late: pointwise monotonicity identity. Y(D(1)) - Y(D(0)) = (Y(1) - Y(0)) · 1_C almost surely.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → ∀ᵐ (ω : P.Ω) ∂P.μ, S.YofDofZ true ω - S.YofDofZ false ω = (S.YofD true ω - S.YofD false ω) * S.complierEvent.indicator (fun x => 1) ω","l":413,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.reduced_form_identity","k":"theorem","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":2,"d":"Step 2 of rem:po-late: reduced-form identity. E[Y | Z=1] - E[Y | Z=0] = E[Y(D(1)) - Y(D(0))].","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POIVSystem P), S.Assumptions → 0 < (P.μ (S.zEvent true)).toReal → 0 < (P.μ (S.zEvent false)).toReal → MeasureTheory.Integrable (S.YofD true) P.μ → MeasureTheory.Integrable (S.YofD false) P.μ → S.condExpYZ true - S.condExpYZ false = ∫ (ω : P.Ω), S.YofDofZ true ω - S.YofDofZ false ω ∂P.μ","l":326,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.treatmentRegime","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and a treatment value, the treatment intervention regime fixes treatment to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.Regime P.V P.X","l":69,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.yUnderD","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and a treatment value, the outcome under the treatment regime is the potential outcome represented together with the intervention fixing treatment to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Causalean.PO.RegimedVar P ℝ","l":169,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the real-valued outcome potential-outcome variable is its outcome node with its real-valued representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P ℝ","l":62,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system and an instrument value, the instrument event is the set of units whose observed instrument equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Bool → Set P.Ω","l":104,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Exact.LATE","a":"PO/ID/Exact","t":1,"d":"For a binary instrumental-variables system, the binary instrument potential-outcome variable is its instrument node with its binary representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POIVSystem P → Causalean.PO.POVar P Bool","l":56,"f":"Causalean/PO/ID/Exact/LATE.lean"},{"n":"Causalean.PO.POLeeSystem","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"The data layer for Lee (2009) bounds: a treatment-selection model in which a binary treatment A affects whether an outcome Y is observed at all, through a binary sample-selection indicator Sel. The outcome Y is only meaningful when Sel = true (e.g. a wage observed only for the employed), so the treatment effect on Y among the always-selected subpopulation is only partially identified — the object the Lee bounds bracket. Formally this packages, inside an ambient PO system P, the three nodes A, Sel, Y, the measurable identifications of their value spaces with Bool/Bool/ℝ, and the fact that the three nodes are distinct (def:po-lee-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":33,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Lee sample-selection baseline assumptions -- def:po-lee-assumptions.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Prop","l":53,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions.integrableY","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee sample-selection assumptions — consistency, pair-level random assignment, positive and finite selected cells, and integrability of both potential outcomes Y(0), Y(1), the factual outcome Y is integrable, obtained from the arm integrability of Y(0), Y(1) via consistency (factualY = Σ_a Y(a)·1{A=a} a.e.), so it need not be assumed separately.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POLeeSystem P}, S.BaseAssumptions → MeasureTheory.Integrable S.factualY P.μ","l":92,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.BaseAssumptions.integrable_YofA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":2,"d":"Binary-folded form of integrableY1 / integrableY0.","s":"∀ {P : Causalean.PO.POSystem} {S : Causalean.PO.POLeeSystem P}, S.BaseAssumptions → ∀ (a : Bool), MeasureTheory.Integrable (S.YofA a) P.μ","l":85,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.LeeTrimWeight","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"A Lee trim weight on a finite outcome support 𝒴 -- def:po-lee-trim.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → Type","l":78,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.MonotoneSelection","k":"structure","m":"Causalean.PO.ID.Partial.Lee.Assumptions","a":"PO/ID/Partial/Lee","t":1,"d":"Monotone sample selection -- def:po-lee-assumptions, item 3. Sel(0) ≤ Sel(1) almost surely (with ≤ interpreted on Bool via the canonical false ≤ true order).","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Prop","l":130,"f":"Causalean/PO/ID/Partial/Lee/Assumptions.lean"},{"n":"Causalean.PO.POLeeSystem.Mw","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, a finite outcome support, and a feasible Lee trim weight on that support, the trimmed mean is the trimming-ratio inverse times the weighted selected-treated outcome sum over that support.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POLeeSystem P) → {𝒴 : Finset ℝ} → S.LeeTrimWeight 𝒴 → ℝ","l":90,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.Mw_alwaysSelectedTrimWeight_eq_condExp_Y1_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.TrimMean","a":"PO/ID/Partial/Lee","t":1,"d":"The constructed always-selected trim weight has the target latent mean. Given the baseline Lee assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, evaluating the trim-weight mean functional Mw at the constructed always-selected trim weight recovers exactly the conditional mean E[Y(1) | alwaysSelected] of the treated potential outcome among always-selected units.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (hA : S.BaseAssumptions) (hMono : S.MonotoneSelection) (𝒴 : Finset ℝ) (hSupp : ∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴), S.Mw (S.alwaysSelectedTrimWeight hA hMono 𝒴 hSupp) = Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true)","l":33,"f":"Causalean/PO/ID/Partial/Lee/TrimMean.lean"},{"n":"Causalean.PO.POLeeSystem.SelOfA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a treatment arm, the potential selection indicator assigns each unit whether its outcome would be selected under that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → P.Ω → Bool","l":82,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.YofA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a treatment arm, the potential outcome assigns each unit its outcome under that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → P.Ω → ℝ","l":89,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.YofA_true_in_finset_ae_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.LatentSupport","a":"PO/ID/Partial/Lee","t":1,"d":"Support transfer. Under the baseline Lee assumptions, if the factual outcome lies a.e. in a finite support set 𝒴, when restricted to the observable selected-treated cell, then the latent treated potential outcome Y(1) lies a.e. in the same support 𝒴, both on the always-selected stratum and on the helped-selected stratum. This feeds the sum_eq field of alwaysSelectedTrimWeight, which needs ∑ y ∈ 𝒴, f1AS y = 1.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.alwaysSelected, S.YofA true ω ∈ 𝒴) ∧ ∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.helpedSelected, S.YofA true ω ∈ 𝒴","l":41,"f":"Causalean/PO/ID/Partial/Lee/LatentSupport.lean"},{"n":"Causalean.PO.POLeeSystem.aEvent","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a treatment arm, the treatment event is the set of units whose observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → Set P.Ω","l":105,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.aVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the binary treatment potential-outcome variable is its treatment node with its binary representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P Bool","l":73,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.alwaysSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system and a Lee potential-outcome system based on it, the always-selected stratum is the set of units whose potential selection indicator equals true both under control and under treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":37,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.alwaysSelectedTrimWeight","k":"def","m":"Causalean.PO.ID.Partial.Lee.TrimWeight","a":"PO/ID/Partial/Lee","t":1,"d":"Given a potential-outcome system, a Lee potential-outcome system based on it, its base assumptions, monotone sample selection, a finite set of real outcome values, and the condition that the observed outcome belongs to that set almost surely under the selected-treated conditional measure, the always-selected trim weight on that finite set is a Lee trim weight.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POLeeSystem P) → S.BaseAssumptions → S.MonotoneSelection → (𝒴 : Finset ℝ) → (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → S.LeeTrimWeight 𝒴","l":42,"f":"Causalean/PO/ID/Partial/Lee/TrimWeight.lean"},{"n":"Causalean.PO.POLeeSystem.disjoint_alwaysSelected_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"alwaysSelected and helpedSelected are disjoint.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Disjoint S.alwaysSelected S.helpedSelected","l":123,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.f1","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and an outcome value, the selected-treated outcome mass function is the event-conditional probability that the observed outcome equals that value among selected treated units.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ → ℝ","l":70,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.f1AS","k":"def","m":"Causalean.PO.ID.Partial.Lee.TrimWeight","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system, a Lee potential-outcome system based on it, and a real outcome value, the always-selected treated-outcome mass at that value is the conditional expectation, given the always-selected stratum, of the indicator that the potential outcome under treatment equals that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ → ℝ","l":30,"f":"Causalean/PO/ID/Partial/Lee/TrimWeight.lean"},{"n":"Causalean.PO.POLeeSystem.factualA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the factual treatment assigns each unit its observed binary treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → Bool","l":96,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.factualSel","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the factual selection indicator assigns each unit its observed selection status.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → Bool","l":99,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the factual outcome assigns each unit its observed real outcome.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → P.Ω → ℝ","l":102,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.harmedSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system and a Lee potential-outcome system based on it, the harmed-selected stratum is the set of units whose potential selection indicator equals true under control and false under treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":55,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.harmedSelected_ae_empty","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"If sample selection is monotone, i.e. selection under control implies selection under treatment almost surely (Sel(0) ≤ Sel(1)), then the harmed-selected stratum — units who would be selected under control but not under treatment — has probability zero.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), (∀ᵐ (ω : P.Ω) ∂P.μ, S.SelOfA false ω ≤ S.SelOfA true ω) → P.μ S.harmedSelected = 0","l":143,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.helpedSelected","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system and a Lee potential-outcome system based on it, the treatment-induced-selected stratum is the set of units whose potential selection indicator equals false under control and true under treatment.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":46,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.lee_bounds_ATT_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Main","a":"PO/ID/Partial/Lee","t":1,"d":"Finite-support Lee bounds -- prop:po-lee-bounds. Under the baseline Lee sample-selection assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, the average treatment effect among always-selected units — those who would be selected for observation whether treated or not — is sandwiched between the lower Lee trimmed mean minus the observable selected-control mean m₀, and the upper Lee trimmed mean minus m₀.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) [MeasureTheory.IsFiniteMeasure P.μ], S.BaseAssumptions → S.MonotoneSelection → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → (S.lowerTrimMean 𝒴 - S.m0 ≤ Causalean.PO.eventCondExp P.μ S.alwaysSelected fun ω => S.YofA true ω - S.YofA false ω) ∧ (Causalean.PO.eventCondExp P.μ S.alwaysSelected fun ω => S.YofA true ω - S.YofA false ω) ≤ S.upperTrimMean 𝒴 - S.m0","l":33,"f":"Causalean/PO/ID/Partial/Lee/Main.lean"},{"n":"Causalean.PO.POLeeSystem.lowerTrimMean","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a finite outcome support, the lower trimmed mean is the infimum of trimmed means over all feasible Lee trim weights on that support.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → ℝ","l":97,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.m0","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the selected-control outcome mean is the event-conditional mean of the observed outcome among selected control units.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":111,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.m0_eq_eventCondExp_Y0_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.ControlMean","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee assumptions (consistency and pair-level random assignment of the factual treatment to each (Y(a), Sel(a))) together with monotone sample selection, Sel(0) ≤ Sel(1) almost surely, the observable selected-control mean m₀ = E[Y | A = false, Sel = true] equals the latent conditional mean E[Y(0) | alwaysSelected] of the control potential outcome among units who would be selected under either treatment arm.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → S.MonotoneSelection → S.m0 = Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA false)","l":50,"f":"Causalean/PO/ID/Partial/Lee/ControlMean.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_aEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Each factual treatment arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), MeasurableSet (S.aEvent a)","l":140,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The always-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.alwaysSelected","l":66,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_harmedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The harmed-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.harmedSelected","l":78,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The helped-selected stratum is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.helpedSelected","l":72,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Each factual selection event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (s : Bool), MeasurableSet (S.selEvent s)","l":144,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selOfAFalseSet","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-under-control latent set is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selOfAFalseSet","l":104,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selOfATrueSet","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-under-treatment latent set is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selOfATrueSet","l":92,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selectedControl","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-control observed cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selectedControl","l":152,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurableSet_selectedTreated","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"The selected-treated observed cell is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), MeasurableSet S.selectedTreated","l":148,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_SelOfA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Counterfactual selection under any fixed arm is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), Measurable (S.SelOfA a)","l":119,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a fixed treatment arm a, the counterfactual outcome Y(a) is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), Measurable (S.YofA a)","l":124,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualA","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualA","l":130,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualSel","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual selection is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualSel","l":133,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":2,"d":"Factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), Measurable S.factualY","l":136,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.p0","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the control-arm selection probability is the conditional selection probability when treatment is zero.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":61,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.p1","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the treated-arm selection probability is the conditional selection probability when treatment is one.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":64,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.pSelGivenA","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a treatment arm, the conditional selection probability is the event-conditional mean of the observed selection indicator among units in that arm.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → ℝ","l":46,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.pSelGivenA_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"The conditional selection probability given a treatment arm is the event-conditional expectation of the selection indicator on that arm's event.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P) (a : Bool), S.pSelGivenA a = Causalean.PO.eventCondExp P.μ (S.aEvent a) (S.selVar.indicator true)","l":54,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.rho","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the trimming ratio is the control-arm selection probability divided by the treated-arm selection probability.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → ℝ","l":67,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.selEvent","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a selection status, the selection event is the set of units whose observed selection indicator equals that status.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Bool → Set P.Ω","l":108,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selOfAFalseSet","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system and a Lee potential-outcome system based on it, the latent selected-under-control set is the set of units whose potential selection indicator would equal true if untreated.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":96,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfAFalseSet_ae_eq_alwaysSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"Under monotone sample selection, {Sel(0)=true} =ᵐ alwaysSelected.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), (∀ᵐ (ω : P.Ω) ∂P.μ, S.SelOfA false ω ≤ S.SelOfA true ω) → S.selOfAFalseSet =ᵐ[P.μ] S.alwaysSelected","l":164,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfATrueSet","k":"def","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":1,"d":"For a potential-outcome system and a Lee potential-outcome system based on it, the latent selected-under-treatment set is the set of units whose potential selection indicator would equal true if treated.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":84,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selOfATrueSet_eq_alwaysSelected_union_helpedSelected","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.PrincipalStrata","a":"PO/ID/Partial/Lee","t":2,"d":"selOfATrueSet = alwaysSelected ∪ helpedSelected as a pure set equality (no a.s. needed -- the two RHS sets are disjoint and cover the LHS by case analysis on SelOfA false ω : Bool).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.selOfATrueSet = S.alwaysSelected ∪ S.helpedSelected","l":108,"f":"Causalean/PO/ID/Partial/Lee/PrincipalStrata.lean"},{"n":"Causalean.PO.POLeeSystem.selVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the binary selection-indicator potential-outcome variable is its selection node with its binary representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P Bool","l":76,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedControl","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the selected-control cell is the set of units with observed treatment equal to zero and observed selection equal to one.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":114,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the selected-treated cell is the set of units with observed treatment and observed selection both equal to one.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Set P.Ω","l":111,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_integral_split","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":2,"d":"Selected-treated integral split — the f = factualY analogue of selectedTreated_measure_split (Step B'). Consistency on {A=true, Sel=true} rewrites factualY to YofA true; pair random assignment A ⫫ (Y(1), Sel(1)) then drops the {A=true} conditioning and introduces the scalar factor (P.μ (S.aEvent true)).toReal; the latent {Sel(1)=true} set then splits as the disjoint union of alwaysSelected and helpedSelected.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∫ (ω : P.Ω) in S.selectedTreated, S.factualY ω ∂P.μ = (P.μ (S.aEvent true)).toReal * (∫ (ω : P.Ω) in S.alwaysSelected, S.YofA true ω ∂P.μ + ∫ (ω : P.Ω) in S.helpedSelected, S.YofA true ω ∂P.μ)","l":60,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_integral_split_indicator","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":2,"d":"Step B for the indicator integrand 1_{factualY = y} — same shape as selectedTreated_integral_split but with the indicator integrand. Used by the le_one and sum_eq fields of alwaysSelectedTrimWeight. Proof mirrors Step B; the only change is the integrand.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → ∀ (y : ℝ), ∫ (ω : P.Ω) in S.selectedTreated, if S.factualY ω = y then 1 else 0 ∂P.μ = (P.μ (S.aEvent true)).toReal * (∫ (ω : P.Ω) in S.alwaysSelected, if S.YofA true ω = y then 1 else 0 ∂P.μ + ∫ (ω : P.Ω) in S.helpedSelected, if S.YofA true ω = y then 1 else 0 ∂P.μ)","l":216,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.selectedTreated_measure_split","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.MixtureIdentity","a":"PO/ID/Partial/Lee","t":1,"d":"Under the baseline Lee sample-selection assumptions, the probability mass of the observable selected-treated cell equals the probability of being treated times the sum of the probability masses of the two latent strata always-selected and helped-selected. This is the analog of selectedTreated_integral_split for the constant function 1, and it is what gives ρ = μ(AS) / μ({Sel(1)=true}).","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → (P.μ S.selectedTreated).toReal = (P.μ (S.aEvent true)).toReal * ((P.μ S.alwaysSelected).toReal + (P.μ S.helpedSelected).toReal)","l":157,"f":"Causalean/PO/ID/Partial/Lee/MixtureIdentity.lean"},{"n":"Causalean.PO.POLeeSystem.trimmed_bounds_condExp_Y1_AS","k":"theorem","m":"Causalean.PO.ID.Partial.Lee.TrimBound","a":"PO/ID/Partial/Lee","t":1,"d":"The Lee trimmed-mean sandwich for the always-selected treated mean. Given the baseline Lee assumptions, monotone sample selection, and almost-sure finite support 𝒴 for the factual outcome on the selected-treated cell, the conditional mean E[Y(1) | alwaysSelected] of the treated potential outcome among always-selected units lies between the lower and upper Lee trimmed means computed over all feasible Lee trim weights on that support.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POLeeSystem P), S.BaseAssumptions → S.MonotoneSelection → ∀ (𝒴 : Finset ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ.restrict S.selectedTreated, S.factualY ω ∈ 𝒴) → S.lowerTrimMean 𝒴 ≤ Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true) ∧ Causalean.PO.eventCondExp P.μ S.alwaysSelected (S.YofA true) ≤ S.upperTrimMean 𝒴","l":31,"f":"Causalean/PO/ID/Partial/Lee/TrimBound.lean"},{"n":"Causalean.PO.POLeeSystem.upperTrimMean","k":"def","m":"Causalean.PO.ID.Partial.Lee.Trim","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system and a finite outcome support, the upper trimmed mean is the supremum of trimmed means over all feasible Lee trim weights on that support.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Finset ℝ → ℝ","l":104,"f":"Causalean/PO/ID/Partial/Lee/Trim.lean"},{"n":"Causalean.PO.POLeeSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.Lee.Setup","a":"PO/ID/Partial/Lee","t":1,"d":"For a Lee sample-selection system, the real-valued outcome potential-outcome variable is its outcome node with its real-valued representation.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POLeeSystem P → Causalean.PO.POVar P ℝ","l":79,"f":"Causalean/PO/ID/Partial/Lee/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"The Manski IV data layer records a discrete instrument, a binary treatment, and a real outcome inside a potential-outcome system.","s":"Causalean.PO.POSystem → (α : Type u_1) → [inst : MeasurableSpace α] → [MeasurableSingletonClass α] → Type (max (max u_1 u_2) u_3)","l":26,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the average treatment effect is the population expectation of the potential outcome under treatment minus the potential outcome under control.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ","l":139,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Discrete-IV ATE-bound baseline assumptions -- common part of def:po-iv-manski-assumptions.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Type","l":31,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.bounded","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Given the baseline Manski assumptions bundle, which fixes bounds lo ≤ hi and asserts that both potential outcomes Y(1) and Y(0) lie a.s. in [lo, hi], the potential outcome Y(d) lies almost surely between lo and hi, uniformly for either treatment arm d — the binary-folded form of the two separate range assumptions.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions) (d : Bool), ∀ᵐ (ω : P.Ω) ∂P.μ, hA.lo ≤ S.YofD d ω ∧ S.YofD d ω ≤ hA.hi","l":50,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.integrable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":2,"d":"Binary-folded form of integrable_Y1 / integrable_Y0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (S.YofD d) P.μ","l":60,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.BaseAssumptions.integrable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":2,"d":"The factual outcome Y is integrable. Derived from arm integrability via consistency (factualY = Σ_d Y(d)·1{D=d} a.e.), so it need not be assumed separately.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] {S : Causalean.PO.POManskiIVSystem P α} (hA : S.BaseAssumptions), MeasureTheory.Integrable S.factualY P.μ","l":67,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MIV","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone instrumental variable -- prop:po-iv-miv.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Type u_1","l":125,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MTR","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone treatment response -- prop:po-iv-mtr, item 1. Y(0) ≤ Y(1) almost surely.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":112,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MTS","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Monotone treatment selection -- prop:po-iv-mts.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":117,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.MeanIndep","k":"structure","m":"Causalean.PO.ID.Partial.Manski.Assumptions","a":"PO/ID/Partial/Manski","t":1,"d":"Mean independence of the potential outcomes from the instrument. Stated directly on eventCondExp — matches def:po-iv-manski-assumptions letter-for-letter and avoids the stronger joint independence used in LATE.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Prop","l":103,"f":"Causalean/PO/ID/Partial/Manski/Assumptions.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, and a binary treatment arm, the potential outcome under that arm assigns each unit its real-valued outcome were treatment set to that arm.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → P.Ω → ℝ","l":70,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_eq_factualY_on_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise consistency on {D = d}: YofD d = factualY on this event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool) {ω : P.Ω}, ω ∈ S.dEvent d → S.YofD d ω = S.factualY ω","l":119,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_indD_plus_lo_le_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise a.e. bound: Y(d) · 1_{D=d} + lo · 1_{D=!d} ≤ Y(d) under lo ≤ Y(d).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool) (lo : ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, lo ≤ S.YofD d ω) → (fun ω => S.YofD d ω * S.dVar.indicator d ω + lo * S.dVar.indicator (!d) ω) ≤ᵐ[P.μ] S.YofD d","l":89,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_le_YofD_indD_plus_hi","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Pointwise a.e. bound: Y(d) ≤ Y(d) · 1_{D=d} + hi · 1_{D=!d} under Y(d) ≤ hi.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool) (hi : ℝ), (∀ᵐ (ω : P.Ω) ∂P.μ, S.YofD d ω ≤ hi) → S.YofD d ≤ᵐ[P.μ] fun ω => S.YofD d ω * S.dVar.indicator d ω + hi * S.dVar.indicator (!d) ω","l":99,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.YofD_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"YofD d · indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (fun ω => S.YofD d ω * S.dVar.indicator d ω) P.μ","l":53,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.boundArm","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, a binary treatment arm, a real-valued outcome bound, and an instrument value, the arm-bound functional is the conditional mean within that instrument stratum of the observed outcome for units on the specified arm and the supplied bound for units on the opposite arm.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → ℝ → α → ℝ","l":153,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.boundArm_lo_le_cond_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on eventCondExp μ (Z=z) (Y(d)) via the unified arm functional boundArm d lo.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (d : Bool) {z : α}, z ∈ S.support → S.boundArm d hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD d)","l":117,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.ciSup_lowerBound0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Sup-over-support form of lowerBound0_le_integral_Y0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ⨆ z, S.lowerBound0 hA.lo ↑z ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":115,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.ciSup_lowerBound1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Sup-over-support form of lowerBound1_le_integral_Y1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ⨆ z, S.lowerBound1 hA.lo ↑z ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":101,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.compl_dEvent_false","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Complement of {D = false} is {D = true} (Bool case-split).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (S.dEvent false)ᶜ = S.dEvent true","l":58,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.compl_dEvent_true","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Complement of {D = true} is {D = false} (Bool case-split).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (S.dEvent true)ᶜ = S.dEvent false","l":51,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_Y0_le_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on the *conditional expectation* of Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false) ≤ S.upperBound0 hA.hi z","l":222,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_Y1_le_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on the *conditional expectation* of Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true) ≤ S.upperBound1 hA.hi z","l":210,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.cond_YofD_le_boundArm_hi","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":1,"d":"Under the baseline Manski assumptions, for any treatment arm d and any instrument value z in the support of the instrument, the conditional mean of the potential outcome Y(d) given Z = z is bounded above by the unified upper-envelope functional boundArm d hi evaluated at z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (d : Bool) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD d) ≤ S.boundArm d hA.hi z","l":158,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.const_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Constant times indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (c : ℝ) (d : Bool), MeasureTheory.Integrable (fun ω => c * S.dVar.indicator d ω) P.μ","l":58,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.dEvent","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, and a binary treatment arm, the corresponding treatment stratum is the set of units whose observed treatment equals that arm.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Bool → Set P.Ω","l":104,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.dVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the binary treatment variable is its treatment represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P Bool","l":58,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.eventCondExp_YofD_eq_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(d) | D = d] = E[Y | D = d] via consistency on the event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), Causalean.PO.eventCondExp P.μ (S.dEvent d) (S.YofD d) = Causalean.PO.eventCondExp P.μ (S.dEvent d) S.factualY","l":125,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualD","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the observed binary treatment assigns each unit its realized treatment arm.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → Bool","l":85,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualY","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the observed outcome assigns each unit its realized real-valued outcome.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → ℝ","l":91,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualY_mul_indD_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"factualY · indicator d is integrable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), MeasureTheory.Integrable (fun ω => S.factualY ω * S.dVar.indicator d ω) P.μ","l":48,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.factualZ","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the observed instrument assigns each unit its realized instrument value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → P.Ω → α","l":79,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y0_le_ciInf_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Inf-over-support form of integral_Y0_le_upperBound0.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ⨅ z, S.upperBound0 hA.hi ↑z","l":122,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y0_le_upperBound0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ S.upperBound0 hA.hi z","l":61,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y1_le_ciInf_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Inf-over-support form of integral_Y1_le_upperBound1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ ⨅ z, S.upperBound1 hA.hi ↑z","l":108,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_Y1_le_upperBound1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level upper bound on E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ S.upperBound1 hA.hi z","l":49,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.integral_YofD_eq_total_law","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Total-law decomposition for YofD d across the binary partition {D=true} ⊔ {D=false}, via the generic Fintype total law.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions) (d : Bool), ∫ (ω : P.Ω), S.YofD d ω ∂P.μ = (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) (S.YofD d) + (P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) (S.YofD d)","l":100,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, a real outcome floor, and an instrument value, the lower bound for the control potential-outcome mean is the arm-bound functional that uses the floor for treated units in that instrument stratum.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":194,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0_le_cond_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on the *conditional expectation* of Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → S.lowerBound0 hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false)","l":216,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → S.lowerBound0 hA.lo z ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":55,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, a real outcome floor, and an instrument value, the lower bound for the treated potential-outcome mean is the arm-bound functional that uses the floor for untreated units in that instrument stratum.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":172,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1_le_cond_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Helpers","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on the *conditional expectation* of Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) {z : α}, z ∈ S.support → S.lowerBound1 hA.lo z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true)","l":204,"f":"Causalean/PO/ID/Partial/Manski/Helpers.lean"},{"n":"Causalean.PO.POManskiIVSystem.lowerBound1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":2,"d":"Stratum-level lower bound on E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z : α}, z ∈ S.support → S.lowerBound1 hA.lo z ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":43,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.mLower0","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, its base assumptions, including an outcome floor, and an instrument value in a linearly ordered instrument space, the lower monotone-instrument envelope for the control potential outcome is the supremum of the control-arm lower bounds over support values no greater than that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":82,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mLower1","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, its base assumptions, including an outcome floor, and an instrument value in a linearly ordered instrument space, the lower monotone-instrument envelope for the treated potential outcome is the supremum of the treated-arm lower bounds over support values no greater than that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":58,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mUpper0","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, its base assumptions, including an outcome ceiling, and an instrument value in a linearly ordered instrument space, the upper monotone-instrument envelope for the control potential outcome is the infimum of the control-arm upper bounds over support values no smaller than that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":94,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mUpper1","k":"def","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, its base assumptions, including an outcome ceiling, and an instrument value in a linearly ordered instrument space, the upper monotone-instrument envelope for the treated potential outcome is the infimum of the treated-arm upper bounds over support values no smaller than that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → (S : Causalean.PO.POManskiIVSystem P α) → S.BaseAssumptions → [LinearOrder α] → α → ℝ","l":70,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of manski_bounds_ATE. Under the baseline Manski assumptions and mean independence of both potential outcomes from the instrument, for any two instrument values z₁, z₀ in the support of the instrument, the average treatment effect lies in the closed interval from the z₁-lower/z₀-upper worst-case bound to the z₁-upper/z₀-lower worst-case bound — the per-stratum-pair Manski no-assumption sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z₁ z₀ : α}, z₁ ∈ S.support → z₀ ∈ S.support → S.ATE ∈ Set.Icc (S.lowerBound1 hA.lo z₁ - S.upperBound0 hA.hi z₀) (S.upperBound1 hA.hi z₁ - S.lowerBound0 hA.lo z₀)","l":35,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_ATE_mem_Icc_ciSup","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of manski_bounds_ATE_ciSup. Under the baseline Manski assumptions, mean independence of both potential outcomes from the instrument, and a nonempty instrument support, the average treatment effect lies in the closed interval from the supremum-of-lowers-minus-infimum-of-uppers bound to the infimum-of-uppers-minus-supremum-of-lowers bound, aggregated over every instrument stratum — the sup/inf-aggregated Manski no-assumption sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → S.ATE ∈ Set.Icc ((⨆ z, S.lowerBound1 hA.lo ↑z) - ⨅ z, S.upperBound0 hA.hi ↑z) ((⨅ z, S.upperBound1 hA.hi ↑z) - ⨆ z, S.lowerBound0 hA.lo ↑z)","l":49,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":1,"d":"Manski bounds for the ATE under an imperfect instrument. For any two instrument strata z₁ and z₀ in the support of the instrument, the average treatment effect is sandwiched between the worst-case lower bound formed from the z₁-stratum lower envelope and z₀-stratum upper envelope, and the corresponding upper bound with the roles reversed — using only bounded outcomes and mean independence of the instrument, with no selection assumptions.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → ∀ {z₁ z₀ : α}, z₁ ∈ S.support → z₀ ∈ S.support → S.lowerBound1 hA.lo z₁ - S.upperBound0 hA.hi z₀ ≤ S.ATE ∧ S.ATE ≤ S.upperBound1 hA.hi z₁ - S.lowerBound0 hA.lo z₀","l":72,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.manski_bounds_ATE_ciSup","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.NonAsp","a":"PO/ID/Partial/Manski","t":1,"d":"Manski bounds for the ATE in sup/inf form. Under the baseline Manski assumptions, mean independence of the instrument, and a nonempty instrument support, the average treatment effect is sandwiched between the supremum of the lower-envelope bounds minus the infimum of the upper-envelope bounds on one side, and the infimum of the upper-envelope bounds minus the supremum of the lower-envelope bounds on the other, aggregated over the whole instrument support.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions), S.MeanIndep → S.support.Nonempty → (⨆ z, S.lowerBound1 hA.lo ↑z) - ⨅ z, S.upperBound0 hA.hi ↑z ≤ S.ATE ∧ S.ATE ≤ (⨅ z, S.upperBound1 hA.hi ↑z) - ⨆ z, S.lowerBound0 hA.lo ↑z","l":129,"f":"Causalean/PO/ID/Partial/Manski/NonAsp.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Each treatment arm event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool), MeasurableSet (S.dEvent d)","l":133,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurableSet_zEvent","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Each instrument stratum event is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (z : α), MeasurableSet (S.zEvent z)","l":129,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a fixed treatment arm d, the counterfactual outcome Y(d) is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (d : Bool), Measurable (S.YofD d)","l":113,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualD","l":122,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualY","l":125,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measurable_factualZ","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":2,"d":"Factual instrument is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), Measurable S.factualZ","l":119,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.measure_mul_eventCondExp_const","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"(μ A).toReal * eventCondExp μ A (fun _ => c) = (μ A).toReal * c. Follows from eventCondExp_mul_measure_toReal with the constant integrand.","s":"∀ {P : Causalean.PO.POSystem} (A : Set P.Ω) (c : ℝ), ((P.μ A).toReal * Causalean.PO.eventCondExp P.μ A fun x => c) = (P.μ A).toReal * c","l":132,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":1,"d":"MIV ATE envelope bounds (prop:po-iv-miv, integrated form). Under the baseline Manski assumptions and a monotone instrumental variable — the conditional mean of each potential outcome is nondecreasing in the instrument value across its support, the average treatment effect is sandwiched between the integrated lower-envelope contrast ∫ (mLower1(Z) − mUpper0(Z)) and the integrated upper-envelope contrast ∫ (mUpper1(Z) − mLower0(Z)).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower1 hA (S.factualZ ω) - S.mUpper0 hA (S.factualZ ω) ∂P.μ ≤ S.ATE ∧ S.ATE ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ","l":356,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_cond_Y0_le_mUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(0) | Z = z] ≤ mUpper0 z for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false) ≤ S.mUpper0 hA z","l":179,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_cond_Y1_le_mUpper1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"E[Y(1) | Z = z] ≤ mUpper1 z for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true) ≤ S.mUpper1 hA z","l":137,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_Y0_le_integral_mUpper0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 0 (upper): ∫ Y(0) ≤ ∫ mUpper0(Z).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ∫ (ω : P.Ω), S.mUpper0 hA (S.factualZ ω) ∂P.μ","l":341,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_Y1_le_integral_mUpper1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 1 (upper): ∫ Y(1) ≤ ∫ mUpper1(Z).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) ∂P.μ","l":311,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_mLower0_le_integral_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 0 (lower): ∫ mLower0(Z) ≤ ∫ Y(0).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower0 hA (S.factualZ ω) ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":326,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_integral_mLower1_le_integral_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"Integrated envelope bound for arm d = 1 (lower): ∫ mLower1(Z) ≤ ∫ Y(1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions) (hMIV : S.MIV), ∫ (ω : P.Ω), S.mLower1 hA (S.factualZ ω) ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":296,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_mLower0_le_cond_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"mLower0 z ≤ E[Y(0) | Z = z] for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → S.mLower0 hA z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD false)","l":158,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.miv_mLower1_le_cond_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MIV","a":"PO/ID/Partial/Manski","t":2,"d":"mLower1 z ≤ E[Y(1) | Z = z] for every z ∈ supp Z.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] (hA : S.BaseAssumptions) (hMIV : S.MIV) {z : α}, z ∈ S.support → S.mLower1 hA z ≤ Causalean.PO.eventCondExp P.μ (S.zEvent z) (S.YofD true)","l":108,"f":"Causalean/PO/ID/Partial/Manski/MIV.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y0_le_E_Y","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 4 of prop:po-iv-mtr: E[Y(0)] ≤ E[Y].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ ∫ (ω : P.Ω), S.factualY ω ∂P.μ","l":146,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y1_le_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 2 of prop:po-iv-mtr: E[Y(1)] ≤ p·E[Y|D=1] + (1-p)·b.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (P.μ (S.dEvent false)).toReal * hA.hi","l":152,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_E_Y_le_E_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 1 of prop:po-iv-mtr: E[Y] ≤ E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → ∫ (ω : P.Ω), S.factualY ω ∂P.μ ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":140,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":1,"d":"The two-sided ATE sandwich from prop:po-iv-mtr. Under the baseline Manski assumptions and monotone treatment response, Y(0) ≤ Y(1) almost surely, the average treatment effect is nonnegative and is upper-bounded by the probability-weighted mix of the observed treated/control means and the range endpoints lo, hi — imputing hi for Y(1) on the control arm and lo for Y(0) on the treated arm.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → 0 ≤ S.ATE ∧ S.ATE ≤ (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (P.μ (S.dEvent false)).toReal * hA.hi - ((P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY + (P.μ (S.dEvent true)).toReal * hA.lo)","l":251,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_lower_le_E_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 3 of prop:po-iv-mtr: (1-p)·E[Y|D=0] + p·a ≤ E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), (P.μ (S.dEvent false)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY + (P.μ (S.dEvent true)).toReal * hA.lo ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":198,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_miv_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of mtr_miv_bounds_ATE. Under the baseline Manski assumptions, monotone treatment response, and a monotone instrumental variable, the average treatment effect lies in the closed interval from 0 to the integrated monotone-instrument envelope contrast ∫ (mUpper1(Z) − mLower0(Z)) — the MTR + MIV sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions), S.MTR → ∀ (hMIV : S.MIV), S.ATE ∈ Set.Icc 0 (∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ)","l":79,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_miv_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Combined","a":"PO/ID/Partial/Manski","t":1,"d":"cor:po-iv-mtr-miv. Under the baseline Manski assumptions, monotone treatment response, Y(0) ≤ Y(1) almost surely, and a monotone instrumental variable — the conditional mean of each potential outcome is nondecreasing in the instrument value across its support, the average treatment effect is nonnegative and is upper-bounded by the integrated monotone-instrument envelope contrast ∫ (mUpper1(Z) − mLower0(Z)).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) [MeasureTheory.IsFiniteMeasure P.μ] [Fintype α] (hA : S.BaseAssumptions), S.MTR → ∀ (hMIV : S.MIV), 0 ≤ S.ATE ∧ S.ATE ≤ ∫ (ω : P.Ω), S.mUpper1 hA (S.factualZ ω) - S.mLower0 hA (S.factualZ ω) ∂P.μ","l":56,"f":"Causalean/PO/ID/Partial/Manski/Combined.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_mts_ATE_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.IntervalForm","a":"PO/ID/Partial/Manski","t":1,"d":"Set.Icc form of mtr_mts_bounds_ATE. Under the baseline Manski assumptions, monotone treatment response, and monotone treatment selection, the average treatment effect lies in the closed interval from 0 to the naive observed contrast E[Y | D=1] − E[Y | D=0] — the MTR + MTS sandwich restated as interval membership.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → S.MTS → S.ATE ∈ Set.Icc 0 (Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY)","l":66,"f":"Causalean/PO/ID/Partial/Manski/IntervalForm.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_mts_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.Combined","a":"PO/ID/Partial/Manski","t":1,"d":"cor:po-iv-mtr-mts. Under the baseline Manski assumptions, monotone treatment response, Y(0) ≤ Y(1) almost surely, and monotone treatment selection — each potential outcome's mean given control is no larger than its mean given treatment, the average treatment effect is nonnegative and is upper-bounded by the naive observed contrast E[Y | D=1] − E[Y | D=0].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → S.MTS → 0 ≤ S.ATE ∧ S.ATE ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":36,"f":"Causalean/PO/ID/Partial/Manski/Combined.lean"},{"n":"Causalean.PO.POManskiIVSystem.mtr_nonneg_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTR","a":"PO/ID/Partial/Manski","t":2,"d":"Step 5 (ATE nonnegativity) of prop:po-iv-mtr: 0 ≤ ATE.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTR → 0 ≤ S.ATE","l":241,"f":"Causalean/PO/ID/Partial/Manski/MTR.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_E_Y0_le_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS upper bound on E[Y(0)]: E[Y(0)] ≤ p · b + (1-p) · E[Y|D=0].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → ∫ (ω : P.Ω), S.YofD false ω ∂P.μ ≤ (P.μ (S.dEvent true)).toReal * hA.hi + (1 - (P.μ (S.dEvent true)).toReal) * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":263,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_E_Y1_le_condY1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS upper bound on E[Y(1)]: E[Y(1)] ≤ E[Y|D=1].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → ∫ (ω : P.Ω), S.YofD true ω ∂P.μ ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY","l":205,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_bounds_ATE","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":1,"d":"MTS bounds for the ATE. Under the baseline Manski assumptions and monotone treatment selection — each potential outcome's mean given control is no larger than its mean given treatment, the average treatment effect is sandwiched between a lower bound built from the probability-weighted observed treated mean plus the range floor lo, minus the probability-weighted mix of the range ceiling hi and the observed control mean, and an upper bound equal to the naive observed contrast E[Y | D=1] − E[Y | D=0] — obtained by subtracting the Y(0) upper/lower bounds from the Y(1) lower/upper bounds.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (1 - (P.μ (S.dEvent true)).toReal) * hA.lo - ((P.μ (S.dEvent true)).toReal * hA.hi + (1 - (P.μ (S.dEvent true)).toReal) * Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY) ≤ S.ATE ∧ S.ATE ≤ Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY - Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY","l":311,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_condY0_le_E_Y0","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS lower bound on E[Y(0)]: E[Y|D=0] ≤ E[Y(0)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → Causalean.PO.eventCondExp P.μ (S.dEvent false) S.factualY ≤ ∫ (ω : P.Ω), S.YofD false ω ∂P.μ","l":236,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.mts_lower_le_E_Y1","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"MTS lower bound on E[Y(1)]: p · E[Y|D=1] + (1-p) · a ≤ E[Y(1)].","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α) (hA : S.BaseAssumptions), S.MTS → (P.μ (S.dEvent true)).toReal * Causalean.PO.eventCondExp P.μ (S.dEvent true) S.factualY + (1 - (P.μ (S.dEvent true)).toReal) * hA.lo ≤ ∫ (ω : P.Ω), S.YofD true ω ∂P.μ","l":143,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.prob_dEvent_true_add_false","k":"theorem","m":"Causalean.PO.ID.Partial.Manski.MTS","a":"PO/ID/Partial/Manski","t":2,"d":"Probability split: P(D=true).toReal + P(D=false).toReal = 1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : MeasurableSingletonClass α] (S : Causalean.PO.POManskiIVSystem P α), (P.μ (S.dEvent true)).toReal + (P.μ (S.dEvent false)).toReal = 1","l":65,"f":"Causalean/PO/ID/Partial/Manski/MTS.lean"},{"n":"Causalean.PO.POManskiIVSystem.support","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the instrument support is the set of instrument values whose observed-instrument stratum has nonzero probability.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Set α","l":146,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.upperBound0","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, a real outcome ceiling, and an instrument value, the upper bound for the control potential-outcome mean is the arm-bound functional that uses the ceiling for treated units in that instrument stratum.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":205,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.upperBound1","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, a real outcome ceiling, and an instrument value, the upper bound for the treated potential-outcome mean is the arm-bound functional that uses the ceiling for untreated units in that instrument stratum.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → ℝ → α → ℝ","l":183,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.yVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the real-valued outcome variable is its outcome represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P ℝ","l":64,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.zEvent","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, a Manski instrumental-variables system based on them, and an instrument value, the corresponding instrument stratum is the set of units whose observed instrument equals that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → α → Set P.Ω","l":97,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POManskiIVSystem.zVar","k":"def","m":"Causalean.PO.ID.Partial.Manski.Setup","a":"PO/ID/Partial/Manski","t":1,"d":"For a potential-outcome system, a measurable instrument-value space whose singleton values are measurable, and a Manski instrumental-variables system based on them, the instrument variable is its instrument represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSingletonClass α] → Causalean.PO.POManskiIVSystem P α → Causalean.PO.POVar P α","l":52,"f":"Causalean/PO/ID/Partial/Manski/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel","k":"structure","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Partially linear model under the backdoor PO framework. On top of the PO substrate, this bundles a measurable covariate function b giving the nonparametric baseline, a homogeneous per-unit treatment effect θ, a measurable structural error term U, the structural restriction that every unit's dose-response is the straight line Y(d) = b(X) + θ·d + U with the same slope for everyone, the standard backdoor (unconfoundedness) assumption that the structural error has zero mean conditional on the observed covariate and treatment, and consistency: the observed outcome is the potential outcome at the realized treatment.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max (max u_1 u_2) u_3) u_4)","l":193,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.causal_homogeneity","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Causal reading of θ. Almost surely, for every pair of dose levels, the difference of the corresponding potential outcomes equals the slope θ times the difference of the doses, so θ is the constant per-unit causal effect of the treatment.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (M : Causalean.PO.POPartialLinearModel P γ), ∀ᵐ (ω : P.Ω) ∂P.μ, ∀ (d d' : ℝ), M.YofD d ω - M.YofD d' ω = M.θ * (d - d')","l":252,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.condExp_U_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The structural error has zero conditional mean given the covariate. Follows from the backdoor assumption E[U | σ(X,D)] = 0 by the tower property, since σ(X) ⊆ σ(X,D).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), P.μ[M.U | M.sigmaX] =ᵐ[P.μ] 0","l":86,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.condExp_resid_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The treatment residual is conditionally mean-zero given the covariate: E[D − m₀(X) | σ(X)] = 0. Immediate from condExp_sub and idempotence of the conditional expectation.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → P.μ[M.resid | M.sigmaX] =ᵐ[P.μ] 0","l":99,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.factualY_eq","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed-data form of the structural model: almost surely Y = b(X) + θ·D + U. Obtained from the homogeneous dose-response evaluated at the realized treatment, using consistency Y = Y(D).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (M : Causalean.PO.POPartialLinearModel P γ), M.factualY =ᵐ[P.μ] fun ω => M.b (M.factualX ω) + M.θ * M.factualD ω + M.U ω","l":236,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearModel.factualY_sub_lReg","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"Partialling-out identity (observed data): Y − ℓ₀(X) = θ·(D − m₀(X)) + U. Algebraic consequence of factualY_eq and lReg_eq.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → (fun ω => M.factualY ω - M.lReg ω) =ᵐ[P.μ] fun ω => M.θ * M.resid ω + M.U ω","l":212,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.integral_U_resid","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"Orthogonality of the structural error to the covariate-treatment residual: E[U·(D − m₀(X))] = 0. Since the residual is σ(X,D)-measurable and E[U | σ(X,D)] = 0, the product integrates to zero (pull the residual out of the conditional expectation, then integrate).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), M.U ω * M.resid ω ∂P.μ = 0","l":121,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.integral_partialled","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The Robinson numerator equals θ times the residual second moment: E[(Y − ℓ₀(X))·(D − m₀(X))] = θ·E[(D − m₀(X))²]. Expand Y − ℓ₀ = θ·V + U and use E[U·V] = 0.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.resid ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), (M.factualY ω - M.lReg ω) * M.resid ω ∂P.μ = M.θ * ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ","l":225,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.lReg","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome model based on them, the outcome regression assigns each unit the conditional mean of its observed outcome given the σ-algebra generated by the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":68,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.lReg_eq","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The outcome regression decomposes as ℓ₀(X) = b(X) + θ·m₀(X). Apply the conditional expectation given σ(X) to the observed-data form Y = b(X) + θ·D + U: the covariate term is σ(X)-measurable, the treatment term contributes θ·m₀(X), and the error term vanishes by condExp_U_sigmaX.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → M.lReg =ᵐ[P.μ] fun ω => M.b (M.factualX ω) + M.θ * M.mReg ω","l":158,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.mReg","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome model based on them, the treatment regression assigns each unit the conditional mean of its observed treatment given the σ-algebra generated by the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":59,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.resid","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome model based on them, the treatment residual assigns each unit its observed treatment minus its treatment regression given the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearModel P γ → P.Ω → ℝ","l":77,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearModel.robinson_estimand_eq_theta","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Identification","a":"PO/ID/Exact/PartialLinear","t":1,"d":"The causal bridge. Given that the factual treatment is integrable, the outcome regression on covariates is integrable, the structural error U is integrable, the squared treatment residual is integrable, and the product of the structural error and the treatment residual is integrable, provided also that the treatment has residual variation, i.e. E[(D − m₀(X))²] ≠ 0, the Robinson partialling-out estimand equals the structural treatment effect θ. This is what makes the partially linear DML estimator target the causal parameter.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] [MeasureTheory.IsFiniteMeasure P.μ] (M : Causalean.PO.POPartialLinearModel P γ), MeasureTheory.Integrable M.factualD P.μ → MeasureTheory.Integrable (fun ω => M.b (M.factualX ω)) P.μ → MeasureTheory.Integrable M.U P.μ → MeasureTheory.Integrable (fun ω => M.resid ω ^ 2) P.μ → MeasureTheory.Integrable (fun ω => M.U ω * M.resid ω) P.μ → ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ ≠ 0 → (∫ (ω : P.Ω), (M.factualY ω - M.lReg ω) * M.resid ω ∂P.μ) / ∫ (ω : P.Ω), M.resid ω ^ 2 ∂P.μ = M.θ","l":243,"f":"Causalean/PO/ID/Exact/PartialLinear/Identification.lean"},{"n":"Causalean.PO.POPartialLinearSystem","k":"structure","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"Continuous-treatment backdoor subsystem. Inside a potential-outcome system, this bundles a real-valued treatment node D, a real-valued outcome node Y, and a covariate variable X taking values in an arbitrary measurable space, subject to the treatment, outcome, and covariate being pairwise distinct. This mirrors the binary backdoor subsystem but the treatment now ranges over the real line, as required by the partially linear model Y = g(X) + θ·D + noise.","s":"Causalean.PO.POSystem → (γ : Type u_1) → [MeasurableSpace γ] → Type (max (max u_1 u_2) u_3)","l":48,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, a partially linear potential-outcome system based on them, and a real-valued treatment dose, the potential outcome at that dose assigns each unit the outcome it would have if treatment were set to that dose.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → ℝ → P.Ω → ℝ","l":92,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.dVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the treatment variable is its treatment represented as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P ℝ","l":77,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the observed treatment level assigns each unit its realized real-valued treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → ℝ","l":97,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the observed covariate assigns each unit its realized covariate value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → γ","l":105,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualXD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the joint observed covariate-and-treatment map assigns each unit the pair consisting of its observed covariate and observed treatment level.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → γ × ℝ","l":141,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the observed outcome assigns each unit its realized outcome.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → P.Ω → ℝ","l":101,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The potential outcome under a fixed dose is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ) (d : ℝ), Measurable (S.YofD d)","l":110,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual treatment level is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualD","l":114,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualD_sigmaXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed treatment is measurable for the joint covariate-treatment sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualD","l":177,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualX","l":120,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The joint observed covariate-and-treatment map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualXD","l":148,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualXD_sigmaXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed covariate-treatment pair is measurable for the joint sigma-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualXD","l":165,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualX_sigmaX","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed covariate is measurable for the covariate sigma-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualX","l":135,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualX_sigmaXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The observed covariate is measurable for the joint covariate-treatment sigma-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualX","l":171,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), Measurable S.factualY","l":117,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the covariate σ-algebra is the σ-algebra on the sample space generated by the observed covariate.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → MeasurableSpace P.Ω","l":124,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaXD","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the joint covariate-treatment σ-algebra is the σ-algebra on the sample space generated jointly by the observed covariate and treatment.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → MeasurableSpace P.Ω","l":153,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaXD_le","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The joint covariate-treatment sigma-algebra is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaXD ≤ inferInstance","l":160,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX_le","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The covariate-generated sigma-algebra is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaX ≤ inferInstance","l":131,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.sigmaX_le_sigmaXD","k":"theorem","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":2,"d":"The covariate σ-algebra is contained in the joint covariate-treatment σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (S : Causalean.PO.POPartialLinearSystem P γ), S.sigmaX ≤ S.sigmaXD","l":183,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.xVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the covariate variable is its covariate represented as a potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P γ","l":87,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POPartialLinearSystem.yVar","k":"def","m":"Causalean.PO.ID.Exact.PartialLinear.Setup","a":"PO/ID/Exact/PartialLinear","t":1,"d":"For a potential-outcome system, a measurable covariate space, and a partially linear potential-outcome system based on them, the outcome variable is its outcome represented as a real-valued potential-outcome variable.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POPartialLinearSystem P γ → Causalean.PO.POVar P ℝ","l":82,"f":"Causalean/PO/ID/Exact/PartialLinear/Setup.lean"},{"n":"Causalean.PO.POProximalSystem","k":"structure","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE system. Bundles six distinguished potential-outcome variables inside a potential-outcome system: an observed covariate, a binary treatment, a treatment-side proxy, an outcome-side proxy, a real-valued outcome, and a latent confounder (def:po-proximal-system).","s":"Causalean.PO.POSystem → (γ_X : Type u_1) → (γ_Z : Type u_2) → (γ_W : Type u_3) → (γ_U : Type u_4) → [MeasurableSpace γ_X] → [MeasurableSpace γ_Z] → [MeasurableSpace γ_W] → [MeasurableSpace γ_U] → Type (max (max (max (max (max u_1 u_2) u_3) u_4) u_5) u_6)","l":29,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.A","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the factual binary treatment assigns to each unit its treatment under the factual, unmanipulated regime.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool","l":63,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.ATE","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system and a measure on its sample space, the average treatment effect is the marginal mean potential outcome under treatment one minus that under treatment zero.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → ℝ","l":175,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the joint treatment, latent-confounder, and covariate map assigns to each unit its factual treatment, latent confounder, and covariate.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_U × γ_X","l":115,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AX","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, the joint treatment-and-covariate map assigns to each unit its factual binary treatment and covariate.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_X","l":47,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AZUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the joint treatment, proxy, latent-confounder, and covariate map assigns to each unit its factual treatment, treatment-side proxy, latent confounder, and covariate.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_Z × γ_U × γ_X","l":123,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.AZX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the joint treatment, treatment-side-proxy, and covariate map assigns to each unit its factual treatment, treatment-side proxy, and covariate.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → Bool × γ_Z × γ_X","l":111,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE assumption bundle (Miao–Geng–Tchetgen Tchetgen proximal identification, def:po-proximal-assumptions). For a proximal system with covariate, binary treatment, treatment-side and outcome-side proxies, outcome, and latent confounder, this packages consistency (SUTVA), latent exchangeability: each potential outcome is independent of treatment given the latent confounder and covariate, the two proxy restrictions that the outcome-side proxy carries no information about the outcome beyond treatment, latent confounder, and covariate and the treatment-side proxy is independent of treatment and the outcome-side proxy given the latent confounder and covariate, a measurable outcome bridge function h satisfying the bridge equation that the outcome minus h evaluated at treatment, outcome-side proxy, and covariate has zero mean conditional on treatment, treatment-side proxy, and covariate, a positivity condition that every latent-confounder-and-covariate- measurable event of positive probability meets each treatment arm with positive probability, a completeness condition that within each treatment arm, functions of the latent confounder and covariate with zero bridge-conditional mean vanish almost surely, and integrability of the two potential outcomes, the composite h(A,W,X), and the bridge function evaluated at each fixed treatment arm.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max u_1 u_3)","l":35,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.Eofyofa_eq_Eh","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Main","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE identification (prop:po-proximal-ate step 1). Under the proximal identifying assumption bundle, including completeness within treatment level and integrability of the potential outcomes and bridge-function values, provided the treatment and outcome are distinct nodes, for each treatment level a the counterfactual mean outcome equals the mean of the proximal bridge function evaluated at that level: E[Y(a)] = E[h(a,W,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω), S.YofA a ω ∂μ = ∫ (ω : P.Ω), HA.h (a, S.W ω, S.X ω) ∂μ","l":53,"f":"Causalean/PO/ID/Exact/Proximal/Main.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.ate_proximal","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Main","a":"PO/ID/Exact/Proximal","t":1,"d":"Proximal ATE identification. Under the proximal identifying assumption bundle and given the treatment and outcome are distinct nodes, the average treatment effect E[Y(1)] − E[Y(0)] equals the bridge-function contrast E[h(1,W,X)] − E[h(0,W,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω), S.YofA true ω ∂μ - ∫ (ω : P.Ω), S.YofA false ω ∂μ = ∫ (ω : P.Ω), HA.h (true, S.W ω, S.X ω) ∂μ - ∫ (ω : P.Ω), HA.h (false, S.W ω, S.X ω) ∂μ","l":484,"f":"Causalean/PO/ID/Exact/Proximal/Main.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.integrable_Y","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":1,"d":"Compatibility projection. Under the proximal identifying assumption bundle, and given the treatment and outcome are distinct nodes, the factual outcome Y is integrable, as a consequence of the consistency assumption together with the integrability of the two potential-outcome cells Y(0) and Y(1).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable S.Y μ","l":169,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.integrable_YofA","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":2,"d":"Arm-uniform potential-outcome integrability. The bundle records integrability of the potential outcome under each of the two treatment arms separately; this states the same fact for a treatment arm left as a variable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), MeasureTheory.Integrable (S.YofA a) μ","l":160,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.integrable_h_arm","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":2,"d":"Arm-uniform bridge integrability. The bundle records integrability of the outcome bridge evaluated at each of the two treatment arms separately; this states the same fact for a treatment arm left as a variable, which is the form every downstream side condition actually needs.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), MeasureTheory.Integrable (fun ω => HA.h (a, S.W ω, S.X ω)) μ","l":149,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.Assumptions.integrable_mul_indicator","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Assumptions","a":"PO/ID/Exact/Proximal","t":2,"d":"The product of an integrable measurable function with a variable indicator is integrable, because the indicator only ever takes the values one and zero.","s":"∀ {P : Causalean.PO.POSystem} {μ : MeasureTheory.Measure P.Ω} {α : Type u_5} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α) {f : P.Ω → ℝ}, MeasureTheory.Integrable f μ → Measurable f → MeasureTheory.Integrable (fun ω => f ω * a.indicator x ω) μ","l":137,"f":"Causalean/PO/ID/Exact/Proximal/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvW","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed lower-envelope function of treatment and covariates, the lower W-proxy envelope condition holds exactly when the function is measurable [step:1] and, for every measurable nonnegative test function of the outcome-side proxy and covariates whose unweighted and envelope-weighted versions are integrable, its stratum-odds-ratio- and envelope-weighted integral over the specified treatment arm is at most its integral over the opposite treatment arm [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":216,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvWZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed lower-envelope function of treatment and covariates, the lower joint W--Z envelope condition holds exactly when the function is measurable [step:1] and, for every pair of measurable nonnegative test functions of respectively the outcome-side and treatment-side proxies with all three displayed products integrable, the envelope times the product of their conditional expectations given treatment and covariates is at most the conditional expectation of their product, almost everywhere in the specified treatment arm [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":283,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsLowerEnvZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed lower-envelope function of treatment and covariates, the lower Z-proxy envelope condition holds exactly when the function is measurable [step:1] and, for almost every unit in the specified treatment arm, the envelope evaluated at that treatment and covariate does not exceed the conditional mean of the factual outcome given treatment, treatment-side proxy, and covariates [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":244,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvW","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed upper-envelope function of treatment and covariates, the upper W-proxy envelope condition holds exactly when the function is measurable [step:1] and, for every measurable nonnegative test function of the outcome-side proxy and covariates whose unweighted and envelope-weighted versions are integrable, its integral over the opposite treatment arm is at most its stratum-odds-ratio- and envelope-weighted integral over the specified treatment arm [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":193,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvWZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed upper-envelope function of treatment and covariates, the upper joint W--Z envelope condition holds exactly when the function is measurable [step:1] and, for every pair of measurable nonnegative test functions of respectively the outcome-side and treatment-side proxies with all three displayed products integrable, the conditional expectation of their product given treatment and covariates is at most the envelope times the product of their conditional expectations, almost everywhere in the specified treatment arm [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":254,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.IsUpperEnvZ","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, a treatment level, and a proposed upper-envelope function of treatment and covariates, the upper Z-proxy envelope condition holds exactly when the function is measurable [step:1] and, for almost every unit in the specified treatment arm, the conditional mean of the factual outcome given treatment, treatment-side proxy, and covariates does not exceed the envelope evaluated at that treatment and covariate [step:2].","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → (Bool × γ_X → ℝ) → Prop","l":233,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.TwoProxyAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the two-proxy partial-identification theorem (Thm 3). Both proxies W and Z are present, but the strong exclusion restrictions W ⟂ A | (U, X) and Y ⟂ Z | (A, U, X) may FAIL — that is the \"invalid proxy\" angle. The only proxy-side conditional independence required is proxy_WZ_indep : W ⟂ Z | (A, X, U) (paper Assumption 6); in particular, no W ⟂ A | (U, X) requirement is bundled. Both bridge functions h and q are kept as fields, with their latent-side defining equations.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max (max (max u_1 u_2) u_3) u_7)","l":260,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.U","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the latent confounder assigns to each unit its unobserved confounder value.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_U","l":75,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.UX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the joint latent-confounder and covariate map assigns to each unit its latent confounder and covariate.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_U × γ_X","l":119,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.W","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the factual outcome-side proxy assigns to each unit its outcome-side proxy value under the factual regime.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_W","l":69,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.WBasedAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the W-proxy partial-identification theorem (Ghassami-Shpitser-Tchetgen Tchetgen 2024, Thm 1).","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max u_1 u_3)","l":49,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.X","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the factual covariate assigns to each unit its covariate under the factual, unmanipulated regime.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_X","l":60,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.Y","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the factual outcome assigns to each unit its real-valued outcome under the factual regime.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → ℝ","l":72,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.YofA","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system and a treatment level, the treatment-specific potential outcome assigns to each unit the real outcome it would have under an intervention setting treatment to that level.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → Bool → P.Ω → ℝ","l":101,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.YofA_essbound_above","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Y(a) inherits Y's a.e. upper bound under latent exchangeability and consistency. Requires overlap_strong: every σ_UX-measurable null-on-{A=a} set is globally null, so the single-arm bound lifts globally.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {a : Bool}, P.Consistency → ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → S.Avar.v ≠ S.Yvar.v → (∀ (s : Set P.Ω), MeasurableSet s → μ (s ∩ {ω | S.A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : P.Ω) ∂μ, S.Y ω ≤ M) → ∀ᵐ (ω : P.Ω) ∂μ, S.YofA a ω ≤ M","l":87,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.YofA_essbound_below","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Y(a) inherits Y's a.e. lower bound. Mirror of YofA_essbound_above.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {a : Bool}, P.Consistency → ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → S.Avar.v ≠ S.Yvar.v → (∀ (s : Set P.Ω), MeasurableSet s → μ (s ∩ {ω | S.A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : P.Ω) ∂μ, M ≤ S.Y ω) → ∀ᵐ (ω : P.Ω) ∂μ, M ≤ S.YofA a ω","l":105,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.Z","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the factual treatment-side proxy assigns to each unit its treatment-side proxy value under the factual regime.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → P.Ω → γ_Z","l":66,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.ZBasedAssumptions","k":"structure","m":"Causalean.PO.ID.Partial.Proxy.Assumptions","a":"PO/ID/Partial/Proxy","t":1,"d":"Assumption bundle for the Z-proxy partial-identification theorem (Thm 2). Drops proxy_WAZ, completeness, and W-related fields.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace P.Ω] → Type (max (max u_1 u_2) u_7)","l":132,"f":"Causalean/PO/ID/Partial/Proxy/Assumptions.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"In the W-proxy bundle, the observed conditional mean of the outcome equals the conditional mean of the bridge function on the matching treatment arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AX]","l":288,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX_core","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"Core observed-data collapse identity (no assumption-bundle dependency).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] [StandardBorelSpace P.Ω] {h : Bool × γ_W × γ_X → ℝ} (a : Bool), MeasureTheory.Integrable S.Y μ → MeasureTheory.Integrable (fun ω => h (S.A ω, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => h (a, S.W ω, S.X ω)) μ → μ[fun ω => S.Y ω - h (S.A ω, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] 0 → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => h (a, S.W ω, S.X ω) | S.σ_AX]","l":123,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_Y_eq_condExp_h_arm_AX_twoProxy","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct. Then, restricted to the stratum {A = a}, the conditional expectation of the observed outcome Y given the σ-algebra generated by treatment and covariates equals the conditional expectation of the outcome bridge function evaluated at (a, W, X) given that same σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AX]","l":303,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_drop_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_YZ : Y ⟂ Z | (A,U,X), conclude E[Y | σ(A,Z,U,X)] =ᵐ[μ] E[Y | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AZUX] =ᵐ[μ] μ[S.Y | S.σ_AUX]","l":81,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_drop_Z'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_YZ : Y ⟂ Z | (A,U,X) and integrability of Y, E[Y | σ(A,Z,U,X)] =ᵐ[μ] E[Y | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω], ProbabilityTheory.CondIndepFun S.σ_AUX (Causalean.PO.POProximalSystem.σ_AUX_le S) S.Y S.Z μ → MeasureTheory.Integrable S.Y μ → μ[S.Y | S.σ_AZUX] =ᵐ[μ] μ[S.Y | S.σ_AUX]","l":53,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_A","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_WAZ : W ⟂ (A,Z) | (U,X), projecting to W ⟂ A | (U,X), E[h(a,W,X) | σ(A,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_UX]","l":206,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_A'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_WA : W ⟂ A | (U,X), measurability and arm-integrability of the bridge h, conclude E[h(a,W,X) | σ(A,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {h_fun : Bool × γ_W × γ_X → ℝ}, ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) S.W S.A μ → Measurable h_fun → ∀ (a : Bool), MeasureTheory.Integrable (fun ω => h_fun (a, S.W ω, S.X ω)) μ → μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AUX] =ᵐ[μ] μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_UX]","l":172,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From proxy_WAZ : W ⟂ (A,Z) | (U,X), for any a : Bool, E[h(a,W,X) | σ(A,Z,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AZUX] =ᵐ[μ] μ[fun ω => HA.h (a, S.W ω, S.X ω) | S.σ_AUX]","l":159,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_h_drop_Z'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from proxy_WAZ : W ⟂ (A,Z) | (U,X), measurability and arm-integrability of the bridge h, conclude E[h(a,W,X) | σ(A,Z,U,X)] =ᵐ[μ] E[h(a,W,X) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] {h_fun : Bool × γ_W × γ_X → ℝ}, ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) S.W (fun ω => (S.A ω, S.Z ω)) μ → Measurable h_fun → ∀ (a : Bool), MeasureTheory.Integrable (fun ω => h_fun (a, S.W ω, S.X ω)) μ → μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AZUX] =ᵐ[μ] μ[fun ω => h_fun (a, S.W ω, S.X ω) | S.σ_AUX]","l":92,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.condExp_q_eq_stratumOddsRatio_arm_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.CondExpQ","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"q-collapse on the on-arm σ(A,X) (TwoProxy). Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct. Under the probability-ratio convention for the treatment-proxy bridge q encoded by likelihoodRatio_swapA_spec, restricted to the stratum {A = a}, the conditional expectation of q(Z, a, X) given the σ-algebra generated by treatment and covariates collapses to the σ(X)-measurable stratum odds ratio p(A≠a∣X)/p(A=a∣X):","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[fun ω => HA.q (S.Z ω, a, S.X ω) | S.σ_AX] =ᵐ[μ.restrict {ω | S.A ω = a}] S.stratumOddsRatio μ a","l":137,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/CondExpQ.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_eq_h_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.BridgeW","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Off-arm bridge-substitution identity (W-proxy). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle — latent exchangeability of the potential outcome Y(a) given treatment conditional on the latent confounder and covariates, independence of the proxy W from treatment given that same conditioning, the outcome-bridge equation, and consistency — together with the treatment and outcome variables being distinct. Then, restricted to the off-arm stratum {A ≠ a}, the average potential outcome Y(a) equals the average of the bridge function h evaluated at (a, W, X): ∫_{A≠a} Y(a) dμ = ∫_{A≠a} h(a, W, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ = ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, HA.h (a, S.W ω, S.X ω) ∂μ","l":40,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/BridgeW.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_eq_hq_armSwap_twoProxy","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.BridgeWZ","a":"PO/ID/Partial/Proxy/Helpers","t":1,"d":"Two-proxy bridge-substitution identity, same-arm form (Ghassami-Shpitser-Tchetgen Tchetgen 2024, Theorem 3, equation (★), in the codebase's probability-ratio q convention). Fix a treatment arm a and assume the two-proxy bridge assumption bundle — consistency, latent exchangeability of Y(a), the likelihood-ratio arm-swap relation linking the off-arm and on-arm measures, the outcome bridge h, the treatment-proxy bridge q, and conditional independence of the two proxies W and Z given treatment and covariates — together with the treatment and outcome variables being distinct. Then the average potential outcome Y(a) over the off-arm stratum {A ≠ a} equals the on-arm average of the product of the outcome bridge h(a, W, X) and the treatment-proxy bridge q(Z, a, X): ∫_{A≠a} Y(a) dμ = ∫_{A=a} h(a, W, X) · q(Z, a, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ = ∫ (ω : P.Ω) in {ω | S.A ω = a}, HA.h (a, S.W ω, S.X ω) * HA.q (S.Z ω, a, S.X ω) ∂μ","l":40,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/BridgeWZ.lean"},{"n":"Causalean.PO.POProximalSystem.condIntYofA_le_envelope_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased.ArmChain","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Off-arm bridge-substitution identity, upper envelope side. Fix a treatment arm a and assume the Z-based assumption bundle — consistency, latent exchangeability, the outcome-proxy independence condition, and the treatment-side bridge q — together with the treatment and outcome variables being distinct; let Uenv be an upper envelope function bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the average potential outcome Y(a) over the off-arm stratum {A ≠ a} is at most the average of the envelope Uenv(a, X) over that same stratum: ∫_{A≠a} Y(a) dμ ≤ ∫_{A≠a} Uenv(a, X) dμ.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ {Uenv : Bool × γ_X → ℝ}, S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ","l":164,"f":"Causalean/PO/ID/Partial/Proxy/ZBased/ArmChain.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, and a treatment level, the opposite-arm conditional potential-outcome mean is the integral of that level's potential outcome over units whose factual treatment differs from that level, divided by the measure of that stratum.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → ℝ","l":159,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_WZ_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.TwoProxy","a":"PO/ID/Partial/Proxy","t":1,"d":"Theorem 3 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, two-proxy partial identification). Fix a treatment arm a and assume the two-proxy assumption bundle — consistency, latent exchangeability, conditional independence of the proxies W and Z given treatment and covariates, the outcome and treatment-proxy bridges, and essential bounds on Y — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the same-arm joint-versus-product density ratio of the two proxies, with the off-arm stratum of positive mass and the envelope-weighted product of the outcome- and treatment-proxy conditional bridge means integrable. Then the conditional target E[Y(a) ∣ A ≠ a] is sandwiched between the essential Y-bounds and the integrated envelope bound, expressed entirely through observable quantities — the stratum odds ratio, the envelope functions, and the observed conditional mean of Y.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvWZ μ a Lenv → S.IsUpperEnvWZ μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)","l":74,"f":"Causalean/PO/ID/Partial/Proxy/TwoProxy.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_WZ_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 3 (condMeanYofA_WZ_bounds). Fix a treatment arm a and assume the two-proxy bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the same-arm joint-versus-product W-Z density ratio, with the off-arm stratum of positive mass and the envelope-weighted product of the two conditional bridge means integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the essential Y-bounds and the joint-WZ envelope clamps, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_WZ_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.TwoProxyAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvWZ μ a Lenv → S.IsUpperEnvWZ μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * μ[fun ω' => HA.h (a, S.W ω', S.X ω') | S.σ_AX] ω * μ[fun ω' => HA.q (S.Z ω', a, S.X ω') | S.σ_AX] ω) μ → S.condMeanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)) (min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ))","l":198,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_W_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.WBased","a":"PO/ID/Partial/Proxy","t":1,"d":"Theorem 1 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, W-only partial identification). Fix a treatment arm a and assume the W-only assumption bundle — consistency, latent exchangeability, independence of the proxy W from treatment given the latent confounder and covariates, the outcome bridge, and essential bounds on Y — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, with the off-arm stratum of positive mass and the envelope-weighted bridge moments integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies between the trivial essential-Y bound and the integrated envelope bound on each side.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)","l":64,"f":"Causalean/PO/ID/Partial/Proxy/WBased.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_W_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 1 (condMeanYofA_W_bounds). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, with the off-arm stratum of positive mass and the envelope-weighted bridge moments integrable. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the essential Y-bounds and the W-proxy envelope clamps, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_W_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → 0 < (μ {ω | S.A ω ≠ a}).toReal → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → S.condMeanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ)) (min (Classical.choose HA.Y_bdd_above) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ))","l":54,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_Z_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Theorem 2 (Ghassami-Shpitser-Tchetgen Tchetgen, arXiv 2304.04374). Fix a treatment arm a and assume the Z-based assumption bundle — consistency, latent exchangeability, the outcome-proxy independence condition Y ⟂ Z ∣ (A, U, X), and the treatment-side bridge q — together with the treatment and outcome variables being distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable, and the off-arm stratum of positive mass. Then the conditional target E[Y(a) ∣ A ≠ a] is sandwiched between the normalised {A ≠ a}-integrals of Lenv(a,X) and Uenv(a,X).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → 0 < (μ {ω | S.A ω ≠ a}).toReal → (μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ ≤ S.condMeanYofA μ a ∧ S.condMeanYofA μ a ≤ (μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ","l":69,"f":"Causalean/PO/ID/Partial/Proxy/ZBased.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_Z_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Theorem 2 (condMeanYofA_Z_bounds). Fix a treatment arm a and assume the Z-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable, and the off-arm stratum of positive mass. Then the conditional target E[Y(a) ∣ A ≠ a] lies in the closed interval spanned by the normalised on-arm integrals of Lenv and Uenv, the Set.Icc membership restatement of the scalar sandwich bound condMeanYofA_Z_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → 0 < (μ {ω | S.A ω ≠ a}).toReal → S.condMeanYofA μ a ∈ Set.Icc ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ) ((μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ)","l":128,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.condMeanYofA_eq","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The conditional potential outcome mean of an arm on the opposite-arm stratum is the integral of that arm's potential outcome over the stratum where treatment differs from the arm, divided by the mass of that stratum.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U) (μ : MeasureTheory.Measure P.Ω) (a : Bool), S.condMeanYofA μ a = (μ {ω | S.A ω ≠ a}).toReal⁻¹ * ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ","l":166,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.consistency_event","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":1,"d":"Factual-outcome bridge on the treatment-arm event. From the consistency assumption in the proximal identifying assumption bundle, and given the treatment and outcome are distinct nodes, on the event where the treatment equals arm a, the conditional expectation of the factual outcome given σ(A,U,X) agrees almost surely with the conditional expectation of the potential outcome Y(a) given the same σ-algebra, since Y = Y(a) pointwise there.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → μ[S.Y | S.σ_AUX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[S.YofA a | S.σ_AUX]","l":299,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.consistency_event'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from consistency, integrability of Y and Y(a), on {A=a} we have E[Y | σ(A,U,X)] =ᵐ[μ.restrict {A=a}] E[Y(a) | σ(A,U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] [StandardBorelSpace P.Ω], P.Consistency → ∀ (a : Bool), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable S.Y μ → MeasureTheory.Integrable (S.YofA a) μ → μ[S.Y | S.σ_AUX] =ᵐ[μ.restrict {ω | S.A ω = a}] μ[S.YofA a | S.σ_AUX]","l":253,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.envelope_le_condIntYofA_arm","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased.ArmChain","a":"PO/ID/Partial/Proxy/ZBased","t":2,"d":"Off-arm bridge-substitution identity, lower envelope side. Mirror of condIntYofA_le_envelope_arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ {Lenv : Bool × γ_X → ℝ}, S.IsLowerEnvZ μ a Lenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ","l":471,"f":"Causalean/PO/ID/Partial/Proxy/ZBased/ArmChain.lean"},{"n":"Causalean.PO.POProximalSystem.eq_zero_globally_of_eq_zero_on_arm","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From positivity_arm a and a σ_UX-measurable function that is μ-a.e. zero on the arm {A=a}, conclude it is μ-a.e. zero globally.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool) {f : P.Ω → ℝ}, Measurable f → f =ᵐ[μ.restrict {ω | S.A ω = a}] 0 → f =ᵐ[μ] 0","l":316,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.latent_exch_to_condExp","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"From latent_exch a : Y(a) ⟂ A | (U,X): E[Y(a) | σ(A,U,X)] =ᵐ[μ] E[Y(a) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.Assumptions μ) (a : Bool), μ[S.YofA a | S.σ_AUX] =ᵐ[μ] μ[S.YofA a | S.σ_UX]","l":241,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.latent_exch_to_condExp'","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Helpers","a":"PO/ID/Exact/Proximal","t":2,"d":"Field-level form: from latent_exch a : Y(a) ⟂ A | (U,X) and integrability of Y(a), conclude E[Y(a) | σ(A,U,X)] =ᵐ[μ] E[Y(a) | σ(U,X)].","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (a : Bool), ProbabilityTheory.CondIndepFun S.σ_UX (Causalean.PO.POProximalSystem.σ_UX_le S) (S.YofA a) S.A μ → MeasureTheory.Integrable (S.YofA a) μ → μ[S.YofA a | S.σ_AUX] =ᵐ[μ] μ[S.YofA a | S.σ_UX]","l":223,"f":"Causalean/PO/ID/Exact/Proximal/Helpers.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, and a treatment level, the marginal potential-outcome mean is the integral of the outcome that each unit would have under that treatment level with respect to that measure.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → ℝ","l":155,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_W_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.WBased","a":"PO/ID/Partial/Proxy","t":1,"d":"Corollary 1 (Ghassami-Shpitser-Tchetgen Tchetgen 2024, W-only marginal version). Fix a treatment arm a and assume the W-only assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, and assume the envelope-weighted bridge moments and the envelope-weighted observed conditional means are integrable. Then the marginal target E[Y(a)] lies between a trivial clamp and an envelope clamp on each side, plus the point-identified on-arm contribution.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → max (Classical.choose HA.Y_bdd_below * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ ≤ S.meanYofA μ a ∧ S.meanYofA μ a ≤ min (Classical.choose HA.Y_bdd_above * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":324,"f":"Causalean/PO/ID/Partial/Proxy/WBased.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_W_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Corollary 1 (meanYofA_W_bounds). Fix a treatment arm a and assume the W-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the W-proxy density ratio, and assume the envelope-weighted bridge moments and the envelope-weighted observed conditional means are integrable. Then the marginal target E[Y(a)] lies in the closed interval spanned by the W-proxy marginal lower and upper clamps, the Set.Icc membership restatement of the scalar sandwich bound meanYofA_W_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.WBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvW μ a Lenv → S.IsUpperEnvW μ a Uenv → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * HA.h (a, S.W ω, S.X ω)) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → MeasureTheory.Integrable (fun ω => S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω) μ → S.meanYofA μ a ∈ Set.Icc (max (Classical.choose HA.Y_bdd_below * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Lenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ) (min (Classical.choose HA.Y_bdd_above * (μ {ω | S.A ω ≠ a}).toReal) (∫ (ω : P.Ω) in {ω | S.A ω = a}, S.stratumOddsRatio μ a ω * Uenv (a, S.X ω) * μ[S.Y | S.σ_AX] ω ∂μ) + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ)","l":87,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_Z_bounds","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.ZBased","a":"PO/ID/Partial/Proxy/ZBased","t":1,"d":"Corollary 2 (Ghassami-Shpitser-Tchetgen Tchetgen, arXiv 2304.04374). Fix a treatment arm a and assume the Z-based assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the marginal target E[Y(a)] is sandwiched between the {A ≠ a}-integral of Lenv(a,X) and of Uenv(a,X), each plus the point-identified {A = a}-integral of Y.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ ≤ S.meanYofA μ a ∧ S.meanYofA μ a ≤ ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":118,"f":"Causalean/PO/ID/Partial/Proxy/ZBased.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_Z_mem_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.IntervalForm","a":"PO/ID/Partial/Proxy","t":1,"d":"Set.Icc form of Corollary 2 (meanYofA_Z_bounds). Fix a treatment arm a and assume the Z-only proximal bridge assumption bundle, with the treatment and outcome variables distinct; let Lenv, Uenv be lower and upper envelope functions bounding the σ(A,Z,X)-conditional mean of the outcome on the on-arm stratum, assumed integrable, with the envelope weighted by the treatment-proxy bridge and the envelope weighted by the likelihood-ratio arm-swap factor both integrable. Then the marginal target E[Y(a)] lies in the closed interval spanned by the Z-proxy marginal envelope integrals, the Set.Icc membership restatement of the scalar sandwich bound meanYofA_Z_bounds.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] [inst_5 : StandardBorelSpace P.Ω] (HA : S.ZBasedAssumptions μ) (a : Bool), S.Avar.v ≠ S.Yvar.v → ∀ (Lenv Uenv : Bool × γ_X → ℝ), S.IsLowerEnvZ μ a Lenv → S.IsUpperEnvZ μ a Uenv → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.q (S.Z ω, a, S.X ω)) μ → MeasureTheory.Integrable (fun ω => Lenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → MeasureTheory.Integrable (fun ω => Uenv (a, S.X ω) * HA.likelihoodRatio_swapA a ω) μ → S.meanYofA μ a ∈ Set.Icc (∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Lenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ) (∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, Uenv (a, S.X ω) ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ)","l":162,"f":"Causalean/PO/ID/Partial/Proxy/IntervalForm.lean"},{"n":"Causalean.PO.POProximalSystem.meanYofA_eq_strata","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"E[Y(a)] = ∫_{A = ¬a} Y(a) dμ + ∫_{A = a} Y dμ. Combines the stratum decomposition μ = μ.restrict {A=a} + μ.restrict {A=¬a} with consistency on the {A = a} arm.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] {S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U} {μ : MeasureTheory.Measure P.Ω}, P.Consistency → ∀ (a : Bool), S.Avar.v ≠ S.Yvar.v → MeasureTheory.Integrable (S.YofA a) μ → S.meanYofA μ a = ∫ (ω : P.Ω) in {ω | S.A ω ≠ a}, S.YofA a ω ∂μ + ∫ (ω : P.Ω) in {ω | S.A ω = a}, S.Y ω ∂μ","l":52,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_A","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.A","l":83,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, latent confounder, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AUX","l":134,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The treatment-covariate tuple map is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AX","l":50,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AZUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, treatment-side proxy, latent confounder, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AZUX","l":144,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AZX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, treatment-side proxy, and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AZX","l":129,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_AZX_σ_AZUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment, treatment-side proxy, and covariate tuple is measurable with respect to the σ-algebra generated by the larger tuple that also carries the latent confounder.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.AZX","l":273,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_A_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment is measurable with respect to the σ-algebra generated by treatment, latent confounder, and covariate, being the first coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.A","l":226,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_A_σ_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The treatment is measurable with respect to the σ-algebra generated by treatment and covariate, being the first coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.A","l":77,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_A_σ_AZX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment is measurable with respect to the σ-algebra generated by treatment, treatment-side proxy, and covariate, being the first coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.A","l":199,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_U","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual latent confounder is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.U","l":95,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_UX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The latent confounder and covariate tuple is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.UX","l":139,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_UX_σ_UX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The latent-confounder-and-covariate tuple is measurable with respect to the σ-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.UX","l":252,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_U_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The latent confounder is measurable with respect to the σ-algebra generated by treatment, latent confounder, and covariate, being the second coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.U","l":234,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_U_σ_UX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The latent confounder is measurable with respect to the σ-algebra generated by the latent confounder and covariate, being the first coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.U","l":257,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_W","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual outcome-side proxy is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.W","l":89,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual covariate is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":80,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The covariate is measurable with respect to the σ-algebra generated by treatment, latent confounder, and covariate, being the third coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":243,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X_σ_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The covariate is measurable with respect to the σ-algebra generated by treatment and covariate, being the second coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":85,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X_σ_AZX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The covariate is measurable with respect to the σ-algebra generated by treatment, treatment-side proxy, and covariate, being the third coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":217,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X_σ_UX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The covariate is measurable with respect to the σ-algebra generated by the latent confounder and covariate, being the second coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":265,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_X_σ_X","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The covariate is measurable with respect to the σ-algebra it generates.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.X","l":73,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_Y","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.Y","l":92,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_YofA","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U) (a : Bool), Measurable (S.YofA a)","l":104,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_Z","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The factual treatment-side proxy is measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.Z","l":86,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.measurable_Z_σ_AZX","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The treatment-side proxy is measurable with respect to the σ-algebra generated by treatment, treatment-side proxy, and covariate, being the second coordinate of that tuple.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), Measurable S.Z","l":207,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.stratumOddsRatio","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, a measure on its sample space, and a treatment level, the stratum odds-ratio function assigns each unit the conditional expected indicator of the opposite treatment arm given its covariates divided by the conditional expected indicator of its own arm given those covariates.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasureTheory.Measure P.Ω → Bool → P.Ω → ℝ","l":181,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by factual treatment, latent confounder, and covariate is the pullback to the sample space of the product σ-algebra for those three measurements.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":157,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, latent confounder, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AUX ≤ inferInstance","l":176,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AUX_le_σ_AZUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The sigma-algebra on treatment, unobserved confounder, and covariates is coarser than the sigma-algebra that also includes the Z proxy.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AUX ≤ S.σ_AZUX","l":119,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by factual treatment and covariate is the pullback to the sample space of their product σ-algebra.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":55,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX_le","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"The sigma-algebra generated by treatment and covariates is a sub-sigma-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AX ≤ inferInstance","l":59,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AX_le_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_AX is coarser than σ_AUX, since (A, X) factors through (A, U, X) via the projection (a, u, x) ↦ (a, x).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AX ≤ S.σ_AUX","l":132,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZUX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by factual treatment, treatment-side proxy, latent confounder, and covariate is the pullback to the sample space of the corresponding product σ-algebra.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":165,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZUX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, treatment-side proxy, latent confounder, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZUX ≤ inferInstance","l":186,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by factual treatment, treatment-side proxy, and covariate is the pullback to the sample space of the product σ-algebra for those three measurements.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":153,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":2,"d":"The sigma-algebra generated by treatment, treatment-side proxy, and covariate is a sub-sigma-algebra of the ambient space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZX ≤ inferInstance","l":171,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_AZX_le_σ_AZUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_AZX ≤ σ_AZUX, since (A, Z, X) factors through (A, Z, U, X).","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_AZX ≤ S.σ_AZUX","l":109,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_UX","k":"def","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by latent confounder and covariate is the pullback to the sample space of the product σ-algebra for those two measurements.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":161,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_UX_le","k":"theorem","m":"Causalean.PO.ID.Exact.Proximal.Setup","a":"PO/ID/Exact/Proximal","t":1,"d":"The σ-algebra generated by the latent confounder and covariate is a sub-σ-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_UX ≤ inferInstance","l":181,"f":"Causalean/PO/ID/Exact/Proximal/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X","k":"def","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"For a proximal potential-outcome system, the σ-algebra generated by the factual covariate is the pullback to the sample space of the covariate's σ-algebra.","s":"{P : Causalean.PO.POSystem} → {γ_X : Type u_1} → {γ_Z : Type u_2} → {γ_W : Type u_3} → {γ_U : Type u_4} → [inst : MeasurableSpace γ_X] → [inst_1 : MeasurableSpace γ_Z] → [inst_2 : MeasurableSpace γ_W] → [inst_3 : MeasurableSpace γ_U] → Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U → MeasurableSpace P.Ω","l":64,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":1,"d":"The σ-algebra generated by the covariates is a sub-σ-algebra of the ambient measurable space.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ inferInstance","l":68,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_AUX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X ≤ σ_AUX, by composition σ_X ≤ σ_AX ≤ σ_AUX.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_AUX","l":149,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_AX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X is coarser than σ_AX: X factors through (A, X) via the second projection.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_AX","l":93,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POProximalSystem.σ_X_le_σ_UX","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Setup","a":"PO/ID/Partial/Proxy","t":2,"d":"σ_X ≤ σ_UX, since X factors through (U, X) via the second projection.","s":"∀ {P : Causalean.PO.POSystem} {γ_X : Type u_1} {γ_Z : Type u_2} {γ_W : Type u_3} {γ_U : Type u_4} [inst : MeasurableSpace γ_X] [inst_1 : MeasurableSpace γ_Z] [inst_2 : MeasurableSpace γ_W] [inst_3 : MeasurableSpace γ_U] (S : Causalean.PO.POProximalSystem P γ_X γ_Z γ_W γ_U), S.σ_X ≤ S.σ_UX","l":102,"f":"Causalean/PO/ID/Partial/Proxy/Setup.lean"},{"n":"Causalean.PO.POSharpRDDSystem","k":"structure","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp regression-discontinuity model in the potential-outcome framework. A unit has a continuous running variable X, a deterministic treatment D that switches on exactly when X crosses the cutoff c, jumping from 0 to 1 at the cutoff, and a real outcome Y, with the outcome and treatment nodes distinct. The jump in the outcome regression at the cutoff identifies the cutoff-local average treatment effect (def:po-sharp-rdd-system).","s":"Causalean.PO.POSystem → Type (max u_1 u_2)","l":29,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.Assumptions","k":"structure","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp RDD assumption bundle (def:po-sharp-rdd-assumptions). For a sharp regression-discontinuity system, this packages consistency (SUTVA) and the deterministic cutoff rule: the treatment indicator agrees almost surely with whether the running variable has reached the cutoff. It supplies treatment-specific latent regression representatives mu and an observable outcome regression representative nu, each certified as a genuine regression function of the corresponding response on the running variable, with the latent representatives continuous at the cutoff; it also assumes the running variable has positive local probability mass on both sides of the cutoff and the observable outcome regression has well-defined one-sided limits at the cutoff.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Type","l":83,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.YofD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system and a binary treatment level, the treatment-specific potential outcome assigns to each unit the outcome it would have under an intervention setting treatment to that level.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Bool → P.Ω → ℝ","l":59,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.aemeasurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The observed running variable is almost-everywhere measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), AEMeasurable S.factualX P.μ","l":224,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.dEvent","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system and a binary treatment level, the factual-treatment event is the set of units whose factual treatment equals that level.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → Bool → Set P.Ω","l":76,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system, the factual treatment assigns to each unit its binary treatment under the factual regime.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → Bool","l":53,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualX","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system, the factual running variable assigns to each unit its real-valued forcing variable under the factual regime.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → ℝ","l":50,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.factualY","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system, the factual outcome assigns to each unit its real-valued outcome under the factual regime.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POSharpRDDSystem P → P.Ω → ℝ","l":56,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurableSet_dEvent","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment event is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (d : Bool), MeasurableSet (S.dEvent d)","l":79,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_YofD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The treatment-specific potential outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (d : Bool), Measurable (S.YofD d)","l":71,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualD","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual treatment is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualD","l":65,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualX","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual running variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualX","l":62,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.measurable_factualY","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The factual outcome is measurable.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P), Measurable S.factualY","l":68,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_left_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system satisfying its sharp-RDD assumptions, the chosen left-hand limit of the observable outcome regression at the cutoff is the real number supplied by the assumption that this one-sided limit exists.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":129,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_left_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any left-limit of ν at c coincides with μ_0 c.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Iio S.c)) (nhds L) → L = hA.mu false S.c","l":304,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_right_limit","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system satisfying its sharp-RDD assumptions, the chosen right-hand limit of the observable outcome regression at the cutoff is the real number supplied by the assumption that this one-sided limit exists.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":122,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.nu_right_limit_eq","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"Any right-limit of ν at c coincides with μ_1 c.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions) {L : ℝ}, Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Ioi S.c)) (nhds L) → L = hA.mu true S.c","l":294,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.rdd_identification","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"Sharp RDD identification at the cutoff (textbook form). Under the sharp-RDD assumption bundle — SUTVA consistency, deterministic treatment assignment D = 1{X ≥ c} almost surely, regression-function representatives for the potential and observed outcomes, continuity of the potential-outcome regression functions at the cutoff, positive local mass of the running variable on each side of c, and existence of both one-sided limits of the observed-outcome regression function at c, the cutoff-local treatment effect equals the difference of the right- and left-hand limits of the observed-outcome regression function at the cutoff:","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), S.tau_RDD hA = S.nu_right_limit hA - S.nu_left_limit hA","l":316,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tau_RDD","k":"def","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":1,"d":"For a sharp regression-discontinuity system satisfying its sharp-RDD assumptions, the cutoff-local regression-discontinuity estimand is the treated treatment-specific regression representative at the cutoff minus the untreated representative there.","s":"{P : Causalean.PO.POSystem} → (S : Causalean.PO.POSharpRDDSystem P) → S.Assumptions → ℝ","l":115,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tendsto_nu_left_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen left-hand observable regression limit is a genuine left-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Iio S.c)) (nhds (S.nu_left_limit hA))","l":142,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSharpRDDSystem.tendsto_nu_right_limit","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.SharpRDD","a":"PO/ID/Exact/RDD","t":2,"d":"The chosen right-hand observable regression limit is a genuine right-hand limit at the cutoff.","s":"∀ {P : Causalean.PO.POSystem} (S : Causalean.PO.POSharpRDDSystem P) (hA : S.Assumptions), Filter.Tendsto hA.nu (nhdsWithin S.c (Set.Ioi S.c)) (nhds (S.nu_right_limit hA))","l":136,"f":"Causalean/PO/ID/Exact/RDD/SharpRDD.lean"},{"n":"Causalean.PO.POSystem","k":"structure","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"A potential-outcome system consists of a finite set of variables, a measurable value space for each variable, a measurable sample space carrying a probability measure, and, for every intervention regime and sample point, a jointly measurable assignment of potential-outcome values to all variables.","s":"Type (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1))","l":28,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system whose sample space is standard Borel, a regimed potential-outcome variable, a finite counterfactual bundle, a regimed conditioning variable, and a finite measure on the sample space, the conditional counterfactual-independence condition says that the first variable's potential outcome is conditionally independent of the bundle's joint potential-outcome vector given the σ-algebra generated by the conditioning variable.","s":"{α : Type u_1} → {γ : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace γ] → (P : Causalean.PO.POSystem) → [StandardBorelSpace P.Ω] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.RegimedVar P γ → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":108,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.component","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Component projection: conditional independence of a.value from a single coordinate of the bundle follows from CondIndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCF a B c μ → ∀ (i : Fin B.n), ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value (fun ω => B.jointValue ω i) μ","l":196,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.project","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"Fix a potential-outcome system in which a regimed variable a is conditionally independent of a counterfactual bundle B given a regimed variable c. Then for any measurable function ψ of the bundle's joint value, the value of a remains conditionally independent, given c, of ψ composed with the bundle's joint value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {ψ : ((i : Fin B.n) → B.type i) → β}, P.CondIndepCF a B c μ → Measurable ψ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value (ψ ∘ B.jointValue) μ","l":179,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCF.toCondIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: CondIndepCF unfolds to CondIndepFun with the comap σ-algebra generated by c.value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCF a B c μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap c.value inferInstance) (Measurable.comap_le (Causalean.PO.RegimedVar.measurable_value c)) a.value B.jointValue μ","l":137,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle","k":"def","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system whose sample space is standard Borel, a measurable outcome space, a potential-outcome variable taking values in that space, a counterfactual bundle, a conditioning counterfactual bundle, and a finite measure on the sample space, the conditional-independence assertion states that the variable and the first bundle are conditionally independent given the σ-algebra generated by jointly observing the conditioning bundle.","s":"(P : Causalean.PO.POSystem) → [StandardBorelSpace P.Ω] → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → Causalean.PO.POCFBundle P → (μ : optParam (MeasureTheory.Measure P.Ω) P.μ) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":289,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle.project","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":1,"d":"Bundle-conditional projection. Given a regimed variable a that is conditionally independent, given the σ-algebra generated by a counterfactual bundle C, of another counterfactual bundle B, then for any measurable function ψ of the joint value of B, the value of a remains conditionally independent, given C's σ-algebra, of ψ composed with the joint value of B.","s":"∀ {P : Causalean.PO.POSystem} [inst : StandardBorelSpace P.Ω] {α : Type u_1} {β : Type u_2} [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {C : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {ψ : ((i : Fin B.n) → B.type i) → β}, P.CondIndepCFBundle a B C μ → Measurable ψ → ProbabilityTheory.CondIndepFun C.sigma (Causalean.PO.POCFBundle.sigma_le C) a.value (ψ ∘ B.jointValue) μ","l":317,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.CondIndepCFBundle.toCondIndepFun","k":"theorem","m":"Causalean.PO.Conditioning.Bundle","a":"PO/Conditioning","t":2,"d":"Bridge: CondIndepCFBundle unfolds to CondIndepFun w.r.t. C.sigma.","s":"∀ {P : Causalean.PO.POSystem} [inst : StandardBorelSpace P.Ω] {α : Type u_1} [inst_1 : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {C : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} [inst_2 : MeasureTheory.IsFiniteMeasure μ], P.CondIndepCFBundle a B C μ → ProbabilityTheory.CondIndepFun C.sigma (Causalean.PO.POCFBundle.sigma_le C) a.value B.jointValue μ","l":308,"f":"Causalean/PO/Conditioning/Bundle.lean"},{"n":"Causalean.PO.POSystem.Consistency","k":"structure","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"A potential-outcome system is consistent when two conditions hold. First, for every intervention regime and every finite set of variables disjoint from the regime's target, a unit's potential outcomes for that set under the regime equal its factual potential outcomes whenever it factually agrees with the regime. Second, for every pair of disjoint regimes and every finite set of variables disjoint from the union of their targets, a unit's potential outcomes for that set under the composed regime equal its potential outcomes under the first regime alone, whenever it agrees with the second regime after the first has been applied.","s":"Causalean.PO.POSystem → Prop","l":47,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.FactualAgrees","k":"def","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, an intervention regime, and a unit in its sample space, factual agreement means that every variable targeted by the regime has, for that unit under the factual regime, exactly the value assigned by the intervention.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → P.Ω → Prop","l":25,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.IndepCF","k":"def","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, a regimed potential-outcome variable, a finite counterfactual bundle, and a measure on the sample space, the counterfactual-independence condition says that the variable's potential outcome is independent of the bundle's joint potential-outcome vector under that measure.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → (P : Causalean.PO.POSystem) → Causalean.PO.RegimedVar P α → Causalean.PO.POCFBundle P → optParam (MeasureTheory.Measure P.Ω) P.μ → Prop","l":100,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.component","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Component projection: independence of a.value from a single coordinate of the bundle follows from IndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF a B μ → ∀ (i : Fin B.n), ProbabilityTheory.IndepFun a.value (fun ω => B.jointValue ω i) μ","l":171,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.ofIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: IndepFun ⇒ IndepCF (reverse direction, also trivial).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, ProbabilityTheory.IndepFun a.value B.jointValue μ → P.IndepCF a B μ","l":132,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.project","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Projection: independence of a.value from any measurable function of the bundle's joint value follows from IndepCF.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω} {ψ : ((i : Fin B.n) → B.type i) → β}, P.IndepCF a B μ → Measurable ψ → ProbabilityTheory.IndepFun a.value (ψ ∘ B.jointValue) μ","l":163,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IndepCF.toIndepFun","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Bridge: IndepCF is by definition IndepFun a.value B.jointValue μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF a B μ → ProbabilityTheory.IndepFun a.value B.jointValue μ","l":127,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.IntermediateAgrees","k":"def","m":"Causalean.PO.Assumptions.Consistency","a":"PO/Assumptions","t":1,"d":"For a potential-outcome system, a first intervention regime, a second intervention regime, and a unit in its sample space, intermediate agreement means that, after the first intervention, every variable targeted by the second has exactly the value assigned by the second.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → Causalean.PO.Regime P.V P.X → P.Ω → Prop","l":36,"f":"Causalean/PO/Assumptions/Consistency.lean"},{"n":"Causalean.PO.POSystem.component","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"For a potential-outcome system, an intervention regime, and a variable, the coordinate potential outcome assigns to each unit the value that variable would take under that intervention.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (v : P.V) → P.Ω → P.X v","l":61,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.component_apply","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"At a given unit, the coordinate potential outcome of a variable under a regime is the system's world evaluation of that variable at that unit under that regime.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (v : P.V) (ω : P.Ω), P.component r v ω = P.eval r ω v","l":70,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.condIndepCF_congr_cond","k":"theorem","m":"Causalean.PO.Assumptions.IndepCF","a":"PO/Assumptions","t":2,"d":"Transport CondIndepCF across equality of the conditioning variables' generated σ-algebras.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {γ : Type u_2} {γ' : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace γ] [inst_2 : MeasurableSpace γ'] [inst_3 : StandardBorelSpace P.Ω] {a : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {c : Causalean.PO.RegimedVar P γ} {c' : Causalean.PO.RegimedVar P γ'} {μ : MeasureTheory.Measure P.Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ], MeasurableSpace.comap c.value inferInstance = MeasurableSpace.comap c'.value inferInstance → P.CondIndepCF a B c μ → P.CondIndepCF a B c' μ","l":149,"f":"Causalean/PO/Assumptions/IndepCF.lean"},{"n":"Causalean.PO.POSystem.counterfactualDist","k":"def","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a potential-outcome system and a finite ordered list of counterfactual queries, the joint counterfactual distribution is the probability measure induced by applying the cross-world evaluation map to a random unit drawn from the system's probability measure.","s":"(P : Causalean.PO.POSystem) → (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) → MeasureTheory.Measure ((i : Fin qs.length) → Causalean.ValuesOn qs[i].2 P.X)","l":61,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.counterfactualDist_eq","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"The joint counterfactual law of a finite list of queries is the pushforward of the system's probability measure along the cross-world evaluation map for those queries.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)), P.counterfactualDist qs = MeasureTheory.Measure.map (P.crossWorldEval qs) P.μ","l":74,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.counterfactualDist_marginal","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a finite list of counterfactual queries qs and an index i into that list, the i-th coordinate marginal of the joint counterfactual distribution over all queries equals the potential-outcome law for query i alone.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) (i : Fin qs.length), MeasureTheory.Measure.map (fun f => f i) (P.counterfactualDist qs) = P.poOperator qs[i].1 qs[i].2","l":98,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.crossWorldEval","k":"def","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":1,"d":"For a potential-outcome system and a finite ordered list of queries, each consisting of an intervention regime and a finite variable set, the cross-world evaluation map assigns to every unit the tuple whose $i$th component is the joint potential outcome for the $i$th query.","s":"(P : Causalean.PO.POSystem) → (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) → P.Ω → (i : Fin qs.length) → Causalean.ValuesOn qs[i].2 P.X","l":25,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.crossWorldEval_apply","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"Reading off the i-th coordinate of the cross-world evaluation of a unit returns the joint potential outcome of the i-th query's variable set under the i-th query's regime, at that same unit.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)) (ω : P.Ω) (i : Fin qs.length), P.crossWorldEval qs ω i = P.poVariable qs[i].1 qs[i].2 ω","l":39,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_eq_integral_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Drop-of-conditioning (quotient form). Suppose a regimed variable rv is independent of a counterfactual bundle B, where h is a measurable function of the bundle's joint value and {x} is a measurable singleton in the range of rv. If the event {rv = x} has positive and finite measure, then the event-level conditional expectation of h composed with the bundle's joint value, given {rv = x}, equals the unconditional integral of the same composite.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {rv : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF rv B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → μ (rv.value ⁻¹' {x}) ≠ 0 → μ (rv.value ⁻¹' {x}) ≠ ⊤ → (Causalean.PO.eventCondExp μ (rv.value ⁻¹' {x}) fun ω => h (B.jointValue ω)) = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":296,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_event_eq_integral_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning on factual events (quotient form). Specialisation of the above to the factual event a.event x.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → μ (a.event x) ≠ 0 → μ (a.event x) ≠ ⊤ → (Causalean.PO.eventCondExp μ (a.event x) fun ω => h (B.jointValue ω)) = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":323,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.eventCondExp_of_consistency_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Drop-of-conditioning on an event, given a bundle relabeling of the integrand. Suppose a factual variable a is independent of a counterfactual bundle B, where h is a measurable function of the bundle's joint value and {x} is a measurable singleton in the range of a. If the factual integrand factualF agrees, almost everywhere on the event {a = x}, with h composed with the bundle's joint value, and that event has positive and finite measure, then the event-level conditional expectation of factualF given {a = x} equals the unconditional integral of h composed with the bundle's joint value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {factualF : P.Ω → ℝ} {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → (factualF =ᵐ[μ.restrict (a.event x)] fun ω => h (B.jointValue ω)) → μ (a.event x) ≠ 0 → μ (a.event x) ≠ ⊤ → Causalean.PO.eventCondExp μ (a.event x) factualF = ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":363,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.factualAgrees_empty","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"FactualAgrees holds vacuously for the empty regime.","s":"∀ {P : Causalean.PO.POSystem} (ω : P.Ω), P.FactualAgrees Causalean.PO.Regime.empty ω","l":46,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POSystem.factualAgrees_sqcup","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Combinator: FactualAgrees for a disjoint union reduces to FactualAgrees for each component.","s":"∀ {P : Causalean.PO.POSystem} {r₁ r₂ : Causalean.PO.Regime P.V P.X} (h : r₁.Disjoint r₂) {ω : P.Ω}, P.FactualAgrees r₁ ω → P.FactualAgrees r₂ ω → P.FactualAgrees (r₁.sqcup r₂ h) ω","l":51,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POSystem.instIsProbabilityMeasureForallValuesOnVSndRegimeXFinsetGetElemListProdFinLengthLtNatValCounterfactualDist","k":"instance","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"For a potential-outcomes system and a finite ordered list of counterfactual queries, each consisting of an intervention regime and a finite variable set, the joint cross-world counterfactual distribution for that list is a probability measure.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)), MeasureTheory.IsProbabilityMeasure (P.counterfactualDist qs)","l":86,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.instIsProbabilityMeasureValuesOnVXPoOperator","k":"instance","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"For a potential-outcomes system, an intervention regime, and a finite set of variables, the distribution of those variables' joint potential outcome under that regime is a probability measure.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V), MeasureTheory.IsProbabilityMeasure (P.poOperator r Y)","l":144,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.integral_event_eq_mul_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning for the factual event. Specialisation of POSystem.integral_restrict_value_eq_mul_of_IndepCF to a factual POVar, using POVar.event directly.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {a : Causalean.PO.POVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF (Causalean.PO.RegimedVar.ofFactual a) B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ (x : α), MeasurableSet {x} → ∫ (ω : P.Ω) in a.event x, h (B.jointValue ω) ∂μ = (μ (a.event x)).toReal * ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":275,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.integral_restrict_value_eq_mul_of_IndepCF","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Drop-of-conditioning (multiplied form). If a regimed variable rv is independent of the counterfactual bundle B, then the integral of h ∘ B.jointValue over the preimage rv.value ⁻¹' {x} factors as (μ (rv.value ⁻¹' {x})).toReal * ∫ h ∘ B.jointValue ∂μ.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] {rv : Causalean.PO.RegimedVar P α} {B : Causalean.PO.POCFBundle P} {μ : MeasureTheory.Measure P.Ω}, P.IndepCF rv B μ → ∀ {h : ((i : Fin B.n) → B.type i) → ℝ}, Measurable h → ∀ (x : α), MeasurableSet {x} → ∫ (ω : P.Ω) in rv.value ⁻¹' {x}, h (B.jointValue ω) ∂μ = (μ (rv.value ⁻¹' {x})).toReal * ∫ (ω : P.Ω), h (B.jointValue ω) ∂μ","l":253,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POSystem.liftRegime","k":"def","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"For a potential-outcome system, a finite collection of its variables, and an intervention regime on that collection, the lifted intervention regime is the regime on the original variable collection that targets the same variables and assigns each its restricted-regime value.","s":"(P : Causalean.PO.POSystem) → (R : Finset P.V) → (Causalean.PO.Regime ↥R fun v => P.X ↑v) → Causalean.PO.Regime P.V P.X","l":28,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_assign","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Reading the lifted assignment at a known sub-regime member returns the original sub-assignment. Membership proofs are propositional, so the ambient hv is irrelevant.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (w : ↥R) (hw : w ∈ r'.target) (hv : ↑w ∈ (P.liftRegime R r').target), (P.liftRegime R r').assign (↑w) hv = r'.assign w hw","l":57,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_disjoint","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Disjoint restricted regimes remain disjoint after lifting them to the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v}, r₁'.Disjoint r₂' → (P.liftRegime R r₁').Disjoint (P.liftRegime R r₂')","l":121,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_empty","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Lifting the empty restricted regime gives the empty regime on the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), P.liftRegime R Causalean.PO.Regime.empty = Causalean.PO.Regime.empty","l":68,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_sqcup","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The lift commutes with Regime.sqcup (full equality).","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v} (h : r₁'.Disjoint r₂'), P.liftRegime R (r₁'.sqcup r₂' h) = (P.liftRegime R r₁').sqcup (P.liftRegime R r₂') (Causalean.PO.POSystem.liftRegime_disjoint P R h)","l":143,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_sqcup_target","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The lift commutes with Regime.sqcup (target equality).","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) {r₁' r₂' : Causalean.PO.Regime ↥R fun v => P.X ↑v} (h : r₁'.Disjoint r₂'), (P.liftRegime R (r₁'.sqcup r₂' h)).target = (P.liftRegime R r₁').target ∪ (P.liftRegime R r₂').target","l":135,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.liftRegime_target","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The target of a lifted restricted regime is the image of the restricted target in the original variable set.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v), (P.liftRegime R r').target = Finset.map { toFun := Subtype.val, inj' := Subtype.val_injective } r'.target","l":50,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.measurable_component","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"For an intervention regime r and a variable v, the coordinate potential outcome of v under r is a measurable function of the unit.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (v : P.V), Measurable (P.component r v)","l":82,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.measurable_crossWorldEval","k":"theorem","m":"Causalean.PO.Core.Counterfactual","a":"PO/Core","t":2,"d":"The cross-world evaluation map for a finite list of counterfactual queries is measurable.","s":"∀ (P : Causalean.PO.POSystem) (qs : List (Causalean.PO.Regime P.V P.X × Finset P.V)), Measurable (P.crossWorldEval qs)","l":52,"f":"Causalean/PO/Core/Counterfactual.lean"},{"n":"Causalean.PO.POSystem.measurable_poVariable","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"The joint potential outcome for any finite set of variables under any intervention regime is measurable.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V), Measurable (P.poVariable r Y)","l":113,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.ofSCM","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model and a background assignment of its fixed variables, the induced potential-outcome system is the potential-outcome system whose units are latent-variable assignments and whose variables are all observed random nodes of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.PO.POSystem","l":254,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POSystem.ofSCM_condIndepCF_of_dSep","k":"theorem","m":"Causalean.PO.Bridge.FromSCMCondIndep","a":"PO/Bridge","t":1,"d":"SCM-to-PO conditional-independence bridge under d-separation. Fix a structural causal model M with a fixed-value assignment s, and SWIG node sets that are each contained in the model's random variables and pairwise disjoint from one another. Suppose the first node set is d-separated from the second by the third in the model's DAG. Suppose further that there is a measurable value-space map aMap for the first node set and a measurable value-space map BMap for the second node set such that, under the latent draw, the potential-outcome value of a regimed variable a equals aMap applied to the projection of the evaluated model state onto the first node set, the joint value of a counterfactual bundle B equals BMap applied to the projection onto the second node set, and the conditioning value of a regimed variable c equals the projection onto the third node set. Then, in the potential-outcome system induced by M at s, a and B are conditionally independent given c.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [inst_5 : StandardBorelSpace (Causalean.PO.POSystem.ofSCM M s).Ω] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s')] {X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), Disjoint X Y → Disjoint X Z → Disjoint Y Z → M.dag.dSep X Y Z → ∀ {α : Type u_1} [inst_9 : MeasurableSpace α] [StandardBorelSpace α] [Nonempty α] (a : Causalean.PO.RegimedVar (Causalean.PO.POSystem.ofSCM M s) α) (B : Causalean.PO.POCFBundle (Causalean.PO.POSystem.ofSCM M s)) [StandardBorelSpace ((i : Fin B.n) → B.type i)] [Nonempty ((i : Fin B.n) → B.type i)] (c : Causalean.PO.RegimedVar (Causalean.PO.POSystem.ofSCM M s) (Causalean.ValuesOn Z (Causalean.swigΩ Ω))) (aMap : Causalean.ValuesOn X (Causalean.swigΩ Ω) → α) (BMap : Causalean.ValuesOn Y (Causalean.swigΩ Ω) → (i : Fin B.n) → B.type i), Measurable aMap → Measurable BMap → (a.value = aMap ∘ Causalean.valuesProjection hX ∘ fun ℓ => M.evalMap s ℓ) → (B.jointValue = BMap ∘ Causalean.valuesProjection hY ∘ fun ℓ => M.evalMap s ℓ) → (c.value = Causalean.valuesProjection hZ ∘ fun ℓ => M.evalMap s ℓ) → (Causalean.PO.POSystem.ofSCM M s).CondIndepCF a B c (Causalean.PO.POSystem.ofSCM M s).μ","l":91,"f":"Causalean/PO/Bridge/FromSCMCondIndep.lean"},{"n":"Causalean.PO.POSystem.ofSCM_consistency","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"For a structural causal model M and an assignment of values to its fixed background variables s, the potential-outcome system induced by M and s satisfies the consistency assumption.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), (Causalean.PO.POSystem.ofSCM M s).Consistency","l":408,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.POSystem.poOperator","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"For a potential-outcome system, an intervention regime, and a finite set of variables, the potential-outcome law is the distribution of those variables' joint potential outcome under that intervention, induced by the system's probability measure.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (Y : Finset P.V) → MeasureTheory.Measure (Causalean.ValuesOn Y P.X)","l":122,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.poOperator_eq","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"The potential-outcome law of a finite variable set under a regime is the pushforward of the system's probability measure along the joint potential outcome for that set and regime.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V), P.poOperator r Y = MeasureTheory.Measure.map (P.poVariable r Y) P.μ","l":132,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.poVariable","k":"def","m":"Causalean.PO.Core.System","a":"PO/Core","t":1,"d":"For a potential-outcome system, an intervention regime, and a finite set of variables, the joint potential outcome assigns to each unit the vector of values that all variables in the set would take under that intervention.","s":"(P : Causalean.PO.POSystem) → Causalean.PO.Regime P.V P.X → (Y : Finset P.V) → P.Ω → Causalean.ValuesOn Y P.X","l":90,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.poVariable_apply","k":"theorem","m":"Causalean.PO.Core.System","a":"PO/Core","t":2,"d":"Reading off one coordinate of the joint potential outcome of a finite variable set under a regime gives the system's world evaluation of that coordinate's variable, at the same unit and under the same regime.","s":"∀ (P : Causalean.PO.POSystem) (r : Causalean.PO.Regime P.V P.X) (Y : Finset P.V) (ω : P.Ω) (v : ↥Y), P.poVariable r Y ω v = P.eval r ω ↑v","l":100,"f":"Causalean/PO/Core/System.lean"},{"n":"Causalean.PO.POSystem.restrict","k":"def","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"For a potential-outcome system and a finite collection of its variables, the restricted potential-outcome system has precisely that collection as its variables, retains the original sample space and probability measure, and evaluates every restricted intervention by lifting it to the original system first.","s":"(P : Causalean.PO.POSystem) → Finset P.V → Causalean.PO.POSystem","l":78,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_V","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The variable type of the restricted potential-outcome system is the chosen finite set of variables.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).V = ↥R","l":93,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_X","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The value space in the restricted system is the original value space at the underlying variable.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (v : ↥R), (P.restrict R).X v = P.X ↑v","l":96,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_component","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"A coordinate potential outcome in the restricted system agrees with the original coordinate after lifting the regime.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (v : ↥R), (P.restrict R).component r' v = P.component (P.liftRegime R r') ↑v","l":112,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_consistency","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":1,"d":"Restriction preserves consistency. If the ambient potential-outcome system P is consistent (SUTVA holds), then the system P restricted to R is consistent as well.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), P.Consistency → (P.restrict R).Consistency","l":196,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_eval","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"Evaluation in the restricted system is evaluation in the original system after lifting the restricted regime.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V) (r' : Causalean.PO.Regime ↥R fun v => P.X ↑v) (ω : P.Ω) (v : ↥R), (P.restrict R).eval r' ω v = P.eval (P.liftRegime R r') ω ↑v","l":105,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_Ω","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The restricted potential-outcome system uses the same sample space as the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).Ω = P.Ω","l":99,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POSystem.restrict_μ","k":"theorem","m":"Causalean.PO.Bridge.Induce","a":"PO/Bridge","t":2,"d":"The restricted potential-outcome system uses the same probability measure as the original system.","s":"∀ (P : Causalean.PO.POSystem) (R : Finset P.V), (P.restrict R).μ = P.μ","l":101,"f":"Causalean/PO/Bridge/Induce.lean"},{"n":"Causalean.PO.POVar","k":"structure","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A potential-outcome variable records a system variable together with a measurable relabeling of its native value space into a chosen analysis scale α.","s":"Causalean.PO.POSystem → (α : Type u_1) → [MeasurableSpace α] → Type (max (max u_1 u_2) u_3)","l":29,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cf","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, a potential-outcome variable, and an intervention regime, the counterfactual value function maps every sample-space unit to that variable's potential outcome under the regime, expressed on the analysis scale.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → Causalean.PO.Regime P.V P.X → P.Ω → α","l":45,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cfCDF","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a potential-outcome system, a real-valued potential-outcome variable, an intervention regime, and a measure on the sample space, the counterfactual cumulative distribution function is the cumulative distribution function of the variable's counterfactual law under that regime and measure.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → StieltjesFunction ℝ","l":48,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfCDF_eq_measureReal","k":"theorem","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For an intervention regime r and a measure μ on the sample space, the cdf of the potential outcome a(r) evaluated at y equals the μ-probability of the event a(r) ≤ y.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X) (μ : MeasureTheory.Measure P.Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : ℝ), ↑(a.cfCDF r μ) y = (a.cfLaw r μ).real (Set.Iic y)","l":61,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfLaw","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a potential-outcome system, a real-valued potential-outcome variable, an intervention regime, and a measure on the sample space, the counterfactual law is the distribution obtained by pushing the measure forward through the variable's potential-outcome function under that regime.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":29,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfQuantile","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a potential-outcome system, a real-valued potential-outcome variable, an intervention regime, a measure on the sample space, and a real index, the counterfactual quantile is the quantile at that index of the variable's counterfactual law under the regime and measure.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → Causalean.PO.Regime P.V P.X → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":55,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnder","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable outcome scale, a measurable intervention-variable scale, an outcome variable, an intervention variable, and a value of that intervention variable, the single-intervention counterfactual value function maps every sample-space unit to the outcome variable's potential outcome when the intervention variable is set to that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → {β : Type u_2} → [inst_1 : MeasurableSpace β] → Causalean.PO.POVar P α → Causalean.PO.POVar P β → β → P.Ω → α","l":111,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cfUnderLaw","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a potential-outcome system, a real-valued outcome variable, a potential-outcome intervention variable on a measurable scale, a value of that intervention variable, and a measure on the sample space, the single-intervention counterfactual law is the distribution of the outcome variable's potential outcome when the intervention variable is set to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → {β : Type u_1} → [inst : MeasurableSpace β] → Causalean.PO.POVar P β → β → MeasureTheory.Measure P.Ω → MeasureTheory.Measure ℝ","l":73,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnderQuantile","k":"def","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":1,"d":"For a potential-outcome system, a real-valued outcome variable, a potential-outcome intervention variable on a measurable scale, a value of that intervention variable, a measure on the sample space, and a real index, the single-intervention counterfactual quantile is the quantile at that index of the outcome variable's potential outcome when the intervention variable is set to that value.","s":"{P : Causalean.PO.POSystem} → Causalean.PO.POVar P ℝ → {β : Type u_1} → [inst : MeasurableSpace β] → Causalean.PO.POVar P β → β → MeasureTheory.Measure P.Ω → ℝ → ℝ","l":79,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnderQuantile_eq","k":"theorem","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":2,"d":"The single-intervention quantile is exactly the quantile computed from the corresponding single-intervention law.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) {β : Type u_1} [inst : MeasurableSpace β] (w : Causalean.PO.POVar P β) (y : β) (μ : MeasureTheory.Measure P.Ω) (τ : ℝ), a.cfUnderQuantile w y μ τ = Causalean.Stat.quantile (a.cfUnderLaw w y μ) τ","l":86,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.cfUnder_eq","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The single-intervention counterfactual value function of an outcome variable at a treatment value is its counterfactual value function under the singleton regime that fixes the treatment variable to that value, read back through the treatment variable's measurable equivalence.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {β : Type u_2} [inst_1 : MeasurableSpace β] (y : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β) (d : β), y.cfUnder w d = y.cf (Causalean.PO.Regime.single w.v (w.equiv.symm d))","l":120,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cf_apply","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"At a given unit, the counterfactual value of a variable under a regime is the system's world evaluation of that variable at that unit under that regime, carried over to the analysis scale by the variable's measurable equivalence.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (r : Causalean.PO.Regime P.V P.X) (ω : P.Ω), a.cf r ω = a.equiv (P.eval r ω a.v)","l":51,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.cf_eq_factual_of_factualAgrees","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":1,"d":"Multi-target consistency. Under the consistency (SUTVA) assumption on the potential-outcome system, for a regimed variable a whose index does not lie in the target of a regime r, if the outcome ω factually agrees with the regime r, then the counterfactual value of a under r at ω equals its factual value at ω.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (r : Causalean.PO.Regime P.V P.X), a.v ∉ r.target → ∀ (ω : P.Ω), P.FactualAgrees r ω → a.cf r ω = a.factual ω","l":201,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.cf_eq_factual_on_event","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Under consistency, changing a distinct variable to a value it already has does not change the counterfactual value of the target variable on that event.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β) (y : β), a.v ≠ w.v → ∀ {ω : P.Ω}, ω ∈ w.event y → a.cfUnder w y ω = a.factual ω","l":99,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.comap_factual_le","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"The conditioning σ-algebra σ(c.factual) is a sub-σ-algebra of the ambient σ-algebra on P.Ω.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ), MeasurableSpace.comap c.factual inferInstance ≤ inferInstance","l":55,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven","k":"def","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system, a conditioning variable on a measurable scale, a real-valued sample-space function, and a measure on the sample space, the conditional expectation given the variable is the conditional expectation of the function given the σ-algebra generated by that variable's factual value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POVar P γ → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":33,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_add","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Linearity of condExpGiven: additivity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven (f + g) μ =ᵐ[μ] c.condExpGiven f μ + c.condExpGiven g μ","l":112,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_add'","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Additivity of condExpGiven with the integrability side conditions discharged automatically.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (MeasureTheory.Integrable f μ) Causalean.PO.POVar.condExpGiven_add'._auto_1 → autoParam (MeasureTheory.Integrable g μ) Causalean.PO.POVar.condExpGiven_add'._auto_3 → c.condExpGiven (f + g) μ =ᵐ[μ] c.condExpGiven f μ + c.condExpGiven g μ","l":133,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_const","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"A constant is its own conditional expectation: on a finite measure, averaging a constant given a potential-outcome variable returns that constant, everywhere (not merely almost everywhere).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (k : ℝ) {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ], c.condExpGiven (fun x => k) μ = fun x => k","l":159,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_eq","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Averaging a real-valued sample-space quantity given a potential-outcome variable is exactly the standard conditional expectation given the σ-algebra generated by that variable's factual value.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) (μ : MeasureTheory.Measure P.Ω), c.condExpGiven g μ = μ[g | MeasurableSpace.comap c.factual inferInstance]","l":42,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_finsetSum'","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpGiven commutes with a finite sum, with the family-wide integrability side condition discharged automatically by fun_prop.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] {ι : Type u_2} {s : Finset ι} (c : Causalean.PO.POVar P γ) {F : ι → P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (∀ i ∈ s, MeasureTheory.Integrable (F i) μ) Causalean.PO.POVar.condExpGiven_finsetSum'._auto_1 → c.condExpGiven (∑ i ∈ s, F i) μ =ᵐ[μ] ∑ i ∈ s, c.condExpGiven (F i) μ","l":167,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_indicator_mul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out version specialised to indicator functions of sets measurable w.r.t. σ(c.factual). If the event is measurable with respect to the conditioning variable's factual σ-algebra, its indicator factors out of the conditional expectation just like any other strongly measurable multiplier.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {s : Set P.Ω} {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSet s → MeasureTheory.Integrable g μ → c.condExpGiven ((s.indicator fun x => 1) * g) μ =ᵐ[μ] (s.indicator fun x => 1) * c.condExpGiven g μ","l":224,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_mul_of_stronglyMeasurable_left","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out-left for condExpGiven: if f is strongly measurable w.r.t. σ(c.factual), then μ[f·g | σ(c.factual)] =ᵐ f · μ[g | σ(c.factual)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable g μ → c.condExpGiven (f * g) μ =ᵐ[μ] f * c.condExpGiven g μ","l":203,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_mul_of_stronglyMeasurable_right","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Pull-out-right for condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable g → MeasureTheory.Integrable (f * g) μ → MeasureTheory.Integrable f μ → c.condExpGiven (f * g) μ =ᵐ[μ] c.condExpGiven f μ * g","l":214,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_neg","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Negation passes through condExpGiven: averaging the negation of a quantity given a variable is the negation of averaging it. No integrability is required.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, c.condExpGiven (-g) μ =ᵐ[μ] -c.condExpGiven g μ","l":152,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_smul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Scalar homogeneity of condExpGiven.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (k : ℝ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, c.condExpGiven (k • g) μ =ᵐ[μ] k • c.condExpGiven g μ","l":126,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_sub","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Linearity of condExpGiven: subtraction.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven (f - g) μ =ᵐ[μ] c.condExpGiven f μ - c.condExpGiven g μ","l":119,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_sub'","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Subtractivity of condExpGiven with the integrability side conditions discharged automatically; see POVar.condExpGiven_add'.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, autoParam (MeasureTheory.Integrable f μ) Causalean.PO.POVar.condExpGiven_sub'._auto_1 → autoParam (MeasureTheory.Integrable g μ) Causalean.PO.POVar.condExpGiven_sub'._auto_3 → c.condExpGiven (f - g) μ =ᵐ[μ] c.condExpGiven f μ - c.condExpGiven g μ","l":145,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_tower","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Tower property between two POVars: if σ(c₁.factual) ≤ σ(c₂.factual), then μ[μ[g|σ(c₂)] | σ(c₁)] =ᵐ μ[g|σ(c₁)].","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] {δ : Type u_2} [inst_1 : MeasurableSpace δ] (c₁ : Causalean.PO.POVar P γ) (c₂ : Causalean.PO.POVar P δ) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasurableSpace.comap c₁.factual inferInstance ≤ MeasurableSpace.comap c₂.factual inferInstance → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c₂))], c₁.condExpGiven (c₂.condExpGiven g μ) μ =ᵐ[μ] c₁.condExpGiven g μ","l":257,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpGiven_tower_of_le","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Tower property over an arbitrary smaller sub-σ-algebra m ≤ σ(c.factual). Requires the usual SigmaFinite hypothesis on the trim of μ to σ(c.factual).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {g : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} {m : MeasurableSpace P.Ω}, m ≤ MeasurableSpace.comap c.factual inferInstance → ∀ [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c))], μ[c.condExpGiven g μ | m] =ᵐ[μ] μ[g | m]","l":247,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpRatio","k":"def","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"For a potential-outcome system, a conditioning variable on a measurable scale, two real-valued sample-space functions, and a measure on the sample space, the conditional-expectation ratio maps each unit to the conditional expectation of the first function divided by that of the second, with both conditional on the σ-algebra generated by the variable's factual value.","s":"{P : Causalean.PO.POSystem} → {γ : Type u_1} → [inst : MeasurableSpace γ] → Causalean.PO.POVar P γ → (P.Ω → ℝ) → (P.Ω → ℝ) → optParam (MeasureTheory.Measure P.Ω) P.μ → P.Ω → ℝ","l":270,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpRatio_def","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"The variable conditional-expectation ratio is, pointwise, the quotient of the two conditional expectations given that variable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g h : P.Ω → ℝ) (μ : MeasureTheory.Measure P.Ω), c.condExpRatio g h μ = fun ω => c.condExpGiven g μ ω / c.condExpGiven h μ ω","l":279,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.condExpRatio_eq_of_mul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Characterisation of condExpRatio via a product identity. If μ[g|σ(c)] = μ[h|σ(c)] · target a.s. and μ[h|σ(c)] ≠ 0 a.s., then condExpRatio c g h =ᵐ target. This is the reusable algebraic step for turning a conditional-expectation product identity into a ratio identity.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {g h target : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, c.condExpGiven g μ =ᵐ[μ] c.condExpGiven h μ * target → (∀ᵐ (ω : P.Ω) ∂μ, c.condExpGiven h μ ω ≠ 0) → c.condExpRatio g h μ =ᵐ[μ] target","l":306,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.event","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, a potential-outcome variable, and a value on that scale, the factual-value event is the set of all sample-space units whose factual value of the variable equals that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → α → Set P.Ω","l":89,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.eventCondExp_cfUnder_eq_factual_on_event","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Consistency-on-event for eventCondExp. Generic consumer of POVar.cf_eq_factual_on_event: on the event {a = a₀}, the counterfactual y.cfUnder a a₀ agrees with y.factual pointwise, so their conditional means on that event coincide. This is the shared rewrite underlying Manski MTR/MTS and (via the finite-partition total law) the integrated MIV bounds.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_2} [inst : MeasurableSpace β], P.Consistency → ∀ (y : Causalean.PO.POVar P ℝ) (a : Causalean.PO.POVar P β) (a₀ : β), MeasurableSet (a.event a₀) → y.v ≠ a.v → ∀ (μ : MeasureTheory.Measure P.Ω), Causalean.PO.eventCondExp μ (a.event a₀) (y.cfUnder a a₀) = Causalean.PO.eventCondExp μ (a.event a₀) y.factual","l":342,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.POVar.event_eq","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The factual-value event of a variable at a value is the preimage, under the variable's factual value function, of the one-point set at that value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), a.event x = a.factual ⁻¹' {x}","l":94,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.factual","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, and a potential-outcome variable, the factual value function maps every sample-space unit to the variable's potential outcome under the empty intervention regime, expressed on the analysis scale.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → P.Ω → α","l":63,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.factualAgrees_single","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Combinator: from a factual equality a.factual ω = x, build FactualAgrees for the singleton regime {a.v ← a.equiv.symm x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α) {ω : P.Ω}, a.factual ω = x → P.FactualAgrees (Causalean.PO.Regime.single a.v (a.equiv.symm x)) ω","l":83,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_eq","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The factual value function of a variable is its counterfactual value function under the empty intervention regime.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α), a.factual = a.cf Causalean.PO.Regime.empty","l":68,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.factual_eq_cfUnder_self_selected","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Under consistency, setting a distinct variable to its realized factual value leaves the target variable at its factual value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β), a.v ≠ w.v → ∀ (ω : P.Ω), a.factual ω = a.cfUnder w (w.factual ω) ω","l":135,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cfUnder_mul_indicator","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Integrated form of consistency: Y · 1_{W=y} = Y(w=y) · 1_{W=y} pointwise, where a plays the role of Y and w the role of the treatment. Used in backdoor-style identification proofs where the factual outcome is replaced by the counterfactual on the event {W = y}.","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β], P.Consistency → ∀ (a : Causalean.PO.POVar P ℝ) (w : Causalean.PO.POVar P β) (y : β), a.v ≠ w.v → (fun ω => a.factual ω * (w.event y).indicator (fun x => 1) ω) = fun ω => a.cfUnder w y ω * (w.event y).indicator (fun x => 1) ω","l":163,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cfUnder_mul_indicator_fn","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Pointwise-function variant of POVar.factual_mul_indicator_eq_cfUnder_mul_indicator, phrased directly in terms of POVar.indicator (rather than Set.indicator on w.event y).","s":"∀ {P : Causalean.PO.POSystem} {β : Type u_1} [inst : MeasurableSpace β] [MeasurableSingletonClass β], P.Consistency → ∀ (y : Causalean.PO.POVar P ℝ) (w : Causalean.PO.POVar P β) (x : β), y.v ≠ w.v → (fun ω => y.factual ω * w.indicator x ω) = fun ω => y.cfUnder w x ω * w.indicator x ω","l":185,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.factual_mul_indicator_eq_cf_mul_indicator","k":"theorem","m":"Causalean.PO.Assumptions.ConsistencyLemmas","a":"PO/Assumptions","t":2,"d":"Multi-target integrated consistency: Y · 1_E = Y(r) · 1_E pointwise, whenever every ω ∈ E factually agrees with r and a.v ∉ r.target.","s":"∀ {P : Causalean.PO.POSystem}, P.Consistency → ∀ (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X), a.v ∉ r.target → ∀ (E : Set P.Ω), (∀ ω ∈ E, P.FactualAgrees r ω) → (fun ω => a.factual ω * E.indicator (fun x => 1) ω) = fun ω => a.cf r ω * E.indicator (fun x => 1) ω","l":225,"f":"Causalean/PO/Assumptions/ConsistencyLemmas.lean"},{"n":"Causalean.PO.POVar.indicator","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, a potential-outcome variable, and a value on that scale, the factual-value indicator maps each sample-space unit to one when the variable's factual value equals that value and to zero otherwise.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → α → P.Ω → ℝ","l":145,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_add_indicator_not","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Binary case: a.indicator true ω + a.indicator false ω = 1.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P Bool) (ω : P.Ω), a.indicator true ω + a.indicator false ω = 1","l":272,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_apply_eq_one","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Pointwise: a.indicator x ω = 1 on {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) {x : α} {ω : P.Ω}, a.factual ω = x → a.indicator x ω = 1","l":160,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_apply_eq_zero","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"Pointwise: a.indicator x ω = 0 off {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) {x : α} {ω : P.Ω}, a.factual ω ≠ x → a.indicator x ω = 0","l":167,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_eq_event_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome variable a and a value x in its range, the real-valued factual indicator a.indicator x equals the set-indicator of the factual event {a = x}.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), a.indicator x = (a.event x).indicator fun x => 1","l":153,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.indicator_eq_one_or_zero","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x ω is always 0 or 1.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α) (ω : P.Ω), a.indicator x ω = 1 ∨ a.indicator x ω = 0","l":264,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.instIsProbabilityMeasureCfLaw","k":"instance","m":"Causalean.PO.Analysis.Quantile","a":"PO/Analysis","t":2,"d":"For a potential-outcomes system, a real-valued potential-outcome variable in that system, an intervention regime, and a measure on the system's sample space, if that sample-space measure is a probability measure, then the distribution of the variable's potential outcome under the regime is a probability measure.","s":"∀ {P : Causalean.PO.POSystem} (a : Causalean.PO.POVar P ℝ) (r : Causalean.PO.Regime P.V P.X) (μ : MeasureTheory.Measure P.Ω) [MeasureTheory.IsProbabilityMeasure μ], MeasureTheory.IsProbabilityMeasure (a.cfLaw r μ)","l":36,"f":"Causalean/PO/Analysis/Quantile.lean"},{"n":"Causalean.PO.POVar.integrable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Mathlib's totalized conditional expectation given the variable's σ-algebra is integrable, even when the original sample-space function is not assumed integrable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable (c.condExpGiven g μ) μ","l":191,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integrable_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is integrable under any finite measure (bounded by 1).","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasureTheory.Integrable (a.indicator x) μ","l":199,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.integrable_indicator_mul","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Multiplying a POVar factual indicator by an integrable function preserves integrability. This is the indicator-on-the-left companion of POVar.integrable_mul_indicator; the two orders are distinct goal shapes for the function-property tactics.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → ∀ {f : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun ω => a.indicator x ω * f ω) μ","l":77,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integrable_indicator_mul_of_singleton","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"On a value space whose one-point sets are measurable, multiplying a POVar factual indicator by an integrable function preserves integrability.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α) {f : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun ω => a.indicator x ω * f ω) μ","l":100,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integrable_indicator_of_singleton","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"On a value space whose one-point sets are measurable, the factual indicator of a potential-outcome variable at any value is integrable under every finite measure on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.IsFiniteMeasure μ] (a : Causalean.PO.POVar P α) (x : α), MeasureTheory.Integrable (a.indicator x) μ","l":255,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.integrable_mul_indicator","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"Multiplying an integrable function by a POVar factual indicator preserves integrability.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → ∀ {f : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun ω => f ω * a.indicator x ω) μ","l":63,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integrable_mul_indicator_of_singleton","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"On a value space whose one-point sets are measurable, multiplying an integrable function by a POVar factual indicator preserves integrability.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_2} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α) {f : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun ω => f ω * a.indicator x ω) μ","l":90,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.integral_sub_eq_integral_sub_of_condExpGiven_ae_eq","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":1,"d":"CATE-to-ATE integrator. Fix a conditioning variable c and integrable functions f and g on the same probability space. If f' is an almost-everywhere representative of the conditional expectation of f given c and g' is an almost-everywhere representative of the conditional expectation of g given c, then the integral of f - g equals the integral of f' - g'.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) {f g f' g' : P.Ω → ℝ} {μ : MeasureTheory.Measure P.Ω} [MeasureTheory.SigmaFinite (μ.trim (Causalean.PO.POVar.comap_factual_le c))], MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → c.condExpGiven f μ =ᵐ[μ] f' → c.condExpGiven g μ =ᵐ[μ] g' → ∫ (ω : P.Ω), f ω - g ω ∂μ = ∫ (ω : P.Ω), f' ω - g' ω ∂μ","l":321,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.measurableSet_event","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The event that a potential-outcome variable's factual value equals a given singleton-measurable value is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasurableSet (a.event x)","l":104,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_cf","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The counterfactual-value function of a potential-outcome variable under any intervention regime is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (r : Causalean.PO.Regime P.V P.X), Measurable (a.cf r)","l":77,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_cfUnder","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The single-intervention counterfactual-value function is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] {β : Type u_2} [inst_1 : MeasurableSpace β] (y : Causalean.PO.POVar P α) (w : Causalean.PO.POVar P β) (d : β), Measurable (y.cfUnder w d)","l":133,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpRatio is measurable (pointwise division of two strongly measurable functions is measurable).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, Measurable (c.condExpRatio g h μ)","l":291,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.measurable_factual","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The factual-value function of a potential-outcome variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α), Measurable a.factual","l":84,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → Measurable (a.indicator x)","l":174,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.measurable_indicator_of_singleton","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"On a value space whose one-point sets are measurable, the factual indicator of a potential-outcome variable at any value is a measurable function of the unit.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α), Measurable (a.indicator x)","l":230,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpGiven","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a strongly measurable function w.r.t. the ambient σ-algebra.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpGiven g μ)","l":185,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpGiven_comap","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpGiven yields a function strongly measurable w.r.t. σ(c.factual).","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpGiven g μ)","l":177,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_condExpRatio","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"condExpRatio is strongly measurable.","s":"∀ {P : Causalean.PO.POSystem} {γ : Type u_1} [inst : MeasurableSpace γ] (c : Causalean.PO.POVar P γ) (g h : P.Ω → ℝ) {μ : MeasureTheory.Measure P.Ω}, MeasureTheory.StronglyMeasurable (c.condExpRatio g h μ)","l":300,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_indicator","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The real-valued factual indicator of a potential-outcome variable at a singleton-measurable value is strongly measurable on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasureTheory.StronglyMeasurable (a.indicator x)","l":222,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_indicator_comap","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"a.indicator x is strongly measurable w.r.t. the σ-algebra generated by a.factual.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (a : Causalean.PO.POVar P α) (x : α), MeasurableSet {x} → MeasureTheory.StronglyMeasurable (a.indicator x)","l":182,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_indicator_comap_of_singleton","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"On a value space whose one-point sets are measurable, the factual indicator of a potential-outcome variable at any value is strongly measurable with respect to the information carried by that variable's factual value.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α), MeasureTheory.StronglyMeasurable (a.indicator x)","l":245,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.POVar.stronglyMeasurable_indicator_of_singleton","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"On a value space whose one-point sets are measurable, the factual indicator of a potential-outcome variable at any value is strongly measurable on the sample space.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : Causalean.PO.POVar P α) (x : α), MeasureTheory.StronglyMeasurable (a.indicator x)","l":237,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RDDLimits.oneSidedLimit_eq_left","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":1,"d":"Left-side limit identification. Symmetric form of oneSidedLimit_eq_right. If a real-valued function f agrees with a reference function g almost everywhere, with respect to a measure π, on the half-line to the left of the cutoff c, g is continuous at c, and every open interval immediately to the left of c has positive π-measure, then any left-hand limit L of f at c must equal g evaluated at c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Iio c)] g → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo (c - ε) c) ≠ 0) → ∀ {L : ℝ}, Filter.Tendsto f (nhdsWithin c (Set.Iio c)) (nhds L) → L = g c","l":122,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.oneSidedLimit_eq_right","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Right-side limit identification. If f =ᵐ[π.restrict (Ici c)] g, g is continuous at c, every right-open neighborhood of c has positive π-mass, and f has any right-side limit L at c, then L = g c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Ici c)] g → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo c (c + ε)) ≠ 0) → ∀ {L : ℝ}, Filter.Tendsto f (nhdsWithin c (Set.Ioi c)) (nhds L) → L = g c","l":89,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.value_eq_of_aeEq_left","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Symmetric pointwise version of value_eq_of_aeEq_right.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Iio c)] g → ContinuousAt f c → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo (c - ε) c) ≠ 0) → f c = g c","l":170,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.RDDLimits.value_eq_of_aeEq_right","k":"theorem","m":"Causalean.PO.ID.Exact.RDD.RDDLimits","a":"PO/ID/Exact/RDD","t":2,"d":"Pointwise equality from a.e. agreement plus continuity. If f and g agree π.restrict (Ici c)-a.e., both are continuous at c, and every right-open neighborhood of c has positive π-mass, then f c = g c.","s":"∀ {π : MeasureTheory.Measure ℝ} {f g : ℝ → ℝ} {c : ℝ}, f =ᵐ[π.restrict (Set.Ici c)] g → ContinuousAt f c → ContinuousAt g c → (∀ ε > 0, π (Set.Ioo c (c + ε)) ≠ 0) → f c = g c","l":158,"f":"Causalean/PO/ID/Exact/RDD/RDDLimits.lean"},{"n":"Causalean.PO.Regime","k":"structure","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"An intervention regime specifies a finite set of targeted variables together with an assigned value in the corresponding value space for each targeted variable.","s":"(V : Type u_3) → [DecidableEq V] → (V → Type u_4) → Type (max u_3 u_4)","l":29,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.Disjoint","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared and their value spaces, two intervention regimes are disjoint exactly when no variable is targeted by both regimes.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X → Causalean.PO.Regime V X → Prop","l":49,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.disjoint_iff","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Two intervention regimes are disjoint exactly when their target sets are disjoint as finite sets.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X), r₁.Disjoint r₂ ↔ Disjoint r₁.target r₂.target","l":71,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.disjoint_single_of_not_mem","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Any regime that does not target a variable is disjoint from the singleton intervention on that variable.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} (x : X v) (r : Causalean.PO.Regime V X), v ∉ r.target → r.Disjoint (Causalean.PO.Regime.single v x)","l":167,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared and their value spaces, the empty intervention regime targets no variable and consequently has no substantive assignments.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X","l":41,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_disjoint_left","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Every regime is disjoint from the empty intervention regime on its right.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r : Causalean.PO.Regime V X), r.Disjoint Causalean.PO.Regime.empty","l":108,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_disjoint_right","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The empty intervention regime is disjoint from every regime on its right.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r : Causalean.PO.Regime V X), Causalean.PO.Regime.empty.Disjoint r","l":104,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.empty_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The empty intervention regime has no target variables.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2}, Causalean.PO.Regime.empty.target = ∅","l":101,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ext","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Extensionality for Regime: equal targets and pointwise-equal assignments.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {r₁ r₂ : Causalean.PO.Regime V X}, r₁.target = r₂.target → (∀ (v : V) (h₁ : v ∈ r₁.target) (h₂ : v ∈ r₂.target), r₁.assign v h₁ = r₂.assign v h₂) → r₁ = r₂","l":130,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.leftBiasedUnion","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared and their value spaces, and two intervention regimes, the left-biased union targets every variable targeted by either regime and uses the first regime's assigned value whenever both assign that variable.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → Causalean.PO.Regime V X → Causalean.PO.Regime V X → Causalean.PO.Regime V X","l":52,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.leftBiasedUnion_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The left-biased union of two regimes targets the union of their target sets.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X), (r₁.leftBiasedUnion r₂).target = r₁.target ∪ r₂.target","l":82,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared, their value spaces, a list of variable--value assignments, and a variable occurring in that list, the list lookup result is the value assigned to that variable by the first matching list entry: the empty-list case follows from the impossible occurrence assertion, while the nonempty-list case returns the head value when its variable matches and otherwise recurses on the tail.","s":"{V : Type u_1} → [DecidableEq V] → {X : V → Type u_2} → (l : List ((v : V) × X v)) → (v : V) → v ∈ List.map Sigma.fst l → X v","l":175,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup_cons_of_ne","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Looking up a different variable skips the head of a dependent assignment list and continues in the tail.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v w : V} {x : X w} {rest : List ((v : V) × X v)}, v ≠ w → ∀ (hv' : v ∈ List.map Sigma.fst rest), Causalean.PO.Regime.listLookup (⟨w, x⟩ :: rest) v (of_eq_true (Eq.trans (Eq.trans (congrFun' (congrArg Membership.mem List.map_cons) v) List.mem_cons._simp_1) (Eq.trans (congrArg (Or (v = w)) (eq_true hv')) (or_true (v = w))))) = Causalean.PO.Regime.listLookup rest v hv'","l":251,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.listLookup_cons_self","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Looking up the head variable of a dependent assignment list returns the head value.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} {x : X v} {rest : List ((v : V) × X v)}, Causalean.PO.Regime.listLookup (⟨v, x⟩ :: rest) v (of_eq_true (Eq.trans (Eq.trans (congrFun' (congrArg Membership.mem List.map_cons) v) List.mem_cons._simp_1) (Eq.trans (congr (congrArg Or (eq_self v)) (Eq.trans List.mem_map._simp_1 (Eq.trans Sigma.exists._simp_1 (Eq.trans (congrArg Exists (funext fun a => exists_and_right._simp_1)) exists_eq_right._simp_1)))) (true_or (∃ x, ⟨v, x⟩ ∈ rest))))) = x","l":245,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared and their value spaces, a list of variable--value assignments, and the condition that no variable appears more than once in that list, the list-built intervention regime targets exactly the listed variables and assigns each its listed value.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (l : List ((v : V) × X v)) → (List.map Sigma.fst l).Nodup → Causalean.PO.Regime V X","l":207,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofListLeftBiased","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared, their value spaces, and a list of variable--value assignments, the left-biased list-built regime targets the variables appearing in the list and assigns each the value at its first occurrence.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → List ((v : V) × X v) → Causalean.PO.Regime V X","l":191,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofListLeftBiased_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The regime built from a list of assignments by taking the first listed value for each variable targets exactly the finite set of listed variables.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (l : List ((v : V) × X v)), (Causalean.PO.Regime.ofListLeftBiased l).target = (List.map Sigma.fst l).toFinset","l":197,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_cons_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The target of a regime built from a nonempty list inserts the head variable into the target from the tail.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} {x : X v} {rest : List ((v : V) × X v)} (h : (List.map Sigma.fst (⟨v, x⟩ :: rest)).Nodup), (Causalean.PO.Regime.ofList (⟨v, x⟩ :: rest) h).target = insert v (List.map Sigma.fst rest).toFinset","l":236,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_eq_ofListLeftBiased","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"On a duplicate-free list of assignments, the regime it determines is the left-biased list-built regime; the duplicate-freeness makes the left bias immaterial.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (l : List ((v : V) × X v)) (h : (List.map Sigma.fst l).Nodup), Causalean.PO.Regime.ofList l h = Causalean.PO.Regime.ofListLeftBiased l","l":212,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_nil","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Building a regime from the empty list gives the empty intervention regime.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2}, Causalean.PO.Regime.ofList [] (of_eq_true (Eq.trans (congrArg List.Nodup List.map_nil) List.nodup_nil._simp_1)) = Causalean.PO.Regime.empty","l":231,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.ofList_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a duplicate-free list of variable-value assignments l, the target of the regime it determines is exactly the finite set of variables listed in l.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (l : List ((v : V) × X v)) (h : (List.map Sigma.fst l).Nodup), (Causalean.PO.Regime.ofList l h).target = (List.map Sigma.fst l).toFinset","l":224,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared, their value spaces, a variable, and a value in that variable's value space, the singleton intervention regime targets exactly that variable and assigns it that value.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (v : V) → X v → Causalean.PO.Regime V X","l":143,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_assign_self","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Evaluating the singleton intervention assignment at its target returns the supplied value.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (v : V) (x : X v), (Causalean.PO.Regime.single v x).assign v (Finset.mem_singleton_self v) = x","l":152,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_disjoint_of_not_mem","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"A singleton intervention regime is disjoint from any regime that does not target its variable.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v : V} (x : X v) (r : Causalean.PO.Regime V X), v ∉ r.target → (Causalean.PO.Regime.single v x).Disjoint r","l":161,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_disjoint_single","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"Singleton intervention regimes on two distinct variables are disjoint.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} {v w : V}, v ≠ w → ∀ (x : X v) (y : X w), (Causalean.PO.Regime.single v x).Disjoint (Causalean.PO.Regime.single w y)","l":156,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.single_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The singleton intervention regime targets exactly the one variable it fixes.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (v : V) (x : X v), (Causalean.PO.Regime.single v x).target = {v}","l":148,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup","k":"def","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":1,"d":"For a collection of variables whose identities can be compared and their value spaces, two intervention regimes, and the condition that they have no target variable in common, their disjoint union is their union regime, targeting every variable targeted by either regime and using its unique component assignment.","s":"{V : Type u_1} → [inst : DecidableEq V] → {X : V → Type u_2} → (r₁ r₂ : Causalean.PO.Regime V X) → r₁.Disjoint r₂ → Causalean.PO.Regime V X","l":63,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_assign_neg","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"sqcup agrees with r₂ whenever v ∉ r₁.target (and hence v ∈ r₂.target).","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂), ∀ v ∉ r₁.target, ∀ (h2 : v ∈ r₂.target), (r₁.sqcup r₂ h).assign v (of_eq_true (Eq.trans Finset.mem_union._simp_1 (Eq.trans (congrArg (Or (v ∈ r₁.target)) (eq_true h2)) (or_true (v ∈ r₁.target))))) = r₂.assign v h2","l":123,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_assign_pos","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"sqcup agrees with r₁ whenever v ∈ r₁.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (h1 : v ∈ r₁.target), (r₁.sqcup r₂ h).assign v (of_eq_true (Eq.trans Finset.mem_union._simp_1 (Eq.trans (congrFun' (congrArg Or (eq_true h1)) (v ∈ r₂.target)) (true_or (v ∈ r₂.target))))) = r₁.assign v h1","l":116,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_eq_leftBiasedUnion","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The disjoint union of two compatible regimes is their left-biased union; disjointness makes the left bias immaterial.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂), r₁.sqcup r₂ h = r₁.leftBiasedUnion r₂","l":91,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.Regime.sqcup_target","k":"theorem","m":"Causalean.PO.Core.Regime","a":"PO/Core","t":2,"d":"The target of the disjoint union of two regimes is the union of their target sets.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂), (r₁.sqcup r₂ h).target = r₁.target ∪ r₂.target","l":112,"f":"Causalean/PO/Core/Regime.lean"},{"n":"Causalean.PO.RegimedVar","k":"structure","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"A regimed variable pairs a potential-outcome variable with the intervention regime under which it should be evaluated.","s":"Causalean.PO.POSystem → (α : Type u_1) → [MeasurableSpace α] → Type (max (max u_1 u_2) u_3)","l":289,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.measurable_value","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The value function of a regimed variable is measurable.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (rv : Causalean.PO.RegimedVar P α), Measurable rv.value","l":317,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.ofFactual","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, and a potential-outcome variable, the factual bundle pairs that variable with the empty intervention regime.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → Causalean.PO.RegimedVar P α","l":322,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.ofSingle","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, a potential-outcome variable, a system variable, and a value in that variable's native value space, the single-intervention bundle pairs the potential-outcome variable with the regime that fixes the system variable to that value.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.POVar P α → (w : P.V) → P.X w → Causalean.PO.RegimedVar P α","l":327,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.value","k":"def","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":1,"d":"For a potential-outcome system, a measurable analysis scale, and a variable paired with an intervention regime, the regimed-variable value function maps every sample-space unit to the paired variable's potential outcome under its paired regime.","s":"{P : Causalean.PO.POSystem} → {α : Type u_1} → [inst : MeasurableSpace α] → Causalean.PO.RegimedVar P α → P.Ω → α","l":303,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.RegimedVar.value_eq","k":"theorem","m":"Causalean.PO.Core.Variable","a":"PO/Core","t":2,"d":"The value function of a regimed variable is the counterfactual value function of its underlying variable under its bundled regime.","s":"∀ {P : Causalean.PO.POSystem} {α : Type u_1} [inst : MeasurableSpace α] (rv : Causalean.PO.RegimedVar P α), rv.value = rv.var.cf rv.regime","l":308,"f":"Causalean/PO/Core/Variable.lean"},{"n":"Causalean.PO.ae_abs_potential_le_of_indep_positive_arm","k":"theorem","m":"Causalean.PO.Assumptions.ArmSupportTransfer","a":"PO/Assumptions","t":1,"d":"Within a measurable positive-mass cell, for a measurable arm event, if the potential outcome is measurable, the arm has positive normalized cell probability, the potential outcome is independent of the arm indicator under the normalized cell law, the observed and potential outcomes agree almost surely on that arm, and the observed outcome obeys an absolute bound there, then the potential outcome obeys the same absolute bound almost surely throughout the cell.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {C A : Set Ω}, MeasurableSet C → 0 < P C → MeasurableSet A → ∀ {Ypot Yobs : Ω → ℝ}, Measurable Ypot → 0 < (Causalean.Mathlib.Probability.normalizedRestrict P C) A → ProbabilityTheory.IndepFun Ypot (Causalean.PO.armIndicator A) (Causalean.Mathlib.Probability.normalizedRestrict P C) → Yobs =ᵐ[(Causalean.Mathlib.Probability.normalizedRestrict P C).restrict A] Ypot → ∀ {R : ℝ}, (∀ᵐ (ω : Ω) ∂(Causalean.Mathlib.Probability.normalizedRestrict P C).restrict A, |Yobs ω| ≤ R) → ∀ᵐ (ω : Ω) ∂P.restrict C, |Ypot ω| ≤ R","l":34,"f":"Causalean/PO/Assumptions/ArmSupportTransfer.lean"},{"n":"Causalean.PO.armIndicator","k":"def","m":"Causalean.PO.Assumptions.ArmSupportTransfer","a":"PO/Assumptions","t":1,"d":"Given a sample space and an event in it, the real-valued arm indicator equals one for sample points in the event and zero for all other sample points.","s":"{Ω : Type u_1} → Set Ω → Ω → ℝ","l":25,"f":"Causalean/PO/Assumptions/ArmSupportTransfer.lean"},{"n":"Causalean.PO.combinedFixed","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model, a background assignment of its fixed variables, and a potential-outcome regime, the combined fixed-variable assignment assigns each fixed coordinate of the intervened model either its background value or, for a newly targeted coordinate, the regime's intervention value.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) → (M.fixSet (Causalean.PO.regimeTargetN M r) (Causalean.PO.regimeTargetN_obs M r) (Causalean.PO.regimeTargetN_notFixed M r)).FixedValues","l":178,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.combinedFixed_new","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"On new intervention coordinates, combinedFixed at ⟨.fixed D, _⟩ equals r.assign v' hv'tgt, both at type Ω D. The internal Classical.choose witness is identified with v' by uniqueness (both have .val = .random D); the cast chain collapses via cast_heq + proof_irrel_heq.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) (v' : Causalean.PO.ObsIdx M) (hv'tgt : v' ∈ r.target) (D : N) (hD : D ∈ Causalean.PO.regimeTargetN M r) (hDval : ↑v' = Causalean.SWIGNode.random D), Causalean.PO.combinedFixed M s r ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩ = cast (congrArg (Causalean.swigΩ Ω) hDval) (r.assign v' hv'tgt)","l":325,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.combinedFixed_old","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"On original fixed coordinates, combinedFixed agrees with s.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)) (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), Causalean.PO.combinedFixed M s r ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed (Causalean.PO.regimeTargetN M r)) hv⟩ = s ⟨v, hv⟩","l":277,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.condIndepFun_finite_of_measureReal_fibers","k":"theorem","m":"Causalean.PO.Conditioning.CondExpTooling","a":"PO/Conditioning","t":2,"d":"For a finite-valued conditioning map, if the conditioning map and the two conditioned maps are measurable and their ordinary real masses factor on every measurable pair of events inside every conditioning fiber, then the two maps are conditionally independent given the σ-algebra generated by the conditioning map.","s":"∀ {Ω : Type u_1} {T : Type u_2} {A : Type u_3} {B : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [Finite T] [inst_3 : MeasurableSpace T] [MeasurableSingletonClass T] [inst_5 : MeasurableSpace A] [inst_6 : MeasurableSpace B] (μ : MeasureTheory.Measure Ω) [inst_7 : MeasureTheory.IsFiniteMeasure μ] (X : Ω → T) (hX : Measurable X) (f : Ω → A), Measurable f → ∀ (g : Ω → B), Measurable g → (∀ (x : T) (s : Set A) (t : Set B), MeasurableSet s → MeasurableSet t → μ.real (f ⁻¹' s ∩ g ⁻¹' t ∩ X ⁻¹' {x}) * μ.real (X ⁻¹' {x}) = μ.real (f ⁻¹' s ∩ X ⁻¹' {x}) * μ.real (g ⁻¹' t ∩ X ⁻¹' {x})) → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap X inferInstance) (Measurable.comap_le hX) f g μ","l":457,"f":"Causalean/PO/Conditioning/CondExpTooling.lean"},{"n":"Causalean.PO.eventCondExp","k":"def","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"For a measurable sample space, a measure on that space, an event in the sample space, and a real-valued sample-space function, the event-level conditional expectation is the integral of the function over the event divided by the event's real-valued measure.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Set Ω → (Ω → ℝ) → ℝ","l":47,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_add","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is additive for integrable summands on the event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {g₁ g₂ : Ω → ℝ}, MeasureTheory.IntegrableOn g₁ A μ → MeasureTheory.IntegrableOn g₂ A μ → Causalean.PO.eventCondExp μ A (g₁ + g₂) = Causalean.PO.eventCondExp μ A g₁ + Causalean.PO.eventCondExp μ A g₂","l":199,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_congr_ae","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"a.e.-equal integrands have equal event-level conditional expectations.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {f g : Ω → ℝ}, f =ᵐ[μ.restrict A] g → Causalean.PO.eventCondExp μ A f = Causalean.PO.eventCondExp μ A g","l":168,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_congr_on","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Equal-on-A integrands have equal event-level conditional expectations. This specialises eventCondExp_congr_ae to a pointwise identity on a measurable set A.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {A : Set Ω}, MeasurableSet A → ∀ {f g : Ω → ℝ}, (∀ ω ∈ A, f ω = g ω) → Causalean.PO.eventCondExp μ A f = Causalean.PO.eventCondExp μ A g","l":175,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_eq","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"The average of a real-valued quantity over an event is its integral restricted to that event, divided by the event's real-valued mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) (g : Ω → ℝ), Causalean.PO.eventCondExp μ A g = (∫ (ω : Ω) in A, g ω ∂μ) / (μ A).toReal","l":55,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_eq_sum_condProb_mul_eventCondExp","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":1,"d":"Totalized finite-partition identity for event-level averages. Fix a measurable event A and a finite family of pairwise disjoint measurable sets C i covering the whole space, such that each intersection A ∩ C i has finite measure, and let f be an integrable function. Then the event-level average of f on A equals the sum, over i, of the event-level average of f on A ∩ C i weighted by the ratio of the measure of A ∩ C i to the measure of A:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) (A : Set Ω) (C : ι → Set Ω), MeasurableSet A → (∀ (i : ι), MeasurableSet (C i)) → Pairwise (Function.onFun Disjoint C) → ⋃ i, C i = Set.univ → (∀ (i : ι), μ (A ∩ C i) ≠ ⊤) → ∀ (f : Ω → ℝ), MeasureTheory.Integrable f μ → Causalean.PO.eventCondExp μ A f = ∑ i, (μ (A ∩ C i)).toReal / (μ A).toReal * Causalean.PO.eventCondExp μ (A ∩ C i) f","l":113,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_mono_ae","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Monotonicity of eventCondExp under an a.e. inequality of integrable functions.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {A : Set Ω} {f g : Ω → ℝ}, MeasureTheory.IntegrableOn f A μ → MeasureTheory.IntegrableOn g A μ → f ≤ᵐ[μ.restrict A] g → Causalean.PO.eventCondExp μ A f ≤ Causalean.PO.eventCondExp μ A g","l":185,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_mul_measure_toReal","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"eventCondExp μ A f · (μ A).toReal = ∫_A f, including the zero-measure case where both sides collapse to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω), μ A ≠ ⊤ → ∀ (f : Ω → ℝ), Causalean.PO.eventCondExp μ A f * (μ A).toReal = ∫ (ω : Ω) in A, f ω ∂μ","l":73,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_of_ae_eq_IndepFun","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Consistency + drop-of-conditioning for plain IndepFun. If z is independent of a counterfactual bundle B, and a factual integrand agrees with a measurable projection h ∘ B a.e. on the cell {z = x}, then its event-level conditional expectation on that cell equals the unconditional integral of the projection.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {α : Type u_2} {β : Type u_3} [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} {z : Ω → α} {B : Ω → β}, ProbabilityTheory.IndepFun z B μ → Measurable z → Measurable B → ∀ {factualF : Ω → ℝ} {h : β → ℝ}, Measurable h → ∀ {x : α}, MeasurableSet {x} → (factualF =ᵐ[μ.restrict (z ⁻¹' {x})] fun ω => h (B ω)) → (μ (z ⁻¹' {x})).toReal ≠ 0 → Causalean.PO.eventCondExp μ (z ⁻¹' {x}) factualF = ∫ (ω : Ω), h (B ω) ∂μ","l":222,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_smul","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is homogeneous with respect to real scalar multiplication.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) (c : ℝ) (g : Ω → ℝ), (Causalean.PO.eventCondExp μ A fun ω => c * g ω) = c * Causalean.PO.eventCondExp μ A g","l":214,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.eventCondExp_sub","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Event-level conditional expectation is additive over subtraction for integrable functions on the event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (A : Set Ω) {g₁ g₂ : Ω → ℝ}, MeasureTheory.IntegrableOn g₁ A μ → MeasureTheory.IntegrableOn g₂ A μ → Causalean.PO.eventCondExp μ A (g₁ - g₂) = Causalean.PO.eventCondExp μ A g₁ - Causalean.PO.eventCondExp μ A g₂","l":206,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.inducedEval","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model, a background assignment of its fixed variables, a potential-outcome regime, and a latent-variable assignment, the induced joint evaluation assigns each observed random node its value in the intervened structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M) → M.LatentValues → (v : Causalean.PO.ObsIdx M) → Causalean.PO.obsValue M v","l":209,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.inducedEval_empty_eq_evalMap","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"inducedEval at Regime.empty equals M.evalMap s at the same latent.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (v : Causalean.PO.ObsIdx M), Causalean.PO.inducedEval M s Causalean.PO.Regime.empty ℓ v = M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩","l":358,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.inducedEval_measurable","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"The induced joint evaluation map of a structural model is measurable in the latent draw.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), Measurable (Causalean.PO.inducedEval M s r)","l":232,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instDecidableEqObsIdx","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"For a finite collection of node names with decidable equality, measurable value spaces indexed by those names, and a structural causal model, equality between observed-node indices in its induced potential-outcome system can be decided. The decision follows from equality of the underlying nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → DecidableEq (Causalean.PO.ObsIdx M)","l":92,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instFintypeObsIdx","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"For a finite collection of node names with decidable equality, measurable value spaces indexed by those names, and a structural causal model, the collection of observed-node indices in its induced potential-outcome system is finite. Its finiteness follows from the model's finite node collection.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Fintype (Causalean.PO.ObsIdx M)","l":88,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.instMeasurableObsValue","k":"instance","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"For a finite collection of node names with decidable equality, measurable value spaces indexed by those names, a structural causal model, and one of its observed-node indices, the value space associated with that index carries the σ-algebra of the corresponding structural-model variable. That σ-algebra is inherited from the corresponding variable.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (v : Causalean.PO.ObsIdx M) → MeasurableSpace (Causalean.PO.obsValue M v)","l":96,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.integral_eq_sum_measure_mul_eventCondExp","k":"theorem","m":"Causalean.PO.Conditioning.EventCondExp","a":"PO/Conditioning","t":2,"d":"Finite-partition total law. For any disjoint covering of univ by a Fintype-indexed family of measurable sets, ∫ f = ∑ i, (μ (A i)).toReal · eventCondExp μ (A i) f.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (A : ι → Set Ω), (∀ (i : ι), MeasurableSet (A i)) → Pairwise (Function.onFun Disjoint A) → ⋃ i, A i = Set.univ → ∀ (f : Ω → ℝ), MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ i, (μ (A i)).toReal * Causalean.PO.eventCondExp μ (A i) f","l":88,"f":"Causalean/PO/Conditioning/EventCondExp.lean"},{"n":"Causalean.PO.obsIdx_mk_random","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model, a node name, and evidence that its random node is observed, the corresponding observed-node index is that observed random node viewed as an index of the induced potential-outcome system.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : N) → Causalean.SWIGNode.random n ∈ M.observed → Causalean.PO.ObsIdx M","l":74,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsIdx_val_injective","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Two elements of ObsIdx M with the same .val are equal (injectivity of the coercion).","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M : Causalean.SCM N Ω} {v w : Causalean.PO.ObsIdx M}, ↑v = ↑w → v = w","l":300,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsName_injective","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"obsName M is injective: obsName v = obsName w → v = w.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Function.Injective (Causalean.PO.obsName✝ M)","l":304,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.obsValue","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model and an observed-node index, the associated observed value space is the structural-model value space attached to that node.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.PO.ObsIdx M → Type uΩ","l":79,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN","k":"def","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":1,"d":"Given a structural causal model and a potential-outcome regime, the structural intervention target set is the finite set of underlying names targeted by that regime whose fixed counterparts are not already fixed in the structural model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M) → Finset N","l":105,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_empty","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"regimeTargetN M Regime.empty = ∅.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Causalean.PO.regimeTargetN M Causalean.PO.Regime.empty = ∅","l":353,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_eq_image_obsName","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"regimeTargetN M r is the image of the structurally eligible regime targets under obsName M.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), Causalean.PO.regimeTargetN M r = Finset.image (Causalean.PO.obsName✝ M) ({v ∈ r.target | Causalean.SWIGNode.fixed (Causalean.PO.obsName✝ M v) ∉ M.fixed})","l":284,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_mem_val","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Every structural variable name targeted by a translated regime comes from an observed-node index whose observed node is the corresponding random node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, ∃ v' ∈ r.target, ↑v' = Causalean.SWIGNode.random D","l":291,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_notFixed","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"fixSet-obligation: .fixed D ∉ M.fixed for every structurally eligible regime target D. The target-name translation filters out observed variables whose fixed counterpart is already in M.fixed.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, Causalean.SWIGNode.fixed D ∉ M.fixed","l":132,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.regimeTargetN_obs","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"fixSet-obligation for regimeTargetN: every target D has .random D ∈ M.observed.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (r : Causalean.PO.Regime (Causalean.PO.ObsIdx M) (Causalean.PO.obsValue M)), ∀ D ∈ Causalean.PO.regimeTargetN M r, Causalean.SWIGNode.random D ∈ M.observed","l":121,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.setIntegral_mul_condExp_of_stronglyMeasurableLeft","k":"theorem","m":"Causalean.PO.ID.Partial.Proxy.Helpers.Common","a":"PO/ID/Partial/Proxy/Helpers","t":2,"d":"A sigma-measurable left factor can be pulled through conditional expectation inside a set integral over a measurable event.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω}, m ≤ mΩ → ∀ {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f g : Ω → ℝ}, MeasureTheory.StronglyMeasurable f → MeasureTheory.Integrable g μ → MeasureTheory.Integrable (f * g) μ → ∀ {s : Set Ω}, MeasurableSet s → ∫ (ω : Ω) in s, f ω * g ω ∂μ = ∫ (ω : Ω) in s, f ω * μ[g | m] ω ∂μ","l":330,"f":"Causalean/PO/ID/Partial/Proxy/Helpers/Common.lean"},{"n":"Causalean.PO.sqcup_assign_left","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Regime.sqcup agrees with r₁ on r₁.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(v : V) → MeasurableSpace (X v)] (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (hv : v ∈ r₁.target), (r₁.sqcup r₂ h).assign v (Finset.mem_union_left r₂.target hv) = r₁.assign v hv","l":308,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.PO.sqcup_assign_right","k":"theorem","m":"Causalean.PO.Bridge.FromSCM","a":"PO/Bridge","t":2,"d":"Regime.sqcup agrees with r₂ on r₂.target.","s":"∀ {V : Type u_1} [inst : DecidableEq V] {X : V → Type u_2} [(v : V) → MeasurableSpace (X v)] (r₁ r₂ : Causalean.PO.Regime V X) (h : r₁.Disjoint r₂) (v : V) (hv : v ∈ r₂.target), (r₁.sqcup r₂ h).assign v (Finset.mem_union_right r₁.target hv) = r₂.assign v hv","l":316,"f":"Causalean/PO/Bridge/FromSCM.lean"},{"n":"Causalean.Panel.AdoptionPath.absorbingTreatment","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon, an adoption path $h$, and a period $t$ in that horizon, the absorbing treatment indicator equals one exactly when $h$ has adopted by $t$, and equals zero otherwise.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → ℝ","l":97,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.absorbingTreatment_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For an adoption date h and period t within a horizon of T periods, the absorbing treatment indicator equals one exactly when adoption has occurred by that period, and zero otherwise.","s":"∀ {T : ℕ} (h : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.absorbingTreatment h t = if h ≤ ↑t then 1 else 0","l":103,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.finite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a panel with a finite horizon of $T$ periods and a finite adoption period $g$, the finite adoption path is the path whose adoption date is $g$, rather than the never-treated date.","s":"{T : ℕ} → Fin T → WithTop (Fin T)","l":35,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.finite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"Embedding a finite adoption period into the shared adoption-path type is definitionally the ordinary finite-period inclusion.","s":"∀ {T : ℕ} (g : Fin T), Causalean.Panel.AdoptionPath.finite g = ↑g","l":39,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isEventuallyTreated","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon and an adoption path $h$, the eventually-treated predicate holds exactly when $h$ has a finite adoption date.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":87,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isEventuallyTreated_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The Sun-Abraham-compatible eventually-treated name is the finite adoption-date predicate.","s":"∀ {T : ℕ} (h : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isEventuallyTreated h = (h ≠ ⊤)","l":91,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isFinite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon and an adoption path $a$, the finite-path predicate holds exactly when its adoption date is not the never-treated date.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":62,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isFinite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"A path is eventually treated exactly when its adoption date is not infinite.","s":"∀ {T : ℕ} (a : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isFinite a = (a ≠ ⊤)","l":66,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isInfinite","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon and an adoption path $a$, the infinite-path predicate holds exactly when its adoption date is the never-treated date.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":70,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isInfinite_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"A path is never treated exactly when its adoption date is infinite.","s":"∀ {T : ℕ} (a : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isInfinite a = (a = ⊤)","l":74,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isNeverTreated","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon and an adoption path $h$, the never-treated predicate holds exactly when $h$ has the never-treated adoption date.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":78,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.isNeverTreated_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The Sun-Abraham-compatible never-treated name is the infinite adoption-date predicate.","s":"∀ {T : ℕ} (h : WithTop (Fin T)), Causalean.Panel.AdoptionPath.isNeverTreated h = (h = ⊤)","l":82,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.le","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon, an adoption path $a$, and a period $t$ in that horizon, the treated-by-$t$ predicate holds exactly when the adoption date is no later than $t$.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":44,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.le_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The treated-by-period predicate is exactly the order comparison with the finite period viewed as an adoption date.","s":"∀ {T : ℕ} (a : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.le a t = (a ≤ ↑t)","l":48,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt","k":"def","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":1,"d":"For a finite-period horizon, an adoption path $a$, and a period $t$ in that horizon, the untreated-at-$t$ predicate holds exactly when $t$ is strictly before the adoption date.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":53,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt_eq","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"The untreated-before-adoption predicate is exactly the strict order comparison with the finite period viewed as an adoption date.","s":"∀ {T : ℕ} (a : WithTop (Fin T)) (t : Fin T), Causalean.Panel.AdoptionPath.lt a t = (↑t < a)","l":57,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.lt_of_isInfinite","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"Never-treated paths are untreated in every finite period.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.AdoptionPath.isInfinite a → Causalean.Panel.AdoptionPath.lt a t","l":116,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.AdoptionPath.not_le_of_lt","k":"theorem","m":"Causalean.Panel.AdoptionPath","a":"Panel","t":2,"d":"If t < A, then adoption has not occurred by t.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.AdoptionPath.lt a t → ¬Causalean.Panel.AdoptionPath.le a t","l":111,"f":"Causalean/Panel/AdoptionPath.lean"},{"n":"Causalean.Panel.BaselineExposure","k":"def","m":"Causalean.Panel.PO.Exposure","a":"Panel/PO","t":1,"d":"For a set of units, a set of periods, and a cell-specific exposure set, a baseline exposure assigns a distinguished exposure to every unit-period cell.","s":"{I : Type u_1} → {T : Type u_2} → (I × T → Type u_3) → Type (max (max u_1 u_2) u_3)","l":37,"f":"Causalean/Panel/PO/Exposure.lean"},{"n":"Causalean.Panel.CellBridge.cellIndicator","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a sample space equipped with a σ-algebra, a set of cell labels, a cell-assignment map $X$, and a cell label $i$, the cell indicator is the real-valued function equal to one exactly on the event that $X$ equals $i$, and zero elsewhere.","s":"{Ω : Type u_1} → {ι : Type u_2} → [MeasurableSpace Ω] → (Ω → ι) → ι → Ω → ℝ","l":72,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMass","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a sample space equipped with a σ-algebra, a set of cell labels, a measure $μ$, a cell-assignment map $G$, and a cell label $g$, the cell mass is the real-valued mass that $μ$ assigns to the event that $G$ equals $g$.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":255,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMean","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a sample space equipped with a σ-algebra, a set of cell labels, a measure $μ$, a real-valued function $F$, a cell-assignment map $G$, and a cell label $g$, the cell mean is the integral of $F$ over the event $G=g$, divided by that event's real-valued mass.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":265,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cellMean_mul_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell mean multiplied by cell mass recovers the indicator-weighted raw integral.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), μ {ω | G ω = g} ≠ ⊤ → Causalean.Panel.CellBridge.cellMean μ F G g * Causalean.Panel.CellBridge.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":334,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Dividing an indicator-weighted cell integral by a nonzero cell mass and multiplying back recovers the numerator; on zero-mass cells the numerator is zero because the indicator is a.e. zero.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), μ {ω | G ω = g} ≠ ⊤ → (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.CellBridge.cellMass μ G g * Causalean.Panel.CellBridge.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":297,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteCellIndicator","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a collection of records, a set of cell labels with decidable equality, a cell map $G$, and a cell label $g$, the finite-cell indicator is the real-valued function that equals one for records assigned label $g$ and zero for all other records.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [DecidableEq 𝒢] → (R → 𝒢) → 𝒢 → R → ℝ","l":49,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a collection of records, a cell map $G$, and a finite set of cell labels with decidable equality, the finite one-axis indicator span is the real vector space generated by the indicators of the cells of $G$.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [Fintype 𝒢] → [DecidableEq 𝒢] → (R → 𝒢) → Submodule ℝ (R → ℝ)","l":54,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.finiteTwoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a collection of records, cell maps $G$ and $T$, and their finite label sets with decidable equality, the finite two-axis indicator span is the real vector space generated jointly by the indicators of the cells of $G$ and of $T$.","s":"{R : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [Fintype 𝒢] → [DecidableEq 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → (R → 𝒢) → (R → 𝒯) → Submodule ℝ (R → ℝ)","l":61,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.indicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a measurable sample space, a finite cell-label set with decidable equality and measurable singleton events, a finite measure $μ$, a measurable cell-assignment map $G$, the one-axis indicator span is the class of square-integrable real functions that agree almost everywhere with a finite linear combination of the indicators of the cells of $G$.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → Measurable G → Causalean.Panel.LinearL2Class μ","l":77,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.indicator_cell_memLp","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell indicators are in L² under a finite measure.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":274,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Cell indicators integrate to their real cell mass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.CellBridge.cellMass μ G g","l":285,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_eq_sum_cell","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"Integrate an integrable function by summing over a finite measurable partition generated by G.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":482,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_mul_indicatorSpan_eq_zero_of_cell","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":2,"d":"If a square-integrable residual is orthogonal to every cell indicator, it is orthogonal to every finite indicator-span member.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp V 2 μ → ∀ (c : 𝒢 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * ∑ g, c g * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":345,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.integral_mul_twoAxisIndicatorSpan_eq_zero_of_axes","k":"theorem","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"If a square-integrable residual V is orthogonal in expectation to the indicator of every level g of one axis G and orthogonal in expectation to the indicator of every level t of the other axis T, then it is orthogonal in expectation to every finite linear combination of those axis indicators, with coefficients cG and cT.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] [inst_4 : Fintype 𝒯] [inst_5 : MeasurableSpace 𝒯] [MeasurableSingletonClass 𝒯] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢) (T : Ω → 𝒯), Measurable G → Measurable T → MeasureTheory.MemLp V 2 μ → ∀ (cG : 𝒢 → ℝ) (cT : 𝒯 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → (∀ (t : 𝒯), ∫ (ω : Ω), V ω * {ω' | T ω' = t}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * (∑ g, cG g * {ω' | G ω' = g}.indicator (fun x => 1) ω + ∑ t, cT t * {ω' | T ω' = t}.indicator (fun x => 1) ω) ∂μ = 0","l":406,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.jointCellMass","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a sample space equipped with a σ-algebra, sets of labels for two cell maps, a measure $μ$, cell-assignment maps $G$ and $T$, and labels $g$ and $t$, the joint cell mass is the real-valued mass that $μ$ assigns to the event that $G=g$ and $T=t$.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → 𝒯) → 𝒢 → 𝒯 → ℝ","l":260,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.CellBridge.twoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.CellBridge","a":"Panel","t":1,"d":"For a measurable sample space, finite cell-label sets with decidable equality and measurable singleton events, a finite measure $μ$, and measurable cell-assignment maps $G$ and $T$, the two-axis indicator span is the class of square-integrable real functions that agree almost everywhere with the sum of a finite linear combination of $G$-cell indicators and a finite linear combination of $T$-cell indicators.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → [inst_7 : MeasurableSpace 𝒯] → [MeasurableSingletonClass 𝒯] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → (T : Ω → 𝒯) → Measurable G → Measurable T → Causalean.Panel.LinearL2Class μ","l":159,"f":"Causalean/Panel/CellBridge.lean"},{"n":"Causalean.Panel.Cells","k":"def","m":"Causalean.Panel.Cells","a":"Panel","t":1,"d":"For finite sets of units and periods with decidable equality, the observed cells of a panel are a finite collection of unit-period pairs equipped with strictly positive weights that sum to one over that collection.","s":"(I : Type u_1) → (T : Type u_2) → [Fintype I] → [Fintype T] → [DecidableEq I] → [DecidableEq T] → Type (max u_2 u_1)","l":56,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.CohortLaw","k":"structure","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"Cohort law on a finite cohort index Fin C: probabilities π(g) ≥ 0 summing to one.","s":"ℕ → Type","l":446,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.H_twfe","k":"def","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":1,"d":"The two-way fixed-effect subspace.","s":"{I : Type u_1} → {T : Type u_2} → Submodule ℝ (Causalean.Panel.Cells.V I T)","l":51,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.H_twfe_eq","k":"theorem","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":1,"d":"The two-way fixed-effect subspace H_twfe equals the generic two-axis additive span over units and periods.","s":"∀ {I : Type u_1} {T : Type u_2}, Causalean.Panel.Cells.H_twfe = Causalean.Panel.Weighted.twoAxisAdditiveSpan I T","l":60,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.H_twfe_finiteDimensional","k":"instance","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":2,"d":"H_twfe is finite-dimensional: inherits from AdditiveSpan.finiteDimensional.","s":"∀ {I : Type u_1} {T : Type u_2} [Finite I] [Finite T], Module.Finite ℝ ↥Causalean.Panel.Cells.H_twfe","l":75,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.H_twfe_orthogonal_iff","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a panel weighting scheme and any panel array g, g is orthogonal, under the weighted inner product, to every member of the two-way fixed-effect subspace H_twfe exactly when it is orthogonal to every unit indicator and every period indicator.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (g : Causalean.Panel.Cells.V I T), (∀ h ∈ Causalean.Panel.Cells.H_twfe, c.ip g h = 0) ↔ (∀ (i₀ : I), (c.ip g fun r => if r.1 = i₀ then 1 else 0) = 0) ∧ ∀ (t₀ : T), (c.ip g fun r => if r.2 = t₀ then 1 else 0) = 0","l":125,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.V","k":"def","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"Cell-array space: scalar-valued arrays on I × T.","s":"Type u_1 → Type u_2 → Type (max u_1 u_2)","l":31,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.balanced","k":"def","m":"Causalean.Panel.Cells","a":"Panel","t":1,"d":"The balanced panel: every cell is observed and every cell carries weight 1 / (|I| · |T|).","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → [Nonempty I] → [Nonempty T] → Causalean.Panel.Cells I T","l":80,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.balanced_observed","k":"theorem","m":"Causalean.Panel.Cells","a":"Panel","t":2,"d":"The balanced panel observes every unit-period cell.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T], Causalean.Panel.Cells.balanced.observed = Finset.univ","l":112,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.balanced_weight","k":"theorem","m":"Causalean.Panel.Cells","a":"Panel","t":1,"d":"For any unit-period cell r, the balanced panel design assigns it weight equal to one divided by the total number of unit-period cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T] (r : I × T), Causalean.Panel.Cells.balanced.weight r = 1 / (↑(Fintype.card I) * ↑(Fintype.card T))","l":116,"f":"Causalean/Panel/Cells.lean"},{"n":"Causalean.Panel.Cells.cohortPeriodCells","k":"def","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"Generic cohort-period Cells instance for a nonempty set of cohorts and a nonempty set of periods, a cohort law (a probability mass function over cohorts, summing to one) and strictly positive cohort masses. Every cell (g, t) ∈ Fin C × Fin S is observed and carries weight π(g) / S.","s":"{C S : ℕ} → [Nonempty (Fin C)] → [Nonempty (Fin S)] → (law : Causalean.Panel.Cells.CohortLaw C) → (∀ (g : Fin C), 0 < ↑(law.pi g)) → Causalean.Panel.Cells (Fin C) (Fin S)","l":454,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.cohortPeriod_ip_period_eq","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"In a cohort-by-period panel with cohort-period weights, the weighted inner product of any outcome array with a period indicator equals the cohort-weighted sum of outcomes in that period.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (Y : Causalean.Panel.Cells.V (Fin C) (Fin S)) (t₁ : Fin S), ((Causalean.Panel.Cells.cohortPeriodCells law hpi).ip Y fun r => if r.2 = t₁ then 1 else 0) = 1 / ↑(Fintype.card (Fin S)) * ∑ g, ↑(law.pi g) * Y (g, t₁)","l":681,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.cohortPeriod_ip_unit_eq","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"In a cohort-by-period panel with cohort-period weights, the weighted inner product of any outcome array with a cohort's unit indicator equals that cohort's weighted sum of outcomes across periods.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (Y : Causalean.Panel.Cells.V (Fin C) (Fin S)) (g₁ : Fin C), ((Causalean.Panel.Cells.cohortPeriodCells law hpi).ip Y fun r => if r.1 = g₁ then 1 else 0) = ↑(law.pi g₁) / ↑(Fintype.card (Fin S)) * ∑ t, Y (g₁, t)","l":659,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.cohort_sum_pi_centered_eq_zero","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"A singleton indicator centered by the selected category's weight has a zero weighted average whenever the weights across categories sum to one.","s":"∀ {α : Type u_3} [inst : Fintype α] [inst_1 : DecidableEq α] (w : α → ℝ), ∑ g, w g = 1 → ∀ (g₀ : α), ∑ g, w g * ((if g = g₀ then 1 else 0) - w g₀) = 0","l":493,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.const_mem_H_twfe","k":"theorem","m":"Causalean.Panel.FixedEffect","a":"Panel/FixedEffect","t":1,"d":"For any constant c₀, the constant array c₀ belongs to the two-way fixed-effect subspace H_twfe.","s":"∀ {I : Type u_1} {T : Type u_2} (c₀ : ℝ), (fun x => c₀) ∈ Causalean.Panel.Cells.H_twfe","l":65,"f":"Causalean/Panel/FixedEffect.lean"},{"n":"Causalean.Panel.Cells.ip","k":"def","m":"Causalean.Panel.InnerProduct","a":"Panel","t":1,"d":"Bare-name alias for c.ip. Defined with the explicit finset-sum body (same as WeightedSupport.ip) so that unfold ip exposes the sum form expected by the pre-refactor proof scripts. Definitionally equal (by rfl) to Causalean.Panel.Weighted.WeightedSupport.ip.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → Causalean.Panel.Cells I T → (I × T → ℝ) → (I × T → ℝ) → ℝ","l":60,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat","k":"def","m":"Causalean.Panel.InnerProduct","a":"Panel","t":1,"d":"Bare-name alias for the matrix-valued panel inner product. Its (j, k) entry is the scalar weighted inner product of the jth array in A with the kth array in B.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → {K : ℕ} → Causalean.Panel.Cells I T → (Fin K → I × T → ℝ) → (Fin K → I × T → ℝ) → Matrix (Fin K) (Fin K) ℝ","l":73,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_add_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is additive in its left tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A A' B : Fin K → I × T → ℝ), c.ipMat (A + A') B = c.ipMat A B + c.ipMat A' B","l":144,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_add_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is additive in its right tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B B' : Fin K → I × T → ℝ), c.ipMat A (B + B') = c.ipMat A B + c.ipMat A B'","l":150,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_apply","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product has entries equal to scalar inner products of the corresponding array columns.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ) (j k : Fin K), c.ipMat A B j k = c.ip (A j) (B k)","l":132,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_eq_weighted","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Cells.ipMat is definitionally equal to WeightedSupport.ipMat.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ), c.ipMat A B = Causalean.Panel.Weighted.WeightedSupport.ipMat c A B","l":80,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_smul_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is homogeneous in its left tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : Fin K → I × T → ℝ), c.ipMat (s • A) B = s • c.ipMat A B","l":156,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_smul_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The matrix-valued panel inner product is homogeneous in its right tuple of arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : Fin K → I × T → ℝ), c.ipMat A (s • B) = s • c.ipMat A B","l":162,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ipMat_transpose","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Swapping the two tuples of arrays transposes the matrix-valued panel inner product.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (A B : Fin K → I × T → ℝ), (c.ipMat A B).transpose = c.ipMat B A","l":138,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_add_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is additive in its left array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A A' B : I × T → ℝ), c.ip (A + A') B = c.ip A B + c.ip A' B","l":101,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_add_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is additive in its right array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B B' : I × T → ℝ), c.ip A (B + B') = c.ip A B + c.ip A B'","l":106,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_def","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product unfolds to the weighted sum over observed unit-period cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = ∑ r ∈ c.observed, c.weight r * A r * B r","l":91,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_eq_weighted","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"Cells.ip is definitionally equal to WeightedSupport.ip.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = Causalean.Panel.Weighted.WeightedSupport.ip c A B","l":67,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_self_eq_zero_iff","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":1,"d":"For a panel cell structure and a panel array A, the self inner product c.ip A A is zero exactly when A vanishes on every observed cell of c.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A : I × T → ℝ), c.ip A A = 0 ↔ ∀ r ∈ c.observed, A r = 0","l":126,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_self_nonneg","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The self inner product of any panel array is nonnegative.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A : I × T → ℝ), 0 ≤ c.ip A A","l":121,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_smul_left","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is homogeneous in its left array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : I × T → ℝ), c.ip (s • A) B = s * c.ip A B","l":111,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_smul_right","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is homogeneous in its right array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (s : ℝ) (A B : I × T → ℝ), c.ip A (s • B) = s * c.ip A B","l":116,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.ip_symm","k":"theorem","m":"Causalean.Panel.InnerProduct","a":"Panel","t":2,"d":"The panel weighted inner product is symmetric in its two arrays.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (A B : I × T → ℝ), c.ip A B = c.ip B A","l":96,"f":"Causalean/Panel/InnerProduct.lean"},{"n":"Causalean.Panel.Cells.proj","k":"def","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"Bare-name alias for c.proj. Definitionally equal to Causalean.Panel.Weighted.WeightedSupport.proj.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → Causalean.Panel.Cells I T → Submodule ℝ (Causalean.Panel.Cells.V I T) → Causalean.Panel.Cells.V I T →ₗ[ℝ] Causalean.Panel.Cells.V I T","l":46,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_apply_eq_of_mem_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Any nuisance-space candidate with the projection orthogonality condition matches the chosen projection on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {Y : Causalean.Panel.Cells.V I T}, Y ∈ H → (∀ h ∈ H, c.ip (X - Y) h = 0) → ∀ r ∈ c.observed, (c.proj H) X r = Y r","l":163,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_apply_of_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Projecting a nuisance-space array returns the same values on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) {Y : Causalean.Panel.Cells.V I T}, Y ∈ H → ∀ r ∈ c.observed, (c.proj H) Y r = Y r","l":156,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel projection alias is definitionally equal to the generic weighted support projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)), c.proj H = Causalean.Panel.Weighted.WeightedSupport.proj c H","l":52,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_idem_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the chosen panel projection twice agrees with applying it once on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), ∀ r ∈ c.observed, (c.proj H) ((c.proj H) X) r = (c.proj H) X r","l":173,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The chosen panel projection of an array lies in the nuisance subspace.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), (c.proj H) X ∈ H","l":144,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.proj_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The projection residual is orthogonal to every nuisance-space array under the panel weighted inner product.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {h : Causalean.Panel.Cells.V I T}, h ∈ H → c.ip (X - (c.proj H) X) h = 0","l":149,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize","k":"def","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"Bare-name alias for c.residualize. Definitionally equal to Causalean.Panel.Weighted.WeightedSupport.residualize.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → Causalean.Panel.Cells I T → Submodule ℝ (Causalean.Panel.Cells.V I T) → Causalean.Panel.Cells.V I T →ₗ[ℝ] Causalean.Panel.Cells.V I T","l":57,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the panel residual maker subtracts the nuisance-space projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), (c.residualize H) X = X - (c.proj H) X","l":110,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel residual-maker alias is definitionally equal to the generic weighted support residual maker.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)), c.residualize H = Causalean.Panel.Weighted.WeightedSupport.residualize c H","l":63,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_idem_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Applying the panel residual maker twice agrees with applying it once on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), ∀ r ∈ c.observed, (c.residualize H) ((c.residualize H) X) r = (c.residualize H) X r","l":136,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_in_orthogonal","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"For an array h lying in the nuisance subspace H, the array X residualized against H is orthogonal to h under the panel weighted inner product c.ip.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T) {h : Causalean.Panel.Cells.V I T}, h ∈ H → c.ip (c.tildeX H X) h = 0","l":122,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.residualize_self_of_mem","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"A nuisance-space array residualizes to zero on observed cells.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) {X : Causalean.Panel.Cells.V I T}, X ∈ H → ∀ r ∈ c.observed, c.tildeX H X r = 0","l":129,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.sum_centered_indicator_mem_real","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Centering the indicator of a finite subset by its uniform average makes its sum across the index set equal to zero.","s":"∀ {α : Type u_3} [inst : Fintype α] [inst_1 : DecidableEq α] (S : Finset α), ∑ x, ((if x ∈ S then 1 else 0) - ↑S.card / ↑(Fintype.card α)) = 0","l":88,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX","k":"def","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"Bare-name alias for c.tildeX. Same body as Causalean.Panel.Weighted.WeightedSupport.tildeX (defined as residualize H X) so that unfold tildeX exposes the residualized form.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → Causalean.Panel.Cells I T → Submodule ℝ (Causalean.Panel.Cells.V I T) → Causalean.Panel.Cells.V I T → Causalean.Panel.Cells.V I T","l":68,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeXVec","k":"def","m":"Causalean.Panel.Subspace","a":"Panel","t":1,"d":"Bare-name alias for c.tildeXVec. Same body as Causalean.Panel.Weighted.WeightedSupport.tildeXVec so that unfold tildeXVec exposes the column-by-column form.","s":"{I : Type u_1} → {T : Type u_2} → [inst : Fintype I] → [inst_1 : Fintype T] → [inst_2 : DecidableEq I] → [inst_3 : DecidableEq T] → {K : ℕ} → Causalean.Panel.Cells I T → Submodule ℝ (Causalean.Panel.Cells.V I T) → (Fin K → Causalean.Panel.Cells.V I T) → Fin K → Causalean.Panel.Cells.V I T","l":83,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeXVec_apply","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Column-wise residualization residualizes each regressor column separately.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Fin K → Causalean.Panel.Cells.V I T) (k : Fin K), c.tildeXVec H X k = c.tildeX H (X k)","l":116,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeXVec_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel column-wise residualization alias is definitionally equal to the generic weighted support column-wise residualization.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] {K : ℕ} (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Fin K → Causalean.Panel.Cells.V I T), c.tildeXVec H X = Causalean.Panel.Weighted.WeightedSupport.tildeXVec c H X","l":90,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_cell_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a target cohort g₀, a target period t₀, and an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the cell indicator 1{g = g₀, t = t₀}, evaluated at r, equals the product of centered cohort and period indicators minus the corresponding cross terms fixed by the gauge normalization ∑_t β(t) = π(g₀).","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (g₀ : Fin C) (t₀ : Fin S) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.1 = g₀ ∧ s.2 = t₀ then 1 else 0) r = ((((if r.1 = g₀ then 1 else 0) * if r.2 = t₀ then 1 else 0) - ↑(law.pi g₀) * if r.2 = t₀ then 1 else 0) - 1 / ↑(Fintype.card (Fin S)) * if r.1 = g₀ then 1 else 0) + ↑(law.pi g₀) / ↑(Fintype.card (Fin S))","l":590,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_const","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"The TWFE residual of a constant array vanishes on every observed cell.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (k : ℝ), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun x => k) r = 0","l":249,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_diagonal_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the diagonal (switch-on) indicator 1{g.val = t.val}, evaluated at r, equals the indicator minus the cohort-conditional mean, minus the period-conditional mean net of the grand mean.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if ↑s.1 = ↑s.2 then 1 else 0) r = (((if ↑r.1 = ↑r.2 then 1 else 0) - if ↑r.1 < S then 1 / ↑(Fintype.card (Fin S)) else 0) - if h : ↑r.2 < C then ↑(law.pi ⟨↑r.2, h⟩) else 0) + 1 / ↑(Fintype.card (Fin S)) * ∑ g, if ↑g < S then ↑(law.pi g) else 0","l":997,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_eq","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"Residualizing a panel array subtracts its nuisance-space projection.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), c.tildeX H X = X - (c.proj H) X","l":104,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_eq_weighted","k":"theorem","m":"Causalean.Panel.Subspace","a":"Panel","t":2,"d":"The panel residualized-array alias is definitionally equal to the generic weighted support residualized array.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (H : Submodule ℝ (Causalean.Panel.Cells.V I T)) (X : Causalean.Panel.Cells.V I T), c.tildeX H X = Causalean.Panel.Weighted.WeightedSupport.tildeX c H X","l":75,"f":"Causalean/Panel/Subspace.lean"},{"n":"Causalean.Panel.Cells.tildeX_period_indicator","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Period indicator is absorbed by H_twfe: ind(t = t₀) lies in H_twfe (via a := 0, b(t) := ind(t = t₀)), so its residual vanishes on every observed cell. Independent of the weights.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (t₀ : T), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.2 = t₀ then 1 else 0) r = 0","l":269,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_product_indicator_balanced","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For a subset of units S_I, a subset of periods S_T, and an evaluation cell r, under balanced (equal) weights, the two-way-fixed-effect residual of the product indicator 1{i ∈ S_I}·1{t ∈ S_T}, evaluated at r, factorizes into the product of the centered unit indicator and the centered period indicator.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] [inst_4 : Nonempty I] [inst_5 : Nonempty T] (S_I : Finset I) (S_T : Finset T) (r : I × T), Causalean.Panel.Cells.balanced.tildeX Causalean.Panel.Cells.H_twfe (fun s => (if s.1 ∈ S_I then 1 else 0) * if s.2 ∈ S_T then 1 else 0) r = ((if r.1 ∈ S_I then 1 else 0) - ↑S_I.card / ↑(Fintype.card I)) * ((if r.2 ∈ S_T then 1 else 0) - ↑S_T.card / ↑(Fintype.card T))","l":376,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_triangular_indicator_cohortPeriod","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":1,"d":"For an evaluation cell r, the cohort-period-weighted two-way-fixed-effect residual of the triangular (continued-treatment) indicator 1{g.val < t.val}, evaluated at r, equals the indicator minus the cohort-conditional mean, minus the period-conditional mean net of the grand mean.","s":"∀ {C S : ℕ} [inst : Nonempty (Fin C)] [inst_1 : Nonempty (Fin S)] (law : Causalean.Panel.Cells.CohortLaw C) (hpi : ∀ (g : Fin C), 0 < ↑(law.pi g)) (r : Fin C × Fin S), (Causalean.Panel.Cells.cohortPeriodCells law hpi).tildeX Causalean.Panel.Cells.H_twfe (fun s => if ↑s.1 < ↑s.2 then 1 else 0) r = (if ↑r.1 < ↑r.2 then 1 else 0) - (∑ s, if ↑r.1 < ↑s then 1 else 0) / ↑(Fintype.card (Fin S)) - ∑ g, (if ↑g < ↑r.2 then 1 else 0) * ↑(law.pi g) + 1 / ↑(Fintype.card (Fin S)) * ∑ g, ∑ s, if ↑g < ↑s then ↑(law.pi g) else 0","l":1058,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.Cells.tildeX_unit_indicator","k":"theorem","m":"Causalean.Panel.FixedEffect.IndicatorClosedForms","a":"Panel/FixedEffect","t":2,"d":"Unit indicator is absorbed by H_twfe: ind(i = i₀) lies in H_twfe (via a(i) := ind(i = i₀), b(t) := 0), so its residual vanishes on every observed cell. Independent of the weights.","s":"∀ {I : Type u_1} {T : Type u_2} [inst : Fintype I] [inst_1 : Fintype T] [inst_2 : DecidableEq I] [inst_3 : DecidableEq T] (c : Causalean.Panel.Cells I T) (i₀ : I), ∀ r ∈ c.observed, c.tildeX Causalean.Panel.Cells.H_twfe (fun s => if s.1 = i₀ then 1 else 0) r = 0","l":258,"f":"Causalean/Panel/FixedEffect/IndicatorClosedForms.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A staggered-adoption event-study population: a probability space carrying a realized adoption cohort G — with every cohort cell measurable — a calendar-time map time that is strictly increasing in the period index, a finite set cohorts of adoption cohorts in the event-study support, and adoption-path–indexed potential outcomes Ypath t h ω = Y_{ωt}(h) satisfying structural no-anticipation: in any period where a path is untreated, its outcome equals the never-treated outcome, for every unit.","s":"ℕ → Type (u_1 + 1)","l":50,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.IW_convex_characterization_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Population interaction-weighted characterization (headline). For a staggered-adoption event-study population E and an interaction-weighted design I on the system it induces, if the induced system satisfies comparison-group parallel trends, the eligibility, baseline, target, and comparison-group support conditions of IWSupport hold, the aggregation weights rho are nonnegative and sum to one over the eligible cohorts, and every eligible cohort's population CATT at the fixed event time lies between bounds lo and hi, then the interaction-weighted estimand nuIW is the rho-weighted convex average of the genuine population effects CATT g ℓ, and in particular lies between lo and hi — with no contamination from other event times.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (I : E.toSystem.IWDesign), E.toSystem.IWComparisonParallelTrends I → E.toSystem.IWSupport I → (∀ g ∈ I.cohortsIW, 0 ≤ I.rho g) → ∑ g ∈ I.cohortsIW, I.rho g = 1 → ∀ {lo hi : ℝ}, (∀ g ∈ I.cohortsIW, lo ≤ E.toSystem.CATT g I.eventTime) → (∀ g ∈ I.cohortsIW, E.toSystem.CATT g I.eventTime ≤ hi) → (∀ g ∈ I.cohortsIW, E.toSystem.Delta I g = E.toSystem.CATT g I.eventTime) ∧ E.toSystem.nuIW I = ∑ g ∈ I.cohortsIW, I.rho g * E.toSystem.CATT g I.eventTime ∧ lo ≤ E.toSystem.nuIW I ∧ E.toSystem.nuIW I ≤ hi","l":297,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.OutcomesIntegrable","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$, the outcome-integrability condition states that the potential outcome under every finite period and every adoption path is integrable with respect to the population measure.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → Prop","l":213,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cell","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$ and an adoption path $h$, the adoption-path cell is the event consisting exactly of units whose realized adoption path is $h$.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → WithTop (Fin T) → Set E.Ω","l":102,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$ and an adoption path $h$, the adoption-path cell mass is the real-valued probability mass of the units whose realized adoption path is $h$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → WithTop (Fin T) → ℝ","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$, a real-valued unit-level function $f$, and an adoption path $h$, the adoption-path cell mean is the conditional mean of $f$ on the event that the realized adoption path equals $h$, with value zero when that event has zero probability.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → (E.Ω → ℝ) → WithTop (Fin T) → ℝ","l":110,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean_congr_on","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Event-level means agree when the integrands agree pointwise on the adoption-path cell.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) {f g : E.Ω → ℝ} (h : WithTop (Fin T)), (∀ ω ∈ E.cell h, f ω = g ω) → E.cellMean f h = E.cellMean g h","l":117,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.cellMean_sub","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Event-level means are additive over subtraction of integrable integrands.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) {f g : E.Ω → ℝ} (h : WithTop (Fin T)), MeasureTheory.IntegrableOn f (E.cell h) E.μ → MeasureTheory.IntegrableOn g (E.cell h) E.μ → E.cellMean (f - g) h = E.cellMean f h - E.cellMean g h","l":124,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.contamination_representation_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Population contamination representation (headline). For a staggered-adoption event-study population E and a conventional design D on the system it induces, if the never-treated potential outcome follows the additive parallel-trends restriction, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then the conventional TWFE event-study coefficient D.mu equals the contamination-weighted sum of genuine population cohort-relative-time effects CATT g e = E[Y_{·t}(g) − Y_{·t}(∞) ∣ G = g]. Consistency and no-anticipation are derived from the potential-outcome structure rather than assumed; this is the Sun-Abraham contamination theorem stated over a genuinely potential-outcome-anchored system.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (D : E.toSystem.ConventionalDesign), E.toSystem.MeanParallelUntreated → E.toSystem.ConventionalFiniteSupport D → E.toSystem.CellGridResidualization D → D.mu = ∑ ge ∈ E.toSystem.admissibleCells D.eventSupport, E.toSystem.omega D ge.1 ge.2 * E.toSystem.CATT ge.1 ge.2","l":271,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.observed","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$, a finite period $t$, and a unit $ω$, the observed outcome is that unit's potential outcome at $t$ under its realized adoption path.","s":"{T : ℕ} → (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) → Fin T → E.Ω → ℝ","l":131,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.observed_eqOn_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"On the cohort cell {G = finitePath g}, the observed outcome equals the own-path potential outcome — the pointwise content of consistency.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (g t : Fin T), ∀ ω ∈ E.cell (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g), E.observed t ω = E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) ω","l":162,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study population $E$, the induced event-study system has the population's calendar-time map and supported cohorts, and defines every cohort share, cell mass, and outcome-mean field as the corresponding adoption-path cell probability or conditional mean.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T","l":137,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_CATT_eq_meanDiff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Integrability makes CATT a genuine expected contrast. Under outcome integrability (assumption H5), the two cohort-cell means combine into a single cell mean of the potential-outcome difference: each summand is E[Y_{·t}(g) − Y_{·t}(∞) ∣ G = g], a genuine expectation of the individual treatment-effect random variable. This is where the integrability hypothesis does real work — cellMean_sub requires each slice to be integrable on the cell, so without H5 the two means could not be merged.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.OutcomesIntegrable → ∀ (g : Fin T) (e : ℤ), E.toSystem.CATT g e = (↑(E.toSystem.targetPeriods g e).card)⁻¹ * ∑ t ∈ E.toSystem.targetPeriods g e, E.cellMean (E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) - E.Ypath t ⊤) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g)","l":250,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_CATT_eq_po_contrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Causal-meaning certificate. For a population satisfying the event-study setup, in the system it induces, the treatment-effect estimand at cohort g and relative time e equals the cohort-cell average of the population potential-outcome contrast E[Y_{·t}(g) ∣ G = g] − E[Y_{·t}(∞) ∣ G = g] over the relevant periods, so the estimand carries genuine causal content rather than a free-standing definition on reals.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T) (g : Fin T) (e : ℤ), E.toSystem.CATT g e = (↑(E.toSystem.targetPeriods g e).card)⁻¹ * ∑ t ∈ E.toSystem.targetPeriods g e, (E.cellMean (E.Ypath t (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g)) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g) - E.cellMean (E.Ypath t ⊤) (Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath g))","l":234,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_causalRestrictions","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Bundle: the induced system satisfies the Sun-Abraham causal restrictions, given the additive parallel-trends hypothesis. Consistency and no-anticipation are derived; only parallel trends is assumed.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.MeanParallelUntreated → E.toSystem.EventStudyCausalRestrictions","l":224,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_consistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Consistency is derived. The observed cohort mean equals the own-path potential-outcome mean, because the observed outcome is the potential outcome under the realized path.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.Consistency","l":172,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_noAnticipation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"No-anticipation is derived from structural no-anticipation: on a pre-adoption period the own-path and never-treated potential-outcome means coincide, so their cohort means do.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.NoAnticipation","l":182,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation.toSystem_pathConsistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.PopulationBridge","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Path-consistency is derived: on a period where comparison path h is untreated, the observed path mean equals the never-treated path mean.","s":"∀ {T : ℕ} (E : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudyPopulation T), E.toSystem.PathConsistency","l":200,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"A finite-cell record of a staggered-adoption event-study design over T periods, where adoption paths are finite periods or the never-treated path. It bundles a numeric encoding of each period used to form relative event time, the finite set of adoption cohorts in the event-study support, each adoption path's population share, and the cohort-period cell mass, together with, by cohort or by comparison adoption path, the factual observed outcome mean, the mean potential outcome under the cohort's own treatment path, and the mean never-treated potential outcome.","s":"ℕ → Type","l":37,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.AdmissibleCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a finite cohort $g$, and an integer relative time $e$, the admissible-cell predicate holds exactly when the cohort belongs to the system's event-study support and at least one finite period has relative event time $e$ for that cohort.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → Prop","l":101,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a cohort $g$, and a relative time $e$, the cohort average treatment effect on the treated is the mean cell contrast for $(g,e)$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":134,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT_eq_sourceCATT_of_injective","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"G1 faithfulness corollary. When the calendar-time map time is injective (the usual one-period-per-relative-time encoding), the cell-averaged CATT g e collapses to the source's *point* CATT_{g,e} at the unique period t realizing relative time e, i.e. treatedMean g t - untreatedMean g t.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), Function.Injective P.time → ∀ {g : Fin T} {e : ℤ} {t : Fin T}, t ∈ P.targetPeriods g e → P.CATT g e = P.treatedMean g t - P.untreatedMean g t","l":278,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CATT_eq_zero_of_noAnticipation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"No anticipation implies zero pre-treatment CATT. If the mean treated and never-treated potential outcomes for every adopting cohort coincide in every period strictly preceding that cohort's own adoption period (no anticipation), then for a cohort g in the finite adoption-cohort support and a relative event time e strictly before adoption ($e < 0$), the cohort-average treatment effect on the treated at cell (g, e) is zero.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.NoAnticipation → ∀ {g : Fin T} {e : ℤ}, g ∈ P.cohorts → e < 0 → P.CATT g e = 0","l":197,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CellGridResidualization","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid residualization input.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":266,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.CellIndex","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the cell index is the collection of admissible cohort--relative-time cells in that design.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Type","l":50,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.Consistency","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, the consistency condition states that every cohort in the event-study support has its factual observed outcome mean equal to its own-treatment-path mean potential outcome in every finite period.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":155,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalDesign","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Conventional event-study finite design for the coefficient on displayedEvent.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Type","l":39,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalFiniteSupport","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Transparent finite support bookkeeping for the conventional cell expansion. It records that the explicit event-time support is the finite universe over which the displayed theorem is expanded, without asserting the headline contamination formula itself.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":133,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ConventionalResidualization","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Finite-cell residualization record for the conventional coefficient.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Prop","l":79,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.DIDContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, an interaction-weighted design, and a cohort, the difference-in- differences contrast is the treated cohort's observed target-minus- baseline mean change minus the corresponding comparison-group mean change.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":100,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.Delta","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, an interaction-weighted design, and a cohort, the interaction- weighted cohort contrast is that cohort's difference-in-differences contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":109,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.EventStudyCausalRestrictions","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Sun-Abraham event-study causal restrictions. Field names mirror the NL artifact's assumption names.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":190,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWComparisonParallelTrends","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Comparison-group parallel trends for the IW DID contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Prop","l":149,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWDesign","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Interaction-weighted finite DID design for a fixed event time.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Type","l":52,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IWSupport","k":"structure","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"IW support restrictions for eligible cohorts and their comparison groups.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Prop","l":124,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IW_Delta_eq_CATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and interaction-weighted design I, if consistency, no-anticipation, and path-consistency hold, every eligible cohort's comparison-group mean change from baseline to target period matches its own untreated-mean change, every eligible cohort is a genuine cohort of the system, and every comparison unit is untreated throughout the baseline and target periods, then for any eligible cohort g, its interaction-weighted DID contrast Delta I g equals the cohort-relative-time average treatment effect CATT g I.eventTime.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (I : P.IWDesign), P.Consistency → P.NoAnticipation → P.PathConsistency → (∀ g ∈ I.cohortsIW, (↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedMean g t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedMean g t = (P.comparisonMass I g)⁻¹ * ∑ h ∈ I.comparisonGroup g, P.cohortShare h * ((↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedPathMean h t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedPathMean h t)) → (∀ g ∈ I.cohortsIW, g ∈ P.cohorts) → (∀ g ∈ I.cohortsIW, ∀ h ∈ I.comparisonGroup g, ∀ t ∈ P.baselinePeriods g, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0) → (∀ g ∈ I.cohortsIW, ∀ h ∈ I.comparisonGroup g, ∀ t ∈ P.targetPeriods g I.eventTime, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0) → ∀ {g : Fin T}, g ∈ I.cohortsIW → P.Delta I g = P.CATT g I.eventTime","l":174,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IW_convex_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and interaction-weighted design I, if consistency, no-anticipation, and path-consistency hold, every eligible cohort's comparison-group mean change from baseline to target period matches its own untreated-mean change, the eligibility, baseline, target, and comparison-group support conditions of IWSupport hold, the aggregation weights rho are nonnegative and sum to one over the eligible cohorts, and every eligible cohort's CATT at the fixed event time lies between bounds lo and hi, then each cohort's DID contrast equals its CATT, the interaction-weighted estimand nuIW I equals the rho-weighted average of those CATTs, and nuIW I itself lies between lo and hi — a genuine convex average with no contamination from other relative times.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (I : P.IWDesign), P.Consistency → P.NoAnticipation → P.PathConsistency → (∀ g ∈ I.cohortsIW, (↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedMean g t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedMean g t = (P.comparisonMass I g)⁻¹ * ∑ h ∈ I.comparisonGroup g, P.cohortShare h * ((↑(P.targetPeriods g I.eventTime).card)⁻¹ * ∑ t ∈ P.targetPeriods g I.eventTime, P.untreatedPathMean h t - (↑(P.baselinePeriods g).card)⁻¹ * ∑ t ∈ P.baselinePeriods g, P.untreatedPathMean h t)) → P.IWSupport I → (∀ g ∈ I.cohortsIW, 0 ≤ I.rho g) → ∑ g ∈ I.cohortsIW, I.rho g = 1 → ∀ {lo hi : ℝ}, (∀ g ∈ I.cohortsIW, lo ≤ P.CATT g I.eventTime) → (∀ g ∈ I.cohortsIW, P.CATT g I.eventTime ≤ hi) → (∀ g ∈ I.cohortsIW, P.Delta I g = P.CATT g I.eventTime) ∧ P.nuIW I = ∑ g ∈ I.cohortsIW, I.rho g * P.CATT g I.eventTime ∧ lo ≤ P.nuIW I ∧ P.nuIW I ≤ hi","l":301,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.IsEventStudyNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, a conventional design, and a cohort-period function, the event-study nuisance condition holds exactly when the function is the sum of a cohort-and-period additive component and a linear combination of the included relative-time indicators other than the displayed indicator, for every cohort in the system and every period.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → (Fin T → Fin T → ℝ) → Prop","l":55,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.MeanParallelUntreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, the mean-parallel-untreated condition states that there exist cohort and period components such that their sum gives an additive cohort-period function and every supported cohort's never-treated mean potential outcome equals that function in every finite period.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":185,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.NoAnticipation","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, the no-anticipation condition states that, for every supported cohort and every finite period strictly before that cohort's adoption date in calendar time, the own-treatment-path and never-treated mean potential outcomes are equal.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":180,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.PathConsistency","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, the path-consistency condition states that, for every adoption path and every finite period in which that path is untreated, the factual observed path mean equals the never-treated potential-outcome path mean.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Prop","l":159,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event study with $T$ periods, an adoption path $h$, and a period $t$, the absorbing treatment indicator equals one exactly when $h$ has adopted by $t$, and equals zero otherwise.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → ℝ","l":83,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.admissibleCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$ and a finite set $E$ of relative times, the admissible-cell support is the finite set of cohort-relative-time pairs that use a supported cohort and a relative time in $E$ and satisfy the admissible-cell predicate.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Finset ℤ → Finset (Fin T × ℤ)","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.apparent_pretrends_from_post_treatment","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and conventional design D, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the residualized displayed-event indicator Rdot is orthogonal in expectation to every function in the event-study nuisance class, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu equals the contamination-weighted sum of cohort-relative-time CATTs restricted to nonnegative relative times, i.e. once every negative-relative-time CATT vanishes under no-anticipation, the displayed lead's coefficient is a weighted sum of post-treatment effects.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.EventStudyCausalRestrictions → (∀ (h : Fin T → Fin T → ℝ), P.IsEventStudyNuisance D h → ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * P.cellAverage h ge.1 ge.2 = 0) → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport with 0 ≤ ge.2, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":394,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.apparent_pretrends_from_post_treatment_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Apparent pretrends (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the included, displayed, and admissible event times all lie within the declared finite event-time support, and the cell-grid weighted-projection residualization input is supplied, then D.mu equals the contamination-weighted sum of cohort-relative-time CATTs restricted to nonnegative relative times — since no-anticipation forces every negative-relative-time CATT to vanish, this exhibits D.mu as a weighted average of post-treatment effects.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.EventStudyCausalRestrictions → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport with 0 ≤ ge.2, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":75,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.baselinePeriods","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$ and a finite cohort $g$, the baseline-period set consists of exactly the target periods at relative event time minus one.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → Finset (Fin T)","l":96,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, a cohort-period function, a cohort, and a relative time, the cell average is the average of that function over the periods in which the cohort has the given relative time.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → (Fin T → Fin T → ℝ) → Fin T → ℤ → ℝ","l":70,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage_eventIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"The cell average of the relative-time indicator 1{relTime = e}, evaluated on a nonempty cell (g, e'), is the cell-level indicator 1{e' = e}.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) {e : ℤ} {g : Fin T} {e' : ℤ}, (P.targetPeriods g e').Nonempty → P.cellAverage (fun g t => Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e (P.relTime g t)) g e' = Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e e'","l":159,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellAverage_mem_cellNuisance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Every cell-evaluated nuisance function lies in the cell-nuisance subspace.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {hCell : Fin T → Fin T → ℝ}, P.IsEventStudyNuisance D hCell → (fun cell => P.cellAverage hCell (↑cell).1 (↑cell).2) ∈ P.cellNuisance D","l":152,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellGrid_mu_eq_conventionalMuRatio","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-grid FWL bridge. Under cell-grid residualization of the event-study design, the conventional event-study coefficient equals the Frisch–Waugh–Lovell residualized ratio computed directly on the cohort × relative-time cell grid.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.CellGridResidualization D → D.mu = P.conventionalMuRatio D","l":310,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellGrid_provides_residualization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a conventional event-study design D over an event-study system P, if every admissible cohort-relative-time cell has strictly positive population mass, the collection of admissible cells is nonempty, and the design's residualized displayed-event indicator Rdot agrees, cell by cell, with the weighted projection residual on the cell grid, then the three finite-cell orthogonality conditions packaged as ConventionalResidualization — derived here from a genuine weighted projection rather than assumed — hold for D.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign} (hCellMassPos : ∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) (hCellNonempty : (P.admissibleCells D.eventSupport).Nonempty), (∀ (cell : P.CellIndex D), D.Rdot (↑cell).1 (↑cell).2 = (P.cellSupport D hCellMassPos hCellNonempty).tildeX (P.cellNuisance D) (P.cellRegressor D) cell) → P.ConventionalResidualization D","l":357,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellMassAtEvent","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a cohort $g$, and a relative time $e$, the cohort-relative-time cell mass is the sum of the cohort-period cell masses over all target periods for $(g,e)$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":117,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the cell-nuisance space is the linear span of all cell-level averages of event-study nuisance functions.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → Submodule ℝ (P.CellIndex D → ℝ)","l":145,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellNuisanceGen","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the generators of the cell-nuisance space are precisely the functions on admissible cells obtained by averaging an event-study nuisance function within each cohort--relative-time cell.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → Set (P.CellIndex D → ℝ)","l":136,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellOutcome","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the cell-grid outcome assigns to each admissible cohort--relative-time cell its observed mean outcome.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → P.CellIndex D → ℝ","l":259,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellRegressor","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the cell-grid regressor assigns one to an admissible cell exactly when its relative time is the displayed event time, and zero otherwise.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → P.CellIndex D → ℝ","l":251,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellSupport","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, if every admissible cohort--relative-time cell has strictly positive population mass and at least one admissible cell exists, the cell-grid weighted support is the finite population of admissible cells, weighted by each cell's mass divided by total admissible cell mass.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → (D : P.ConventionalDesign) → (∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) → (P.admissibleCells D.eventSupport).Nonempty → Causalean.Panel.Weighted.WeightedSupport (P.CellIndex D)","l":62,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cellTotalMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the total admissible cell mass is the sum of the population masses of all admissible cohort--relative-time cells.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":56,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.cohortsAtEvent","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a finite set $E$ of relative times, and a relative time $e$, the cohorts at event time $e$ are exactly the supported cohorts for which $e$ belongs to $E$ and the cohort-relative-time cell is admissible.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Finset ℤ → ℤ → Finset (Fin T)","l":112,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.comparisonMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, an interaction-weighted design, and a cohort, the comparison mass is the total population share of the cohorts in that cohort's comparison group.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":81,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.comparisonMeanChange","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, an interaction-weighted design, and a cohort, the comparison mean change is the comparison-group population-share-weighted average of each comparison path's change from the cohort's baseline periods to its target periods.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":89,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Conventional Sun-Abraham contamination representation. For an event-study system P and conventional design D, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated potential outcome follows an additive parallel-trends restriction, the residualized displayed-event indicator Rdot is orthogonal in expectation to every function in the event-study nuisance class, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu equals the Sun-Abraham contamination-weighted sum of cohort-relative-time CATTs over every admissible cell.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.Consistency → P.MeanParallelUntreated → (∀ (h : Fin T → Fin T → ℝ), P.IsEventStudyNuisance D h → ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * P.cellAverage h ge.1 ge.2 = 0) → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":195,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Contamination representation (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated potential outcome follows an additive parallel-trends restriction, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then D.mu equals the Sun-Abraham contamination-weighted sum of cohort-relative-time CATTs over every admissible cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.Consistency → P.MeanParallelUntreated → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ ge ∈ P.admissibleCells D.eventSupport, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":35,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_split","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system P and conventional design D, if observed outcomes equal the potential outcome under the realized treatment path (consistency), the never-treated outcome satisfies additive parallel trends, the design satisfies the finite-cell orthogonality conditions ConventionalResidualization, the residualized denominator is strictly positive, the coefficient D.mu equals its FWL residualized-ratio form, and the included, displayed, and admissible event times all lie within the declared finite support, then D.mu splits as the displayed-event-time contamination term plus the contamination-weighted sum over every other admissible cohort-relative-time cell.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), P.Consistency → P.MeanParallelUntreated → P.ConventionalResidualization D → 0 < P.residualDenom D → D.mu = P.conventionalMuRatio D → P.ConventionalFiniteSupport D → D.mu = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent * P.CATT g D.displayedEvent + ∑ ge ∈ P.admissibleCells D.eventSupport with ge.2 ≠ D.displayedEvent, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":321,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.contamination_representation_split_of_cellGrid","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Contamination","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Contamination split (genuine). For the conventional event-study coefficient D.mu of a design D on system P, if the consistency, mean-parallel-trends, and no-anticipation causal restrictions hold, the included, displayed, and admissible event times all lie within the declared finite support, and the cell-grid weighted-projection residualization input is supplied, then D.mu splits as the displayed-event-time contamination term plus the contamination-weighted sum over every other admissible cohort-relative-time cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign}, P.EventStudyCausalRestrictions → P.ConventionalFiniteSupport D → P.CellGridResidualization D → D.mu = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent * P.CATT g D.displayedEvent + ∑ ge ∈ P.admissibleCells D.eventSupport with ge.2 ≠ D.displayedEvent, P.omega D ge.1 ge.2 * P.CATT ge.1 ge.2","l":54,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Contamination.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.conventionalMuRatio","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the conventional coefficient ratio is the residualized numerator divided by the residualized denominator.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":126,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.desired_event_weights_sum_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Desired-event-time weights sum to one.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign), ∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * D.Rdot ge.1 ge.2 * Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator D.displayedEvent ge.2 = ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.cellMassAtEvent g D.displayedEvent * D.Rdot g D.displayedEvent → 0 < P.residualDenom D → ∑ g ∈ P.cohortsAtEvent D.eventSupport D.displayedEvent, P.omega D g D.displayedEvent = 1","l":152,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For two relative times, the event-time indicator equals one when they are equal and zero otherwise.","s":"ℤ → ℤ → ℝ","l":34,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator_mem_cellNuisance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"A non-displayed included relative-time indicator lies in the cell-nuisance subspace (it is the cell average of the corresponding event-study nuisance).","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {e : ℤ}, e ∈ D.includedEvents → e ≠ D.displayedEvent → (fun cell => Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e (↑cell).2) ∈ P.cellNuisance D","l":178,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.finitePath","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event study with $T$ periods and a finite adoption cohort $g$, the finite adoption path is the path with adoption date $g$.","s":"{T : ℕ} → Fin T → WithTop (Fin T)","l":69,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.ip_cellSupport","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"ip → cell-sum. The cell-grid weighted inner product of two cell functions reproduces the finite-cell sum (divided by the total mass Z). This is the bridge that turns weighted-projection facts into the finite-cell orthogonality conditions.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) (hpos : ∀ ge ∈ P.admissibleCells D.eventSupport, 0 < P.cellMassAtEvent ge.1 ge.2) (hne : (P.admissibleCells D.eventSupport).Nonempty) (A B : Fin T × ℤ → ℝ), ((P.cellSupport D hpos hne).ip (fun cell => A ↑cell) fun cell => B ↑cell) = (∑ ge ∈ P.admissibleCells D.eventSupport, P.cellMassAtEvent ge.1 ge.2 * A ge * B ge) / P.cellTotalMass D","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.isEventuallyTreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event study with $T$ periods and an adoption path $h$, the eventually-treated predicate holds exactly when $h$ has a finite adoption date.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":75,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.isNeverTreated","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event study with $T$ periods and an adoption path $h$, the never-treated predicate holds exactly when $h$ is the never-treated path.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":72,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.meanCellContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a cohort $g$, and a relative time $e$, the mean cell contrast is the arithmetic average, over all target periods for $(g,e)$, of the treated-path mean potential outcome minus the never-treated mean potential outcome; it is zero when there are no such periods.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":128,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.nuIW","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and an interaction-weighted design, the interaction-weighted event-study estimand is the sum, over all eligible cohorts, of each aggregation weight times that cohort's interaction-weighted contrast.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → ℝ","l":117,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedBaselineMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a cohort, the observed baseline mean is the average observed outcome of that cohort over its baseline periods, defined at relative time $-1$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℝ","l":72,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedCellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a cohort $g$, and a relative time $e$, the observed cell mean is the arithmetic average of that cohort's factual observed outcome means over all target periods for $(g,e)$; it is zero when there are no such periods.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → ℝ","l":122,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedCellMean_eq_fixedEffects_add_CATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"Cell-mean decomposition into additive untreated fixed effects and CATT. If the factual observed outcome mean on a cohort's own periods equals its mean treated potential outcome (consistency) and the mean never-treated potential outcome admits an additive cohort/period fixed-effects representation (mean-parallel untreated paths), then for any cohort g in the finite adoption-cohort support, the average observed outcome mean over the target periods of cell (g, e) decomposes as the average, over those periods, of the additive fixed effects alpha g + lambda t plus the cohort-average treatment effect on the treated CATT g e.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.Consistency → P.MeanParallelUntreated → ∀ {g : Fin T} {e : ℤ}, g ∈ P.cohorts → ∃ alpha lambda, P.observedCellMean g e = (↑(P.targetPeriods g e).card)⁻¹ * ∑ t ∈ P.targetPeriods g e, (alpha g + lambda t) + P.CATT g e","l":220,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.observedTargetMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, an interaction-weighted design, and a cohort, the observed target mean is the average observed outcome of that cohort over the periods at the design's specified relative time.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.IWDesign → Fin T → ℝ","l":63,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.omega","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, a conventional design, a cohort, and a relative time, the Sun--Abraham contamination weight is that cell's mass times its design residual, divided by the residualized denominator.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → Fin T → ℤ → ℝ","l":144,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.other_included_event_weights_sum_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Other included-event-time weights sum to zero.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) {e : ℤ}, ∑ g ∈ P.cohortsAtEvent D.eventSupport e, P.cellMassAtEvent g e * D.Rdot g e = 0 → 0 < P.residualDenom D → ∑ g ∈ P.cohortsAtEvent D.eventSupport e, P.omega D g e = 0","l":177,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathBaselineMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, an adoption path $h$, and a treated cohort $g$, the path baseline mean is the arithmetic average of $h$'s factual observed outcome mean over cohort $g$'s baseline periods; it is zero when that set is empty.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → WithTop (Fin T) → Fin T → ℝ","l":149,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathConsistency_observed_eq_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Path-consistency, applied to an untreated comparison period, yields the observed-equals-untreated path-mean bridge used by the IW comparison-group argument.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), P.PathConsistency → ∀ {h : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.absorbingTreatment h t = 0 → P.observedPathMean h t = P.untreatedPathMean h t","l":171,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.pathTargetMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, an adoption path $h$, a treated cohort $g$, and a relative time $e$, the path target mean is the arithmetic average of $h$'s factual observed outcome mean over the target periods for $(g,e)$; it is zero when that set is empty.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → WithTop (Fin T) → Fin T → ℤ → ℝ","l":143,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.relTime","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, an adoption cohort $g$, and a period $t$, relative event time is the integer calendar time of $t$ minus the integer calendar time of $g$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → Fin T → ℤ","l":78,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.residualDenom","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the residualized denominator is the sum over admissible cells of cell mass times the design residual times the indicator for the displayed event time.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":106,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.residualNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Conventional","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For a finite time horizon, an event-study system, and a conventional design, the residualized numerator is the sum over admissible cells of cell mass times the design residual times the observed cell mean.","s":"{T : ℕ} → (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) → P.ConventionalDesign → ℝ","l":116,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Conventional.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.sum_admissible_mul_eventIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Cell-grid filtering identity. Weighting an admissible-cell sum by the relative-time indicator 1{e = e'} collapses it to a sum over the cohorts observed at relative time e'. Pure finite algebra; no residualization needed. This is the engine behind hDisplayedExpansion and hOtherIncludedOrthogonal.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T) (D : P.ConventionalDesign) (e' : ℤ) (f : Fin T → ℤ → ℝ), ∑ ge ∈ P.admissibleCells D.eventSupport, f ge.1 ge.2 * Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.eventIndicator e' ge.2 = ∑ g ∈ P.cohortsAtEvent D.eventSupport e', f g e'","l":207,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.targetPeriods","k":"def","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":1,"d":"For an event-study system $P$, a finite cohort $g$, and an integer relative time $e$, the target-period set consists of exactly the finite periods whose relative event time for cohort $g$ is $e$.","s":"{T : ℕ} → Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T → Fin T → ℤ → Finset (Fin T)","l":90,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.targetPeriods_subsingleton_of_injective","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.Setup","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Under an injective calendar-time encoding, at most one finite period can realize a given relative time, so the target-period set of any cell is a subsingleton.","s":"∀ {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T), Function.Injective P.time → ∀ (g : Fin T) (e : ℤ), (↑(P.targetPeriods g e)).Subsingleton","l":263,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/Setup.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem.tildeX_eq_Rdot","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.CellGrid","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Under cell-grid residualization, the residualized displayed-event regressor equals the design residual at every admissible cohort--relative-time cell.","s":"∀ {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.EventStudyContamination.EventStudySystem T} {D : P.ConventionalDesign} (h : P.CellGridResidualization D), (P.cellSupport D h.hCellMassPos h.hCellNonempty).tildeX (P.cellNuisance D) (P.cellRegressor D) = fun cell => D.Rdot (↑cell).1 (↑cell).2","l":302,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/CellGrid.lean"},{"n":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.sum_convex_mem_Icc","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.EventStudyContamination.InteractionWeighted","a":"Panel/EstimandCharacterization/EventStudyContamination","t":2,"d":"Finite convex-combination bound. If the weights w are nonnegative and sum to one, then any nonnegative-weighted average of values lying in [lo, hi] again lies in [lo, hi]. This is the algebraic content of \"convex combination\" used to certify that the IW estimand is a genuine convex average of the target CATTs (no contamination).","s":"∀ {ι : Type u_1} (s : Finset ι) (w f : ι → ℝ) {lo hi : ℝ}, (∀ i ∈ s, 0 ≤ w i) → ∑ i ∈ s, w i = 1 → (∀ i ∈ s, lo ≤ f i) → (∀ i ∈ s, f i ≤ hi) → lo ≤ ∑ i ∈ s, w i * f i ∧ ∑ i ∈ s, w i * f i ≤ hi","l":25,"f":"Causalean/Panel/EstimandCharacterization/EventStudyContamination/InteractionWeighted.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.AggregateWeights","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Nonnegative aggregate weights summing to one on the target treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max u_1 u_2)","l":108,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a staggered-adoption cell system, conditional parallel trends in additive form means that there exist cohort-by-covariate and period-by- covariate functions whose sum equals the untreated potential-outcome mean for every cohort, period, and covariate cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":123,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On top of a staggered-cell design P and its saturated untreated-outcome regression S, this structure packages three families of treated-cell coefficients — an imputation coefficient, a pooled-least-squares (POLS) coefficient, and an extended two-way-fixed-effects (ETWFE) coefficient — together with the conditions pinning them down: on every treated cell the imputation coefficient equals the covariate-weighted imputation residual mean, and the POLS and ETWFE coefficients each solve the finite-cell covariate-weighted residual normal equation.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Type (max u_1 u_2)","l":479,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.etwfe_cell_eq_pols","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Compatibility alias: ETWFE/POLS equality is now derived by solving both finite-cell normal equations, rather than stored as a field.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaETWFE g t = E.thetaPOLS g t","l":554,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.pols_cell_eq_imputation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Compatibility alias: the POLS/imputation equality is now derived from the POLS normal equation, rather than stored as a field.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaPOLS g t = E.thetaImp g t","l":544,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.thetaETWFE_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"ETWFE cell coefficients equal imputation. On any treated cohort-time cell (g,t), the extended two-way-fixed-effects (ETWFE) cell coefficient equals the imputation residual mean at that cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaETWFE g t = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t","l":528,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands.thetaPOLS_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"POLS cell coefficients equal imputation. On any treated cohort-time cell (g,t), the flexible POLS cell coefficient equals the imputation residual mean at that cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaPOLS g t = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t","l":513,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsCellAdditive","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a cohort-period-covariate function, the cell-additivity condition means that there exist cohort-by-covariate and period-by- covariate functions whose sum equals that function at every cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → (Cohort → Time → Covar → ℝ) → Prop","l":133,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units, periods, unit-level controls, and period-level controls, a scalar regressor, unit-level control functions, period-level control functions, and a candidate nuisance function, the two-way Mundlak nuisance condition holds exactly when the candidate is a constant plus arbitrary multiples of the regressor's unit means and period means and linear combinations of the supplied unit-level and period-level controls.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {Z : Type u_3} → {M : Type u_4} → [Fintype Z] → [Fintype M] → (Unit → Time → ℝ) → (Z → Unit → ℝ) → (M → Time → ℝ) → (Unit → Time → ℝ) → Prop","l":148,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsVectorTwoWayMundlakNuisance","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units, periods, regressor coordinates, unit-level controls, and period-level controls, a vector regressor, unit-level control functions, period-level control functions, and a candidate nuisance function, the vector two-way Mundlak nuisance condition holds exactly when the candidate is a constant plus linear combinations of every regressor coordinate's unit and period means and of the supplied unit-level and period-level controls.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → [Fintype K] → {Z : Type u_4} → {M : Type u_5} → [Fintype Z] → [Fintype M] → (Unit → Time → K → ℝ) → (Z → Unit → ℝ) → (M → Time → ℝ) → (Unit → Time → ℝ) → Prop","l":182,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a staggered-adoption cell system, no anticipation means that, for every cohort, every period designated untreated, and every covariate cell, that cohort's potential outcome equals its untreated potential outcome.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":115,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A finite-cell weighted least-squares fit of the untreated-outcome mean for a staggered cell design P, restricted to the untreated observations. It bundles a fitted untreated-outcome mean that is additive in cohort and time given the covariate cell, projection weights that are strictly positive on the untreated design, the requirement that the fit solves the covariate/cell-weighted normal equations against every additive test function, summed over the untreated design, full-rank identification of the additive class from vanishing on the untreated design alone, and a positive cohort share on every treated cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max (max u_1 u_2) u_3)","l":159,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.recovers_target_Y0","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Saturated untreated regression recovers the untreated potential outcome on treated cells. If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then on any treated cohort-time cell (g,t) covered by the saturated untreated regression S, the fitted value S.m0 g t c equals the mean untreated potential outcome Y0Mean g t c, for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.treatedCell g t → ∀ (c : Covar), S.m0 g t c = P.Y0Mean g t c","l":320,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.toUntreatedFitWitness","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, the staggered-adoption cell system underlying the regression, and a saturated untreated-outcome regression, the untreated-fit witness retains its fitted untreated mean, additivity, untreated-cell weights, positivity condition, and untreated normal equations, while omitting its target-support and design-identification conditions.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness P","l":225,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression.untreatedFit","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The weighted projection m0 reproduces the factual cohort-g outcome mean on every untreated cell.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.untreatedCell g t → ∀ (c : Covar), S.m0 g t c = P.YgMean g t c","l":242,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A scalar two-way-fixed-effects regression problem on a finite balanced panel of units and time periods, given a scalar outcome and a scalar regressor, where the sum of squared double-demeaned regressor values is strictly positive — the scalar full-rank condition ensuring the two-way within estimator is well defined.","s":"(Unit : Type u_3) → (Time : Type u_4) → [Fintype Unit] → [Fintype Time] → Type (max u_3 u_4)","l":33,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units and periods and a scalar two-way-fixed-effects regression problem, the scalar two-way-fixed- effects coefficient is the residualized numerator divided by the residualized-design denominator.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":59,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE_normalEq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The closed-form coefficient satisfies the scalar TWFE normal equation by dividing through the positive residualized sum of squares.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time), P.twfeNormalEq P.betaTWFE","l":73,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.betaTWFE_unique","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Scalar full-rank uniqueness of the TWFE normal-equation solution.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) {β : ℝ}, P.twfeNormalEq β → β = P.betaTWFE","l":105,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.toVector","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit and time-period sets and a scalar two-way-fixed-effects problem, the associated one-coordinate vector two-way-fixed-effects problem has the same panel and outcome, uses the scalar regressor as its sole coordinate, and has a nonsingular residualized Gram matrix because the scalar double-demeaned regressor has a strictly positive sum of squares.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time (Fin 1)","l":174,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.toVector_betaTWFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The singleton-coordinate vector TWFE coefficient recovers the scalar TWFE coefficient, so the scalar theorem is the K = Fin 1 case of the vector one.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time), P.toVector.betaTWFE 0 = P.betaTWFE","l":197,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeDenominator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units and periods and a scalar two-way-fixed-effects regression problem, the residualized-design denominator is the sum of squared doubly demeaned regressor values over all unit--period observations.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":45,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units and periods, a scalar two-way- fixed-effects regression problem, and a proposed coefficient, the scalar two-way-fixed-effects normal equation states that the sum of the doubly demeaned regressor times the corresponding residual is zero.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ → Prop","l":66,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem.twfeNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units and periods and a scalar two-way-fixed-effects regression problem, the residualized numerator is the sum, over all unit--period observations, of the doubly demeaned regressor times the doubly demeaned outcome.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → ℝ","l":52,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A scalar two-way Mundlak fit of the TWFE problem P against optional time-constant controls Zvar and time-only controls Mvar, stated by normal equations rather than by a particular coding of the nuisance regressors. It bundles a scalar coefficient on the regressor and a nuisance function lying in the two-way Mundlak span, subject to the pooled normal equation against the regressor and the pooled normal equation against every nuisance function in that span.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {Z : Type u_3} → {M : Type u_4} → [Fintype Z] → [Fintype M] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time → (Z → Unit → ℝ) → (M → Time → ℝ) → Type (max u_1 u_2)","l":178,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Finite staggered-adoption cell system: cohort shares and within-cohort covariate weights over treated and untreated cohort-time cells, with cell-level means of the untreated and cohort-specific treated potential outcomes. It requires that every treated cell's cohort has positive share, that the covariate weights are nonnegative and sum to one within each cohort, and that the observed cell mean coincides with the treated mean on treated cells and with the untreated mean on untreated cells.","s":"(Cohort : Type u_4) → (Time : Type u_5) → (Covar : Type u_6) → [Fintype Cohort] → [Fintype Time] → [Fintype Covar] → Type (max (max u_4 u_5) u_6)","l":37,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofMeasure","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite cohort, time-period, and covariate sets, a measurable sample space with a measure, cohort-time-covariate cell events, untreated, cohort-specific, and observed population outcome functions, treated and untreated cell indicators, cohort shares, and within-cohort covariate weights, if every cell event is measurable, every cell has strictly positive measure, each of the three outcome functions is integrable on every cell, cohort shares are strictly positive on treated cells, covariate weights are nonnegative and sum to one within every cohort, and the observed outcome agrees pointwise with the cohort-specific outcome on treated cells and with the untreated outcome on untreated cells, the finite staggered-DID cell system uses the supplied shares, weights, and indicators and defines each outcome mean as its population event-conditional mean.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {Ω : Type u_4} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (cellEvent : Cohort → Time → Covar → Set Ω) → (Y0pop Ygpop Yobspop : Ω → ℝ) → (treatedCell untreatedCell : Cohort → Time → Prop) → (cohortShare : Cohort → ℝ) → (covarWeight : Cohort → Covar → ℝ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) → (∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) → (∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) → (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar","l":58,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite cohort, time-period, and covariate sets, a measurable sample space with a probability measure, cohort-time-covariate cell events, untreated, cohort-specific, and observed population outcome functions, treated and untreated cell indicators, cohort shares, and within-cohort covariate weights, if every cell event is measurable, every cell has strictly positive probability, each outcome function is integrable on every cell, cohort shares are strictly positive on treated cells, covariate weights are nonnegative and sum to one within every cohort, and pointwise consistency holds on treated and untreated cells, the finite staggered-DID cell system is the system constructed from the same data by the measure-based constructor.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → {Ω : Type u_4} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (cellEvent : Cohort → Time → Covar → Set Ω) → (Y0pop Ygpop Yobspop : Ω → ℝ) → (treatedCell untreatedCell : Cohort → Time → Prop) → (cohortShare : Cohort → ℝ) → (covarWeight : Cohort → Covar → ℝ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) → (∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) → (∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) → (∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) → (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar","l":121,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.tauAgg","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, and a cohort-period weighting function, the aggregate average treatment effect on the treated is the weighted sum of cell average treatment effects over all treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → (Cohort → Time → ℝ) → ℝ","l":98,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.tauCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a cohort, and a period, the cell average treatment effect on the treated is the covariate- weighted average, within that cohort, of the treated potential outcome minus the untreated potential outcome in that period.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Cohort → Time → ℝ","l":89,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.treatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a staggered-adoption cell system, the treated-cell set is the finite set of all cohort--period pairs designated as treated by that system.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Finset (Cohort × Time)","l":73,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.untreatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a staggered-adoption cell system, the untreated-cell set is the finite set of all cohort--period pairs designated as untreated and used to fit the untreated-outcome regression.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Finset (Cohort × Time)","l":81,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedDesignIdentifies","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells and a staggered-adoption cell system, the untreated-design identification condition means that every additive cohort-period- covariate function which vanishes at every untreated cell also vanishes at every treated cell.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Prop","l":141,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.UntreatedFitWitness","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"The part of an untreated-regression witness needed to prove exact fit on the untreated design.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar → Type (max (max u_1 u_2) u_3)","l":209,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A K-vector two-way-fixed-effects regression problem on a finite balanced panel of units and time periods, given a scalar outcome and a K-vector of regressors, where the residualized Gram matrix of the double-demeaned regressors is nonsingular — the vector full-rank condition ensuring the two-way within estimator is well defined.","s":"(Unit : Type u_4) → (Time : Type u_5) → [Fintype Unit] → [Fintype Time] → (K : Type u_6) → [Fintype K] → [DecidableEq K] → Type (max (max u_4 u_5) u_6)","l":61,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit, time-period, and regressor-coordinate sets and a vector two-way-fixed-effects problem, the closed-form vector TWFE coefficient is the inverse residualized Gram matrix multiplied by the residualized outcome-regressor numerator vector.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → K → ℝ","l":75,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE_normalEq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Existence of a TWFE solution. For a vector two-way-fixed-effects problem, whose residualized Gram matrix is nonsingular by assumption, the closed-form coefficient P.betaTWFE solves the matrix normal equation defining the TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K), P.vecTwfeNormalEq P.betaTWFE","l":149,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.betaTWFE_unique","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Full-rank uniqueness of the vector TWFE coefficient. For a vector TWFE problem P with nonsingular residualized Gram matrix, if a coefficient vector β satisfies the coordinate-wise TWFE normal equation — in every coordinate the double-demeaned regressor is orthogonal to the double-demeaned residual, then β equals the closed-form vector TWFE coefficient P.betaTWFE.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) {β : K → ℝ}, P.vecTwfeNormalEq β → β = P.betaTWFE","l":158,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem.vecTwfeNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit, time-period, and regressor-coordinate sets, a vector two-way-fixed-effects problem, and a candidate coefficient vector, the vector TWFE normal-equation condition requires that, for every regressor coordinate, the finite inner product of its double-demeaned regressor with the double-demeaned outcome residual is zero.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → (K → ℝ) → Prop","l":81,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit","k":"structure","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"A coding-free K-vector two-way Mundlak fit of the vector TWFE problem P against optional time-constant controls Zvar and time-only controls Mvar, stated by normal equations rather than by a particular coding of the nuisance regressors. It bundles a coefficient vector and a nuisance function lying in the vector two-way Mundlak span, subject to the pooled normal equation against every regressor coordinate and the pooled normal equation against every nuisance function in that span.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → {K : Type u_3} → [inst_2 : Fintype K] → [inst_3 : DecidableEq K] → {Z : Type u_4} → {M : Type u_5} → [Fintype Z] → [Fintype M] → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K → (Z → Unit → ℝ) → (M → Time → ℝ) → Type (max (max u_1 u_2) u_3)","l":223,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts and periods, a cohort, and a period, the saturated treated-cell indicator is one at that cohort--period pair and zero at every other pair.","s":"{Cohort : Type u_1} → {Time : Type u_2} → [DecidableEq Cohort] → [DecidableEq Time] → Cohort → Time → Cohort → Time → ℝ","l":425,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator_normalEq_eq_cellResidual","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Saturated block-diagonalization for treated-cell indicators.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] [inst_3 : DecidableEq Cohort] [inst_4 : DecidableEq Time] (outcome fitted : Cohort → Time → Covar → ℝ) (covarWeight : Cohort → Covar → ℝ) (theta : ℝ) (g : Cohort) (t : Time), ∑ g', ∑ t', ∑ c, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator g t g' t' * covarWeight g' c * (outcome g' t' c - fitted g' t' c - Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellIndicator g t g' t' * theta) = 0 ↔ ∑ c, covarWeight g c * (outcome g t c - fitted g t c - theta) = 0","l":432,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellResidualNormalEq","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a saturated untreated-outcome regression, a proposed cell coefficient, a cohort, and a period, the cell residual normal equation states that the within-cohort covariate-weighted mean of observed outcome minus fitted untreated outcome minus that coefficient is zero.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → ℝ → Cohort → Time → Prop","l":383,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.cellResidualNormalEq_eq_imputationTheta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"A finite-cell residual normal equation identifies the coefficient with the imputation residual mean.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Covar] (outcome fitted : Cohort → Time → Covar → ℝ) (covarWeight : Cohort → Covar → ℝ), (∀ (g : Cohort), ∑ c, covarWeight g c = 1) → ∀ {theta : ℝ} {g : Cohort} {t : Time}, ∑ c, covarWeight g c * (outcome g t c - fitted g t c - theta) = 0 → theta = ∑ c, covarWeight g c * (outcome g t c - fitted g t c)","l":395,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit, time-period, and regressor-coordinate sets, a vector-valued regressor array, a unit, a time period, and a regressor coordinate, the componentwise double-demeaned regressor is the scalar double demean of that coordinate over the finite balanced panel.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Unit → Time → K → ℝ","l":43,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_aggregate_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Aggregate characterization: every weighting of imputation, POLS, and ETWFE equals the weighted ATT aggregate. If no anticipation holds and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then for any treated-cell weighting function a, the a-weighted aggregates of the imputation, POLS, and ETWFE cell estimands all equal the a-weighted ATT aggregate.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ (a : Cohort → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE P E a = P.tauAgg a","l":638,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_cell_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Cell-level characterization: imputation, POLS, and ETWFE agree with the ATT cell. If no anticipation holds, conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, and (g,t) is a treated cohort-time cell covered by the untreated-regression witness S and the imputation/POLS/ETWFE estimands E, then the imputation, POLS, and ETWFE cell coefficients at (g,t) all equal the ATT cell τ_gt.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaImp g t = P.tauCell g t ∧ E.thetaPOLS g t = P.tauCell g t ∧ E.thetaETWFE g t = P.tauCell g t","l":614,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.flexible_did_scaffold_characterization","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Headline finite-cell characterization (Wooldridge, Theorem B). If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then, given the saturated untreated regression S and the POLS/ETWFE finite-cell residual normal equations carried by E, on every treated cohort-time cell the flexible imputation, POLS, and ETWFE estimands all equal the ATT cell, and consequently every treated-cell weighted aggregate of the three estimands equals the correspondingly weighted ATT aggregate.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → (∀ ⦃g : Cohort⦄ ⦃t : Time⦄, P.treatedCell g t → E.thetaImp g t = P.tauCell g t ∧ E.thetaPOLS g t = P.tauCell g t ∧ E.thetaETWFE g t = P.tauCell g t) ∧ ∀ (a : Cohort → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS P E a = P.tauAgg a ∧ Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE P E a = P.tauAgg a","l":679,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit, time-period, and regressor-coordinate sets and a vector-valued regressor array, the residualized Gram matrix has entry $(j,k)$ equal to the finite sum, over units and time periods, of the product of the double-demeaned $j$-th and $k$-th regressor coordinates.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Matrix K K ℝ","l":49,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units, periods, and regressor coordinates and a residualized vector regressor, the residualized Gram matrix has as its coordinate pair entry the sum over unit--period observations of the product of the corresponding two regressor coordinates.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → Matrix K K ℝ","l":37,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram_eq_gramOf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"gram is the residualized instance of the generic version.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} (X : Unit → Time → K → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram X = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X)","l":52,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a saturated untreated-outcome regression, a cohort, and a period, the imputation residual mean is the within-cohort covariate-weighted average of the observed cell mean minus the fitted untreated mean.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P → Cohort → Time → ℝ","l":367,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The finite imputation residual mean equals a population conditional expectation.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (g : Cohort) (t : Time) (cohortEvent : Set Ω) (covarCell : Covar → Set Ω) (Δ : Ω → ℝ), MeasurableSet cohortEvent → (∀ (c : Covar), MeasurableSet (covarCell c)) → Pairwise (Function.onFun Disjoint covarCell) → ⋃ c, covarCell c = Set.univ → MeasureTheory.Integrable Δ μ → (∀ (c : Covar), P.covarWeight g c = (μ (cohortEvent ∩ covarCell c)).toReal / (μ cohortEvent).toReal) → (∀ (c : Covar), P.observedMean g t c - S.m0 g t c = Causalean.PO.eventCondExp μ (cohortEvent ∩ covarCell c) Δ) → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta P S g t = Causalean.PO.eventCondExp μ cohortEvent Δ","l":46,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.imputationTheta_eq_tauCell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Imputation recovers the ATT cell once the saturated untreated prediction equals the untreated potential-outcome mean in target cells.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → E.thetaImp g t = P.tauCell g t","l":599,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_treated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"On a treated cell, the fitted untreated mean equals the population conditional mean of the untreated potential outcome.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (c : Covar) (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop : Ω → ℝ), P.Y0Mean g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop → S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":130,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_treated_ofPopulation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On a treated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any treated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop Ygpop Yobspop : Ω → ℝ) (treatedCell untreatedCell : Cohort → Time → Prop) (cohortShare : Cohort → ℝ) (covarWeight : Cohort → Covar → ℝ) (hmeas : ∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) (hcell_pos : ∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) (hY0_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) (hYg_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) (hYobs_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) (cohortShare_pos_on_treated : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) (covarWeight_nonneg : ∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) (covarWeight_sum_one : ∀ (g : Cohort), ∑ c, covarWeight g c = 1) (hcons_tr : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) (hcons_ut : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar}, P = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation μ cellEvent Y0pop Ygpop Yobspop treatedCell untreatedCell cohortShare covarWeight hmeas hcell_pos hY0_int hYg_int hYobs_int cohortShare_pos_on_treated covarWeight_nonneg covarWeight_sum_one hcons_tr hcons_ut → ∀ (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (c : Covar), S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":148,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"On an untreated cell, the fitted untreated mean equals the population conditional mean of the untreated potential outcome.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar} (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.untreatedCell g t → ∀ (c : Covar) (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop : Ω → ℝ), P.Y0Mean g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop → S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":149,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.m0_eq_eventCondExp_untreated_ofPopulation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationOrigin","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"On an untreated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any untreated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (cellEvent : Cohort → Time → Covar → Set Ω) (Y0pop Ygpop Yobspop : Ω → ℝ) (treatedCell untreatedCell : Cohort → Time → Prop) (cohortShare : Cohort → ℝ) (covarWeight : Cohort → Covar → ℝ) (hmeas : ∀ (g : Cohort) (t : Time) (c : Covar), MeasurableSet (cellEvent g t c)) (hcell_pos : ∀ (g : Cohort) (t : Time) (c : Covar), 0 < (μ (cellEvent g t c)).toReal) (hY0_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Y0pop (cellEvent g t c) μ) (hYg_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Ygpop (cellEvent g t c) μ) (hYobs_int : ∀ (g : Cohort) (t : Time) (c : Covar), MeasureTheory.IntegrableOn Yobspop (cellEvent g t c) μ) (cohortShare_pos_on_treated : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → 0 < cohortShare g) (covarWeight_nonneg : ∀ (g : Cohort) (c : Covar), 0 ≤ covarWeight g c) (covarWeight_sum_one : ∀ (g : Cohort), ∑ c, covarWeight g c = 1) (hcons_tr : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, treatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Ygpop ω) (hcons_ut : ∀ ⦃g : Cohort⦄ ⦃t : Time⦄, untreatedCell g t → ∀ (c : Covar), ∀ ω ∈ cellEvent g t c, Yobspop ω = Y0pop ω) {P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar}, P = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells.ofPopulation μ cellEvent Y0pop Ygpop Yobspop treatedCell untreatedCell cohortShare covarWeight hmeas hcell_pos hY0_int hYg_int hYobs_int cohortShare_pos_on_treated covarWeight_nonneg covarWeight_sum_one hcons_tr hcons_ut → ∀ (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.NoAnticipation P → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ConditionalParallelTrendsAdditive P → ∀ {g : Cohort} {t : Time}, P.untreatedCell g t → ∀ (c : Covar), S.m0 g t c = Causalean.PO.eventCondExp μ (cellEvent g t c) Y0pop","l":208,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationOrigin.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.matrix_fwl_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Matrix Frisch-Waugh-Lovell handoff. If a coefficient vector β and nuisance term Hβ satisfy the finite normal equations against the raw vector regressor D and a nuisance class H, while each coordinate of the residualized regressor Dtilde is orthogonal to H and the residualized Gram matrix is nonsingular, then β is the residualized matrix coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] (H : (Unit → Time → ℝ) → Prop) {Y Yproj Ytilde : Unit → Time → ℝ} {D Dproj Dtilde : Unit → Time → K → ℝ} {Hβ : Unit → Time → ℝ} {β : K → ℝ}, (∀ (i : Unit) (t : Time), Y i t = Yproj i t + Ytilde i t) → (∀ (i : Unit) (t : Time) (k : K), D i t k = Dproj i t k + Dtilde i t k) → (∀ (k : K), H fun i t => Dproj i t k) → H Hβ → (∀ (k : K) (h : Unit → Time → ℝ), H h → ∑ i, ∑ t, Dtilde i t k * h i t = 0) → (∀ (k : K), ∑ i, ∑ t, Dtilde i t k * Yproj i t = 0) → IsUnit (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf Dtilde).det → (∀ (k : K), ∑ i, ∑ t, D i t k * (Y i t - ∑ j, D i t j * β j - Hβ i t) = 0) → (∀ (h : Unit → Time → ℝ), H h → ∑ i, ∑ t, h i t * (Y i t - ∑ j, D i t j * β j - Hβ i t) = 0) → β = (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gramOf Dtilde)⁻¹.mulVec (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf Dtilde Ytilde)","l":83,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.mundlak_nuisance_unit_time","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Mundlak nuisance functions are unit/time additive, so optional time-constant and time-only controls lie inside the same orthogonality class.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (X : Unit → Time → ℝ) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) {h : Unit → Time → ℝ}, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h","l":164,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite unit, time-period, and regressor-coordinate sets, a vector-valued regressor array, and an outcome array, the residualized numerator vector has coordinate $k$ equal to the finite sum of the product of the double-demeaned $k$-th regressor and the double-demeaned outcome.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → (Unit → Time → ℝ) → K → ℝ","l":55,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of units, periods, and regressor coordinates, a residualized vector regressor, and a residualized outcome, the residualized numerator vector has as each coordinate the sum over unit--period observations of that regressor coordinate times the outcome.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → {K : Type u_3} → (Unit → Time → K → ℝ) → (Unit → Time → ℝ) → K → ℝ","l":44,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer_eq_numerOf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"numer is the residualized instance of the generic version.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} (X : Unit → Time → K → ℝ) (Y : Unit → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer X Y = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numerOf (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X) (Causalean.Panel.UniformTwoWayPanel.ddot Y)","l":56,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiETWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a saturated untreated-outcome regression, a flexible-DID estimand collection, and a cohort-period weighting function, the aggregate extended two-way- fixed-effects estimand is the weighted sum of extended two-way-fixed- effects coefficients over treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":588,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiImp","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a saturated untreated-outcome regression, a flexible-DID estimand collection, and a cohort-period weighting function, the aggregate imputation estimand is the weighted sum of imputation coefficients over treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":567,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.psiPOLS","k":"def","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.DID","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"For finite sets of cohorts, periods, and covariate cells, a staggered-adoption cell system, a saturated untreated-outcome regression, a flexible-DID estimand collection, and a cohort-period weighting function, the aggregate pooled-least-squares estimand is the weighted sum of pooled-least-squares coefficients over treated cells.","s":"{Cohort : Type u_1} → {Time : Type u_2} → {Covar : Type u_3} → [inst : Fintype Cohort] → [inst_1 : Fintype Time] → [inst_2 : Fintype Covar] → (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) → {S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P} → Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S → (Cohort → Time → ℝ) → ℝ","l":577,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/DID.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.sum_dotRegressor","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Reshuffle: a residualized regressor against a β-combination of regressors factors through the cross-Gram.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] (Dt D : Unit → Time → K → ℝ) (β : K → ℝ) (k : K), ∑ i, ∑ t, Dt i t k * ∑ j, D i t j * β j = ∑ j, (∑ i, ∑ t, Dt i t k * D i t j) * β j","l":61,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.sum_ite_one_mul","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Selecting a single coordinate via a 0/1 indicator collapses the coordinate sum to that coordinate's value.","s":"∀ {K : Type u_3} [inst : Fintype K] [inst_1 : DecidableEq K] (k : K) (f : K → ℝ), ∑ k', (if k' = k then 1 else 0) * f k' = f k","l":214,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.thetaImp_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.PopulationBridge","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"The imputation estimand equals a population conditional expectation. On a treated cohort-time cell (g,t), suppose the cohort event is measurable, each covariate cell is measurable, the covariate cells are pairwise disjoint, the covariate cells cover the whole sample space, the treatment-effect integrand Δ is integrable, each finite covariate weight equals the conditional probability of that covariate cell given the cohort event, and each finite cell residual — the observed mean minus the fitted untreated mean — equals the within-cell conditional mean of Δ given the cohort event and that covariate cell. Then the imputation cell estimand thetaImp g t equals the population conditional expectation E[Δ | cohortEvent].","s":"∀ {Cohort : Type u_1} {Time : Type u_2} {Covar : Type u_3} [inst : Fintype Cohort] [inst_1 : Fintype Time] [inst_2 : Fintype Covar] {Ω : Type u_4} [inst_3 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.StaggeredATTCells Cohort Time Covar) (S : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.SaturatedUntreatedRegression P) (E : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.FlexibleDIDEstimands P S) {g : Cohort} {t : Time}, P.treatedCell g t → ∀ (cohortEvent : Set Ω) (covarCell : Covar → Set Ω) (Δ : Ω → ℝ), MeasurableSet cohortEvent → (∀ (c : Covar), MeasurableSet (covarCell c)) → Pairwise (Function.onFun Disjoint covarCell) → ⋃ c, covarCell c = Set.univ → MeasureTheory.Integrable Δ μ → (∀ (c : Covar), P.covarWeight g c = (μ (cohortEvent ∩ covarCell c)).toReal / (μ cohortEvent).toReal) → (∀ (c : Covar), P.observedMean g t c - S.m0 g t c = Causalean.PO.eventCondExp μ (cohortEvent ∩ covarCell c) Δ) → E.thetaImp g t = Causalean.PO.eventCondExp μ cohortEvent Δ","l":82,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_equivalence","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Wooldridge finite-panel scalar TWFE-two-way-Mundlak equivalence. For a scalar two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient on the regressor equals the two-way-fixed-effects coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) (fit : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar Mvar), fit.beta = P.betaTWFE","l":213,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_optional_controls_invariant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Adding or removing optional time-constant or time-only controls does not change the scalar coefficient, because both fits equal the TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z₁ : Type u_5} {M₁ : Type u_6} {Z₂ : Type u_7} {M₂ : Type u_8} [inst_2 : Fintype Z₁] [inst_3 : Fintype M₁] [inst_4 : Fintype Z₂] [inst_5 : Fintype M₂] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar₁ : Z₁ → Unit → ℝ) (Mvar₁ : M₁ → Time → ℝ) (Zvar₂ : Z₂ → Unit → ℝ) (Mvar₂ : M₂ → Time → ℝ) (fit₁ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar₁ Mvar₁) (fit₂ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWMFit P Zvar₂ Mvar₂), fit₁.beta = fit₂.beta","l":270,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.twfe_twm_residual_common","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.TWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Residualizing the scalar regressor against the two-way Mundlak nuisance span leaves the same residual as double demeaning.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {Z : Type u_3} {M : Type u_4} [inst_2 : Fintype Z] [inst_3 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ScalarTWFEProblem Unit Time) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ), (Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive fun i t => P.X i t - Causalean.Panel.UniformTwoWayPanel.ddot P.X i t) ∧ ∀ (h : Unit → Time → ℝ), Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsTwoWayMundlakNuisance P.X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.ddot P.X) h = 0","l":195,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/TWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vecNormalEq_iff_mulVec","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFE","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"The coordinate-wise normal equation is equivalent to the matrix normal equation Q_{\\ddot X} β = Σ_it ddot X ddot Y.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] (X : Unit → Time → K → ℝ) (Y : Unit → Time → ℝ) (β : K → ℝ), (∀ (k : K), ∑ i, ∑ t, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X i t k * (Causalean.Panel.UniformTwoWayPanel.ddot Y i t - ∑ j, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.ddotVec X i t j * β j) = 0) ↔ (Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.gram X).mulVec β = Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.numer X Y","l":92,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorTWFE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vec_twfe_twm_equivalence","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":1,"d":"Wooldridge finite-panel K-vector TWFE-two-way-Mundlak equivalence (Theorem A). For a K-vector two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient vector on the regressors equals the K-vector two-way-fixed-effects coefficient vector.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] {Z : Type u_4} {M : Type u_5} [inst_4 : Fintype Z] [inst_5 : Fintype M] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) (fit : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar Mvar), fit.beta = P.betaTWFE","l":243,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vec_twfe_twm_optional_controls_invariant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Adding or removing optional time-constant or time-only controls does not change the K-vector Mundlak coefficient, since both fits equal the vector TWFE coefficient.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] [inst_3 : DecidableEq K] {Z₁ : Type u_6} {M₁ : Type u_7} {Z₂ : Type u_8} {M₂ : Type u_9} [inst_4 : Fintype Z₁] [inst_5 : Fintype M₁] [inst_6 : Fintype Z₂] [inst_7 : Fintype M₂] (P : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWFEProblem Unit Time K) (Zvar₁ : Z₁ → Unit → ℝ) (Mvar₁ : M₁ → Time → ℝ) (Zvar₂ : Z₂ → Unit → ℝ) (Mvar₂ : M₂ → Time → ℝ) (fit₁ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar₁ Mvar₁) (fit₂ : Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorTWMFit P Zvar₂ Mvar₂), fit₁.beta = fit₂.beta","l":312,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.vector_mundlak_nuisance_unit_time","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.VectorMundlak","a":"Panel/EstimandCharacterization/FlexibleDIDMundlak","t":2,"d":"Vector two-way Mundlak nuisance terms are unit/time additive, so the optional controls lie inside the same orthogonality class as for the scalar case.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] {K : Type u_3} [inst_2 : Fintype K] {Z : Type u_4} {M : Type u_5} [inst_3 : Fintype Z] [inst_4 : Fintype M] (X : Unit → Time → K → ℝ) (Zvar : Z → Unit → ℝ) (Mvar : M → Time → ℝ) {h : Unit → Time → ℝ}, Causalean.Panel.EstimandCharacterization.FlexibleDIDMundlak.IsVectorTwoWayMundlakNuisance X Zvar Mvar h → Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h","l":199,"f":"Causalean/Panel/EstimandCharacterization/FlexibleDIDMundlak/VectorMundlak.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"A finite de Chaisemartin-D'Haultfoeuille group-time panel: it bundles group-time cell weights, a binary treatment indicator, the observed outcome, the untreated potential outcome, cell-level treatment effects, and a residualized-treatment witness, subject to strict positivity and unit sum of the weights, the treatment indicator taking only the values zero and one, potential-outcome consistency — the observed outcome equals the untreated outcome plus the treatment indicator times the treatment effect, the residualized witness differing from the treatment indicator by a group-plus-time additive function, its orthogonality, in the weighted inner product, to every group-plus-time additive function, and a strictly positive weighted sum of its squares.","s":"(G : Type u_1) → (T : Type u_2) → [Fintype G] → [Fintype T] → Type (max u_1 u_2)","l":87,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.DWeightedTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the all-cell treatment-effect component is the finite sum of each cell's treatment effect weighted by its cell weight, residualized treatment, and binary treatment indicator, normalized by the denominator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":146,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.DWeightedTau_eq_treatedWeightedTau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"The all-cell D-weighted component is the same as the treated-cell sum.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.DWeightedTau = P.treatedWeightedTau","l":229,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.SD","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the residualized-treatment denominator is the weighted finite sum of squared residualized treatment values over all group-time cells.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":118,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the finite TWFE coefficient is the weighted finite inner product of residualized treatment and observed outcome, divided by the residualized-treatment denominator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":123,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.inner_Dtilde_D_eq_SD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Orthogonality of the residualized treatment against D - Dtilde gives the DCDH denominator identity.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), ∑ g, ∑ t, P.pi g t * P.Dtilde g t * P.D g t = P.SD","l":184,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.inner_Dtilde_D_eq_SD_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Orthogonality of the residualized treatment against D - Dtilde gives the DCDH denominator identity.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (pi D Dtilde : G → T → ℝ), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE fun g t => D g t - Dtilde g t) → (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, pi g t * Dtilde g t * h g t = 0) → ∑ g, ∑ t, pi g t * Dtilde g t * D g t = ∑ g, ∑ t, pi g t * Dtilde g t ^ 2","l":164,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_neg_iff_residual_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Dividing a positive cell mass times a residual by a positive normalizer preserves whether the residual is negative.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (pi * Dtilde / SD < 0 ↔ Dtilde < 0)","l":289,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_pos_iff_residual_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Dividing a positive cell mass times a residual by a positive normalizer preserves whether the residual is positive.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (0 < pi * Dtilde / SD ↔ 0 < Dtilde)","l":315,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.normalized_weight_zero_iff_residual_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"A positive cell mass and positive normalizer make the normalized weight zero exactly when its residual is zero.","s":"∀ {pi Dtilde SD : ℝ}, 0 < pi → 0 < SD → (pi * Dtilde / SD = 0 ↔ Dtilde = 0)","l":343,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.ofPopulation","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.PopulationBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets, a measurable sample space with a probability measure, group-time cell events, factual, untreated-potential, and treated-potential outcome functions, a cell-level treatment array, and a residualized-treatment array, if the cells are measurable, pairwise disjoint, and cover the sample space, every cell has strictly positive probability, treatment is binary in every cell, pointwise potential-outcome consistency holds in every cell, treatment minus the residualized array is additive in group and time, the residualized array is weighted-orthogonal to every group-plus-time additive array, and its weighted sum of squares is strictly positive, the finite cell partition supplies cell probabilities and cell means to the finite DCDH panel, whose treatment effects are treated-minus-untreated population cell means.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → {Ω : Type u_3} → [inst_2 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (cellEvent : G → T → Set Ω) → (Yfac Y0pop Y1pop : Ω → ℝ) → (d Dtilde : G → T → ℝ) → (∀ (g : G) (t : T), MeasurableSet (cellEvent g t)) → Pairwise (Function.onFun Disjoint fun p => cellEvent p.1 p.2) → ⋃ p, cellEvent p.1 p.2 = Set.univ → (∀ (g : G) (t : T), 0 < (μ (cellEvent g t)).toReal) → (∀ (g : G) (t : T), d g t = 0 ∨ d g t = 1) → (∀ (g : G) (t : T), ∀ ω ∈ cellEvent g t, Yfac ω = Y0pop ω + d g t * (Y1pop ω - Y0pop ω)) → (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE fun g t => d g t - Dtilde g t) → (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, (μ (cellEvent g t)).toReal * Dtilde g t * h g t = 0) → 0 < ∑ g, ∑ t, (μ (cellEvent g t)).toReal * Dtilde g t ^ 2 → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T","l":45,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.omega","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets, a finite DCDH group-time panel, a group, and a time period, the normalized DCDH cell weight is that cell's weight times its residualized treatment, divided by the residualized-treatment denominator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → G → T → ℝ","l":135,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treatedCells","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the set of treated cells consists exactly of the group-time pairs whose binary treatment indicator equals one.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → Finset (G × T)","l":141,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treatedWeightedTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the treated-cell weighted treatment-effect component is the finite sum, over treated cells only, of each normalized DCDH cell weight times that cell's treatment effect.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":152,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_neg_iff_Dtilde_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Sign equivalence for the DCDH weight (negative direction). For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is negative exactly when the residualized treatment D̃_gt is negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, P.omega g t < 0 ↔ P.Dtilde g t < 0","l":303,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_pos_iff_Dtilde_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Positive-weight direction of the DCDH sign characterization. For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is positive exactly when the residualized treatment D̃_gt is positive.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, 0 < P.omega g t ↔ 0 < P.Dtilde g t","l":330,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH weights sum to one. For a DCDH panel, the normalized DCDH weights sum to one over all treated cells.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), ∑ gt ∈ P.treatedCells, P.omega gt.1 gt.2 = 1","l":262,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.treated_omega_zero_iff_Dtilde_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Zero-weight direction of the DCDH sign characterization. For a DCDH panel and any cohort-period cell, the normalized weight ω_gt is zero exactly when the residualized treatment D̃_gt is zero.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T) {g : G} {t : T}, P.omega g t = 0 ↔ P.Dtilde g t = 0","l":360,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_treated_weighted_tau_of_zeroUntreatedContrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Zero untreated bias implies TWFE equals the treated-cell weighted effect. If the untreated-outcome residual contrast vanishes — the residualized-treatment-weighted average of the untreated potential outcome Y0 over the whole panel is zero, then the finite two-way fixed-effects (TWFE) coefficient equals the treated-cell weighted sum of treatment effects.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.zeroUntreatedResidualContrast → P.betaTWFE = P.treatedWeightedTau","l":249,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_untreatedBias_add_DWeightedTau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH finite TWFE decomposition (all-cell weighting). For a DCDH panel, the finite two-way fixed-effects (TWFE) coefficient decomposes as the sum of the untreated bias and the all-cell D-weighted treatment-effect component.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.betaTWFE = P.untreatedBias + P.DWeightedTau","l":210,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.twfe_eq_untreatedBias_add_treated_weighted_tau","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"DCDH finite TWFE decomposition (treated-cell weighting). For a DCDH panel, the finite TWFE coefficient decomposes as the sum of the untreated bias and the normalized treated-cell weighted sum of treatment effects.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.betaTWFE = P.untreatedBias + P.treatedWeightedTau","l":242,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.untreatedBias","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the untreated residual contrast is the weighted finite inner product of residualized treatment and the untreated potential outcome, divided by the residualized-treatment denominator.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → ℝ","l":129,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.weighted_residual_contrast_eq_zero_of_isGTFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"A weighted residual is orthogonal to every additive array, so its contrast with an additive untreated-outcome array vanishes.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (pi Dtilde Y0 : G → T → ℝ), (∀ (h : G → T → ℝ), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE h → ∑ g, ∑ t, pi g t * Dtilde g t * h g t = 0) → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE Y0 → ∑ g, ∑ t, pi g t * Dtilde g t * Y0 g t = 0","l":191,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.zeroUntreatedResidualContrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets and a finite DCDH group-time panel, the zero untreated residual-contrast condition holds exactly when the weighted finite inner product of residualized treatment and the untreated potential outcome is zero.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T → Prop","l":158,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel.zeroUntreatedResidualContrast_of_Y0_mem_gtfe","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Group-plus-period untreated means imply the zero untreated residual contrast.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE P.Y0 → P.zeroUntreatedResidualContrast","l":201,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.IsGTFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For group and time index sets and a group-by-time array, the group-time fixed-effect condition holds exactly when the array can be written as the sum of a group-specific component and a time-specific component.","s":"{G : Type u_1} → {T : Type u_2} → (G → T → ℝ) → Prop","l":76,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.exists_panel_with_positive_treated_effects_twfe_negative_of_negative_component","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"Panel-level DCDH sign reversal. If the untreated-outcome residual contrast vanishes and the treated cells with negative normalized DCDH weight carry a strictly negative total weight, then there exists another finite DCDH panel — sharing the same cell weights, treatment indicator, untreated potential outcomes, and residualized treatment, but with strictly positive treatment effects on every cell and zero untreated residual contrast — whose two-way fixed-effects (TWFE) coefficient is strictly negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), P.zeroUntreatedResidualContrast → ∑ gt ∈ P.treatedCells with P.omega gt.1 gt.2 < 0, P.omega gt.1 gt.2 < 0 → ∃ P', (∀ (g : G) (t : T), P'.pi g t = P.pi g t) ∧ (∀ (g : G) (t : T), P'.D g t = P.D g t) ∧ (∀ (g : G) (t : T), P'.Y0 g t = P.Y0 g t) ∧ (∀ (g : G) (t : T), P'.Dtilde g t = P.Dtilde g t) ∧ (∀ (g : G) (t : T), 0 < P'.tau g t) ∧ P'.zeroUntreatedResidualContrast ∧ P'.betaTWFE < 0","l":464,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.exists_positive_effects_negative_weighted_sum_of_negative_component","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Finite signed-average construction: if normalized weights have a strictly negative component, some strictly positive effects have a negative weighted sum.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (w : ι → ℝ), ∑ i with w i < 0, w i < 0 → ∃ tau, (∀ (i : ι), 0 < tau i) ∧ ∑ i, w i * tau i < 0","l":393,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.neg_component_of_exists_Dtilde_neg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"If some treated cell has negative residualized treatment, then the total weight over negatively weighted treated cells is strictly negative.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (P : Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T), (∃ g t, P.D g t = 1 ∧ P.Dtilde g t < 0) → ∑ gt ∈ P.treatedCells with P.omega gt.1 gt.2 < 0, P.omega gt.1 gt.2 < 0","l":435,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For finite group and time sets, a balanced panel with at least two groups and two time periods, treatment, observed-outcome, untreated-outcome, and treatment-effect arrays, if treatment is binary in every cell, the observed outcome equals the untreated outcome plus treatment times the treatment effect in every cell, and the weighted sum of squared double-demeaned treatment is strictly positive, then the uniform balanced-panel DCDH structure assigns equal cell weights and uses double-demeaned treatment as its residualized treatment.","s":"{G : Type u_1} → {T : Type u_2} → [inst : Fintype G] → [inst_1 : Fintype T] → Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T → (D Y Y0 tau : G → T → ℝ) → (∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) → (∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) → 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2 → Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.DCDHPanel G T","l":56,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel_Dtilde","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"The residualized treatment of ofTwoWayPanel is exactly the double-demeaned treatment D̈.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (balanced : Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T) (D Y Y0 tau : G → T → ℝ) (hD_binary : ∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) (hconsistency : ∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) (hSD : 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel balanced D Y Y0 tau hD_binary hconsistency hSD).Dtilde = Causalean.Panel.UniformTwoWayPanel.ddot D","l":129,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel_betaTWFE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"The DCDH TWFE coefficient of the FWL-derived panel equals the uniform-panel Frisch–Waugh–Lovell residualized coefficient of Y on the double-demeaned treatment. For a uniform balanced panel and outcome, treatment, untreated-outcome, and effect arrays Y, D, Y0, tau, suppose the treatment is binary, the observed outcome equals the untreated outcome plus the treatment times the effect (consistency), and the double-demeaned treatment has strictly positive uniformly-weighted variation. Then the two-way fixed-effects (TWFE) coefficient of the DCDH panel built from this data by ofTwoWayPanel equals the Frisch–Waugh–Lovell residualized coefficient of Y regressed on the double-demeaned treatment ddot D.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] (balanced : Causalean.Panel.UniformTwoWayPanel.BalancedPanel G T) (D Y Y0 tau : G → T → ℝ) (hD_binary : ∀ (g : G) (t : T), D g t = 0 ∨ D g t = 1) (hconsistency : ∀ (g : G) (t : T), Y g t = Y0 g t + D g t * tau g t) (hSD : 0 < ∑ g, ∑ t, Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T * Causalean.Panel.UniformTwoWayPanel.ddot D g t ^ 2), (Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.ofTwoWayPanel balanced D Y Y0 tau hD_binary hconsistency hSD).betaTWFE = Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient (Causalean.Panel.UniformTwoWayPanel.ddot D) Y","l":140,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.two_cell_signed_weights_positive_effects_negative_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FinitePanel","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"Two-cell signed-weight calculation from the DCDH sign-reversal example.","s":"∀ {c ε M : ℝ}, 0 < c → 0 < ε → (1 + c) * ε / c < M → 0 < ε ∧ 0 < M ∧ (1 + c) * ε + -c * M < 0","l":374,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":1,"d":"For a finite group set and a finite time set, the uniform group-time cell weight is the reciprocal of the product of their cardinalities.","s":"(G : Type u_3) → (T : Type u_4) → [Fintype G] → [Fintype T] → ℝ","l":41,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.FWLBridge","a":"Panel/EstimandCharacterization/HeterogeneousTWFE","t":2,"d":"For finite nonempty group and time types, the uniform unit-period cell weight is strictly positive.","s":"∀ {G : Type u_1} {T : Type u_2} [inst : Fintype G] [inst_1 : Fintype T] [Nonempty G] [Nonempty T], 0 < Causalean.Panel.EstimandCharacterization.HeterogeneousTWFE.uniformWeight G T","l":46,"f":"Causalean/Panel/EstimandCharacterization/HeterogeneousTWFE/FWLBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A finite treated/untreated cell design for the BJS imputation decomposition, storing treated cells' regressor rows and untreated cells' regressor rows, an arbitrary — possibly signed and non-normalized — target weight on treated cells, the observed-law means on treated and untreated cells, the untreated-potential-outcome means on treated and untreated cells, a nuisance vector for the untreated-outcome model, and treated-cell causal effects.","s":"(Treated : Type u_4) → (Untreated : Type u_5) → (Regressor : Type u_6) → [Fintype Treated] → [Fintype Untreated] → [Fintype Regressor] → Type (max (max u_4 u_5) u_6)","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ImputationWeights","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Witness-form prediction span / imputation weights.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Type (max u_1 u_2)","l":112,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A linear functional of treated and untreated observed cell outcomes.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Type (max u_1 u_2)","l":213,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.HasImputationRepresentation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Explicit imputation representation witness for the linear-unbiased representation helper.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Type (max u_1 u_2)","l":273,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.modelValue","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, a linear estimator, a nuisance coefficient vector, and an arbitrary treated-effect vector, the model-implied linear-estimator value is its value when treated outcomes equal the regressor prediction plus the effect and untreated outcomes equal the regressor prediction.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → (Regressor → ℝ) → (Treated → ℝ) → ℝ","l":236,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.nuisance_coord","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Nuisance unbiasedness for every beta gives the BJS left-null-space coordinate constraint aᵀ Q_T + vUᵀ Q_U = 0 (row by row). Hoisted for the efficiency layer.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (r : Regressor), ∑ c, P.a c * P.qT c r + ∑ u, L.vU u * P.qU u r = 0","l":263,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.observedValue","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel and a linear estimator, the observed-law linear-estimator value is its value at the panel's treated and untreated observed means.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → ℝ","l":231,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.unbiasedForAllTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel and a linear estimator, the unbiasedness condition for all treated effects requires that, for every nuisance coefficient vector and every treated-effect vector, the estimator's model-implied value equals the corresponding target value.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Prop","l":246,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.vT_eq_a","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Unrestricted treatment-effect heterogeneity forces the treated-cell weights to equal the target weights (vT = a). Proved by an indicator-tau test. Hoisted here so the efficiency layer can reuse it without re-running the argument.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (c : Treated), L.vT c = P.a c","l":253,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.LinearEstimator.value","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, a linear estimator, an arbitrary array of treated cell outcomes, and an arbitrary array of untreated cell outcomes, the linear-estimator value is the sum of treated outcomes weighted by treated coefficients plus untreated outcomes weighted by untreated coefficients.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → (Treated → ℝ) → (Untreated → ℝ) → ℝ","l":224,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.PredictionIdentified","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, the target-relevant prediction-identification condition holds exactly when at least one imputation-weight system represents every treated regressor row with nonzero target weight as a weighted combination of untreated regressor rows.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":125,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.TreatmentEffectFixed","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, the fixed-treatment-effect condition requires that in every treated cell the observed mean equals the untreated potential-outcome mean plus that cell's stored treatment effect.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":106,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.UntreatedOutcomeModel","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, the untreated-outcome model condition requires that every treated cell's untreated potential-outcome mean equals its regressor-row dot product with the nuisance vector, every untreated cell's untreated potential-outcome mean equals its regressor-row dot product with that vector, and every untreated cell's observed mean equals its untreated potential-outcome mean.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Prop","l":100,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_imputation_identification","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Population BJS imputation identification. Suppose the untreated outcome mean follows a deterministic linear model in the regressors, and each untreated cell's observed mean equals that untreated model mean (untreated outcome model), there exist imputation weights whose row identity expresses every target-relevant treated cell's regressor row as a weighted combination of untreated-cell rows (target-relevant prediction span), and each treated cell's observed mean equals its untreated potential-outcome mean plus its treatment effect (treatment effect fixed). Then there is an imputation-weight witness for which the observed-law imputation functional psiImp equals the target weighted sum of treatment effects theta.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor), P.UntreatedOutcomeModel → P.PredictionIdentified → P.TreatmentEffectFixed → ∃ h, P.psiImp h.weight = P.theta","l":142,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_linear_unbiased_iff_imputation_form","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Combined iff characterization of the BJS linear-unbiased imputation class. Given a target-relevant prediction-span witness and at least one treated cell with nonzero target weight, a linear estimator L is unbiased for every value of the treatment-effect vector if and only if it admits a BJS imputation-weight representation: its treated coefficients match the target weights, its untreated coefficients equal the negative weighted imputation sum, its value equals the imputation contrast for every pair of treated/untreated observed outcomes, and its observed value equals the population imputation functional psiImp.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), P.PredictionIdentified → (∃ c, P.a c ≠ 0) → (L.unbiasedForAllTau ↔ ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight)","l":360,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.bjs_ols_imputation_min_variance_spherical","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"BJS efficiency: OLS imputation is BLUE under spherical errors. For a BJS event-study panel P and a family of cell outcomes Y on a probability space, suppose each cell outcome is square-integrable, the cell outcomes form a spherical family with common variance σ² — equal variances and zero cross-covariances, and the full event-study design matrix designFull P has full column rank. Then for any linear estimator L unbiased for every value of the treatment-effect vector, the variance of the OLS imputation estimator built from designFull P and the target functional cFull P is no larger than the variance of L.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] [inst_4 : DecidableEq Regressor] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} {Ω : Type u_4} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Treated ⊕ Untreated → Ω → ℝ), (∀ (i : Treated ⊕ Untreated), MeasureTheory.MemLp (Y i) 2 μ) → ∀ {σ : ℝ}, Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (P.designFull.transpose * P.designFull).det → ∀ (L : P.LinearEstimator), L.unbiasedForAllTau → ProbabilityTheory.variance (fun ω => ∑ i, Causalean.GaussMarkov.olsWeight P.designFull P.cFull i * Y i ω) μ ≤ ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf L i * Y i ω) μ","l":85,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.cFull","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS event-study panel with finite treated cells, untreated cells, and regressors, the target functional in full-design coordinates assigns the panel's target weight to each treated-cell-effect coordinate and zero to every regressor coordinate.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Treated ⊕ Regressor → ℝ","l":46,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.columnSpan","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS event-study panel with finite treated cells, untreated cells, and regressors, the untreated-cell regressor column span is the real linear span of all vectors of untreated-cell regressor values, one vector for each regressor.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Submodule ℝ (Untreated → ℝ)","l":70,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.columnSpan_ip_orthogonal_iff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Keystone bridge. A vector over the untreated cells is ip-orthogonal to the whole regressor column span iff it satisfies the BJS left-null-space condition ∀ r, ∑_u v_u q_{ur} = 0. This is the panel-substrate restatement of \"v lies in the left null space of Q_U\".","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ), (∀ h ∈ P.columnSpan, (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v h = 0) ↔ ∀ (r : Regressor), ∑ u, v u * P.qU u r = 0","l":104,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS event-study panel with finite treated cells, untreated cells, and regressors, the full event-study design matrix has one row for each treated or untreated observed cell and one column for each treated-cell effect or regressor. A treated-cell row has its own effect indicator and its treated-cell regressor row, whereas an untreated-cell row has zero effect indicators and its untreated-cell regressor row.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → [DecidableEq Treated] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Matrix (Treated ⊕ Untreated) (Treated ⊕ Regressor) ℝ","l":37,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull_col_eq_cellIndicator","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Each treated-cell fixed-effect column of designFull is the panel cell indicator of treatedClassifier.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (d : Treated), (fun i => P.designFull i (Sum.inl d)) = Causalean.Panel.Weighted.cellIndicator Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier (some d)","l":127,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.designFull_col_mem_indicatorSpan","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The treated-cell fixed-effect block of designFull lies in the panel IndicatorSpan of the treated-cell classifier: the BJS event-study FE design is the panel indicator-span substrate.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (d : Treated), (fun i => P.designFull i (Sum.inl d)) ∈ Causalean.Panel.Weighted.indicatorSpan Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier","l":142,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.exists_imputationWeights_of_gap_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Left-null-space row adjustment (audit M3), phrased through the panel substrate. Given base imputation weights H0, a nonzero target weight at c0, and a target untreated-coefficient vector vU whose gap gap u = vU u + ∑_c a_c · H0.weight c u is ip-orthogonal to the regressor column span (equivalently: gap lies in the left null space of Q_U), one nonzero target row of H0 can absorb the gap without disturbing any target-relevant row identity. The resulting imputation weights H represent the untreated coefficients: ∑_c a_c · H.weight c u = - vU u for every u.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (vU : Untreated → ℝ) {c0 : Treated}, P.a c0 ≠ 0 → ∀ (H0 : P.ImputationWeights), (∀ h ∈ P.columnSpan, (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip (fun u => vU u + ∑ c, P.a c * H0.weight c u) h = 0) → ∃ H, ∀ (u : Untreated), ∑ c, P.a c * H.weight c u = -vU u","l":130,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ip_regressorColumn_eq_zero_iff","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"ip-orthogonality to a single regressor column ⟺ that column's BJS left-null-space coordinate vanishes.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ) (r : Regressor), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v (P.regressorColumn r) = 0 ↔ ∑ u, v u * P.qU u r = 0","l":89,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.ip_untreatedSupport_regressorColumn","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The panel inner product of an untreated-cell vector v with a regressor column is the BJS left-null-space sum, rescaled by the uniform weight.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Untreated] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) [inst_4 : Nonempty Untreated] (v : Untreated → ℝ) (r : Regressor), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport Untreated).ip v (P.regressorColumn r) = (↑(Fintype.card Untreated))⁻¹ * ∑ u, v u * P.qU u r","l":78,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_imputation_representation","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Universal unbiasedness supplies the treated-coefficient identity required by linear_unbiased_of_imputation_representation_of_vT_eq_a.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), L.unbiasedForAllTau → ∀ (hImputationWitness : L.HasImputationRepresentation), ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":397,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_imputation_representation_of_vT_eq_a","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Witness-based BJS linear-estimator representation from its treated coefficients and an imputation witness, exposed as an estimator identity for arbitrary outcome arrays.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), (∀ (c : Treated), L.vT c = P.a c) → ∀ (hImputationWitness : L.HasImputationRepresentation), ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":291,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.linear_unbiased_of_prediction_identified","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"BJS linear-unbiased representation from the primitive prediction-span condition.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor) (L : P.LinearEstimator), L.unbiasedForAllTau → P.PredictionIdentified → (∃ c, P.a c ≠ 0) → ∃ H, (∀ (c : Treated), L.vT c = P.a c) ∧ (∀ (u : Untreated), L.vU u = -∑ c, P.a c * H.weight c u) ∧ (∀ (YT : Treated → ℝ) (YU : Untreated → ℝ), L.value YT YU = ∑ c, P.a c * (YT c - ∑ u, H.weight c u * YU u)) ∧ L.observedValue = P.psiImp H.weight","l":247,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.psiImp","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel and an array of imputation weights from treated to untreated cells, the observed-law imputation functional is the target-weighted sum of each treated observed mean less its imputed untreated observed mean.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → (Treated → Untreated → ℝ) → ℝ","l":131,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.regressorColumn","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS event-study panel with finite treated cells, untreated cells, and regressors and a regressor, the corresponding untreated-cell regressor column assigns to each untreated cell its entry in that regressor's panel row.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → Regressor → Untreated → ℝ","l":67,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.targetForTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel and an arbitrary treated-cell effect vector, the corresponding target value is the finite sum of that vector weighted by the panel's target weights.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → (Treated → ℝ) → ℝ","l":94,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.theta","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For finite treated-cell, untreated-cell, and regressor sets and a BJS panel, the target estimand is the finite sum of each treated cell's causal effect multiplied by its supplied target weight; the weights need not be positive or sum to one.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor → ℝ","l":85,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.treatedClassifier","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"The treated-cell classifier maps each treated observed cell to its own treated-cell label and maps every untreated observed cell to no label.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → Treated ⊕ Untreated → Option Treated","l":119,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a linear estimator of a BJS event-study panel with finite cell sets, the weight vector over all observed cells equals its treated-cell weights on treated cells and its untreated-cell weights on untreated cells.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} → P.LinearEstimator → Treated ⊕ Untreated → ℝ","l":52,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf_vecMul_designFull","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Efficiency","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Unbiasedness bridge. A linear estimator is unbiased for every tau iff its weight vector satisfies the design constraint w ᵥ* designFull = cFull. (Only the forward direction is needed below, but it captures both vT = a and the nuisance left-null constraint.)","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] [inst_3 : DecidableEq Treated] {P : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor} (L : P.LinearEstimator), L.unbiasedForAllTau → Matrix.vecMul (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel.weightOf L) P.designFull = P.cFull","l":58,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Efficiency.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation","k":"structure","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"A population for the BJS imputation design: a probability space with a treated/untreated cell classifier and potential outcomes Y0 (untreated), Y1 (treated), and observed Yobs, related by cell-level consistency. The design rows qT, qU, target weights a, and nuisance vector beta0 are carried through to the induced panel unchanged.","s":"(Treated : Type u_4) → (Untreated : Type u_5) → (Regressor : Type u_6) → [Fintype Treated] → [Fintype Untreated] → [Fintype Regressor] → Type (max (max (max u_4 u_5) u_6) (u_7 + 1))","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.bjs_imputation_identification_population","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Population BJS imputation identification (headline). For a population BJS design E, suppose the conditional mean of the untreated potential outcome on each treated cell equals a linear function q_T · β₀ of the treated-cell regressors and likewise, on each untreated cell, the conditional mean of the untreated potential outcome equals q_U · β₀ — jointly the additive untreated-outcome (parallel-trends) model — together with a target-relevant prediction-span witness for the induced panel. Then the observed-law imputation functional identifies the target ∑ a_c · (E[Y(1) ∣ cell c] − E[Y(0) ∣ cell c]): there is an imputation-weight witness for which the population imputation functional psiImp equals the target theta.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), (∀ (c : Treated), E.cells.mean E.Y0 (Sum.inl c) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qT c) E.beta0) → (∀ (u : Untreated), E.cells.mean E.Y0 (Sum.inr u) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qU u) E.beta0) → E.toPanel.PredictionIdentified → ∃ h, E.toPanel.psiImp h.weight = E.toPanel.theta","l":164,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.cells","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS population with finite treated-cell, untreated-cell, and regressor collections, the treated-and-untreated cell partition partitions its sample space according to the population's treated/untreated cell classifier, using the population probability measure.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) → Causalean.Panel.PO.CellPartition E.μ (Treated ⊕ Untreated)","l":98,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a BJS population with finite treated-cell, untreated-cell, and regressor collections, the induced BJS panel retains its regressor rows, target weights, and nuisance vector, and defines each outcome mean as the corresponding treated- or untreated-cell conditional mean. For every treated cell, its treatment effect is the conditional mean of the treated potential outcome minus that of the untreated potential outcome.","s":"{Treated : Type u_1} → {Untreated : Type u_2} → {Regressor : Type u_3} → [inst : Fintype Treated] → [inst_1 : Fintype Untreated] → [inst_2 : Fintype Regressor] → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel Treated Untreated Regressor","l":102,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_EY_T_eq_mean_Y1","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"On a treated cell the observed mean equals the treated potential-outcome mean.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (c : Treated), E.toPanel.EY_T c = E.cells.mean E.Y1 (Sum.inl c)","l":114,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_EY_U_eq_mean_Y0","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"On an untreated cell the observed mean equals the untreated potential-outcome mean.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (u : Untreated), E.toPanel.EY_U u = E.cells.mean E.Y0 (Sum.inr u)","l":123,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_tau_eq_po_contrast","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Causal-meaning certificate. In the induced panel, tau c is literally the population treatment-effect contrast E[Y(1) ∣ cell c] − E[Y(0) ∣ cell c], so the estimand carries genuine causal content.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor) (c : Treated), E.toPanel.tau c = E.cells.mean E.Y1 (Sum.inl c) - E.cells.mean E.Y0 (Sum.inl c)","l":132,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_treatmentEffectFixed","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Treatment-effect-fixed is derived. EY_T = EY0_T + tau holds because on treated cells the observed mean is the treated potential-outcome mean and tau is the treated-minus-untreated contrast.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), E.toPanel.TreatmentEffectFixed","l":139,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation.toPanel_untreatedModel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PopulationBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"Untreated-outcome model is derived from the linear untreated-mean hypotheses plus consistency. The two linear-model conjuncts E[Y(0) ∣ cell] = q · β₀ are the genuine modeling hypotheses hLinT/hLinU (additive fixed-effect / parallel-trends form); the observed-equals-untreated conjunct is derived from untreated-cell consistency.","s":"∀ {Treated : Type u_1} {Untreated : Type u_2} {Regressor : Type u_3} [inst : Fintype Treated] [inst_1 : Fintype Untreated] [inst_2 : Fintype Regressor] (E : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPopulation Treated Untreated Regressor), (∀ (c : Treated), E.cells.mean E.Y0 (Sum.inl c) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qT c) E.beta0) → (∀ (u : Untreated), E.cells.mean E.Y0 (Sum.inr u) = Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (E.qU u) E.beta0) → E.toPanel.UntreatedOutcomeModel","l":148,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For an adoption-time path for the units over a finite number of periods, the treated cells are precisely the unit-period pairs for which the unit's adoption time is no later than the period.","s":"{I : Type u_1} → {T : ℕ} → (I → WithTop (Fin T)) → Type u_1","l":48,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For an adoption-time path for the units over a finite number of periods, the untreated cells are precisely the unit-period pairs whose period is strictly before the unit's adoption time. They include every period of a never-treated unit and every pre-adoption period of a treated unit.","s":"{I : Type u_1} → {T : ℕ} → (I → WithTop (Fin T)) → Type u_1","l":52,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.Imputation","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a finite regressor index set, a regressor row, and a nuisance coefficient vector, the finite dot product is the sum, over regressors, of their coordinatewise products.","s":"{Regressor : Type u_3} → [Fintype Regressor] → (Regressor → ℝ) → (Regressor → ℝ) → ℝ","l":42,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/Imputation.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot_feRow","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The two-way FE row evaluates the additive fixed-effect model: q_{(i,t)} · (α, λ) = α_i + λ_t.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (c : I × Fin T) (α : I → ℝ) (lam : Fin T → ℝ), Causalean.Panel.EstimandCharacterization.ImputationEventStudy.dot (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.feRow c) (Sum.elim α lam) = α c.1 + lam c.2","l":91,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.feRow","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a unit-period cell with a finite number of periods and equality-comparable unit labels, the two-way fixed-effect design row assigns one to that cell's unit coordinate and period coordinate, and zero to all other unit and period coordinates.","s":"{I : Type u_1} → [DecidableEq I] → {T : ℕ} → I × Fin T → I ⊕ Fin T → ℝ","l":86,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instDecidablePredProdFinLeFstSnd","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"For an adoption-time path for the units over a finite number of periods, a decision procedure for the treated-cell condition determines, for every unit-period cell, whether the unit's adoption time is no later than that period.","s":"{I : Type u_1} → {T : ℕ} → (g : I → WithTop (Fin T)) → DecidablePred fun c => Causalean.Panel.AdoptionPath.le (g c.1) c.2","l":56,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instDecidablePredProdFinLtFstSnd","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"For an adoption-time path for the units over a finite number of periods, a decision procedure for the untreated-cell condition determines, for every unit-period cell, whether the period is strictly before the unit's adoption time.","s":"{I : Type u_1} → {T : ℕ} → (g : I → WithTop (Fin T)) → DecidablePred fun c => Causalean.Panel.AdoptionPath.lt (g c.1) c.2","l":60,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instFintypeTreatedCell","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"For an adoption-time path for a finite population over a finite number of periods, a finite enumeration of the treated cells is available.","s":"{I : Type u_1} → [Fintype I] → {T : ℕ} → (g : I → WithTop (Fin T)) → Fintype (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g)","l":64,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.instFintypeUntreatedCell","k":"instance","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"For an adoption-time path for a finite population over a finite number of periods, a finite enumeration of the untreated cells is available.","s":"{I : Type u_1} → [Fintype I] → {T : ℕ} → (g : I → WithTop (Fin T)) → Fintype (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell g)","l":66,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For an adoption-time path for finitely many equality-comparable units over a finite number of periods, unit effects, period effects, target weights, and treated-cell effects, the staggered-adoption two-way-fixed-effects BJS panel has treated and untreated cells given by that path and regressors given by the unit and period indicators. Its untreated potential-outcome mean is the sum of the relevant unit and period effects, and its treated observed mean adds the treated-cell effect.","s":"{I : Type u_1} → [inst : Fintype I] → [DecidableEq I] → {T : ℕ} → (g : I → WithTop (Fin T)) → (I → ℝ) → (Fin T → ℝ) → (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ) → (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ) → Causalean.Panel.EstimandCharacterization.ImputationEventStudy.BJSPanel (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g) (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.UntreatedCell g) (I ⊕ Fin T)","l":100,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE_treatmentFixed","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The grounded panel satisfies the BJS fixed-effect equation E[Y_T] = E[Y_T(0)] + τ.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (g : I → WithTop (Fin T)) (α : I → ℝ) (lam : Fin T → ℝ) (a tau : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE g α lam a tau).TreatmentEffectFixed","l":128,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE_untreatedModel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"Untreated-outcome model holds by construction. For unit fixed effects α, period fixed effects lam, and target weights and treated-cell effects a, tau, the staggered-adoption grounded panel ofStaggeredTWFE g α lam a tau satisfies the BJS untreated-outcome model: treated and untreated cell means both equal the two-way fixed-effect model α_i + λ_t, and untreated cells exhibit no anticipation.","s":"∀ {I : Type u_1} [inst : Fintype I] [inst_1 : DecidableEq I] {T : ℕ} (g : I → WithTop (Fin T)) (α : I → ℝ) (lam : Fin T → ℝ) (a tau : Causalean.Panel.EstimandCharacterization.ImputationEventStudy.TreatedCell g → ℝ), (Causalean.Panel.EstimandCharacterization.ImputationEventStudy.ofStaggeredTWFE g α lam a tau).UntreatedOutcomeModel","l":114,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.treated_not_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The cell partition is exclusive: no cell is both adopted-by-t and untreated-at-t.","s":"∀ {I : Type u_1} {T : ℕ} (g : I → WithTop (Fin T)) {c : I × Fin T}, Causalean.Panel.AdoptionPath.le (g c.1) c.2 → Causalean.Panel.AdoptionPath.lt (g c.1) c.2 → False","l":70,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.treated_or_untreated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelGrounding","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":2,"d":"The cell partition is exhaustive: every cell is treated or untreated.","s":"∀ {I : Type u_1} {T : ℕ} (g : I → WithTop (Fin T)) (c : I × Fin T), Causalean.Panel.AdoptionPath.le (g c.1) c.2 ∨ Causalean.Panel.AdoptionPath.lt (g c.1) c.2","l":77,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelGrounding.lean"},{"n":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.untreatedSupport","k":"def","m":"Causalean.Panel.EstimandCharacterization.ImputationEventStudy.PanelBridge","a":"Panel/EstimandCharacterization/ImputationEventStudy","t":1,"d":"For a nonempty finite collection of untreated cells, the uniform untreated-cell weighted support assigns every untreated cell weight $1/|U|$ and regards every such cell as observed.","s":"(Untreated : Type u_4) → [inst : Fintype Untreated] → [inst_1 : DecidableEq Untreated] → [Nonempty Untreated] → Causalean.Panel.Weighted.WeightedSupport Untreated","l":45,"f":"Causalean/Panel/EstimandCharacterization/ImputationEventStudy/PanelBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","k":"structure","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"A finite covariate partition for Słoczyński's saturated-OLS weight decomposition, carrying cell probabilities, within-cell treated shares, and within-cell conditional treatment effects, together with the side conditions used by the saturated-OLS algebra: the cell probabilities are nonnegative and sum to one, the treated shares lie in the unit interval, and the overlap denominator Σ π_g p_g (1−p_g) is strictly positive.","s":"(𝒢 : Type u_1) → [Fintype 𝒢] → Type u_1","l":65,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.cellOverlap","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition and one of its cells, the within-cell treatment overlap is $p_g(1-p_g)$, where $p_g$ is that cell's treated share.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":91,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.cellOverlap_eq_perWeights_mul","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The cell overlap variance is the product of the two per-observation leverage factors: p_g (1−p_g) = perUntreatedWeight_g · perTreatedWeight_g. This links the finite-cell denominator directly to the paper's per-unit weight interpretation.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) (g : 𝒢), P.cellOverlap g = P.perUntreatedWeight g * P.perTreatedWeight g","l":200,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.equal_groups_collapses","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Equal-cell-share collapse to unweighted average (finite half of prop:po-estimand-sloczynski-ols-equal-groups). If every cell has an equal treated share p_g = 1/2, then the saturated-OLS estimand β_sat equals the probability-weighted average of the cell treatment effects Σ π_g τ_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), (∀ (g : 𝒢), P.p g = 1 / 2) → P.overlapWeightedATE = ∑ g, P.π g * P.τ g","l":152,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.equal_groups_weight_eq_pi","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Equal-cell-share weight collapse: if p_g = 1/2 for every cell, the normalized overlap weight reduces to the cell probability π_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), (∀ (g : 𝒢), P.p g = 1 / 2) → ∀ (g : 𝒢), P.overlapWeight g = P.π g","l":136,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.finite_weights_eq_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Finite-partition OLS weights (prop:po-estimand-sloczynski-ols-finite-weights). The saturated-OLS estimand equals the sum of normalized overlap weights times cell effects, Σ_g ω_g τ_g.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), P.overlapWeightedATE = ∑ g, P.overlapWeight g * P.τ g","l":127,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.homogeneous_collapses","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Homogeneous-effect collapse (saturated half of prop:po-estimand-sloczynski-ols-homogeneous). If the cell-level treatment effect equals a common constant τ₀ in every cell, then the saturated-OLS estimand β_sat equals τ₀.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {τ₀ : ℝ}, (∀ (g : 𝒢), P.τ g = τ₀) → P.overlapWeightedATE = τ₀","l":164,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.omega_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The overlap weights sum to one. Direct from positivity of the denominator and pulling the common denominator out of the sum.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), ∑ g, P.overlapWeight g = 1","l":119,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapDenominator","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition, the overlap-weighted denominator is $\\sum_g \\pi_g p_g(1-p_g)$, the probability-weighted sum of within-cell treatment overlap.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":99,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapDenominator_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The denominator is positive. Restated from overlap_pos.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢), 0 < P.overlapDenominator","l":115,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapNumerator","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition, the overlap-weighted numerator is $\\sum_g \\pi_g p_g(1-p_g)\\tau_g$, the sum of each cell effect weighted by its probability and treatment overlap.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":95,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition and one of its cells, the normalized overlap weight is $\\pi_g p_g(1-p_g)/\\sum_h\\pi_h p_h(1-p_h)$.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":103,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.overlapWeightedATE","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition, the saturated ordinary-least-squares estimand is $\\sum_g\\pi_g p_g(1-p_g)\\tau_g / \\sum_g\\pi_g p_g(1-p_g)$.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → ℝ","l":108,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perTreatedWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition and one of its cells, the per-treated-observation leverage factor is $1-p_g$, the untreated share in that cell.","s":"{𝒢 : Type u_2} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":178,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perTreatedWeight_antitone","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Headline monotonicity — smaller treated group gets larger per-treated weight (rem:po-estimand-sloczynski-ols-group-size). The per-treated-observation leverage factor 1 − p_g is antitone in the treated share: if cell g has a treated share no larger than cell h's, then g's per-treated-observation leverage factor is at least h's: perTreatedWeight P h ≤ perTreatedWeight P g.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {g h : 𝒢}, P.p g ≤ P.p h → P.perTreatedWeight h ≤ P.perTreatedWeight g","l":210,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perUntreatedWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For a finite covariate partition and one of its cells, the per-untreated-observation leverage factor is $p_g$, the treated share in that cell.","s":"{𝒢 : Type u_2} → [inst : Fintype 𝒢] → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢 → 𝒢 → ℝ","l":190,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition.perUntreatedWeight_monotone","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Monotonicity — larger treated group gives larger per-untreated weight (rem:po-estimand-sloczynski-ols-group-size, untreated side). The per-untreated-observation leverage factor p_g is monotone in the treated share: if cell g has a treated share no larger than cell h's, then h's per-untreated-observation leverage factor is at least g's: perUntreatedWeight P g ≤ perUntreatedWeight P h.","s":"∀ {𝒢 : Type u_2} [inst : Fintype 𝒢] (P : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢) {g h : 𝒢}, P.p g ≤ P.p h → P.perUntreatedWeight g ≤ P.perUntreatedWeight h","l":232,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr","k":"structure","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Słoczyński's two-component ATT/ATU representation (ass:po-estimand-sloczynski-ols-opposite-group).","s":"Type","l":250,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.equal_groups_eq_ATE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Equal-group-size collapse to the ATE (top half of prop:po-estimand-sloczynski-ols-equal-groups). If the treated share ρ equals one half, then the opposite-group OLS coefficient β_ols equals the ATE τ_ATE = ρ·τ_ATT + (1−ρ)·τ_ATU.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.ρ = 1 / 2 → R.β_ols = R.tau_ATE","l":320,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.homogeneous_eq_constant","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Homogeneous-effect collapse for the opposite-group representation (ATT/ATU half of prop:po-estimand-sloczynski-ols-homogeneous). If the ATT equals a common constant τ₀ and the ATU equals that same constant τ₀, then the opposite-group OLS coefficient β_ols equals τ₀, regardless of the treated share ρ.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr) {τ₀ : ℝ}, R.τ_ATT = τ₀ → R.τ_ATU = τ₀ → R.β_ols = τ₀","l":328,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.represents","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Słoczyński opposite-group identity (prop:po-estimand-sloczynski-ols-opposite-group) — derived rather than assumed. Substituting the equal-dispersion weights into the two-component representation shows that the OLS coefficient equals (1−ρ)·τ_ATT + ρ·τ_ATU: the treated-group effect receives the untreated share as its weight, and vice versa.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.β_ols = (1 - R.ρ) * R.τ_ATT + R.ρ * R.τ_ATU","l":313,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.tau_ATE","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"For an opposite-group representation, the average treatment effect is $\\rho\\tau_{ATT}+(1-\\rho)\\tau_{ATU}$, the treated-share-weighted average of the effects on treated and untreated groups.","s":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr → ℝ","l":302,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr.w0_eq_rho","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The untreated weight in the equal-dispersion case is the treated share ρ, forced by w₁ + w₀ = 1 and w₁ = 1 − ρ.","s":"∀ (R : Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OppositeGroupRepr), R.w₀ = R.ρ","l":307,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/FinitePartition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Y_memLp_of_consistency","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The observed outcome Y = D · Y1 + (1 − D) · Y0 is in MemLp 2 μ under consistency, binary treatment, and MemLp 2 μ assumptions for both potential outcomes. The pointwise bound |Y| ≤ |Y0| + |Y1| (a.e.) plus closure of MemLp 2 under sums gives the result. Stated separately because the bridge theorem takes Y_memLp as a hypothesis but downstream consumers (e.g. tests) may need to derive it.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {D Y Y0 Y1 : Ω → ℝ}, Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp Y0 2 μ → MeasureTheory.MemLp Y1 2 μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → MeasureTheory.MemLp Y 2 μ","l":139,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.bridge_finite_residualized_eq_overlap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.OverlapWeightedATE","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Finite-cell saturated-OLS bridge to the overlap-weighted average treatment effect. For a probability space (Ω, μ) with a measurable binary treatment D, an outcome Y, square-integrable potential outcomes Y0, Y1, and a measurable finite covariate G, suppose treatment is binary almost everywhere, the observed outcome is consistent — Y equals D·Y1 + (1−D)·Y0 almost everywhere, the integrated-indicator finite-cell bridge condition holds, a finite-cell operational substitute for full conditional-mean independence of the potential outcomes given treatment and the covariate, and the covariate cells have nondegenerate treatment overlap. Then the saturated-OLS residualized coefficient of Y on D given G equals the overlap-weighted average treatment effect Σ_g ω_g · E[Y(1)−Y(0) ∣ G=g] built from the potential outcomes Y0, Y1:","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (B_meas_G : Measurable G) (B_meas_D : Measurable D) (B_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_Y0_L2 : MeasureTheory.MemLp Y0 2 μ) (B_Y1_L2 : MeasureTheory.MemLp Y1 2 μ) (B_consis : Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω), (∀ (d : ℝ) (g : 𝒢), d = 0 ∨ d = 1 → ∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = (∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) * if d = 1 then Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g else 1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g) → ∀ (B_overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))), Causalean.Panel.residualizedCoefficient μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G B_meas_G) (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessY μ Y G B_meas_G (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Y_memLp_of_consistency μ B_meas_D B_binary B_Y0_L2 B_Y1_L2 B_consis)) (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessD μ D G B_meas_G B_meas_D B_binary) = (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G B_meas_G B_binary B_overlap).overlapWeightedATE","l":37,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/OverlapWeightedATE.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a measure on a sample space, a covariate map, and a covariate cell, the cell mass is the measure of the event that the covariate map equals that cell, expressed as a real number.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":51,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell mass is nonnegative — (μ S).toReal ≥ 0.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":135,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass_sum_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell masses sum to 1 under IsProbabilityMeasure μ and a measurable G valued in a finite type with MeasurableSingletonClass. The {G = g} family partitions Ω (up to μ-null sets) and the μ-mass of each is finite. This supplies the π_sum_one field for the finite-partition algebra.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (G : Ω → 𝒢), Measurable G → ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = 1","l":141,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMean_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and covariate space, a probability measure μ on Ω, an integrand F, a measurable covariate map G, and a covariate cell g, the CellBridge indicator-weighted cell mean of F on {G = g} equals the shared event-level conditional expectation of F given {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.CellBridge.cellMean μ F G g = Causalean.PO.eventCondExp μ {ω | G ω = g} F","l":78,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a measure on a sample space, a treatment-valued function, a covariate map, and a covariate cell, the cell-wise treated share is the indicator-weighted integral of treatment over that cell divided by its mass. On a zero-mass cell, its value is zero.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":56,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"OLS treated share is a shared population cell mean. cellShare μ D G g = E[D | G = g] in the shared eventCondExp operator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g = Causalean.PO.eventCondExp μ {ω | G ω = g} D","l":96,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_le_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-wise treated share is at most 1 when D lies a.e. between zero and one. The integrand D · 𝟙{G = g} ≤ 𝟙{G = g} a.e., so the numerator is at most cellMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → (∀ᵐ (ω : Ω) ∂μ, 0 ≤ D ω ∧ D ω ≤ 1) → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g ≤ 1","l":179,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-wise treated share is nonnegative when D is a.e. nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢), (∀ᵐ (ω : Ω) ∂μ, 0 ≤ D ω) → ∀ (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g","l":166,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a measure on a sample space, an untreated potential-outcome function, a treated potential-outcome function, a covariate map, and a covariate cell, the cell-wise treatment effect is the indicator-weighted mean of $Y(1)-Y(0)$ in that cell. On a zero-mass cell, its value is zero.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":61,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"OLS cell treatment effect is a shared population cell mean. cellTau μ Y0 Y1 G g = E[Y(1) − Y(0) | G = g] in the shared eventCondExp operator — a genuine potential-outcome contrast.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g = Causalean.PO.eventCondExp μ {ω | G ω = g} fun ω => Y1 ω - Y0 ω","l":104,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell-effect numerator divided by cell mass and multiplied back recovers the raw effect numerator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = ∫ (ω : Ω), (Y1 ω - Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":65,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Dividing an indicator-weighted cell integral by a nonzero cell mass and multiplying back recovers the numerator; on zero-mass cells the numerator is zero because the indicator is a.e. zero.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":52,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_D_bounds","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"A real-valued variable that is zero or one almost surely has conditional expectation, given a sub-σ-algebra, between zero and one almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → (∀ᵐ (ω : Ω) ∂μ, 0 ≤ μ[D | m] ω) ∧ ∀ᵐ (ω : Ω) ∂μ, μ[D | m] ω ≤ 1","l":78,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_D_norm_le","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The conditional expectation of a real-valued variable that is zero or one almost surely has absolute value at most one almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ᵐ (ω : Ω) ∂μ, ‖μ[D | m] ω‖ ≤ 1","l":93,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.condExp_one_sub","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"The conditional expectation of one minus an integrable real-valued variable, given a sub-σ-algebra, equals one minus that variable's conditional expectation almost surely.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ {D : Ω → ℝ}, MeasureTheory.Integrable D μ → μ[fun ω => 1 - D ω | m] =ᵐ[μ] fun ω => 1 - μ[D | m] ω","l":72,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.denom_identity","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Denominator identity (A). For binary D, ∫ (D − μ[D | m])² ∂μ = ∫ μ[D | m]·(1 − μ[D | m]) ∂μ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":102,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.denom_per_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Per-cell denominator identity. On the cell {G = g}, (D − propensity)(ω) = D ω − cellShare μ D G g; squaring and using D ∈ {0, 1} plus the defining identity of cellShare gives","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))","l":31,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.indicator_cell_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Indicator helper: Set.indicator {G = g} 1 is in MemLp 2 μ for a finite measure (it is bounded by 1 and finite measure ⇒ MemLp p for every p). Stated as a separate lemma because it appears repeatedly in orthogonality and per-cell calculations.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":32,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integrable_mul_indicator_D_G","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Product of two singleton indicators against an L² function is integrable under a finite measure.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F D : Ω → ℝ) (G : Ω → 𝒢), Measurable D → Measurable G → MeasureTheory.MemLp F 2 μ → ∀ (d : ℝ) (g : 𝒢), MeasureTheory.Integrable (fun ω => F ω * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω) μ","l":76,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Cell indicators integrate to the corresponding real cell mass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":43,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_eq_sum_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and finite covariate space, a measure μ, a measurable covariate map G, and an integrable function F, the integral of F against μ equals the sum, over covariate cells g, of the integral of F restricted to the cell {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":163,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.integral_mul_saturated_eq_zero_of_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"If a square-integrable residual is orthogonal to every cell indicator, it is orthogonal to every saturated finite linear combination.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp V 2 μ → ∀ (c : 𝒢 → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * ∑ g, c g * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":145,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a measure on a sample space, an outcome-valued function, and a finite covariate map, the saturated mean-regression function assigns to every sample point the indicator-weighted mean outcome of its covariate cell.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → Ω → ℝ","l":122,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg_eq_cellMean_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"On its own cell, the saturated mean-regression representative equals the corresponding cell mean.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) {g : 𝒢} {ω : Ω}, G ω = g → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G ω = (∫ (ω' : Ω), Y ω' * {ω' | G ω' = g}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g","l":126,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg_mem_saturatedClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The mean-regression meanReg μ Y G lies in saturatedClass μ G G_meas. Take the coefficient map c g := (∫ Y · 𝟙{G = g} dμ) / cellMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas).mem (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G)","l":231,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.num_per_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Per-cell numerator identity. For a probability space (Ω, μ) with measurable treatment D, an outcome Y, square-integrable potential outcomes Y0, Y1, and a measurable finite covariate G, suppose treatment is binary almost everywhere, the observed outcome is consistent — Y equals D·Y1 + (1−D)·Y0 almost everywhere, and the finite-cell conditional-mean-independence bridge condition holds, the integrated finite-cell substitute for E[Y(d) ∣ D, G] = E[Y(d) ∣ G]. Then, for any cell g, the propensity-residual-weighted, cell-indicator-integrated outcome ∫ (D − propensity) · Y · 𝟙{G = g} dμ equals cellMass g · cellShare g · (1 − cellShare g) · cellTau g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D Y Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp Y0 2 μ → MeasureTheory.MemLp Y1 2 μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (∀ (d : ℝ) (g : 𝒢), d = 0 ∨ d = 1 → ∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | D ω' = d}.indicator (fun x => 1) ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = (∫ (ω : Ω), (if d = 1 then Y1 ω else Y0 ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) * if d = 1 then Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g else 1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * Y ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g)) * Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellTau μ Y0 Y1 G g","l":110,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.numer_identity","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Numerator identity (B). Under consistency and the factored conditional-mean-independence hypotheses, ∫ (D − μ[D | m])·Y ∂μ = ∫ μ[D | m]·(1 − μ[D | m])·(μ[Y1 | m] − μ[Y0 | m]) ∂μ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → ∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ = ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ","l":159,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Given a probability measure, a treatment variable, the untreated and treated potential-outcome variables, and a measurable finite cell classifier, with treatment equal to zero or one almost surely and a strictly positive sum of cell masses times treated-share variances, the finite Słoczyński partition consists of the cell masses, treated shares, and within-cell treatment effects.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (Ω → ℝ) → (Ω → ℝ) → (G : Ω → 𝒢) → Measurable G → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g)) → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.FinitePartition 𝒢","l":32,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf_p_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Population-cell certificate for partitionOf. For a probability space (Ω, μ) with a measurable finite covariate G and treatment/potential-outcome data D, Y0, Y1, suppose treatment is binary almost everywhere and the covariate cells have nondegenerate treatment overlap. Then, for any cell g, the treated share p_g of the Słoczyński partition partitionOf equals the shared population cell mean E[D ∣ G = g] (via eventCondExp).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))) (g : 𝒢), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G G_meas D_binary overlap).p g = Causalean.PO.eventCondExp μ {ω | G ω = g} D","l":59,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf_tau_eq_eventCondExp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"Population-cell certificate for partitionOf. The cell treatment effect τ_g of the Słoczyński partition is the shared population potential-outcome contrast E[Y(1) − Y(0) | G = g] (via eventCondExp), so the overlap-weighted estimand is built from genuine potential-outcome cell means.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y0 Y1 : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (overlap : 0 < ∑ g, Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellMass μ G g * (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g * (1 - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g))) (g : 𝒢), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.partitionOf μ D Y0 Y1 G G_meas D_binary overlap).τ g = Causalean.PO.eventCondExp μ {ω | G ω = g} fun ω => Y1 ω - Y0 ω","l":77,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a measure on a sample space, a treatment-valued function, and a finite covariate map, the saturated propensity function assigns to every sample point the treated share of its covariate cell.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → Ω → ℝ","l":114,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity_eq_cellShare_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"On its own cell, the saturated propensity representative equals the corresponding cell share.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢) {g : 𝒢} {ω : Ω}, G ω = g → Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω = Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.cellShare μ D G g","l":110,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity_mem_saturatedClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The pointwise representative propensity μ D G lies in saturatedClass μ G G_meas. Take the coefficient map c g := cellShare μ D G g; equality holds pointwise (and so a.e.).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (G_meas : Measurable G), (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas).mem (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G)","l":219,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residD_cell_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":2,"d":"The treatment residual is orthogonal to each saturated cell indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢), Measurable G → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.propensity μ D G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":29,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessD","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Given a probability measure, a treatment variable, and a measurable finite cell classifier, where the treatment is measurable and equals zero or one almost surely, the residualization witness for the treatment variable decomposes treatment into its saturated-cell propensity and an orthogonal residual.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (G : Ω → 𝒢) → (G_meas : Measurable G) → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas) D","l":92,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residWitnessY","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"Given a probability measure, an outcome variable, a measurable finite cell classifier, and a finite second moment for the outcome, the residualization witness for the outcome variable decomposes the outcome into its saturated-cell mean regression and an orthogonal residual.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (Y : Ω → ℝ) → (G : Ω → 𝒢) → (G_meas : Measurable G) → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass μ G G_meas) Y","l":162,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.residY_cell_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For an ambient probability space and finite covariate space, a square-integrable outcome Y, a measurable covariate map G, and a covariate cell g, the regression residual of Y on the saturated cell-mean model is orthogonal to the indicator of cell {G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.MemLp Y 2 μ → ∀ (g : 𝒢), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.meanReg μ Y G ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":87,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturatedClass","k":"def","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/OLSWeightDecomposition/Support","t":1,"d":"For a finite measure on a sample space, a finite covariate-valued map, and the measurability of that map, the saturated linear square-integrable control class consists of functions that agree almost everywhere with a linear combination of the indicators of the covariate cells.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → Measurable G → Causalean.Panel.LinearL2Class μ","l":34,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturated_ols_overlap_weighted_ate","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":1,"d":"Saturated-control OLS overlap-weighted ATE identity. Fix a covariate σ-algebra m at most as fine as the ambient one, and outcome, treatment, and potential-outcome functions D, Y, Y0, Y1. Suppose treatment is binary almost everywhere, D is integrable, D·Y1 is integrable, and (1−D)·Y0 is integrable. Suppose also consistency: the observed outcome equals D·Y1 + (1−D)·Y0 almost everywhere, and the factored conditional-mean-independence conditions μ[D·Y1 ∣ m] equals μ[D∣m]·μ[Y1∣m] almost everywhere and μ[(1−D)·Y0 ∣ m] equals μ[1−D∣m]·μ[Y0∣m] almost everywhere. Then the saturated-control OLS residualized treatment-outcome ratio E[(D−μ[D∣m])·Y] / E[(D−μ[D∣m])²] equals the overlap-weighted average of conditional treatment effects E[μ[D∣m]·(1−μ[D∣m])·(μ[Y1∣m]−μ[Y0∣m])] / E[μ[D∣m]·(1−μ[D∣m])], using Lean's totalized division convention so the identity also covers the zero-denominator case. This is the paper-facing statement corresponding to LaTeX thm:po-estimand-sloczynski-ols-overlap-weighted-ate.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → (∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ) / ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = (∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ) / ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":278,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.saturated_ols_overlap_weighted_ate_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.OLSWeightDecomposition.GeneralCondExp","a":"Panel/EstimandCharacterization/OLSWeightDecomposition","t":2,"d":"Positivity-free core of the saturated-control OLS overlap-weighted ratio identity. Both ratios are totalized, so the algebraic equality also holds when the common denominator is zero.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], m ≤ mΩ → ∀ (D Y Y0 Y1 : Ω → ℝ), (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.Integrable D μ → MeasureTheory.Integrable (fun ω => D ω * Y1 ω) μ → MeasureTheory.Integrable (fun ω => (1 - D ω) * Y0 ω) μ → (Y =ᵐ[μ] fun ω => D ω * Y1 ω + (1 - D ω) * Y0 ω) → (μ[fun ω => D ω * Y1 ω | m] =ᵐ[μ] fun ω => μ[D | m] ω * μ[Y1 | m] ω) → (μ[fun ω => (1 - D ω) * Y0 ω | m] =ᵐ[μ] fun ω => μ[fun ω => 1 - D ω | m] ω * μ[Y0 | m] ω) → (∫ (ω : Ω), (D ω - μ[D | m] ω) * Y ω ∂μ) / ∫ (ω : Ω), (D ω - μ[D | m] ω) ^ 2 ∂μ = (∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) * (μ[Y1 | m] ω - μ[Y0 | m] ω) ∂μ) / ∫ (ω : Ω), μ[D | m] ω * (1 - μ[D | m] ω) ∂μ","l":257,"f":"Causalean/Panel/EstimandCharacterization/OLSWeightDecomposition/GeneralCondExp.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For the never-treated potential-outcome schedule, the own-adoption potential-outcome schedule, a cohort, and a set of periods, the window-specific average treatment effect on the treated is the average over the specified periods of the own-adoption potential outcome minus the never-treated potential outcome for that cohort; it is defined as zero when the set is empty.","s":"{𝒢 : Type u_1} → {T : ℕ} → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → 𝒢 → Finset (Fin T) → ℝ","l":100,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For an adoption date, the eventually-treated condition holds exactly when the date is a finite panel period rather than the never-adopting value.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":91,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For an adoption date, the never-treated condition holds exactly when the date is the never-adopting value.","s":"{T : ℕ} → WithTop (Fin T) → Prop","l":94,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For an adoption date and a panel period, the adopted-by-period condition holds exactly when the adoption date is no later than that period; a never-adopting cohort does not satisfy it.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":85,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For an adoption date and a panel period, the untreated-at-period condition holds exactly when the period precedes the adoption date, including every period for a never-adopting cohort.","s":"{T : ℕ} → WithTop (Fin T) → Fin T → Prop","l":88,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions","k":"structure","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Causal-side assumptions for the Goodman-Bacon decomposition (LaTeX ass:po-estimand-goodman-bacon-causal), given a cohort panel P and potential-outcome maps Y0 (the never-treated path) and Y1 (each cohort's own adoption-date path). It packages consistency on treated cells — the factual outcome equals the post-adoption potential outcome once the cohort has adopted, consistency on untreated cells — the factual outcome equals the never-treated potential outcome before adoption, no anticipation — the pre-adoption potential outcomes under the two paths coincide, and pairwise untreated parallel trends for the treated-versus-never comparison, the early-versus-late-before-late comparison, and the late-versus-early-after-early comparison.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → Prop","l":105,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.AdoptionDate.lt_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Helper: every period is strictly less than ⊤ in WithTop (Fin T).","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf a → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt a t","l":167,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.AdoptionDate.not_le_of_lt","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Helper: AdoptionDate.lt (P.A g) t rules out AdoptionDate.le (P.A g) t.","s":"∀ {T : ℕ} {a : WithTop (Fin T)} {t : Fin T}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt a t → ¬Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le a t","l":161,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_add_ATT_of_treated","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On a window where every cell is treated for cohort g, the factual Ybar decomposes as the never-treated mean plus the window ATT.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 Y1 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le (P.A g) t → P.Y g t = Y1 g t) → ∀ (g : 𝒢) (S : Finset (Fin T)), (∀ t ∈ S, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.le (P.A g) t) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P g S = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 g S + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 g S","l":206,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S0_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S0_EL P e = {t : t < A_e} equals the never-treated Ybar0 when A_e < A_ℓ.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e)","l":260,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S0_LE","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S0_LE P e ℓ = {A_e ≤ t < A_ℓ} equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ)","l":279,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_late_on_S1_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The late-cohort factual Ybar on S1_EL P e ℓ = {A_e ≤ t < A_ℓ} equals the never-treated Ybar0: each cell satisfies t < A_ℓ.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ)","l":245,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_of_inf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On any window S where every period is untreated for cohort u, the factual Ybar equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ {u : 𝒢} (S : Finset (Fin T)), (∀ t ∈ S, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A u) t) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P u S = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 u S","l":191,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_on_S0_EL","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On the early-cohort untreated window S0_EL P e = {t : t < A_e}, the factual Ybar of cohort e equals the never-treated Ybar0.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (e : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL P e)","l":230,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions.Ybar_eq_Ybar0_on_S0_TN","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On the untreated window S0_TN P g = {t : t < A_g}, the factual Ybar of cohort g equals the never-treated Ybar0: by consistencyUntreated on each cell.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} {P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T} {Y0 : 𝒢 → Fin T → ℝ}, (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.lt (P.A g) t → P.Y g t = Y0 g t) → ∀ (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar P g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN P g) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0 Y0 g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN P g)","l":175,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel","k":"structure","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"A staggered-adoption cohort panel: a cell-statistics record carrying, per cohort, a population share, an adoption date — a finite period, or ⊤ for the never-treated case, and the cohort-period factual outcome mean, subject to a positive number of periods, strictly positive cohort shares, and cohort shares summing to one.","s":"(𝒢 : Type u_1) → ℕ → [Fintype 𝒢] → Type u_1","l":64,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","k":"inductive","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"The three-category comparison label classifies each admissible two-by-two comparison as a treated cohort against a never-treated cohort, an early-adopting cohort against a later-adopting cohort before the latter adopts, or a later-adopting cohort against an early-adopting cohort after the latter adopts.","s":"Type","l":57,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a cohort, and a period, the treatment indicator equals one if the cohort has adopted by that period and zero otherwise, and is therefore binary and absorbing.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":102,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_eq_of_A_eq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Cohorts that share the same adoption date have identical treatment status in every time period.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, P.A g = P.A u → ∀ (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P u t","l":255,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_eq_zero_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a cohort panel, a cohort that is never treated has a zero treatment indicator in every period.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → ∀ (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P u t = 0","l":142,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D_sq_eq_D","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a cohort panel, the binary treatment indicator for any cohort and time period equals its own square.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t ^ 2 = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t","l":160,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a cohort, and a period, the residualized treatment is the treatment indicator minus its cohort mean and its cross-cohort period mean, plus the grand mean.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":133,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde_eq","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a cohort-period cell, the double-demeaned residualized treatment Dtilde P g t equals the original Goodman-Bacon closed form: the raw treatment minus the cohort mean, minus the period cross-cohort mean, plus the grand mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g t - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g - ∑ g', P.p g' * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P g' t + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort P","l":138,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde_eq_centeredD_sub_weighted_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The explicit double-demeaning formula for Dtilde is the weighted centering, across cohorts, of the cohort-demeaned treatment path.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD P g t - ∑ h, P.p h * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD P h t","l":374,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a cohort classifier, and a period classifier, the balanced-panel-law condition states that every cohort-period cell has mass equal to the product of its cohort marginal mass and its period marginal mass.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → Fin T) → Prop","l":39,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the aggregate raw-weight denominator is the sum of treated-versus-never raw weights for finite-versus-never-treated pairs and the two timing-comparison raw weights for ordered finite adoption-date pairs.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":235,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda_eq_gap_sums","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The raw denominator of the staggered-adoption two-way fixed-effects decomposition equals the sum of treated-versus-never comparison terms and ordered early-versus-late comparison terms.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P = (∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u) else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ) else 0","l":229,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and an early-adopting cohort, the early-versus-late untreated window is the set of all periods before the early cohort's adoption date.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":176,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the late-versus-early early-treated window is the set of periods from the early cohort's adoption through the period before the late cohort's adoption.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → Finset (Fin T)","l":186,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a cohort, the treated-versus-never untreated window is the set of all panel periods before that cohort's adoption date.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":166,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the early-versus-late treated window is the set of periods from the early cohort's adoption through the period before the late cohort's adoption.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → Finset (Fin T)","l":181,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a late cohort, the late-versus-early both-treated window is the set of all periods at or after the late cohort's adoption date.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":191,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a cohort, the treated-versus-never treated window is the set of all panel periods at or after that cohort's adoption date.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T)","l":171,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TN_pair_contribution_eq_lambda_delta","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"TN numerator pair. For a treated cohort g with finite adoption date and a never-treated cohort u with infinite adoption date, the sum of the two ordered pairwise-covariance contributions between g and u in the finite cohort panel P equals the product of the TN comparison weight and the treated-versus-never contrast, λ_TN P g u · Δ_TN P g u.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P g u + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P u g = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u","l":1003,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TN_pair_vd_contribution_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"TN denominator pair: combining the two ordered pairwise-variance contributions gives the treated-vs-never raw denominator factor.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P g u + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P u g = P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u)","l":961,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TT_pair_contribution_eq_lambda_delta_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Treated-treated numerator pair: combining the two ordered pairwise-covariance contributions splits into the EL and LE comparison windows.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {e ℓ : 𝒢}, P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P ℓ e = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ","l":1029,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.TT_pair_vd_contribution_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Treated-treated denominator pair: combining the two ordered pairwise-variance contributions gives the timing-pair raw denominator factor.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {e ℓ : 𝒢}, P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P ℓ e = P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ)","l":982,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the residualized-treatment variance is the sum, over cohorts and periods, of the cohort share divided by the number of periods times squared residualized treatment.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":149,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD_eq_pairwise_centeredD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the residualized-treatment variance VD P equals the sum, over all ordered pairs of cohorts, of their pairwise centered-treatment contribution vdPairContribution P g u.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P = ∑ g, ∑ u, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution P g u","l":384,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a cohort, and a set of periods, the factual-outcome window mean is the average factual outcome for that cohort over the specified periods, and is defined as zero when the set is empty.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Finset (Fin T) → ℝ","l":159,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Ybar0","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a never-treated potential-outcome schedule, a cohort, and a set of periods, the never-treated potential-outcome window mean is the average of that cohort's never-treated potential outcomes over the specified periods; it is defined as zero when the set is empty.","s":"{𝒢 : Type u_1} → {T : ℕ} → (𝒢 → Fin T → ℝ) → 𝒢 → Finset (Fin T) → ℝ","l":96,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.admissible","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a tagged ordered cohort pair, the admissibility condition holds precisely when a treated-versus-never tag pairs an eventually treated cohort with a never-treated cohort, or either timing-comparison tag orders two finitely adopting cohorts by adoption date, and in every case both cohort shares are positive.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → Prop","l":256,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.adoption_pair_sum_decomp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When a pairwise cohort contribution is zero for cohorts sharing an adoption date, its total over all cohort pairs decomposes into the four possible ordered timing comparisons.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → 𝒢 → ℝ), (∀ (g u : 𝒢), P.A g = P.A u → f g u = 0) → ∑ g, ∑ u, f g u = (((∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then f g u else 0) + ∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A u) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A g) then f g u else 0) + ∑ g, ∑ u, if P.A g < P.A u ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A u) then f g u else 0) + ∑ g, ∑ u, if P.A u < P.A g ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) then f g u else 0","l":372,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a cohort, the cohort treatment share is that cohort's treatment indicator averaged over all panel periods.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → ℝ","l":108,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD_eq_unitMean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Goodman-Bacon's cohort treatment share is the shared unit mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g = Causalean.Panel.WeightedTwoWayPanel.unitMean (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P) g","l":122,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD_eq_zero_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"A never-treated cohort has zero average treatment over the panel.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.barD P g = 0","l":202,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the finite-cell population two-way-fixed-effects coefficient is the weighted covariance of residualized treatment and factual outcomes divided by residualized-treatment variance; it is defined even when that denominator is zero.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":153,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.binary_time_cov_filter_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For an indicator of a nonempty set of periods, the time-average product of its centered value and another quantity equals its variance times the difference between the selected-period and unselected-period averages of that quantity.","s":"∀ {T : ℕ}, 0 < T → ∀ (p : Fin T → Prop) [inst : DecidablePred p] (z : Fin T → ℝ), ↑(Finset.filter p Finset.univ).card ≠ 0 → (↑T)⁻¹ * ∑ t, ((if p t then 1 else 0) - (↑T)⁻¹ * ∑ t, if p t then 1 else 0) * z t = ((↑T)⁻¹ * ∑ t, if p t then 1 else 0) * (1 - (↑T)⁻¹ * ∑ t, if p t then 1 else 0) * ((↑(Finset.filter p Finset.univ).card)⁻¹ * ∑ t with p t, z t - (↑{t | ¬p t}.card)⁻¹ * ∑ t with ¬p t, z t)","l":206,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.binary_time_variance","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For a binary quantity observed over the panel's time periods, its average squared deviation from its time mean equals that mean times one minus that mean.","s":"∀ {T : ℕ}, 0 < T → ∀ (x : Fin T → ℝ), (∀ (t : Fin T), x t ^ 2 = x t) → (↑T)⁻¹ * ∑ t, (x t - (↑T)⁻¹ * ∑ t, x t) ^ 2 = ((↑T)⁻¹ * ∑ t, x t) * (1 - (↑T)⁻¹ * ∑ t, x t)","l":168,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_Dtilde_sq_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the population second moment of the residualized-treatment witness — the FWL denominator ∫ Vtilde² dμ — equals the finite-cell Goodman-Bacon denominator VD of the induced finite panel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → ∫ (ω : Ω), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω * (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum)","l":59,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_VD_pos_iff_Dtilde_sq_pos","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the finite-cell Goodman-Bacon denominator VD is strictly positive if and only if the population FWL denominator ∫ Vtilde² dμ is strictly positive.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → (0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) ↔ 0 < ∫ (ω : Ω), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω * (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced).Vtilde ω ∂μ)","l":126,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.bridge_finite_residualized_eq_twfe","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.PopulationBridge","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"On a probability space carrying cohort label G, period label T_rv, treatment D, and outcome Y, assume D is binary almost everywhere, there is at least one time period, every cohort has strictly positive population mass, the cohort masses sum to one, each cohort's mass is split evenly across periods (a balanced panel law), on each cohort-period cell D is almost-everywhere equal to its own cell mean (cell-measurability of D), and the population per-cell residualized treatment panelDtilde agrees with the finite-panel residualized treatment Dtilde. Then the population residualized regression coefficient of D on Y, controlling for the cohort/period additive class panelClass, equals the finite-cell Goodman-Bacon two-way-fixed-effects coefficient betaTWFE of the induced finite panel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsProbabilityMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (A : 𝒢 → WithTop (Fin T)) (G_meas : Measurable G) (T_meas : Measurable T_rv) (D_meas : Measurable D) (D_binary : ∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) (Y_memLp : MeasureTheory.MemLp Y 2 μ) (B_balanced : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv) (hT_pos : 0 < T) (hp_pos : ∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) (hp_sum : ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1), (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g / ↑T) → (∀ (g : 𝒢) (t : Fin T), ∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → (∀ (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum) g t) → Causalean.Panel.residualizedCoefficient μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessY_panel μ Y G T_rv G_meas T_meas Y_memLp B_balanced) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel μ D G T_rv G_meas T_meas D_meas D_binary B_balanced) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf μ Y G T_rv A hT_pos hp_pos hp_sum)","l":173,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/PopulationBridge.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a cohort classifier, a period classifier, a cohort, and a period, the cohort-period cell mass is the real-valued mass of observations classified in that cohort-period cell.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":63,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cell mass is nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (g : 𝒢) (t : Fin T), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t","l":206,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, an outcome variable, a cohort classifier, a period classifier, a cohort, and a period, the cohort-period cell mean is the outcome integral over that cell divided by its mass, and is zero when the cell has zero mass.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":76,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean_eq_cellBridge","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The panel cell mean for a cohort and period equals the generic cell-mean operator applied to the joint cohort-period cell.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (g : 𝒢) (t : Fin T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean μ Y G T_rv g t = Causalean.Panel.CellBridge.cellMean μ Y (fun ω => (G ω, T_rv ω)) (g, t)","l":84,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.centeredD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a cohort, and a period, the cohort-demeaned treatment path is that cohort's treatment indicator in the period minus its average treatment indicator across all periods.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → Fin T → ℝ","l":119,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a treatment variable, a cohort classifier, and a cohort, the cohort mean treatment share is the treatment integral over that cohort's cell divided by its mass, and is zero when the cell has zero mass.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → 𝒢 → ℝ","l":99,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD_mul_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"cohortBarD g · cohortMass g = ∫ D · 𝟙{G=g} dμ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortBarD μ D G g * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = ∫ (ω : Ω), D ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":188,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a cohort classifier, and a cohort, the cohort mass is the real-valued mass of observations classified in that cohort.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → 𝒢) → 𝒢 → ℝ","l":68,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cohort mass is nonnegative.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (g : 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g","l":211,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortWeights","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the cohort weight system assigns each cohort its population share and carries the panel's positivity and unit-sum conditions.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.WeightedTwoWayPanel.UnitWeights 𝒢","l":116,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohort_integral_div_mul_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Defining identity for cohortBarD: dividing by cohort mass and multiplying back recovers the cohort-indicator-weighted integral. On zero-mass cohorts both sides are zero (a.e.-vanishing indicator).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (g : 𝒢), (∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":116,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a tagged ordered cohort pair, the unified two-by-two contrast is the contrast associated with that pair's comparison tag when the pair is admissible, and zero otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":285,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a finite collection of cohorts whose members can be compared for equality, a natural-number panel length, a cohort panel, two potential-outcome paths, respectively under no treatment and under treatment at the cohort's own adoption date, and a labelled ordered pair of cohorts, the causal two-by-two contrast is zero when that comparison is inadmissible; otherwise, it is the relevant window average treatment effect for a treated-versus-never-treated or early-versus-late comparison, and for a late-versus-early comparison it is the late cohort's window average treatment effect minus the early cohort's corresponding treated-period effect net of its pre-period effect.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → (𝒢 → Fin T → ℝ) → (𝒢 → Fin T → ℝ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":47,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast_eq_contrastCausal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"On an admissible comparison, the algebraic contrast equals the causal potential-outcome contrast, by the Layer C corollaries.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ {k : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.admissible P k → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal P Y0 Y1 k","l":67,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.denom_per_cell_panel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Per-cell denominator identity. On the cell {G = g ∧ T_rv = t}, (D - panelPropensity)(ω) = panelDtilde μ D G T_rv g t; squaring and integrating over the cell gives","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t ^ 2","l":50,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.disjoint_union_mean_eq_card_weighted_mean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When two finite sets are disjoint and the second is nonempty, the mean of a quantity over their union is the cardinality-weighted combination of its means over the two sets.","s":"∀ {α : Type u_2} [inst : DecidableEq α] (A B : Finset α) (z : α → ℝ), Disjoint A B → ↑B.card ≠ 0 → (↑(A ∪ B).card)⁻¹ * ∑ t ∈ A ∪ B, z t = ↑A.card / ↑(A ∪ B).card * ((↑A.card)⁻¹ * ∑ t ∈ A, z t) + (1 - ↑A.card / ↑(A ∪ B).card) * ((↑B.card)⁻¹ * ∑ t ∈ B, z t)","l":600,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.gap_sums_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The total of cohort-share products times each pair's treatment-rate gap and one minus that gap, over treated-versus-never and ordered early-versus-later pairs, equals the residualized treatment variance.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ((∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u) else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ) else 0) = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P","l":483,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_cohort_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cohort indicator 𝟙{G = g} is in MemLp 2 μ (bounded + finite measure).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g}.indicator (fun x => 1) ω) 2 μ","l":41,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_panel_cell_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Joint cell indicator 𝟙{G=g ∧ T_rv=t} is in MemLp 2 μ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), MeasureTheory.MemLp (fun ω => {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω) 2 μ","l":56,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.indicator_period_memLp","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period indicator 𝟙{T_rv = t} is in MemLp 2 μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (T_rv : Ω → Fin T), Measurable T_rv → ∀ (t : Fin T), MeasureTheory.MemLp (fun ω => {ω' | T_rv ω' = t}.indicator (fun x => 1) ω) 2 μ","l":48,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.instDecidableEqCompTag","k":"instance","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"","s":"DecidableEq Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","l":58,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.instFintypeCompTag","k":"instance","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"A finite enumeration of the three comparison labels consists exactly of the treated-versus-never, early-versus-late, and late-versus-early categories.","s":"Fintype Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag","l":60,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_cohort_indicator_one_eq_cohortMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{G = g} dμ = cohortMass μ G g.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢), Measurable G → ∀ (g : 𝒢), ∫ (ω : Ω), {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g","l":77,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_cohort","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over cohort cells: ∫ F dμ = ∑_g ∫ F · 𝟙{G = g} dμ.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢), Measurable G → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∫ (ω : Ω), F ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ","l":304,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_panel_cell","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over cohort × period cells.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ g, ∑ t, ∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":325,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_eq_sum_period","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Integrate by summing over period cells.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (F : Ω → ℝ) (T_rv : Ω → Fin T), Measurable T_rv → MeasureTheory.Integrable F μ → ∫ (ω : Ω), F ω ∂μ = ∑ t, ∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":315,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_mul_panelClass_eq_zero_of_axes","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"If a square-integrable random variable V has zero mean product against every cohort indicator 𝟙{G = g} and zero mean product against every period indicator 𝟙{T_rv = t}, then V is orthogonal, in the L²(μ) sense, to every unit/period-additive combination ∑_g cG(g)·𝟙{G = g} + ∑_t cT(t)·𝟙{T_rv = t}, i.e. to every member of the additive nuisance class panelClass.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (V : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp V 2 μ → ∀ (cG : 𝒢 → ℝ) (cT : Fin T → ℝ), (∀ (g : 𝒢), ∫ (ω : Ω), V ω * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0) → (∀ (t : Fin T), ∫ (ω : Ω), V ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0) → ∫ (ω : Ω), V ω * (∑ g, cG g * {ω' | G ω' = g}.indicator (fun x => 1) ω + ∑ t, cT t * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω) ∂μ = 0","l":276,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_panel_cell_indicator_one_eq_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{G = g ∧ T_rv = t} dμ = cellMass μ G T_rv g t.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), ∫ (ω : Ω), {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t","l":94,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.integral_period_indicator_one_eq_periodMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"∫ 𝟙{T_rv = t} dμ = periodMass μ T_rv t.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (T_rv : Ω → Fin T), Measurable T_rv → ∀ (t : Fin T), ∫ (ω : Ω), {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t","l":85,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the early-versus-late raw weight is the product of their shares, their treatment-share gap, one minus that gap, and the comparison splitting fraction.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":226,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL_add_lambdaLE_eq_gap","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The two ordered raw comparison weights between two cohorts sum to the product of their cohort shares, the gap in their average treatment rates, and one minus that gap.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (e ℓ : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ = P.p e * P.p ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P e ℓ)","l":219,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the late-versus-early raw weight is the product of their shares, their treatment-share gap, one minus that gap, and one minus the comparison splitting fraction.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":230,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a treated cohort, and a never-treated cohort, the treated-versus-never raw weight is the product of their population shares and the treated cohort's average treatment indicator times one minus that indicator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":214,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN_eq_gap_of_isInf","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"When one cohort is never treated, its raw comparison weight with another cohort equals the product of their cohort shares, the gap in their average treatment rates, and one minus that gap.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) {g u : 𝒢}, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u = P.p g * P.p u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u * (1 - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q P g u)","l":209,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a tagged ordered cohort pair, the unified raw weight is the raw weight associated with that pair's comparison tag when the pair is admissible, and zero otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":295,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.memLp_two_of_binary","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability measure, a measurable real-valued variable that equals either zero or one almost surely has a finite second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ), Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → MeasureTheory.MemLp D 2 μ","l":52,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.mu","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the comparison splitting fraction is one minus the early cohort's treatment share divided by one minus their treatment-share gap, with the usual zero-denominator convention.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":222,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and two cohorts, the ordered-pair contribution to the two-way-fixed-effects numerator is one half of their share product divided by the number of periods, multiplied by the sum of the product of their demeaned-treatment difference and factual-outcome difference.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":133,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.num_per_cell_panel","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"Per-cell numerator identity. On the cohort-period cell (g, t), if the treatment D is almost-everywhere equal, on that cell, to its own cell mean (cell-measurability of D), then the integral of the propensity-residual D - panelPropensity times the outcome Y over the cell equals the cell mass times the cell's residualized treatment panelDtilde times the outcome's cell mean:","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (D Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∀ᵐ (ω : Ω) ∂μ.restrict {ω' | G ω' = g ∧ T_rv ω' = t}, D ω = (∫ (ω' : Ω), D ω' * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω' ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t) → ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * Y ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde μ D G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMean μ Y G T_rv g t","l":122,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the overall treatment share is the sum of each cohort's population share times its average treatment indicator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → ℝ","l":112,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort_eq_grandMean","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Goodman-Bacon's overall treatment share is the shared weighted grand mean.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pCohort P = Causalean.Panel.WeightedTwoWayPanel.grandMean (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortWeights P) (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.D P)","l":128,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.pairwise_sum_normalize","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a finite cohort panel, averaging half the ordered-pair weighted total at each period equals the ordered-pair weighted total of the time sums with the same normalization.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → 𝒢 → Fin T → ℝ), (↑T)⁻¹ * ∑ t, 1 / 2 * ∑ g, ∑ u, P.p g * P.p u * f g u t = ∑ g, ∑ u, P.p g * P.p u / (2 * ↑T) * ∑ t, f g u t","l":99,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"Given a finite measure, a cohort classifier, a period classifier, a measurable cohort classifier, and a measurable period classifier, the saturated cohort-and-period function class consists of functions that agree almost everywhere with a linear combination of cohort and period indicator functions.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → [DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → Measurable G → Measurable T_rv → Causalean.Panel.LinearL2Class μ","l":48,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelDtilde","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.PerCell","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measurable sample space, a finite cohort label space whose members can be compared for equality and whose singleton sets are measurable, a natural-number panel length, a measure on that sample space, a treatment variable, a cohort variable, a period variable, a cohort label, and a period, the cell-specific residualized treatment value is the treatment integral over the cohort-period cell divided by that cell's mass, minus the additive cohort-and-period fitted treatment value for that cell.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":31,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/PerCell.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, an outcome variable, a cohort classifier, and a period classifier, the saturated cohort-and-period outcome regression assigns each observation its cohort mean outcome plus its period mean outcome minus the overall mean outcome.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → Ω → ℝ","l":146,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanRegHat","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measurable sample space, a finite cohort label space whose members can be compared for equality and whose singleton sets are measurable, a natural-number panel length, a measure, an outcome variable, a cohort variable, a period variable, a cohort label, and a period, the cell-specific additive outcome fit is the cohort-specific mean outcome plus the period-specific mean outcome minus the overall mean outcome.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":217,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_cohort_axis_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a balanced panel law, the residual from the panel mean regression has zero average product with the indicator of any fixed cohort.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp F 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (F ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ F G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":375,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_eq_hat_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"On the cell {G = g ∧ T_rv = t}, panelMeanReg evaluates to panelMeanRegHat g t.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) {g : 𝒢} {t : Fin T} {ω : Ω}, G ω = g → T_rv ω = t → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanRegHat μ Y G T_rv g t","l":253,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_mem_panelClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a finite measure μ, an outcome variable Y, a measurable cohort map G, and a measurable period map T_rv, the pointwise saturated cohort-and-period mean regression of Y belongs to the two-axis additive (cohort-plus-period) function class.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (G_meas : Measurable G) (T_meas : Measurable T_rv), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas).mem (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv)","l":183,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg_period_axis_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a balanced panel law, the residual from the panel mean regression has zero average product with the indicator of any fixed period.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp F 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (F ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ F G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":487,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelOf","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"Given a measure, an outcome variable, a cohort classifier, a period classifier, an adoption-date schedule, a positive number of periods, strictly positive cohort masses, and cohort masses summing to one, the cohort panel constructed from the law has those cohort masses as shares, the supplied adoption dates, and cohort-period outcome means.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → (Ω → ℝ) → (G : Ω → 𝒢) → (Ω → Fin T) → (𝒢 → WithTop (Fin T)) → 0 < T → (∀ (g : 𝒢), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g) → ∑ g, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cohortMass μ G g = 1 → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T","l":107,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a treatment variable, a cohort classifier, and a period classifier, the saturated cohort-and-period propensity regression assigns each observation its cohort mean treatment plus its period mean treatment minus the overall mean treatment.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype 𝒢] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → Ω → ℝ","l":125,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensityHat","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measurable sample space, a finite cohort label space whose members can be compared for equality and whose singleton sets are measurable, a natural-number panel length, a measure, a treatment variable, a cohort variable, a period variable, a cohort label, and a period, the cell-specific additive treatment fit is the cohort-specific mean treatment plus the period-specific mean treatment minus the overall mean treatment.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → 𝒢) → (Ω → Fin T) → 𝒢 → Fin T → ℝ","l":204,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity_eq_hat_of_mem","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"On the cell {G = g ∧ T_rv = t}, panelPropensity evaluates to panelPropensityHat g t. Pointwise (no a.e. needed) by single-cell membership selecting one term in each finite sum.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) {g : 𝒢} {t : Fin T} {ω : Ω}, G ω = g → T_rv ω = t → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensityHat μ D G T_rv g t","l":231,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity_mem_panelClass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a finite measure μ, a treatment variable D, a measurable cohort map G, and a measurable period map T_rv, the pointwise saturated cohort-and-period propensity regression of D belongs to the two-axis additive (cohort-plus-period) function class.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : DecidableEq 𝒢] [inst_3 : MeasurableSpace 𝒢] [inst_4 : MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [inst_5 : MeasureTheory.IsFiniteMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T) (G_meas : Measurable G) (T_meas : Measurable T_rv), (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas).mem (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv)","l":162,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panel_cell_integral_div_mul_cellMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Cell analogue (joint cohort × period cell).","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → ∀ (g : 𝒢) (t : Fin T), (∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.cellMass μ G T_rv g t = ∫ (ω : Ω), F ω * {ω' | G ω' = g ∧ T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":139,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a measure, a period classifier, and a period, the period mass is the real-valued mass of observations classified in that period.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {T : ℕ} → MeasureTheory.Measure Ω → (Ω → Fin T) → Fin T → ℝ","l":72,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Basic","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period mass is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) (T_rv : Ω → Fin T) (t : Fin T), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t","l":215,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Basic.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.period_centered_sum_eq_zero","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability distribution, period-specific means of an integrable outcome, centered by the overall mean and weighted by their period probabilities, sum to zero.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (F : Ω → ℝ) (T_rv : Ω → Fin T), Measurable T_rv → MeasureTheory.Integrable F μ → ∑ t, ((∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t - ∫ (ω : Ω), F ω ∂μ) * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = 0","l":341,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.period_integral_div_mul_periodMass","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Integrals","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Period analogue.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (F : Ω → ℝ) (T_rv : Ω → Fin T) (t : Fin T), (∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ) / Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = ∫ (ω : Ω), F ω * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ","l":128,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Integrals.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.q","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the treatment-share gap is the early cohort's average treatment indicator minus the late cohort's average treatment indicator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":218,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.raw_weight_sum_eq_VD","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Prop A5.2 (raw_weight_sum_eq_VD). The aggregate raw-weight denominator equals the residualized treatment variance: Λ P = V_D P. Key denominator identity in thm:po-estimand-goodman-bacon-decomposition.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P","l":514,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residD_cohort_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"For a treatment indicator D that is binary almost everywhere, the residual between D and its cohort-period propensity score panelPropensity is orthogonal, in the L²(μ) sense, to every cohort indicator 𝟙{G = g}.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":624,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residD_period_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The treatment residual is orthogonal to every period indicator 𝟙{T_rv = t}. Requires B_balanced to make the cross-cohort sum cancel.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (D : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (D ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelPropensity μ D G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":665,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessD_panel","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"On a measurable sample space with a finite cohort label space whose members can be compared for equality and whose singleton sets are measurable and a natural-number panel length, let the probability measure, the treatment variable, the cohort variable, and the period variable be given. If the cohort variable is measurable, the period variable is measurable, the treatment variable is measurable, the treatment variable equals either zero or one almost everywhere, and each cohort-period cell has probability equal to the product of its cohort and period marginal probabilities, then the additive cohort-and-period treatment fit and its residual constitute a residualization witness for treatment with respect to the class of additive cohort-and-period functions.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (D : Ω → ℝ) → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → (G_meas : Measurable G) → (T_meas : Measurable T_rv) → Measurable D → (∀ᵐ (ω : Ω) ∂μ, D ω = 0 ∨ D ω = 1) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) D","l":31,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residWitnessY_panel","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Partition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":1,"d":"On a measurable sample space with a finite cohort label space whose members can be compared for equality and whose singleton sets are measurable and a natural-number panel length, let the probability measure, the outcome variable, the cohort variable, and the period variable be given. If the cohort variable is measurable, the period variable is measurable, the outcome has a finite second moment, and each cohort-period cell has probability equal to the product of its cohort and period marginal probabilities, then the additive cohort-and-period outcome fit and its residual constitute a residualization witness for the outcome with respect to the class of additive cohort-and-period functions.","s":"{Ω : Type u_1} → {𝒢 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : Fintype 𝒢] → [inst_2 : DecidableEq 𝒢] → [inst_3 : MeasurableSpace 𝒢] → [inst_4 : MeasurableSingletonClass 𝒢] → {T : ℕ} → (μ : MeasureTheory.Measure Ω) → [inst_5 : MeasureTheory.IsProbabilityMeasure μ] → (Y : Ω → ℝ) → (G : Ω → 𝒢) → (T_rv : Ω → Fin T) → (G_meas : Measurable G) → (T_meas : Measurable T_rv) → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → Causalean.Panel.ResidualizationWitness μ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelClass μ G T_rv G_meas T_meas) Y","l":103,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Partition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residY_cohort_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The outcome residual is orthogonal to every cohort indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (g : 𝒢), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω) * {ω' | G ω' = g}.indicator (fun x => 1) ω ∂μ = 0","l":649,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.residY_period_orthogonal","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"The outcome residual is orthogonal to every period indicator.","s":"∀ {Ω : Type u_1} {𝒢 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype 𝒢] [inst_2 : MeasurableSpace 𝒢] [MeasurableSingletonClass 𝒢] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (Y : Ω → ℝ) (G : Ω → 𝒢) (T_rv : Ω → Fin T), Measurable G → Measurable T_rv → MeasureTheory.MemLp Y 2 μ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.IsBalancedPanelLaw μ G T_rv → ∀ (t : Fin T), ∫ (ω : Ω), (Y ω - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.panelMeanReg μ Y G T_rv ω) * {ω' | T_rv ω' = t}.indicator (fun x => 1) ω ∂μ = 0","l":687,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_lambdaWeight_eq_Lambda","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Across all admissible comparison types and cohort pairs, the raw comparison weights sum to the aggregate normalizing denominator.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight P k = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Lambda P","l":570,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_periodMass_eq_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Support.Orthogonality","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support","t":2,"d":"Under a probability distribution, the probabilities assigned to every period by a measurable finite-valued period variable sum to one.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {T : ℕ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (T_rv : Ω → Fin T), Measurable T_rv → ∑ t, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.periodMass μ T_rv t = 1","l":321,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Support/Orthogonality.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.sum_weight_over_T_commute","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"In a finite cohort panel, a cohort-weighted sum over cohorts and periods with equal period weight equals the average over periods of the corresponding cohort-weighted sums.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (f : 𝒢 → Fin T → ℝ), ∑ g, ∑ t, P.p g / ↑T * f g t = (↑T)⁻¹ * ∑ t, ∑ g, P.p g * f g t","l":88,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_eq_weighted_avg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Theorem A5.5 (twfe_eq_weighted_avg, thm:po-estimand-goodman-bacon-decomposition). For a cohort panel, the two-way fixed-effects (TWFE) coefficient, under the totalized zero-variance convention, equals the weighted sum of admissible two-by-two DID contrasts across comparison groups.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k","l":701,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_eq_weighted_avg_core","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"The positivity-free algebraic core of the Goodman-Bacon decomposition: the totalized TWFE ratio equals the totalized weighted sum of admissible two-by-two DID contrasts, including in zero-variance cases.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrast P k","l":664,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_numerator_eq_lambda_delta_sum","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Prop A5.3 (twfe_numerator_eq_lambda_delta_sum). The TWFE numerator decomposes by unordered cohort pairs into raw-weight times 2x2 DID contrast contributions.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ g, ∑ t, P.p g / ↑T * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t * P.Y g t = (∑ g, ∑ u, if Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) then Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaTN P g u * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u else 0) + ∑ e, ∑ ℓ, if P.A e < P.A ℓ ∧ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) then Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaEL P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ + Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaLE P e ℓ * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ else 0","l":524,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_numerator_eq_pairwise_centeredD_Y","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"Pairwise representation of the TWFE numerator.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), ∑ g, ∑ t, P.p g / ↑T * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Dtilde P g t * P.Y g t = ∑ g, ∑ u, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.numPairContribution P g u","l":419,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.twfe_po_decomposition","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Fused causal Goodman-Bacon decomposition. Fix a cohort panel P and potential-outcome maps Y0 (never-treated path) and Y1 (own-adoption-date path). Assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the treated-versus-never, early-versus-late, and late-versus-early comparison types; then the two-way fixed-effects coefficient betaTWFE P equals the sum, over admissible pairwise comparisons, of each comparison's Goodman-Bacon weight times its potential-outcome window contrast — the treated-versus-never and early-versus-late window-specific ATTs, or the late-versus-early bad-comparison adjustment.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.betaTWFE P = ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k * Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.contrastCausal P Y0 Y1 k","l":84,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/CausalDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.vdPairContribution","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and two cohorts, the ordered-pair contribution to residualized-treatment variance is one half of their share product divided by the number of periods, multiplied by the sum of squared differences between their cohort-demeaned treatment paths.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":123,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the normalized early-versus-late weight is its raw weight divided by the aggregate raw-weight denominator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":246,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the normalized late-versus-early weight is its raw weight divided by the aggregate raw-weight denominator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":250,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.w_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a treated cohort, and a never-treated cohort, the normalized treated-versus-never weight is its raw weight divided by the aggregate raw-weight denominator.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":242,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel and a tagged ordered cohort pair, the unified normalized weight is the normalized weight associated with that pair's comparison tag when the pair is admissible, and zero otherwise.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢 → ℝ","l":275,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weighted_center_cov_uncentered_right","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Pairwise","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":2,"d":"For a finite collection with weights summing to one, the weighted sum of deviations of one quantity from its weighted mean times another quantity equals one half of the weighted sum of pairwise differences in the two quantities.","s":"∀ {ι : Type u_2} [inst : Fintype ι] (p x y : ι → ℝ), ∑ i, p i = 1 → ∑ i, p i * (x i - ∑ j, p j * x j) * y i = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j)","l":40,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Pairwise.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weights_nonneg","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Prop A5.1 (weights_nonneg). For a cohort panel and any comparison-type index, the raw Goodman-Bacon comparison weight is nonnegative.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (k : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢), 0 ≤ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.lambdaWeight P k","l":57,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weights_sum_one","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AlgebraicDecomposition","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Prop A5.4 (weights_sum_one). For a finite Goodman–Bacon cohort panel P, if the residualized-treatment variance VD P is strictly positive, then the normalized comparison weights sum to one over all admissible 2×2 comparisons: Σ_{k ∈ 𝒦 P} weight P k = 1.","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] [DecidableEq 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T), 0 < Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.VD P → ∑ k ∈ Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦 P, Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.weight P k = 1","l":638,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/AlgebraicDecomposition.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the early-versus-late two-by-two difference-in-differences contrast is the early cohort's outcome change from its untreated to its treated-before-late window minus the late cohort's change over those same windows.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":202,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL_eq_ATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 2 — EL identifies the early cohort's ATT. Fix a cohort panel P and potential-outcome maps Y0, Y1, and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For an early cohort e whose adoption date strictly precedes that of a late cohort ℓ, with ℓ's adoption date finite, the early-versus-late contrast Δ_EL equals e's window-specific average treatment effect on the treated over the window running from e's own adoption date up to ℓ's adoption date:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_EL P e ℓ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_EL P e ℓ)","l":340,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, an early cohort, and a late cohort, the late-versus-early two-by-two difference-in-differences contrast is the late cohort's outcome change from the early-treated window to the both-treated window minus the early cohort's change over those same windows.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":207,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE_eq_bad_comparison","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 3 — LE has a bad-comparison term. Fix a cohort panel P and potential-outcome maps Y0, Y1, and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For an early cohort e whose adoption date strictly precedes that of a late cohort ℓ, with ℓ's adoption date finite, the late-versus-early contrast Δ_LE equals ℓ's window-specific average treatment effect on the treated over its post-adoption window, minus the \"bad-comparison\" gap between e's own treatment effects on the two comparison windows:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (e ℓ : 𝒢), P.A e < P.A ℓ → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A ℓ) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_LE P e ℓ = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 ℓ (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE P ℓ) - (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_LE P ℓ) - Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 e (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S0_LE P e ℓ))","l":383,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, a treated cohort, and a never-treated comparison cohort, the treated-versus-never two-by-two difference-in-differences contrast is the treated cohort's outcome change between its treated and untreated windows minus the comparison cohort's change over those same windows.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → 𝒢 → 𝒢 → ℝ","l":197,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN_eq_ATT","k":"theorem","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Causal","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"Layer C corollary 1 — TN identifies ATT. Fix a cohort panel P and potential-outcome maps Y0 (never-treated path) and Y1 (own-adoption-date path), and assume consistency on treated and untreated cells, no anticipation, and pairwise untreated parallel trends across the three comparison types. For a treated cohort g whose adoption date is finite compared against a never-treated cohort u whose adoption date is infinite, the treated-versus-never 2x2 difference-in-differences contrast Δ_TN equals g's window-specific average treatment effect on the treated over its post-adoption window:","s":"∀ {𝒢 : Type u_1} [inst : Fintype 𝒢] {T : ℕ} (P : Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T) (Y0 Y1 : 𝒢 → Fin T → ℝ), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CausalAssumptions P Y0 Y1 → ∀ (g u : 𝒢), Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isFin (P.A g) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.AdoptionDate.isInf (P.A u) → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.Δ_TN P g u = Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.ATT_window Y0 Y1 g (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.S1_TN P g)","l":296,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/Causal.lean"},{"n":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.𝒦","k":"def","m":"Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.FinitePanel","a":"Panel/EstimandCharacterization/StaggeredTWFEDecomposition","t":1,"d":"For a cohort panel, the comparison index set is the finite set of every tagged ordered cohort pair satisfying the admissibility condition.","s":"{𝒢 : Type u_1} → [inst : Fintype 𝒢] → {T : ℕ} → Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CohortPanel 𝒢 T → Finset (Causalean.Panel.EstimandCharacterization.StaggeredTWFEDecomposition.CompTag × 𝒢 × 𝒢)","l":270,"f":"Causalean/Panel/EstimandCharacterization/StaggeredTWFEDecomposition/FinitePanel.lean"},{"n":"Causalean.Panel.Exposure","k":"def","m":"Causalean.Panel.PO.Exposure","a":"Panel/PO","t":1,"d":"For a set of units, a set of periods, a treatment-value set, and a cell-specific exposure set, an exposure mapping assigns to every unit-period cell and every realized treatment path an exposure in that cell's exposure set.","s":"(I : Type u_1) → (T : Type u_2) → Type u_3 → (I × T → Type u_4) → Type (max (max (max u_1 u_2) u_3) u_4)","l":31,"f":"Causalean/Panel/PO/Exposure.lean"},{"n":"Causalean.Panel.FWLInstanceL2.fwlCoef_eq_of_original_minimizer","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":1,"d":"FWL uniqueness on L²(μ). Fix square-integrable regressors X and outcome Y, and a finite-dimensional nuisance subspace H of L²(μ), and assume the residualized regressor Gram matrix is invertible. If the nuisance term h lies in H and the pair (β, h) minimizes the original least-squares objective jointly over all coefficient vectors and nuisance terms in H, then β equals the Frisch–Waugh–Lovell coefficient computed by residualizing against H. Specialization of Causalean.Mathlib.FWL.fwlCoef_eq_of_original_minimizer.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ H → (∀ (γ : Fin K → ℝ) {g : ↥(MeasureTheory.Lp ℝ 2 μ)}, g ∈ H → Causalean.Mathlib.FWL.originalObjective X Y β h ≤ Causalean.Mathlib.FWL.originalObjective X Y γ g) → β = Causalean.Mathlib.FWL.fwlCoef H X Y","l":107,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.fwlCoef_original_minimizes","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"FWL least-squares optimality on L²(μ). The FWL coefficient (paired with the nuisance projection of its raw residual) minimizes the original least-squares objective over coefficients and nuisance terms in H. Specialization of Causalean.Mathlib.FWL.fwlCoef_original_minimizes.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → ∀ (β : Fin K → ℝ) {h : ↥(MeasureTheory.Lp ℝ 2 μ)}, h ∈ H → Causalean.Mathlib.FWL.originalObjective X Y (Causalean.Mathlib.FWL.fwlCoef H X Y) (Submodule.orthogonalProjectionFn (Y - Causalean.Mathlib.FWL.fittedValue X (Causalean.Mathlib.FWL.fwlCoef H X Y))) ≤ Causalean.Mathlib.FWL.originalObjective X Y β h","l":94,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.gramResid_mulVec_fwlCoef","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"Residualized normal equations on L²(μ). The FWL coefficient solves gramResid H X *ᵥ fwlCoef H X Y = residInnerVec H X Y whenever the residualized Gram matrix is invertible. Specialization of Causalean.Mathlib.FWL.gramResid_mulVec_fwlCoef.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {K : ℕ} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [inst_1 : H.HasOrthogonalProjection] (X : Fin K → ↥(MeasureTheory.Lp ℝ 2 μ)) (Y : ↥(MeasureTheory.Lp ℝ 2 μ)), IsUnit (Causalean.Mathlib.FWL.gramResid H X).det → (Causalean.Mathlib.FWL.gramResid H X).mulVec (Causalean.Mathlib.FWL.fwlCoef H X Y) = Causalean.Mathlib.FWL.residInnerVec H X Y","l":84,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.hasOrthogonalProjection_of_finiteDimensional","k":"instance","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"For a measurable sample space, a measure on that sample space, and a finite-dimensional real linear subspace of the square-integrable real functions under that measure, an orthogonal projection onto that subspace exists.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} (H : Submodule ℝ ↥(MeasureTheory.Lp ℝ 2 μ)) [FiniteDimensional ℝ ↥H], H.HasOrthogonalProjection","l":59,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.FWLInstanceL2.inner_eq_integral","k":"theorem","m":"Causalean.Panel.FWLInstanceL2","a":"Panel","t":2,"d":"L² inner product = integral pairing. For two square-integrable real random variables, the Hilbert-space inner product equals the integral of their product, independent of the chosen representatives. This is the bridge between abstract FWL inner products and the population second moments used in estimand papers.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} (f g : ↥(MeasureTheory.Lp ℝ 2 μ)), inner ℝ f g = ∫ (a : α), ↑↑f a * ↑↑g a ∂μ","l":67,"f":"Causalean/Panel/FWLInstanceL2.lean"},{"n":"Causalean.Panel.LinearL2Class","k":"structure","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"A linear square-integrable class is a nuisance space of real-valued functions, cut out by a membership predicate mem, that is closed under the zero function, closed under addition, and closed under real scalar multiplication, with every member square-integrable.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → Type u_1","l":33,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.PO.CellPartition","k":"structure","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"A finite measurable partition of a probability space (Ω, μ) into positive-mass cells indexed by a finite type ι: an assignment of a cell to each index such that every cell is measurable, distinct cells are pairwise disjoint, the cells cover the whole space, and every cell has strictly positive probability mass.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι : Type u_2) → [Fintype ι] → Type (max u_1 u_2)","l":46,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.cellPartitionOfClassifier_cell","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_3} [inst_1 : Fintype ι] (μ : MeasureTheory.Measure Ω) (g : Ω → ι) (hmeas : ∀ (i : ι), MeasurableSet (g ⁻¹' {i})) (hpos : ∀ (i : ι), 0 < (μ (g ⁻¹' {i})).toReal) (i : ι), (Causalean.Panel.PO.cellPartitionOfClassifier μ g hmeas hpos).cell i = g ⁻¹' {i}","l":85,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.integral_eq_sum_mass_mul_mean","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Finite-partition total law. The integral of f is the mass-weighted sum of its cell means: ∫ f = ∑ i, π_i · E[f ∣ cell i].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) [MeasureTheory.IsFiniteMeasure μ] (f : Ω → ℝ), MeasureTheory.Integrable f μ → ∫ (ω : Ω), f ω ∂μ = ∑ i, P.mass i * P.mean f i","l":124,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"For a measurable sample space, a measure, a finite index set, a cell partition, and one of its indices, the cell mass is the real-valued measure of that index's cell.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [inst_1 : Fintype ι] → Causalean.Panel.PO.CellPartition μ ι → ι → ℝ","l":91,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_ne_zero","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Every cell has nonzero mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι), P.mass i ≠ 0","l":106,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_pos","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Every cell has strictly positive mass.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι), 0 < P.mass i","l":103,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mass_sum_one","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Masses sum to one. The cell probabilities of a partition of a probability space add to 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) [MeasureTheory.IsProbabilityMeasure μ], ∑ i, P.mass i = 1","l":110,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"For a measurable sample space, a measure, a finite index set, a cell partition, a real-valued variable on the sample space, and one of the partition's indices, the cell-conditional mean is that variable's integral over the indexed cell divided by the cell's real-valued mass.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [inst_1 : Fintype ι] → Causalean.Panel.PO.CellPartition μ ι → (Ω → ℝ) → ι → ℝ","l":96,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_add","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are additive over sums of integrable integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), MeasureTheory.IntegrableOn f (P.cell i) μ → MeasureTheory.IntegrableOn g (P.cell i) μ → P.mean (f + g) i = P.mean f i + P.mean g i","l":152,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_congr_ae","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means agree for a.e.-equal integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), f =ᵐ[μ] g → P.mean f i = P.mean g i","l":140,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_congr_on","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means agree when the integrands agree pointwise on the cell. This is the workhorse \"consistency descent\" step: on a cell where an observed quantity equals a potential-outcome slice, their cell means coincide.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), (∀ ω ∈ P.cell i, f ω = g ω) → P.mean f i = P.mean g i","l":132,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_consistency","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"Cell-mean consistency descent. On a cell where the treatment value d is binary, d ∈ {0,1}, and pointwise potential-outcome consistency Y = Y0 + d·(Y1 − Y0) holds throughout the cell, the cell-conditional means satisfy the same identity: E[Y ∣ cell] = E[Y0 ∣ cell] + d·(E[Y1 ∣ cell] − E[Y0 ∣ cell]).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (i : ι) (Y Y0 Y1 : Ω → ℝ) {d : ℝ}, d = 0 ∨ d = 1 → (∀ ω ∈ P.cell i, Y ω = Y0 ω + d * (Y1 ω - Y0 ω)) → P.mean Y i = P.mean Y0 i + d * (P.mean Y1 i - P.mean Y0 i)","l":163,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_smul","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are homogeneous under real scalar multiplication.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) (c : ℝ) (f : Ω → ℝ) (i : ι), P.mean (fun ω => c * f ω) i = c * P.mean f i","l":158,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.CellPartition.mean_sub","k":"theorem","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":2,"d":"Cell means are additive over subtraction of integrable integrands.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] (P : Causalean.Panel.PO.CellPartition μ ι) {f g : Ω → ℝ} (i : ι), MeasureTheory.IntegrableOn f (P.cell i) μ → MeasureTheory.IntegrableOn g (P.cell i) μ → P.mean (f - g) i = P.mean f i - P.mean g i","l":146,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PO.Mobius.coeff_sum","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"Coefficient sum: for A ⊆ B, ∑ S, A ⊆ S ⊆ B, (-1)^(|S|-|A|) = if A = B then 1 else 0.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] (A B : Finset ι), A ⊆ B → ∑ S ∈ B.powerset with A ⊆ S, (-1) ^ (S.card - A.card) = if A = B then 1 else 0","l":97,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.delta","k":"def","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"For a coordinate label space whose elements can be compared for equality, a real-valued response defined for every binary treatment history, and a finite subset of coordinate labels, the subset interaction coefficient is the alternating sum of the response evaluated at every sub-subset's indicator history, with sign determined by the difference in subset sizes.","s":"{ι : Type u_1} → [DecidableEq ι] → ((ι → Fin 2) → ℝ) → Finset ι → ℝ","l":61,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.indicator","k":"def","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"For a coordinate label space whose elements can be compared for equality and a finite subset of coordinate labels, the subset-indicator treatment history assigns treatment status one to labels in the subset and zero to all other labels.","s":"{ι : Type u_1} → [DecidableEq ι] → Finset ι → ι → Fin 2","l":48,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.indicator_empty","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"The indicator history of the empty subset is the all-zero treatment history.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι], Causalean.Panel.PO.Mobius.indicator ∅ = fun x => 0","l":55,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.mobius_expansion","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":1,"d":"Any binary finite-memory response τ normalized to zero at the all-zero history can be recovered as the sum of its nonempty inclusion-exclusion interaction coefficients, with each interaction contributing only when every one of its lags is active in the history.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] [inst_1 : Fintype ι] (τ : (ι → Fin 2) → ℝ), (τ fun x => 0) = 0 → ∀ (h : ι → Fin 2), τ h = ∑ S ∈ Finset.univ.powerset with S.Nonempty, Causalean.Panel.PO.Mobius.delta τ S * ∏ k ∈ S, ↑↑(h k)","l":177,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.Mobius.prod_indicator_eq","k":"theorem","m":"Causalean.Panel.PO.Mobius","a":"Panel/PO","t":2,"d":"Indicator product: ∏ k ∈ S, ((h k).val : ℝ) equals 1 if S ⊆ B and 0 otherwise, where B = {k : h k = 1}.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] (h : ι → Fin 2) (B : Finset ι), (∀ (k : ι), k ∈ B ↔ h k = 1) → ∀ (S : Finset ι), ∏ k ∈ S, ↑↑(h k) = if S ⊆ B then 1 else 0","l":69,"f":"Causalean/Panel/PO/Mobius.lean"},{"n":"Causalean.Panel.PO.cellPartitionOfClassifier","k":"def","m":"Causalean.Panel.PO.PopulationCells","a":"Panel/PO","t":1,"d":"For a measurable sample space, a finite index set, a measure, a classifier assigning each sample point an index, measurable classifier level sets, and strictly positive real mass for every level set, the classifier-induced cell partition assigns each index its classifier level set.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {ι : Type u_2} → [inst_1 : Fintype ι] → (μ : MeasureTheory.Measure Ω) → (g : Ω → ι) → (∀ (i : ι), MeasurableSet (g ⁻¹' {i})) → (∀ (i : ι), 0 < (μ (g ⁻¹' {i})).toReal) → Causalean.Panel.PO.CellPartition μ ι","l":63,"f":"Causalean/Panel/PO/PopulationCells.lean"},{"n":"Causalean.Panel.PanelConsistency","k":"def","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, the panel consistency condition requires that, for every observed unit-period cell and every sample point, the factual observed outcome equals the potential outcome at that cell under the exposure realized at that sample point.","s":"Causalean.Panel.PanelPOSystem → Prop","l":41,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.PanelPOSystem","k":"structure","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"A panel potential-outcomes system pairs finite unit and time index sets with a finite collection of observed unit-period cells carrying positive normalized weights, a treatment-path-valued random variable D that is measurable in every unit-period coordinate, a cell-specific exposure mapping e together with a baseline exposure e0, a family of potential outcomes Y at each cell and exposure that is measurable and integrable, and a family of factual observed outcomes Yobs stored independently of the potential-outcome schedule that is likewise measurable and integrable.","s":"Type (max (max (max (max (u_1 + 1) (u_2 + 1)) (u_3 + 1)) (u_4 + 1)) (u_5 + 1))","l":43,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.DSigmaAlg","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, the treatment-path σ-algebra is the smallest σ-algebra on the sample space that makes the treatment assigned to every unit in every period measurable.","s":"(P : Causalean.Panel.PanelPOSystem) → MeasurableSpace P.Ω","l":128,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.DSigmaAlg_le","k":"theorem","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"The σ-algebra generated by the treatment path D, σ(D), is a sub-σ-algebra of the ambient σ-algebra on the outcome space Ω.","s":"∀ (P : Causalean.Panel.PanelPOSystem), P.DSigmaAlg ≤ inferInstance","l":137,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.m0","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system and an observed unit-period cell, the baseline conditional mean, evaluated at each sample point, is the conditional expectation of that cell's potential outcome under its baseline exposure given the treatment-path σ-algebra.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":144,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedExposure","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, an observed unit-period cell, and a sample point, the realized exposure is the cell's exposure mapping applied to the treatment path realized at that sample point.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → (hr : r ∈ P.cells.observed) → P.Ω → P.E r hr","l":112,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedY","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, an observed unit-period cell, and a sample point, the factual observed outcome is the outcome stored for that cell and sample point.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":119,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.observedY_eq_potentialOutcome","k":"def","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, an observed unit-period cell, and a sample point, pointwise consistency is the assertion that the factual observed outcome equals the potential outcome at that cell under the exposure realized at that sample point.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → Prop","l":31,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.PanelPOSystem.realizedTau","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, an observed unit-period cell, and a sample point, the realized exposure response is that cell's exposure response at the exposure induced by the treatment path realized at the sample point.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → r ∈ P.cells.observed → P.Ω → ℝ","l":161,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.PanelPOSystem.tau","k":"def","m":"Causalean.Panel.PO.CellPO","a":"Panel/PO","t":1,"d":"For a panel potential-outcomes system, an observed unit-period cell, and an exposure available at that cell, the cell-level exposure response, evaluated at each sample point, is the conditional expectation given the treatment-path σ-algebra of the potential outcome under that exposure minus the potential outcome under the cell's baseline exposure.","s":"(P : Causalean.Panel.PanelPOSystem) → (r : P.I × P.T) → (hr : r ∈ P.cells.observed) → P.E r hr → P.Ω → ℝ","l":151,"f":"Causalean/Panel/PO/CellPO.lean"},{"n":"Causalean.Panel.ResidualizationWitness","k":"structure","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"A residualization witness decomposes a real-valued function V into an in-class projection VH and a square-integrable residual Vtilde such that V equals their sum almost everywhere and the residual is orthogonal in expectation to every member of the nuisance class.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → Causalean.Panel.LinearL2Class μ → (Ω → ℝ) → Type u_1","l":52,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.TreatmentPath","k":"def","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Given a set of units, a set of time periods, and a set of treatment actions, a treatment path assigns one treatment action to every unit-period pair.","s":"Type u_1 → Type u_2 → Type u_3 → Type (max (max u_1 u_2) u_3)","l":22,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.BinaryHistory","k":"def","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Binary finite-memory history with the conventional zero baseline.","s":"{I : Type u_1} → {T₀ : ℕ} → (p : ℕ) → Causalean.Panel.TreatmentPath I (Fin T₀) (Fin 2) → I → Fin T₀ → Fin (p + 1) → Fin 2","l":56,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.BinaryHistory_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":2,"d":"Lag 0 of the binary zero-baseline history is the contemporaneous treatment.","s":"∀ {I : Type u_1} {T₀ : ℕ} (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) (Fin 2)) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.BinaryHistory p D i t 0 = D i t","l":76,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.History","k":"def","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Finite-memory history of length p+1 ending at time t, with the boundary convention \"out-of-range = baseline\" for an explicit baseline treatment value a0 : A. Lag k : Fin (p+1) returns D i ⟨t - k, _⟩ when k ≤ t.val, and a0 otherwise. Periods are indexed 0, …, T₀ - 1. In the binary case (A := Fin 2, baseline 0) this is H_{it}^{(p)} = (D_{it}, D_{i,t-1}, …, D_{i,t-p}) with the convention that treatment before the first period is 0; in one-based notation s = t.val + 1, this is D_{is} = 0 for s ∉ {1, …, T₀}.","s":"{I : Type u_1} → {A : Type u_2} → {T₀ : ℕ} → A → (p : ℕ) → Causalean.Panel.TreatmentPath I (Fin T₀) A → I → Fin T₀ → Fin (p + 1) → A","l":31,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.HistoryDefault","k":"def","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Compatibility wrapper for code that deliberately wants the typeclass-provided default value as the finite-history boundary treatment.","s":"{I : Type u_1} → {A : Type u_2} → {T₀ : ℕ} → [Inhabited A] → (p : ℕ) → Causalean.Panel.TreatmentPath I (Fin T₀) A → I → Fin T₀ → Fin (p + 1) → A","l":49,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.HistoryDefault_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":2,"d":"Lag 0 of the default-boundary history is the contemporaneous treatment.","s":"∀ {I : Type u_1} {A : Type u_2} {T₀ : ℕ} [inst : Inhabited A] (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) A) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.HistoryDefault p D i t 0 = D i t","l":70,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.TreatmentPath.History_zero","k":"theorem","m":"Causalean.Panel.PO.TreatmentPath","a":"Panel/PO","t":1,"d":"Lag 0 of the finite-memory treatment history equals the unit's contemporaneous treatment value.","s":"∀ {I : Type u_1} {A : Type u_2} {T₀ : ℕ} (a0 : A) (p : ℕ) (D : Causalean.Panel.TreatmentPath I (Fin T₀) A) (i : I) (t : Fin T₀), Causalean.Panel.TreatmentPath.History a0 p D i t 0 = D i t","l":62,"f":"Causalean/Panel/PO/TreatmentPath.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.BalancedPanel","k":"structure","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Balanced panel. The side conditions from the source definition that a panel counts as balanced: the unit index type has at least two elements and the time index type has at least two elements.","s":"(Unit : Type u_3) → (Time : Type u_4) → [Fintype Unit] → [Fintype Time] → Prop","l":47,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a set of units, a set of periods, and a real-valued unit-period array, the unit-time additive condition holds precisely when there exist a real-valued unit function and a real-valued period function whose sum equals the array at every unit-period pair.","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":296,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, a real-valued unit-period array, a unit, and a period, the double-demeaned value equals the array value minus its unit mean and period mean plus its grand mean.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → Unit → Time → ℝ","l":96,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Uniform double-demeaning is weighted double-demeaning with uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.UniformTwoWayPanel.ddot V i t = Causalean.Panel.WeightedTwoWayPanel.ddot (Causalean.Panel.UniformTwoWayPanel.uniformWeights (Fintype.card_pos_iff.mpr (Nonempty.intro i))) V i t","l":139,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_time","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal to arbitrary time-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → ∀ (V : Unit → Time → ℝ) (b : Time → ℝ), ∑ i, ∑ t, Causalean.Panel.UniformTwoWayPanel.ddot V i t * b t = 0","l":361,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_unit","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal to arbitrary unit-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → 0 < Fintype.card Time → ∀ (V : Unit → Time → ℝ) (a : Unit → ℝ), ∑ i, ∑ t, Causalean.Panel.UniformTwoWayPanel.ddot V i t * a i = 0","l":333,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_orthogonal_unit_time","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Assume at least one unit and at least one period. Then for any array h of unit/time additive form h_it = a_i + b_t, the double-demeaned array ddot V is orthogonal to h under the unnormalized uniform panel inner product.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time], 0 < Fintype.card Unit → 0 < Fintype.card Time → ∀ (V h : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive h → Causalean.Panel.UniformTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.ddot V) h = 0","l":388,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.ddot_reconstruct","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Reconstruction identity V_it = ddot V_it + unitMean V_i + timeMean V_t - grandMean V.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.UniformTwoWayPanel.ddot V i t + Causalean.Panel.UniformTwoWayPanel.unitMean V i + Causalean.Panel.UniformTwoWayPanel.timeMean V t - Causalean.Panel.UniformTwoWayPanel.grandMean V = V i t","l":311,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, and a residualized regressor and residualized outcome array, the finite residualized coefficient is their unnormalized inner product divided by the regressor's unnormalized self-inner-product.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":192,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.finite_residualized_coefficient_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"Finite scalar Frisch–Waugh–Lovell handoff. Suppose the outcome and regressor decompose as Y = Yproj + Ytilde and D = Dproj + Dtilde, where Dproj and the fitted nuisance term Hβ both satisfy the nuisance predicate H, Dtilde is orthogonal to every array satisfying H, Dtilde is orthogonal to Yproj, and Dtilde has strictly positive self-inner-product (a nondegenerate residualized regressor). If the coefficient β and nuisance fit Hβ satisfy the normal equation against the raw regressor D and the normal equation against its nuisance component Dproj, then β equals the finite residualized coefficient inner Dtilde Ytilde / inner Dtilde Dtilde.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (H : (Unit → Time → ℝ) → Prop) {Y D Yproj Ytilde Dproj Dtilde Hβ : Unit → Time → ℝ} {β : ℝ}, (∀ (i : Unit) (t : Time), Y i t = Yproj i t + Ytilde i t) → (∀ (i : Unit) (t : Time), D i t = Dproj i t + Dtilde i t) → H Dproj → H Hβ → (∀ (h : Unit → Time → ℝ), H h → Causalean.Panel.UniformTwoWayPanel.inner Dtilde h = 0) → Causalean.Panel.UniformTwoWayPanel.inner Dtilde Yproj = 0 → 0 < Causalean.Panel.UniformTwoWayPanel.inner Dtilde Dtilde → (Causalean.Panel.UniformTwoWayPanel.inner D fun i t => Y i t - D i t * β - Hβ i t) = 0 → (Causalean.Panel.UniformTwoWayPanel.inner Dproj fun i t => Y i t - D i t * β - Hβ i t) = 0 → β = Causalean.Panel.UniformTwoWayPanel.finiteResidualizedCoefficient Dtilde Ytilde","l":200,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.grandMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, and a real-valued unit-period array, the grand mean is the arithmetic average of its values over all unit-period pairs.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → ℝ","l":89,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.grandMean_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel with uniform unit weights, the usual grand mean equals the grand mean computed under those weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (V : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.grandMean V = Causalean.Panel.WeightedTwoWayPanel.grandMean (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V","l":117,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.inner","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, and two real-valued unit-period arrays, the unnormalized uniform inner product is the sum, over all unit-period pairs, of the product of their values.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":102,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.inner_eq_card_smul_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"The unnormalized uniform inner product is the unit count times the weighted inner product under uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (V W : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.inner V W = ↑(Fintype.card Unit) * Causalean.Panel.WeightedTwoWayPanel.inner (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V W","l":181,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.sub_ddot_eq_unitTimeProjection","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"Pointwise residual decomposition V - ddot V into the unit/time projection.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ) (i : Unit) (t : Time), V i t - Causalean.Panel.UniformTwoWayPanel.ddot V i t = Causalean.Panel.UniformTwoWayPanel.unitTimeProjection V i t","l":326,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.sum_eq_card_mul_uniform_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel, the unweighted sum across all unit-period cells equals the number of units times the corresponding sum under uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (hU : 0 < Fintype.card Unit) (F : Unit → Time → ℝ), ∑ i, ∑ t, F i t = ↑(Fintype.card Unit) * ∑ i, ∑ t, (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU).p i * F i t","l":150,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.timeMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, a real-valued unit-period array, and a period, the time mean is the arithmetic average of that period's array values over all units.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → (Unit → Time → ℝ) → Time → ℝ","l":83,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.timeMean_eq_weighted","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"In a finite balanced panel with uniformly weighted units, the usual time mean in any period equals the time mean computed under the uniform unit weights.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] (hU : 0 < Fintype.card Unit) (V : Unit → Time → ℝ) (t : Time), Causalean.Panel.UniformTwoWayPanel.timeMean V t = Causalean.Panel.WeightedTwoWayPanel.timeMean (Causalean.Panel.UniformTwoWayPanel.uniformWeights hU) V t","l":109,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.uniformWeights","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units whose cardinality is strictly positive, the uniform unit-weight vector assigns every unit the reciprocal of the number of units.","s":"{Unit : Type u_1} → [inst : Fintype Unit] → 0 < Fintype.card Unit → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit","l":57,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitMean","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, a real-valued unit-period array, and a unit, the unit mean is the arithmetic average of that unit's array values over all periods.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Time] → (Unit → Time → ℝ) → Unit → ℝ","l":77,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitTimeProjection","k":"def","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, a real-valued unit-period array, a unit, and a period, the unit-time projection is that unit's mean plus that period's mean minus the grand mean.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Unit] → [Fintype Time] → (Unit → Time → ℝ) → Unit → Time → ℝ","l":305,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.UniformTwoWayPanel.unitTimeProjection_additive","k":"theorem","m":"Causalean.Panel.UniformTwoWayPanel","a":"Panel","t":2,"d":"The removed component is itself unit/time additive.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (V : Unit → Time → ℝ), Causalean.Panel.UniformTwoWayPanel.IsUnitTimeAdditive (Causalean.Panel.UniformTwoWayPanel.unitTimeProjection V)","l":318,"f":"Causalean/Panel/UniformTwoWayPanel.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"Given an index set, two feature sets, and two maps from the index set to those feature sets, the additive span is the real linear subspace of real-valued arrays on the index set consisting exactly of arrays that can be written as the sum of a real-valued function of the first feature and a real-valued function of the second feature.","s":"{R : Type u_1} → {A : Type u_2} → {B : Type u_3} → (R → A) → (R → B) → Submodule ℝ (R → ℝ)","l":58,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan.const_mem","k":"theorem","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"The constant function equal to c₀ belongs to the additive span generated by two feature maps f₁ and f₂; it is witnessed by taking the f₁-component function constantly c₀ and the f₂-component function constantly 0.","s":"∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} (f₁ : R → A) (f₂ : R → B) (c₀ : ℝ), (fun x => c₀) ∈ Causalean.Panel.Weighted.AdditiveSpan f₁ f₂","l":87,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan.finiteDimensional","k":"instance","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":2,"d":"Under [Finite R] the additive span sits inside the finite-dimensional ambient space R → ℝ, hence is finite-dimensional.","s":"∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {f₁ : R → A} {f₂ : R → B} [Finite R], Module.Finite ℝ ↥(Causalean.Panel.Weighted.AdditiveSpan f₁ f₂)","l":95,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.AdditiveSpan.mem_iff","k":"theorem","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":2,"d":"Membership unfolding for AdditiveSpan.","s":"∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {f₁ : R → A} {f₂ : R → B} {h : R → ℝ}, h ∈ Causalean.Panel.Weighted.AdditiveSpan f₁ f₂ ↔ ∃ a b, ∀ (r : R), h r = a (f₁ r) + b (f₂ r)","l":81,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"Given a set of units, a set of periods, and a real-valued unit-period array, the unit-time additive condition holds precisely when there exist a real-valued unit function and a real-valued period function whose sum equals the array at every unit-period pair.","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":112,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight","k":"def","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":1,"d":"For a finite index set, a field of scalars, a scalar-valued raw weight function, and an index, the normalized finite weight is that index's raw weight divided by the sum of all raw weights.","s":"{ι : Type u_3} → [Fintype ι] → {K : Type u_4} → [Field K] → (ι → K) → ι → K","l":57,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Nonnegativity of normalized weights from nonnegative raw weights and a positive normalizing sum.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] [inst_2 : LinearOrder K] [IsStrictOrderedRing K] (a : ι → K), (∀ (i : ι), 0 ≤ a i) → 0 < ∑ i, a i → ∀ (i : ι), 0 ≤ Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight a i","l":63,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.pairwise_cov_right","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"When finite weights sum to one, the weighted sum of pairwise products of differences equals twice the weighted cross-moment minus twice the product of the weighted means.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : CommRing K] (p x y : ι → K), ∑ i, p i = 1 → ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j) = 2 * (∑ i, p i * x i * y i - (∑ j, p j * x j) * ∑ j, p j * y j)","l":125,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.sum_normalizedWeight_eq_one","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Normalized finite weights sum to one when the normalizing sum is nonzero.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (a : ι → K), ∑ i, a i ≠ 0 → ∑ i, Causalean.Panel.Weighted.NormalizedWeights.normalizedWeight a i = 1","l":72,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_cov","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":1,"d":"For weights p summing to one over a field where 2 is nonzero, the weighted centered covariance of x and y equals half the average, weighted by p ⊗ p, of the pairwise cross-products of their differences: Σᵢ pᵢ (xᵢ − x̄)(yᵢ − ȳ) = 1/2 Σᵢ Σⱼ pᵢpⱼ (xᵢ − xⱼ)(yᵢ − yⱼ).","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (p x y : ι → K), ∑ i, p i = 1 → 2 ≠ 0 → ∑ i, p i * (x i - ∑ j, p j * x j) * (y i - ∑ j, p j * y j) = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) * (y i - y j)","l":158,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_cov_left","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"When finite weights sum to one, the weighted covariance of two centered variables equals their weighted cross-moment minus the product of their weighted means.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : CommRing K] (p x y : ι → K), ∑ i, p i = 1 → ∑ i, p i * (x i - ∑ j, p j * x j) * (y i - ∑ j, p j * y j) = ∑ i, p i * x i * y i - (∑ j, p j * x j) * ∑ j, p j * y j","l":80,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.NormalizedWeights.weighted_center_var","k":"theorem","m":"Causalean.Panel.Weighted.NormalizedWeights","a":"Panel/Weighted","t":2,"d":"Weighted centered variance as half the average pairwise squared gap.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {K : Type u_4} [inst_1 : Field K] (p x : ι → K), ∑ i, p i = 1 → 2 ≠ 0 → ∑ i, p i * (x i - ∑ j, p j * x j) ^ 2 = 1 / 2 * ∑ i, ∑ j, p i * p j * (x i - x j) ^ 2","l":170,"f":"Causalean/Panel/Weighted/NormalizedWeights.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport","k":"structure","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":1,"d":"A finite weighted index together with positive normalized weights: a nonempty observed subset of indices together with a weight function that is strictly positive on every observed index, vanishes off the observed set, and sums to one over the observed indices.","s":"(R : Type u_1) → [Fintype R] → [DecidableEq R] → Type u_1","l":53,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"For a finite record set and a nonnegative integer $K$, a weighted support, a family of $K$ regressors, and a nuisance subspace of real-valued arrays determine the residualized regressor Gram matrix, whose $(j,k)$ entry is the weighted inner product of regressor $j$ and regressor $k$ after both are residualized against the nuisance subspace.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → Matrix (Fin K) (Fin K) ℝ","l":66,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX_apply","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Each entry of the residualized Gram matrix is the weighted inner product of the corresponding residualized regressors.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (j k : Fin K), c.Q_XX X H j k = c.ip (c.tildeX H (X j)) (c.tildeX H (X k))","l":73,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.Q_XX_mulVec_thetaHat","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Under nonsingularity, thetaHat solves the residualized normal equations.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (Y : R → ℝ), IsUnit (c.Q_XX X H).det → (c.Q_XX X H).mulVec (c.thetaHat X H Y) = c.rhsVec X H Y","l":107,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.RankCondition","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"For a finite record set and a nonnegative integer $K$, a weighted support, a nuisance subspace of real-valued arrays, and a family of $K$ regressors satisfy the residualized-regressor rank condition exactly when the determinant of their residualized weighted Gram matrix is nonzero.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (Fin K → R → ℝ) → Prop","l":100,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, a classifier from records to distinguishable cells, and a chosen cell, the cell mass is the sum of the support weights of all observed records classified into that cell.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → [DecidableEq 𝒢] → Causalean.Panel.Weighted.WeightedSupport R → (R → 𝒢) → 𝒢 → ℝ","l":167,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass_def","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The cell mass unfolds to the weighted sum of the corresponding cell indicator over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (g : 𝒢), c.cellMass G g = ∑ r ∈ c.observed, c.weight r * Causalean.Panel.Weighted.cellIndicator G g r","l":174,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMass_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Every classifier cell has nonnegative total weight.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (g : 𝒢), 0 ≤ c.cellMass G g","l":180,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMean","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, an outcome array, a classifier from records to distinguishable cells, a chosen cell, and the condition that this cell has strictly positive mass, the cell mean is the weighted average of the outcome over observed records in that cell.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → [inst_2 : DecidableEq 𝒢] → (c : Causalean.Panel.Weighted.WeightedSupport R) → (R → ℝ) → (G : R → 𝒢) → (g : 𝒢) → 0 < c.cellMass G g → ℝ","l":191,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.cellMean_mul_cellMass","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"On a classifier cell g with strictly positive total weight, multiplying the cell mean of F by the cell's mass recovers the weighted sum of F over observed records in that cell.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} [inst_2 : DecidableEq 𝒢] (c : Causalean.Panel.Weighted.WeightedSupport R) (F : R → ℝ) (G : R → 𝒢) (g : 𝒢) (hpos : 0 < c.cellMass G g), c.cellMean F G g hpos * c.cellMass G g = ∑ r ∈ c.observed, c.weight r * F r * Causalean.Panel.Weighted.cellIndicator G g r","l":202,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.fwl_identity","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"Frisch–Waugh–Lovell at the WeightedSupport level. Assume the residualized-regressor Gram matrix Q_XX is invertible (the rank condition). Then whenever a coefficient vector β together with a nuisance term α ∈ H jointly minimizes the weighted least-squares objective c.ip (Y − ∑ₖ βₖ·Xₖ − α) (Y − ∑ₖ βₖ·Xₖ − α) over all coefficient/nuisance pairs, β must equal the short-regression residualized coefficient thetaHat c X H Y.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : Fin K → R → ℝ) (Y : R → ℝ), c.RankCondition H X → ∀ (β : Fin K → ℝ), ∀ α ∈ H, (∀ (β' : Fin K → ℝ), ∀ α' ∈ H, c.ip (Y - ∑ k, β k • X k - α) (Y - ∑ k, β k • X k - α) ≤ c.ip (Y - ∑ k, β' k • X k - α') (Y - ∑ k, β' k • X k - α')) → β = c.thetaHat X H Y","l":161,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip","k":"def","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, and two real-valued arrays on the records, the weighted inner product is the sum, over observed records, of each support weight times the product of the two arrays at that record.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → (R → ℝ) → (R → ℝ) → ℝ","l":65,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat","k":"def","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"For a finite record set, an index set, a weighted support, and two families of real-valued arrays indexed by that set, the matrix-valued weighted inner product is the matrix whose $(j,k)$ entry is the weighted inner product of array $j$ in the first family and array $k$ in the second family.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {J : Type u_2} → Causalean.Panel.Weighted.WeightedSupport R → (J → R → ℝ) → (J → R → ℝ) → Matrix J J ℝ","l":158,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_add_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the left tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A A' B : J → R → ℝ), c.ipMat (A + A') B = c.ipMat A B + c.ipMat A' B","l":177,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_add_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the right tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B B' : J → R → ℝ), c.ipMat A (B + B') = c.ipMat A B + c.ipMat A B'","l":185,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_apply","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Each entry of the matrix-valued inner product is the scalar weighted inner product of the corresponding two columns.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : J → R → ℝ) (j k : J), c.ipMat A B j k = c.ip (A j) (B k)","l":165,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_smul_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the left tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : J → R → ℝ), c.ipMat (s • A) B = s • c.ipMat A B","l":193,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_smul_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the right tuple argument (entrywise).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : J → R → ℝ), c.ipMat A (s • B) = s • c.ipMat A B","l":201,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ipMat_transpose","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"The transpose of ⟨A, B⟩_ω is ⟨B, A⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : J → R → ℝ), (c.ipMat A B).transpose = c.ipMat B A","l":171,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_add_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the left argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A A' B : R → ℝ), c.ip (A + A') B = c.ip A B + c.ip A' B","l":84,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_add_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Additivity in the right argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B B' : R → ℝ), c.ip A (B + B') = c.ip A B + c.ip A B'","l":93,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_def","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"The weighted inner product unfolds to its finite weighted sum over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : R → ℝ), c.ip A B = ∑ r ∈ c.observed, c.weight r * A r * B r","l":71,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_eq_integral","k":"theorem","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":1,"d":"For a finite probability measure μ, the weighted inner product induced by μ on any two functions A and B equals the L²(μ) integral pairing ∫ A·B dμ.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] [inst_2 : MeasurableSpace R] [inst_3 : MeasurableSingletonClass R] (μ : MeasureTheory.Measure R) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (A B : R → ℝ), (Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure μ).ip A B = ∫ (r : R), A r * B r ∂μ","l":111,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_eq_zero_of_zero_on_observed","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"If h vanishes on every observed index, then c.ip A h = 0 for any A.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A h : R → ℝ), (∀ r ∈ c.observed, h r = 0) → c.ip A h = 0","l":117,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_eq_zero_iff","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":1,"d":"For a weighted-support system c, the weighted self inner product ⟨A, A⟩_ω of an array A vanishes if and only if A is zero at every index in the observed set. Arrays that differ only off c.observed are thus identified by this inner product.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), c.ip A A = 0 ↔ ∀ r ∈ c.observed, A r = 0","l":127,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Positivity: ⟨A, A⟩_ω ≥ 0.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), 0 ≤ c.ip A A","l":121,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_sub_le_of_orth","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"If p ∈ H and the residual X - p is c.ip-orthogonal to all of H, then p minimizes the WLS objective over H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X p : R → ℝ), p ∈ H → (∀ h ∈ H, c.ip (X - p) h = 0) → ∀ {h : R → ℝ}, h ∈ H → c.ip (X - p) (X - p) ≤ c.ip (X - h) (X - h)","l":65,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_self_summand_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Each summand of ⟨A, A⟩_ω is nonnegative.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ), ∀ r ∈ c.observed, 0 ≤ c.weight r * A r * A r","l":112,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_smul_left","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the left argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : R → ℝ), c.ip (s • A) B = s * c.ip A B","l":98,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_smul_right","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Scalar homogeneity in the right argument.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (s : ℝ) (A B : R → ℝ), c.ip A (s • B) = s * c.ip A B","l":107,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sub_smul_expand","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":2,"d":"Quadratic-in-t expansion used in the perturbation argument: ⟨X - p - t • h, X - p - t • h⟩_ω = ⟨X-p, X-p⟩_ω - 2 t · ⟨X-p, h⟩_ω + t² · ⟨h, h⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (X p h : R → ℝ) (t : ℝ), c.ip (X - p - t • h) (X - p - t • h) = c.ip (X - p) (X - p) - 2 * t * c.ip (X - p) h + t ^ 2 * c.ip h h","l":126,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sum_left_finset","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"For a weighted inner product, the inner product of a finite sum of functions with another function equals the corresponding finite sum of inner products.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) {ι : Type u_2} (s : Finset ι) (f : ι → R → ℝ) (B : R → ℝ), c.ip (∑ i ∈ s, f i) B = ∑ i ∈ s, c.ip (f i) B","l":121,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_sum_right","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"The weighted inner product of a function with a finite sum of functions equals the sum of its weighted inner products with the summands.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (A : R → ℝ) (f : Fin K → R → ℝ), c.ip A (∑ k, f k) = ∑ k, c.ip A (f k)","l":137,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_symm","k":"theorem","m":"Causalean.Panel.Weighted.InnerProduct","a":"Panel/Weighted","t":2,"d":"Symmetry of the weighted inner product.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (A B : R → ℝ), c.ip A B = c.ip B A","l":76,"f":"Causalean/Panel/Weighted/InnerProduct.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_tildeX_eq_ip_tildeX_residual","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"A regressor residualized against a control space has the same weighted inner product with any function as with that function after residualizing it against the same control space.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : Fin K → R → ℝ) (A : R → ℝ) (k : Fin K), c.ip (c.tildeX H (X k)) A = c.ip (c.tildeX H (X k)) (c.tildeX H A)","l":146,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ip_tildeX_self","k":"theorem","m":"Causalean.Panel.Weighted.ScalarFWL","a":"Panel/Weighted","t":2,"d":"⟨X̃, X⟩_ω = ⟨X̃, X̃⟩_ω: the residualized regressor sees X and its residual identically, because the projection part lies in H and is orthogonal to the residual.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), c.ip (c.tildeX H X) X = c.ip (c.tildeX H X) (c.tildeX H X)","l":53,"f":"Causalean/Panel/Weighted/ScalarFWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.jointCellMass","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, a first classifier into distinguishable cells, a second classifier into distinguishable cells, a chosen cell of the first classifier, and a chosen cell of the second classifier, the joint cell mass is the sum of the support weights of observed records lying in both chosen cells.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [DecidableEq 𝒢] → [DecidableEq 𝒯] → Causalean.Panel.Weighted.WeightedSupport R → (R → 𝒢) → (R → 𝒯) → 𝒢 → 𝒯 → ℝ","l":213,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.jointCellMass_def","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The joint cell mass unfolds to the weighted sum of the two corresponding cell indicators over observed records.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {𝒢 : Type u_2} {𝒯 : Type u_3} [inst_2 : DecidableEq 𝒢] [inst_3 : DecidableEq 𝒯] (c : Causalean.Panel.Weighted.WeightedSupport R) (G : R → 𝒢) (T : R → 𝒯) (g : 𝒢) (t : 𝒯), c.jointCellMass G T g t = ∑ r ∈ c.observed, c.weight r * Causalean.Panel.Weighted.cellIndicator G g r * Causalean.Panel.Weighted.cellIndicator T t r","l":221,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure","k":"def","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":1,"d":"For a finite measurable record space in which every singleton is measurable and a probability measure, the weighted support induced by that probability measure has as observed records exactly those with strictly positive point mass and assigns each record its point mass as weight.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → [inst_2 : MeasurableSpace R] → [MeasurableSingletonClass R] → (μ : MeasureTheory.Measure R) → [MeasureTheory.IsProbabilityMeasure μ] → Causalean.Panel.Weighted.WeightedSupport R","l":49,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure_weight","k":"theorem","m":"Causalean.Panel.Weighted.OfProbabilityMeasure","a":"Panel/Weighted","t":2,"d":"The induced weighted support assigns each record the point mass of that record under the probability measure.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] [inst_2 : MeasurableSpace R] [inst_3 : MeasurableSingletonClass R] (μ : MeasureTheory.Measure R) [inst_4 : MeasureTheory.IsProbabilityMeasure μ] (r : R), (Causalean.Panel.Weighted.WeightedSupport.ofProbabilityMeasure μ).weight r = (μ {r}).toReal","l":106,"f":"Causalean/Panel/Weighted/OfProbabilityMeasure.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, and a nuisance subspace of real-valued arrays, the weighted orthogonal projection onto that subspace maps every array to an array in the nuisance subspace whose residual is weighted-orthogonal to every array in that subspace.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) →ₗ[ℝ] R → ℝ","l":100,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_apply_eq_of_mem_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Projection uniqueness on observed indices.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {Y : R → ℝ}, Y ∈ H → (∀ h ∈ H, c.ip (X - Y) h = 0) → ∀ r ∈ c.observed, (c.proj H) X r = Y r","l":134,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_apply_of_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"If Y ∈ H then c.proj H Y agrees with Y on every observed index.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) {Y : R → ℝ}, Y ∈ H → ∀ r ∈ c.observed, (c.proj H) Y r = Y r","l":121,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_eq_argmin","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":1,"d":"The weighted orthogonal projection minimizes the WLS objective over H. Given a weighted support on a finite index set and a submodule H of candidate real-valued functions, for any competitor h lying in H, the weighted sum of squared residuals of the projection of a target function onto H is at most the weighted sum of squared residuals of h.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X h : R → ℝ), h ∈ H → c.ip (X - (c.proj H) X) (X - (c.proj H) X) ≤ c.ip (X - h) (X - h)","l":91,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_idem_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Idempotence of the projection on the observed indices.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), ∀ r ∈ c.observed, (c.proj H) ((c.proj H) X) r = (c.proj H) X r","l":165,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The chosen weighted projection always lands in the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), (c.proj H) X ∈ H","l":107,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.proj_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residual from the chosen weighted projection is orthogonal to every element of the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {h : R → ℝ}, h ∈ H → c.ip (X - (c.proj H) X) h = 0","l":112,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, and a nuisance subspace of real-valued arrays, the residual-maker linear map subtracts the weighted orthogonal projection onto the nuisance subspace from each array.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) →ₗ[ℝ] R → ℝ","l":173,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Applying the residual maker subtracts the weighted projection from the original array.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), (c.residualize H) X = X - (c.proj H) X","l":180,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_idem_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Idempotence of M_H on the observed indices: M_H (M_H X) = M_H X on c.observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), ∀ r ∈ c.observed, (c.residualize H) ((c.residualize H) X) r = (c.residualize H) X r","l":219,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_in_orthogonal","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"Key orthogonality. For any array h lying in the nuisance subspace H, the residualized array X̃ = M_H X is orthogonal to h under the weighted inner product c.ip.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ) {h : R → ℝ}, h ∈ H → c.ip (c.tildeX H X) h = 0","l":201,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_orth_iff_argmin","k":"theorem","m":"Causalean.Panel.Weighted.WLS","a":"Panel/Weighted","t":1,"d":"First-order WLS optimality. For a weighted support and a submodule H of candidate functions, fix a candidate function p belonging to H; then the residual X - p being weighted-orthogonal to every element of H is equivalent to p attaining the minimal weighted sum of squared residuals over H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X p : R → ℝ), p ∈ H → ((∀ h ∈ H, c.ip (X - p) h = 0) ↔ ∀ h ∈ H, c.ip (X - p) (X - p) ≤ c.ip (X - h) (X - h))","l":146,"f":"Causalean/Panel/Weighted/WLS.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.residualize_self_of_mem","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"If X ∈ H then X̃ = M_H X vanishes on c.observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) {X : R → ℝ}, X ∈ H → ∀ r ∈ c.observed, c.tildeX H X r = 0","l":210,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.rhsVec","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"For a finite record set and a nonnegative integer $K$, a weighted support, a family of $K$ regressors, a nuisance subspace of real-valued arrays, and an outcome array determine the residualized right-hand-side vector, whose coordinate $j$ is the weighted inner product of regressor $j$ residualized against the nuisance subspace with the outcome.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → (R → ℝ) → Fin K → ℝ","l":80,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.rhsVec_apply","k":"theorem","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":2,"d":"Each entry of the residualized right-hand side is the weighted inner product of a residualized regressor with the outcome.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {K : ℕ} (c : Causalean.Panel.Weighted.WeightedSupport R) (X : Fin K → R → ℝ) (H : Submodule ℝ (R → ℝ)) (Y : R → ℝ) (j : Fin K), c.rhsVec X H Y j = c.ip (c.tildeX H (X j)) Y","l":87,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.scalar_fwl_of_normalEqs","k":"theorem","m":"Causalean.Panel.Weighted.ScalarFWL","a":"Panel/Weighted","t":1,"d":"Scalar FWL from the normal equations. Suppose the nuisance term α lies in H, the H-residualized regressor X̃ has nonzero weighted self-inner-product, and the coefficient β with nuisance term α satisfies the weighted normal equation against the raw regressor X and the weighted normal equation against every element of the nuisance space H. Then β equals the residualized coefficient ratio ⟨X̃, Y⟩_ω / ⟨X̃, X̃⟩_ω.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X Y : R → ℝ) (β : ℝ), ∀ α ∈ H, c.ip (c.tildeX H X) (c.tildeX H X) ≠ 0 → c.ip (Y - β • X - α) X = 0 → (∀ h ∈ H, c.ip (Y - β • X - α) h = 0) → β = c.ip (c.tildeX H X) Y / c.ip (c.tildeX H X) (c.tildeX H X)","l":69,"f":"Causalean/Panel/Weighted/ScalarFWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":1,"d":"For a weighted-support system c, the weights sum to one over the observed index set (restated from the structure's normalization field for convenient reuse).","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r ∈ c.observed, c.weight r = 1","l":102,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight_univ","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"Summing ω over the full type R equals summing over observed, because ω vanishes outside observed.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r, c.weight r = ∑ r ∈ c.observed, c.weight r","l":93,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.sum_weight_univ_eq_one","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"Summing the weights over Finset.univ gives 1.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R), ∑ r, c.weight r = 1","l":108,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.thetaHat","k":"def","m":"Causalean.Panel.Weighted.FWL","a":"Panel/Weighted","t":1,"d":"For a finite record set and a nonnegative integer $K$, a weighted support, a family of $K$ regressors, a nuisance subspace of real-valued arrays, and an outcome array determine the residualized weighted least-squares coefficient vector, obtained by multiplying the inverse of the residualized regressor Gram matrix by the residualized right-hand-side vector.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {K : ℕ} → Causalean.Panel.Weighted.WeightedSupport R → (Fin K → R → ℝ) → Submodule ℝ (R → ℝ) → (R → ℝ) → Fin K → ℝ","l":93,"f":"Causalean/Panel/Weighted/FWL.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeX","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"For a finite record set, a weighted support, a nuisance subspace of real-valued arrays, and a real-valued array, the residualized array is the result of applying the residual maker associated with that nuisance subspace to the array.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (R → ℝ) → R → ℝ","l":187,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeXVec","k":"def","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":1,"d":"For a finite record set, an index set, a weighted support, a nuisance subspace of real-valued arrays, and a family of real-valued arrays, the residualized array family assigns to each member of the family its residual after projection onto the nuisance subspace.","s":"{R : Type u_1} → [inst : Fintype R] → [inst_1 : DecidableEq R] → {J : Type u_2} → Causalean.Panel.Weighted.WeightedSupport R → Submodule ℝ (R → ℝ) → (J → R → ℝ) → J → R → ℝ","l":253,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeXVec_apply","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residualized vector array applies scalar residualization to the chosen column.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] {J : Type u_2} (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : J → R → ℝ) (k : J), c.tildeXVec H X k = c.tildeX H (X k)","l":260,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.tildeX_eq","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"The residualized scalar array is the original array minus its weighted projection onto the nuisance subspace.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)) (X : R → ℝ), c.tildeX H X = X - (c.proj H) X","l":194,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.weight_nonneg","k":"theorem","m":"Causalean.Panel.Weighted.Support","a":"Panel/Weighted","t":2,"d":"The weight function ω is nonnegative everywhere on R.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (r : R), 0 ≤ c.weight r","l":87,"f":"Causalean/Panel/Weighted/Support.lean"},{"n":"Causalean.Panel.Weighted.WeightedSupport.weighted_orthogonal_projection_exists","k":"theorem","m":"Causalean.Panel.Weighted.Subspace","a":"Panel/Weighted","t":2,"d":"Existence of a c.ip-orthogonal projection onto a subspace H.","s":"∀ {R : Type u_1} [inst : Fintype R] [inst_1 : DecidableEq R] (c : Causalean.Panel.Weighted.WeightedSupport R) (H : Submodule ℝ (R → ℝ)), ∃ P, (∀ (X : R → ℝ), P X ∈ H) ∧ ∀ (X h : R → ℝ), h ∈ H → c.ip (X - P X) h = 0","l":59,"f":"Causalean/Panel/Weighted/Subspace.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For a classifier from records to cells and a chosen cell, the cell-indicator array assigns one to each record in the chosen cell and zero to every other record.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [DecidableEq 𝒢] → (R → 𝒢) → 𝒢 → R → ℝ","l":53,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator_apply","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The cell indicator evaluates by checking whether the record belongs to the chosen classifier cell.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : DecidableEq 𝒢] (G : R → 𝒢) (g : 𝒢) (r : R), Causalean.Panel.Weighted.cellIndicator G g r = if G r = g then 1 else 0","l":60,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.cellIndicator_mem_indicatorSpan","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Each singleton cell indicator belongs to the span generated by all cells of the same classifier.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] (G : R → 𝒢) (g : 𝒢), Causalean.Panel.Weighted.cellIndicator G g ∈ Causalean.Panel.Weighted.indicatorSpan G","l":115,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For a finite collection of distinguishable cells and a classifier from records to those cells, the one-axis indicator span is the real vector space of arrays that are weighted sums of the classifier's singleton cell indicators, with one real coefficient for each cell.","s":"{R : Type u_1} → {𝒢 : Type u_2} → [Fintype 𝒢] → [DecidableEq 𝒢] → (R → 𝒢) → Submodule ℝ (R → ℝ)","l":69,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan_le_twoAxisIndicatorSpan","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The first classifier's indicator span is contained in the two-axis indicator span.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] [inst_2 : Fintype 𝒯] [inst_3 : DecidableEq 𝒯] (G : R → 𝒢) (T : R → 𝒯), Causalean.Panel.Weighted.indicatorSpan G ≤ Causalean.Panel.Weighted.twoAxisIndicatorSpan G T","l":147,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.indicatorSpan_le_twoAxisIndicatorSpan_right","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"The second classifier's indicator span is contained in the two-axis indicator span.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} {𝒯 : Type u_3} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] [inst_2 : Fintype 𝒯] [inst_3 : DecidableEq 𝒯] (G : R → 𝒢) (T : R → 𝒯), Causalean.Panel.Weighted.indicatorSpan T ≤ Causalean.Panel.Weighted.twoAxisIndicatorSpan G T","l":154,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.mem_indicatorSpan_iff","k":"theorem","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":2,"d":"Membership in the one-axis indicator span is equivalent to having coefficients on the classifier cells whose weighted sum reproduces the array.","s":"∀ {R : Type u_1} {𝒢 : Type u_2} [inst : Fintype 𝒢] [inst_1 : DecidableEq 𝒢] (G : R → 𝒢) (f : R → ℝ), f ∈ Causalean.Panel.Weighted.indicatorSpan G ↔ ∃ c, ∀ (r : R), f r = ∑ g, c g * Causalean.Panel.Weighted.cellIndicator G g r","l":107,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.Weighted.twoAxisAdditiveSpan","k":"def","m":"Causalean.Panel.Weighted.AdditiveSpan","a":"Panel/Weighted","t":1,"d":"Given a set of units and a set of periods, the two-axis additive span is the real linear subspace of unit-period arrays that can be written as the sum of a unit-specific real-valued function and a period-specific real-valued function.","s":"(I : Type u_1) → (T : Type u_2) → Submodule ℝ (I × T → ℝ)","l":106,"f":"Causalean/Panel/Weighted/AdditiveSpan.lean"},{"n":"Causalean.Panel.Weighted.twoAxisIndicatorSpan","k":"def","m":"Causalean.Panel.Weighted.IndicatorSpan","a":"Panel/Weighted","t":1,"d":"For two finite collections of distinguishable cells, a first classifier and a second classifier from records to their respective cells, the two-axis indicator span is the real vector space generated jointly by the singleton indicators of both classifiers.","s":"{R : Type u_1} → {𝒢 : Type u_2} → {𝒯 : Type u_3} → [Fintype 𝒢] → [DecidableEq 𝒢] → [Fintype 𝒯] → [DecidableEq 𝒯] → (R → 𝒢) → (R → 𝒯) → Submodule ℝ (R → ℝ)","l":139,"f":"Causalean/Panel/Weighted/IndicatorSpan.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a set of units, a set of periods, and a panel array indexed by units and periods, the unit-time additive property holds exactly when there exist a real-valued unit-specific function and a real-valued time-specific function whose sum equals the array at every unit-period pair.","s":"{Unit : Type u_1} → {Time : Type u_2} → (Unit → Time → ℝ) → Prop","l":93,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.UnitWeights","k":"structure","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Unit weights. A weight function p assigning each unit a share that forms a probability vector: every unit's weight is strictly positive and the weights sum to one across units.","s":"(Unit : Type u_3) → [Fintype Unit] → Type u_3","l":53,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For finite, nonempty, distinguishable sets of units and periods and unit weights that are strictly positive and sum to one, the cell-indexed weighted support for the panel treats every unit-period pair as observed and assigns pair $(i,t)$ the weight given by unit $i$'s weight divided by the number of periods.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [inst_1 : Fintype Time] → [inst_2 : DecidableEq Unit] → [inst_3 : DecidableEq Time] → [Nonempty Unit] → [Nonempty Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → Causalean.Panel.Weighted.WeightedSupport (Unit × Time)","l":281,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport_observed","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Every unit-period cell is observed in the cell-support bridge.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit), (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).observed = Finset.univ","l":316,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.cellSupport_weight","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The cell-support bridge assigns each unit-period cell its unit weight divided equally across periods.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (r : Unit × Time), (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).weight r = w.p r.1 / ↑(Fintype.card Time)","l":311,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, unit weights that are strictly positive and sum to one, a panel array indexed by units and periods, a unit, and a period, the double-demeaned value is that array value minus its unit mean and weighted time mean plus its weighted grand mean.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Unit → Time → ℝ","l":81,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_eq_residualize","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Concrete ↔ abstract bridge. For unit weights w and an outcome array V, the closed-form two-way double-demeaned residual ddot w V i t equals the generic weighted-support residual against the two-axis additive span, evaluated at cell (i, t). Every cell is observed, so the identity holds pointwise.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), ((Causalean.Panel.WeightedTwoWayPanel.cellSupport w).residualize (Causalean.Panel.Weighted.twoAxisAdditiveSpan Unit Time)) (fun r => V r.1 r.2) (i, t) = Causalean.Panel.WeightedTwoWayPanel.ddot w V i t","l":355,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_time","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal (in the p-weighted inner product) to arbitrary time-only functions: ∑_i ∑_t p_i · ddot V_{it} · b_t = 0. Per-period the p-weighted unit sum of ddot vanishes (∑ p = 1 cancels the time mean against the grand mean).","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (b : Time → ℝ), ∑ i, ∑ t, w.p i * (Causalean.Panel.WeightedTwoWayPanel.ddot w V i t * b t) = 0","l":198,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_unit","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Double-demeaned arrays are orthogonal (in the p-weighted inner product) to arbitrary unit-only functions.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (a : Unit → ℝ), ∑ i, ∑ t, w.p i * (Causalean.Panel.WeightedTwoWayPanel.ddot w V i t * a i) = 0","l":184,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_orthogonal_unit_time","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"Double-demeaned arrays are orthogonal to every unit/time additive nuisance function. For a probability-weighted panel of units and periods, any array that decomposes as the sum of a unit-specific term and a time-specific term is orthogonal, under the p-weighted inner product, to the double-demeaned residual of any panel array V: their weighted inner product is zero.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V h : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive h → Causalean.Panel.WeightedTwoWayPanel.inner w (Causalean.Panel.WeightedTwoWayPanel.ddot w V) h = 0","l":241,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.ddot_reconstruct","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Reconstruction identity V_it = ddot V_it + unitMean + timeMean - grandMean.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), Causalean.Panel.WeightedTwoWayPanel.ddot w V i t + Causalean.Panel.WeightedTwoWayPanel.unitMean V i + Causalean.Panel.WeightedTwoWayPanel.timeMean w V t - Causalean.Panel.WeightedTwoWayPanel.grandMean w V = V i t","l":106,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.grandMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, unit weights that are strictly positive and sum to one, and a panel array indexed by units and periods, the weighted grand mean is the weighted average across units of their arithmetic means over periods.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → ℝ","l":75,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.inner","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, unit weights that are strictly positive and sum to one, and two panel arrays, the weighted panel inner product is the sum over every unit and period of the unit weight times the product of the arrays; no uniform-period normalizing factor is included.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → (Unit → Time → ℝ) → ℝ","l":87,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.inner_eq_card_smul_ip","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The p-weighted panel inner product is |Time| times the generic weighted-support inner product on the cell support.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] [inst_2 : DecidableEq Unit] [inst_3 : DecidableEq Time] [inst_4 : Nonempty Unit] [inst_5 : Nonempty Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V W : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.inner w V W = ↑(Fintype.card Time) * (Causalean.Panel.WeightedTwoWayPanel.cellSupport w).ip (fun r => V r.1 r.2) fun r => W r.1 r.2","l":336,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.mem_twoAxisAdditiveSpan_iff","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Membership in the two-axis additive span is exactly the panel's IsUnitTimeAdditive predicate after uncurrying.","s":"∀ {Unit : Type u_1} {Time : Type u_2} {h : Unit × Time → ℝ}, h ∈ Causalean.Panel.Weighted.twoAxisAdditiveSpan Unit Time ↔ Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive fun i t => h (i, t)","l":322,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.sub_ddot_eq_unitTimeProjection","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"Pointwise residual decomposition V - ddot V into the unit/time projection.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ) (i : Unit) (t : Time), V i t - Causalean.Panel.WeightedTwoWayPanel.ddot w V i t = Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection w V i t","l":120,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.timeMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a set of periods, unit weights that are strictly positive and sum to one, a panel array indexed by units and periods, and a period, the weighted time mean is the weighted average across units of the array at that period.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Time → ℝ","l":69,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitMean","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a set of units, a finite set of periods, a panel array indexed by units and periods, and a unit, the unit mean is the arithmetic average of that unit's values over all periods.","s":"{Unit : Type u_1} → {Time : Type u_2} → [Fintype Time] → (Unit → Time → ℝ) → Unit → ℝ","l":63,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection","k":"def","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":1,"d":"For a finite set of units, a finite set of periods, unit weights that are strictly positive and sum to one, a panel array indexed by units and periods, a unit, and a period, the unit-time component removed by double demeaning is the unit mean plus the weighted time mean minus the weighted grand mean.","s":"{Unit : Type u_1} → {Time : Type u_2} → [inst : Fintype Unit] → [Fintype Time] → Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit → (Unit → Time → ℝ) → Unit → Time → ℝ","l":99,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection_additive","k":"theorem","m":"Causalean.Panel.WeightedTwoWayPanel","a":"Panel","t":2,"d":"The removed component is itself unit/time additive.","s":"∀ {Unit : Type u_1} {Time : Type u_2} [inst : Fintype Unit] [inst_1 : Fintype Time] (w : Causalean.Panel.WeightedTwoWayPanel.UnitWeights Unit) (V : Unit → Time → ℝ), Causalean.Panel.WeightedTwoWayPanel.IsUnitTimeAdditive (Causalean.Panel.WeightedTwoWayPanel.unitTimeProjection w V)","l":112,"f":"Causalean/Panel/WeightedTwoWayPanel.lean"},{"n":"Causalean.Panel.panelConsistency_holds","k":"theorem","m":"Causalean.Panel.PO.Consistency","a":"Panel/PO","t":1,"d":"If panel consistency holds for the panel potential-outcome system P, then the factual observed outcome equals the potential outcome evaluated at the realized exposure, for every observed unit-period cell and every sample point.","s":"∀ (P : Causalean.Panel.PanelPOSystem), Causalean.Panel.PanelConsistency P → ∀ (r : P.I × P.T) (hr : r ∈ P.cells.observed) (ω : P.Ω), P.observedY r hr ω = P.Y r hr (P.observedExposure r hr ω) ω","l":48,"f":"Causalean/Panel/PO/Consistency.lean"},{"n":"Causalean.Panel.residualizedCoefficient","k":"def","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"On a measurable sample space equipped with a measure, let a linear class of square-integrable real-valued functions be given. Given a residualization witness for a real-valued outcome and a residualization witness for a real-valued treatment—each supplying an in-class component and an orthogonal, square-integrable residual—the residualized population coefficient is the integral of the product of the treatment and outcome residuals divided by the integral of the squared treatment residual.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → (H : Causalean.Panel.LinearL2Class μ) → {Y D : Ω → ℝ} → Causalean.Panel.ResidualizationWitness μ H Y → Causalean.Panel.ResidualizationWitness μ H D → ℝ","l":78,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.Panel.residualizedCoefficient_eq_of_normalEqs","k":"theorem","m":"Causalean.Panel.Analysis.Residualization","a":"Panel/Analysis","t":1,"d":"Population Frisch-Waugh-Lovell theorem (prop:po-estimand-fwl). Let wY and wD be residualization witnesses decomposing outcome Y and treatment D each into a component of a common linear square-integrable nuisance class H plus an orthogonal residual. If the second moment of the treatment residual is strictly positive, and a nuisance-class element Hβ together with a coefficient β are such that the combined residual Y − β·D − Hβ is square-integrable, orthogonal in expectation to D, and orthogonal in expectation to every member of H, then β equals the ratio of the covariance-like integral of the outcome and treatment residuals Ỹ and D̃ to the second moment of D̃, i.e. the residualized coefficient residualizedCoefficient μ H wY wD.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (H : Causalean.Panel.LinearL2Class μ) {Y D : Ω → ℝ} (wY : Causalean.Panel.ResidualizationWitness μ H Y) (wD : Causalean.Panel.ResidualizationWitness μ H D), 0 < ∫ (ω : Ω), wD.Vtilde ω * wD.Vtilde ω ∂μ → ∀ (β : ℝ) (Hβ : Ω → ℝ), H.mem Hβ → MeasureTheory.MemLp (fun ω => Y ω - β * D ω - Hβ ω) 2 μ → ∫ (ω : Ω), (Y ω - β * D ω - Hβ ω) * D ω ∂μ = 0 → (∀ ⦃h : Ω → ℝ⦄, H.mem h → ∫ (ω : Ω), (Y ω - β * D ω - Hβ ω) * h ω ∂μ = 0) → β = Causalean.Panel.residualizedCoefficient μ H wY wD","l":92,"f":"Causalean/Panel/Analysis/Residualization.lean"},{"n":"Causalean.PartialID.ConicProgram","k":"structure","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Primal data of a conic linear program over real Hilbert spaces: minimize ⟪c, x⟫ subject to A x = b and x ∈ K.","s":"(E : Type u_3) → (F : Type u_4) → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [CompleteSpace E] → [inst : NormedAddCommGroup F] → [InnerProductSpace ℝ F] → [CompleteSpace F] → Type (max u_3 u_4)","l":44,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.DualFeasible","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"In complete real inner-product decision and constraint spaces, for a conic program and a candidate dual multiplier, the dual-feasibility predicate holds precisely when the program's objective direction minus the adjoint constraint operator applied to that multiplier belongs to the dual of the constraint cone.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → F → Prop","l":70,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.PrimalFeasible","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"In complete real inner-product decision and constraint spaces, for a conic program and a candidate decision vector, the primal-feasibility predicate holds precisely when the vector satisfies the program's linear equality constraint and belongs to its constraint cone.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → E → Prop","l":62,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.augmentedImage","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"In complete real inner-product decision and constraint spaces, for a conic program, the augmented image is the set of pairs consisting of the constraint-operator image and objective inner product of each decision vector in its constraint cone.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → Set (F × ℝ)","l":137,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.dualValue","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"In complete real inner-product decision and constraint spaces, for a conic program, its dual optimal value is the supremum of the inner products between its right-hand side and all dual-feasible multipliers.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → ℝ","l":87,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.dualValue_le_primalValue","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":2,"d":"Weak duality (value form). When both programs are feasible, dualValue ≤ primalValue. (Boundedness is not needed: the pointwise bound exhibits primalValue as an explicit upper bound for the dual values.)","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → {y | P.DualFeasible y}.Nonempty → P.dualValue ≤ P.primalValue","l":109,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.farkas","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Farkas alternative / feasibility engine. For a target point b, b lies in the closed image cone A(K) if and only if every dual direction y whose pullback Aᵀ y lies in the dual cone of K pairs nonnegatively with b.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F) {b : F}, b ∈ ProperCone.map P.A P.K ↔ ∀ (y : F), (ContinuousLinearMap.adjoint P.A) y ∈ ProperCone.innerDual ↑P.K → 0 ≤ inner ℝ b y","l":122,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.primalValue","k":"def","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"In complete real inner-product decision and constraint spaces, for a conic program, its primal optimal value is the infimum of the objective inner products over all primal-feasible decision vectors.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : CompleteSpace E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : CompleteSpace F] → Causalean.PartialID.ConicProgram E F → ℝ","l":80,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.strong_duality_primal_attained","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Strong duality I — primal attainment (closedness CQ). For a conic program that is primal feasible and whose feasible objective values are bounded below, if the augmented image cone {(Ax, ⟪c,x⟫) : x ∈ K} is closed — the constraint qualification separating attained optima from mere infima — then the primal optimum is attained: some primal-feasible point x achieves the objective value ⟪c,x⟫ = primalValue exactly. This is the \"there is an extremal data-generating distribution\" half of sharpness.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → BddBelow ((fun x => inner ℝ P.c x) '' {x | P.PrimalFeasible x}) → IsClosed P.augmentedImage → ∃ x, P.PrimalFeasible x ∧ inner ℝ P.c x = P.primalValue","l":147,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.strong_duality_zero_gap","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":1,"d":"Strong duality II — zero gap (closedness CQ). For a conic program that is primal feasible and whose feasible objective values are bounded below, if the augmented image cone is closed — the same constraint qualification as primal attainment — then there is no duality gap: the primal optimal value equals the dual optimal value, primalValue = dualValue.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F), {x | P.PrimalFeasible x}.Nonempty → BddBelow ((fun x => inner ℝ P.c x) '' {x | P.PrimalFeasible x}) → IsClosed P.augmentedImage → P.primalValue = P.dualValue","l":177,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.ConicProgram.weak_duality","k":"theorem","m":"Causalean.PO.ID.Partial.LP.ConicDuality","a":"PO/ID/Partial/LP","t":2,"d":"Weak duality (pointwise). Any dual-feasible objective value lower-bounds any primal-feasible objective value.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] (P : Causalean.PartialID.ConicProgram E F) {x : E} {y : F}, P.PrimalFeasible x → P.DualFeasible y → inner ℝ P.b y ≤ inner ℝ P.c x","l":94,"f":"Causalean/PO/ID/Partial/LP/ConicDuality.lean"},{"n":"Causalean.PartialID.CriterionSet.LinearMinorant","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"For a parameter space with a pseudometric, a criterion function, and a real modulus, the linear-minorant condition holds precisely when, for every parameter value, the modulus times its distance from the criterion's zero set is no greater than its criterion value.","s":"{Θ : Type u_1} → [PseudoMetricSpace Θ] → (Θ → ℝ) → ℝ → Prop","l":99,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_eq_zero_of_subset","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"If A ⊆ B then the directed Hausdorff distance from A to B is 0.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {A B : Set Θ}, A ⊆ B → Causalean.PartialID.RandomSet.directedHausdorff A B = 0","l":89,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_le_of_forall","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"If every point of A is within M of B (and M ≥ 0), the directed Hausdorff distance from A to B is at most M.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {A B : Set Θ} {M : ℝ}, 0 ≤ M → (∀ a ∈ A, Metric.infDist a B ≤ M) → Causalean.PartialID.RandomSet.directedHausdorff A B ≤ M","l":83,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.directedHausdorff_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The directed Hausdorff distance is nonnegative (a sup of nonnegative point-to-set distances).","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] (A B : Set Θ), 0 ≤ Causalean.PartialID.RandomSet.directedHausdorff A B","l":74,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.hausdorffDist_levelSet_le","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"Deterministic CHT rate bound. Fix a positive linear-minorant modulus δ, a nonnegative sup-norm error bound ε, and a cutoff c at least ε. If the population criterion Q satisfies a linear minorant of modulus δ relative to its identified set — δ times the distance to the identified set never exceeds Q, and the sample criterion Qn is within ε of Q in sup norm at every point, then the level-set estimator {Qn ≤ c} is within Hausdorff distance (c + ε) / δ of the population identified set {Q = 0}:","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {Q Qn : Θ → ℝ} {δ c ε : ℝ}, 0 < δ → 0 ≤ ε → ε ≤ c → Causalean.PartialID.CriterionSet.LinearMinorant Q δ → (∀ (θ : Θ), |Qn θ - Q θ| ≤ ε) → Causalean.PartialID.RandomSet.hausdorffDist (Causalean.PartialID.CriterionSet.levelSet Qn c) (Causalean.PartialID.CriterionSet.identifiedSet Q) ≤ (c + ε) / δ","l":111,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.hausdorffDist_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The symmetric Hausdorff distance is nonnegative.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] (A B : Set Θ), 0 ≤ Causalean.PartialID.RandomSet.hausdorffDist A B","l":79,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"For a parameter space and a criterion function, its identified set is the set of all parameter values at which the criterion equals zero.","s":"{Θ : Type u_1} → (Θ → ℝ) → Set Θ","l":23,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet_eq_levelSet_zero_of_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"For a nonnegative criterion, the identified set is the level-0 set of the same criterion (since Q θ = 0 ↔ Q θ ≤ 0 under nonnegativity).","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ}, (∀ (θ : Θ), 0 ≤ Q θ) → Causalean.PartialID.CriterionSet.identifiedSet Q = Causalean.PartialID.CriterionSet.levelSet Q 0","l":58,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.identifiedSet_subset_levelSet_self","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The identified set sits inside any nonnegative-cutoff level set of the same criterion.","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ} {c : ℝ}, 0 ≤ c → Causalean.PartialID.CriterionSet.identifiedSet Q ⊆ Causalean.PartialID.CriterionSet.levelSet Q c","l":66,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.levelSet","k":"def","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"For a parameter space, a sample criterion function, and a real cutoff, the level-set estimator is the set of all parameter values whose criterion value is at most that cutoff.","s":"{Θ : Type u_1} → (Θ → ℝ) → ℝ → Set Θ","l":33,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.levelSet_mono","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"The level set is monotone in the cutoff c.","s":"∀ {Θ : Type u_1} {Qn : Θ → ℝ} {c₁ c₂ : ℝ}, c₁ ≤ c₂ → Causalean.PartialID.CriterionSet.levelSet Qn c₁ ⊆ Causalean.PartialID.CriterionSet.levelSet Qn c₂","l":54,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.mem_identifiedSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":1,"d":"For a criterion function Q and a candidate parameter θ, θ belongs to the identified set of Q if and only if Q vanishes at θ.","s":"∀ {Θ : Type u_1} {Q : Θ → ℝ} {θ : Θ}, θ ∈ Causalean.PartialID.CriterionSet.identifiedSet Q ↔ Q θ = 0","l":44,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.mem_levelSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Basic","a":"PO/ID/Partial/CriterionSet","t":2,"d":"Membership in a level set is the same as having criterion value below the cutoff.","s":"∀ {Θ : Type u_1} {Qn : Θ → ℝ} {c : ℝ} {θ : Θ}, θ ∈ Causalean.PartialID.CriterionSet.levelSet Qn c ↔ Qn θ ≤ c","l":50,"f":"Causalean/PO/ID/Partial/CriterionSet/Basic.lean"},{"n":"Causalean.PartialID.CriterionSet.tendsto_hausdorffDist_levelSet","k":"theorem","m":"Causalean.PO.ID.Partial.CriterionSet.Consistency","a":"PO/ID/Partial/CriterionSet","t":1,"d":"Hausdorff consistency of the criterion-set estimator. Fix a positive linear-minorant modulus δ such that the population criterion Q satisfies a linear minorant of modulus δ. Suppose each sup-norm error tolerance is nonnegative, each cutoff is at least the corresponding error tolerance, the sample criterion Qn n is within tolerance ε n of Q in sup norm at every sample size n, the cutoffs tend to zero, and the error tolerances tend to zero. Then the Hausdorff distance between the level-set estimator {Qn n ≤ c n} and the population identified set {Q = 0} tends to zero as the sample size grows: H(levelSet Qnₙ cₙ, identifiedSet Q) → 0.","s":"∀ {Θ : Type u_1} [inst : PseudoMetricSpace Θ] {Q : Θ → ℝ} {Qn : ℕ → Θ → ℝ} {δ : ℝ} {c ε : ℕ → ℝ}, 0 < δ → Causalean.PartialID.CriterionSet.LinearMinorant Q δ → (∀ (n : ℕ), 0 ≤ ε n) → (∀ (n : ℕ), ε n ≤ c n) → (∀ (n : ℕ) (θ : Θ), |Qn n θ - Q θ| ≤ ε n) → Filter.Tendsto c Filter.atTop (nhds 0) → Filter.Tendsto ε Filter.atTop (nhds 0) → Filter.Tendsto (fun n => Causalean.PartialID.RandomSet.hausdorffDist (Causalean.PartialID.CriterionSet.levelSet (Qn n) (c n)) (Causalean.PartialID.CriterionSet.identifiedSet Q)) Filter.atTop (nhds 0)","l":147,"f":"Causalean/PO/ID/Partial/CriterionSet/Consistency.lean"},{"n":"Causalean.PartialID.IdentifiedInterval","k":"def","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"For a parameter space, an objective function, and a feasibility condition on its parameter values, the sharp identified interval is the set of all objective values attained by feasible parameters.","s":"{α : Type u_1} → (α → ℝ) → (α → Prop) → Set ℝ","l":25,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.Inference.RandomCoversIcc","k":"def","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"For a sample space, a lower endpoint function, an upper endpoint function, a real lower bound, and a real upper bound, the identified-set coverage event is the set of sample outcomes at which the random interval with those endpoints contains the entire closed interval from the lower to the upper bound.","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → ℝ → ℝ → Set Ω","l":93,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.RandomCoversPoint","k":"def","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"For a sample space, a lower endpoint function, an upper endpoint function, and a fixed real parameter value, the parameter-coverage event is the set of sample outcomes at which the random interval with those endpoints contains that value.","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → ℝ → Set Ω","l":78,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.bernstein_honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the parameter (Bernstein). Variance-adaptive version: with measurable and integrable bounding statistics f_L, f_U, deviation bounds |f_L − ∫f_L| ≤ cL and |f_U − ∫f_U| ≤ cU holding a.s. for nonnegative constants cL, cU, and variance proxies σL, σU > 0 dominating the respective second central moments, for a positive sample size n and failure probabilities δ_L, δ_U in (0, 1], if the true parameter θ₀ lies in the population sandwich [∫ f_L dP, ∫ f_U dP], then widening the sample means by the corresponding Bernstein half-widths yields a random interval covering θ₀ with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → MeasureTheory.Integrable fL P → MeasureTheory.Integrable fU P → ∀ {cL σL cU σU : ℝ}, 0 ≤ cL → 0 < σL → 0 ≤ cU → 0 < σU → (∀ᵐ (x : X') ∂P, |fL x - ∫ (y : X'), fL y ∂P| ≤ cL) → (∀ᵐ (x : X') ∂P, |fU x - ∫ (y : X'), fU y ∂P| ≤ cU) → ∫ (x : X'), (fL x - ∫ (y : X'), fL y ∂P) ^ 2 ∂P ≤ σL ^ 2 → ∫ (x : X'), (fU x - ∫ (y : X'), fU y ∂P) ^ 2 ∂P ≤ σU ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → ∀ {θ₀ : ℝ}, θ₀ ∈ Set.Icc (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P) → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.bernsteinCIHalfWidth cL σL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.bernsteinCIHalfWidth cU σU n δU) θ₀)","l":136,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.farEnd_vanishes_of_tendsto_atBot","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":2,"d":"Far-endpoint failure vanishes (the Δ > 0 regime). If the normalized upper-endpoint statistic Sᵤ ⇒ N(0,1) and the (signed) threshold drifts to −∞, then the probability that Sᵤ,ₙ falls below it tends to 0. Concretely tₙ = −(c + bₙ) with bₙ → ∞ (the positive-width regime); the proof bounds the moving tail by a fixed Gaussian tail N(0,1)((-∞,-R]) made arbitrarily small via finite_measure_halfline_tails_small, then transported by portmanteau.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Su : ℕ → Ω → ℝ} (hSu : ∀ (n : ℕ), Measurable (Su n)), (Causalean.Stat.Tendsto_dist Su (Causalean.Stat.gaussianMeasure 0 1) μ fun n => Measurable.aemeasurable (hSu n)) → ∀ {t : ℕ → ℝ}, Filter.Tendsto t Filter.atTop Filter.atBot → Filter.Tendsto (fun n => (μ {ω | Su n ω < t n}).toReal) Filter.atTop (nhds 0)","l":106,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.gaussianMeasure_zero_one_frontier_Iic","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":2,"d":"The standard normal gives zero mass to the boundary {c} of the closed half-line (-∞, c]; the continuity-point fact needed to apply portmanteau to a one-sided threshold.","s":"∀ (c : ℝ), (Causalean.Stat.gaussianMeasure 0 1) (frontier (Set.Iic c)) = 0","l":98,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.hoeffding_honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the parameter (Hoeffding). The parameter-coverage corollary of hoeffding_honest_ci_set_cover: under the same setup — measurable bounding statistics f_L, f_U a.s. confined to [aL, bL] and [aU, bU] respectively, a positive sample size n, and failure probabilities δ_L, δ_U in (0, 1] — if the true scalar parameter θ₀ lies in the population sandwich [∫ f_L dP, ∫ f_U dP], then the same widened random interval covers θ₀ with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → ∀ {aL bL aU bU : ℝ}, aL < bL → aU < bU → (∀ᵐ (x : X') ∂P, fL x ∈ Set.Icc aL bL) → (∀ᵐ (x : X') ∂P, fU x ∈ Set.Icc aU bU) → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → ∀ {θ₀ : ℝ}, θ₀ ∈ Set.Icc (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P) → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.hoeffdingCIHalfWidth aL bL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.hoeffdingCIHalfWidth aU bU n δU) θ₀)","l":104,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.hoeffding_honest_ci_set_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.IntervalCI","a":"PO/ID/Partial/Inference","t":1,"d":"Finite-sample honest CI for the identified set (Hoeffding). Given lower and upper bounding statistics f_L, f_U that are measurable, each confined a.s. to a known bounded range — f_L ∈ [aL, bL] with aL < bL, f_U ∈ [aU, bU] with aU < bU — at any positive sample size n and any pair of failure probabilities δ_L, δ_U in (0, 1], widening the sample means X̄ₙ(f_L) and X̄ₙ(f_U) outward by the matching Hoeffding half-widths produces a random interval that covers the whole identified interval [∫ f_L dP, ∫ f_U dP] with probability at least 1 − δ_L − δ_U.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {fL fU : X' → ℝ}, Measurable fL → Measurable fU → ∀ {aL bL aU bU : ℝ}, aL < bL → aU < bU → (∀ᵐ (x : X') ∂P, fL x ∈ Set.Icc aL bL) → (∀ᵐ (x : X') ∂P, fU x ∈ Set.Icc aU bU) → ∀ (n : ℕ), 0 < n → ∀ {δL δU : ℝ}, 0 < δL → δL ≤ 1 → 0 < δU → δU ≤ 1 → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => S.sampleMean fL n ω - Causalean.Stat.Concentration.hoeffdingCIHalfWidth aL bL n δL) (fun ω => S.sampleMean fU n ω + Causalean.Stat.Concentration.hoeffdingCIHalfWidth aU bU n δU) (∫ (x : X'), fL x ∂P) (∫ (x : X'), fU x ∂P))","l":71,"f":"Causalean/PO/ID/Partial/Inference/IntervalCI.lean"},{"n":"Causalean.PartialID.Inference.honest_ci_point_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Abstract honest confidence interval (parameter coverage). Let lo, hi be random lower/upper endpoint estimators of a population interval [L, U], with lo measurable and hi measurable. Suppose the true value θ lies in [L, U], the probability that the widened lower endpoint lo − wL overshoots L is at most δL, and the probability that the widened upper endpoint hi + wU undershoots U is at most δU. Then the widened random interval [lo − wL, hi + wU] covers θ with probability at least 1 − δL − δU.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {lo hi : Ω → ℝ}, Measurable lo → Measurable hi → ∀ {L U wL wU δL δU θ : ℝ}, θ ∈ Set.Icc L U → μ.real {ω | L < lo ω - wL} ≤ δL → μ.real {ω | hi ω + wU < U} ≤ δU → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversPoint (fun ω => lo ω - wL) (fun ω => hi ω + wU) θ)","l":198,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.honest_ci_set_cover","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":1,"d":"Abstract honest confidence interval (set coverage). Let lo, hi be the (random) lower/upper endpoint estimators of the population bounds L, U, with lo measurable and hi measurable. Suppose the probability that the widened lower endpoint lo − wL overshoots L is at most δL, and the probability that the widened upper endpoint hi + wU undershoots U is at most δU. Then the widened random interval [lo − wL, hi + wU] covers the entire identified set [L, U] with probability at least 1 − δL − δU.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {lo hi : Ω → ℝ}, Measurable lo → Measurable hi → ∀ {L U wL wU δL δU : ℝ}, μ.real {ω | L < lo ω - wL} ≤ δL → μ.real {ω | hi ω + wU < U} ≤ δU → 1 - δL - δU ≤ μ.real (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => lo ω - wL) (fun ω => hi ω + wU) L U)","l":155,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.imbensManski_pointwise_coverage","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.ImbensManski","a":"PO/ID/Partial/Inference","t":1,"d":"Lower-protected Imbens-Manski pointwise coverage. Let Sl, Su be the normalized lower- and upper-endpoint statistics for an interval-identified scalar parameter, with Sl n measurable for every sample size n, Su n measurable for every sample size n, and Sl converging in distribution to the standard normal law. Fix a one-sided critical value c. Suppose the normalized lower offset aL n, of the target point inside the identified interval, is nonnegative for every n, and the far-endpoint failure probability — that Su n falls below -(c + bU n) — tends to zero as n → ∞. Then for any positive tolerance ε, eventually, as n → ∞, the probability that the Imbens-Manski interval covers the target point is at least Φ(c) − ε, where Φ is the standard-normal distribution function.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Sl Su : ℕ → Ω → ℝ} (hSl : ∀ (n : ℕ), Measurable (Sl n)), (∀ (n : ℕ), Measurable (Su n)) → (Causalean.Stat.Tendsto_dist Sl (Causalean.Stat.gaussianMeasure 0 1) μ fun n => Measurable.aemeasurable (hSl n)) → ∀ (c : ℝ) {aL bU : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ aL n) → Filter.Tendsto (fun n => (μ {ω | Su n ω < -(c + bU n)}).toReal) Filter.atTop (nhds 0) → ∀ {ε : ℝ}, 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ((Causalean.Stat.gaussianMeasure 0 1) (Set.Iic c)).toReal - ε ≤ (μ {ω | Sl n ω ≤ c + aL n ∧ -(c + bU n) ≤ Su n ω}).toReal","l":171,"f":"Causalean/PO/ID/Partial/Inference/ImbensManski.lean"},{"n":"Causalean.PartialID.Inference.lowerOvershoot_subset_absMiss","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"A lower endpoint overshoot after widening implies the corresponding two-sided absolute-deviation miss.","s":"∀ {Ω : Type u_1} {lo : Ω → ℝ} {L wL : ℝ}, {ω | L < lo ω - wL} ⊆ {ω | wL ≤ |lo ω - L|}","l":134,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.randomCoversIcc_compl_subset","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"The complement of the set-coverage event is contained in the union of the two one-sided endpoint-failure events: lower overshoot after widening or upper undershoot after widening.","s":"∀ {Ω : Type u_1} {lo hi : Ω → ℝ} {L U wL wU : ℝ}, (Causalean.PartialID.Inference.RandomCoversIcc (fun ω => lo ω - wL) (fun ω => hi ω + wU) L U)ᶜ ⊆ {ω | L < lo ω - wL} ∪ {ω | hi ω + wU < U}","l":119,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.randomCoversIcc_subset_randomCoversPoint","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"Set coverage ⟹ point coverage (the easy half of Imbens–Manski 2004, Lemma 1). If the true value θ lies in the identified set [L, U], then every sample realization whose random interval covers all of [L, U] also covers θ. Consequently a confidence interval that is honest for the *set* is automatically honest for the *parameter* — which is why parameter-coverage intervals can be no larger, and generically strictly smaller.","s":"∀ {Ω : Type u_1} (A B : Ω → ℝ) {L U θ : ℝ}, θ ∈ Set.Icc L U → Causalean.PartialID.Inference.RandomCoversIcc A B L U ⊆ Causalean.PartialID.Inference.RandomCoversPoint A B θ","l":104,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.Inference.upperUndershoot_subset_absMiss","k":"theorem","m":"Causalean.PO.ID.Partial.Inference.Basic","a":"PO/ID/Partial/Inference","t":2,"d":"An upper endpoint undershoot after widening implies the corresponding two-sided absolute-deviation miss.","s":"∀ {Ω : Type u_1} {hi : Ω → ℝ} {U wU : ℝ}, {ω | hi ω + wU < U} ⊆ {ω | wU ≤ |hi ω - U|}","l":144,"f":"Causalean/PO/ID/Partial/Inference/Basic.lean"},{"n":"Causalean.PartialID.RandomSet.IsBody","k":"structure","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"A body: a nonempty, compact, convex subset of E — the value type 𝒦ₖ(E) of a Beresteanu–Molinari set-valued random variable. Compactness makes supportFn C d well-defined (the linear functional attains its sup); convexity is what lets the support function characterise the set.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → Prop","l":63,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.IsBody.bddAbove","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"The support function of a body is bounded above, hence meaningful.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E}, Causalean.PartialID.RandomSet.IsBody C → ∀ (d : E), BddAbove ((fun x => inner ℝ d x) '' C)","l":82,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.IsSelection","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a sample space equipped with a measurable structure, a lower endpoint function, an upper endpoint function, and a real-valued function on the sample space, the everywhere measurable-selection condition holds precisely when the function is measurable and at every sample outcome its value lies in the closed interval between the endpoint values.","s":"{Ω : Type u_1} → [MeasurableSpace Ω] → (Ω → ℝ) → (Ω → ℝ) → (Ω → ℝ) → Prop","l":69,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.artstein_supportFn_neg_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"Artstein identity, d = 1, direction −1. For measurable lower and upper endpoint functions L, U of an interval-valued random set that are integrable and satisfy L pointwise at most U, the support function of the Aumann (selection) expectation of the random interval [L, U] in the direction −1, namely the negated lower endpoint, equals the expectation of the support function of the random interval itself in that direction: s(−1, E[F]) = E[s(−1, F)].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.selectionExpectation L U μ) (-1) = ∫ (ω : Ω), Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.randomInterval L U ω) (-1) ∂μ","l":113,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.artstein_supportFn_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"Artstein identity, d = 1, direction +1. For measurable lower and upper endpoint functions L, U of an interval-valued random set that are integrable and satisfy L pointwise at most U, the support function of the Aumann (selection) expectation of the random interval [L, U] in the direction +1, namely the upper endpoint, equals the expectation of the support function of the random interval itself in that direction: s(+1, E[F]) = E[s(+1, F)].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.selectionExpectation L U μ) 1 = ∫ (ω : Ω), Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.randomInterval L U ω) 1 ∂μ","l":97,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.bddAbove_inner_image","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"On a compact set the linear functional ⟪d, ·⟫ is bounded above.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E}, IsCompact C → ∀ (d : E), BddAbove ((fun x => inner ℝ d x) '' C)","l":72,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_dirStat","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"The directed-Hausdorff endpoint functional is continuous.","s":"Continuous Causalean.PartialID.RandomSet.dirStat","l":188,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_maxAbs","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The max-absolute-value functional on endpoint deviations is continuous.","s":"Continuous Causalean.PartialID.RandomSet.maxAbs","l":80,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.continuous_maxAbsK","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"The grid supremum statistic is continuous.","s":"∀ {k : ℕ} [inst : NeZero k], Continuous Causalean.PartialID.RandomSet.maxAbsK","l":72,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.coverage_event_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Coverage event identity (Beresteanu–Molinari coverage corollary). Fix the population identified interval E[Y] = [μL, μU] with μL ≤ μU, the sample-mean interval Ȳₙ(ω) = [yl(ω), yu(ω)] with yl(ω) ≤ yu(ω) for every ω, a sample size of at least one, and a nonnegative critical value c. With the BM bandwidth r = c/√n, the (one-sided) coverage event {E[Y] ⊆ Uₙ} equals the event {√n · dᴴ(E[Y], Ȳₙ) ≤ c} on the directed Hausdorff statistic. Hence the coverage probability is μ {ω | √n · dᴴ(E[Y], Ȳₙ(ω)) ≤ c}.","s":"∀ {Ω : Type u_1} (μL μU : ℝ) (yl yu : Ω → ℝ) (n : ℕ), 1 ≤ n → ∀ (c : ℝ), 0 ≤ c → μL ≤ μU → (∀ (ω : Ω), yl ω ≤ yu ω) → {ω | Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc (yl ω) (yu ω)) (c / √↑n)} = {ω | √↑n * Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc μL μU) (Set.Icc (yl ω) (yu ω)) ≤ c}","l":140,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dilate","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a set of real numbers and a real radius, the Minkowski dilation is the set of all sums $a+t$ such that $a$ belongs to the original set and $|t|\\le r$.","s":"Set ℝ → ℝ → Set ℝ","l":66,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dilate_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Dilation of an interval (Beresteanu–Molinari Uₙ). For r ≥ 0, [a,b] ⊕ [−r,r] = [a−r, b+r]: every endpoint is pushed out by the radius.","s":"∀ {a b r : ℝ}, a ≤ b → 0 ≤ r → Causalean.PartialID.RandomSet.dilate (Set.Icc a b) r = Set.Icc (a - r) (b + r)","l":73,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dirStat","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a two-dimensional endpoint-deviation vector, the directed-Hausdorff functional is $\\max\\{0,w_0,-w_1\\}$. On the centered and normalized endpoint sum, it equals $\\sqrt n$ times the directed Hausdorff distance from the population interval to the sample-mean interval.","s":"EuclideanSpace ℝ (Fin 2) → ℝ","l":181,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.dirStat_normalizedSum_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed Hausdorff bridge. dirStat of the centered endpoint normalised sum equals √n · dᴴ(E[Y], Ȳₙ).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n ω) = √↑n * Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P)) (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω))","l":254,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.directedHausdorff","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"In a pseudo-metric space, for a set $A$ and a set $B$, the directed, one-sided Hausdorff distance from $A$ to $B$ is the supremum, over points of $A$, of their distance to $B$; it is defined to be zero when $A$ is empty.","s":"{α : Type u_1} → [PseudoMetricSpace α] → Set α → Set α → ℝ","l":59,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.directedHausdorff_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed Hausdorff distance between intervals. dᴴ([a,b], [c,d]) = max 0 (max (c − a) (b − d)) (for a ≤ b, c ≤ d): the worst over-reach of [a,b] beyond [c,d], achieved at the endpoints.","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc a b) (Set.Icc c d) = max 0 (max (c - a) (b - d))","l":139,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.directedRegion_coverage","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Asymptotic coverage of the directed confidence region. For an i.i.d. sample with interval endpoints yL, yU satisfying the lower endpoint pointwise at most the upper endpoint and both integrable, assume the centered endpoint influence function is measurable with finite second moment and mean zero, its normalized partial sums are almost-everywhere measurable at every sample size, and fix a nonnegative bandwidth constant c that is a continuity point of the directed-Hausdorff Gaussian limit law. With population identified interval E[Y] = [E y_L, E y_U], sample-mean interval Ȳₙ, and the BM bandwidth c/√n, the coverage probability of the whole identified set converges to the limit-law mass of (-∞, c]:","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (hψ : Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ) → ∀ {c : ℝ}, 0 ≤ c → (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) {c} = 0 → Filter.Tendsto (fun n => μ {ω | Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P) ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (c / √↑n)}) Filter.atTop (nhds ((MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) (Set.Iic c)))","l":276,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.eucl₂","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a pair of real endpoint deviations, the associated two-dimensional Euclidean vector is that pair viewed as a vector in Euclidean space.","s":"(Fin 2 → ℝ) → EuclideanSpace ℝ (Fin 2)","l":66,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.eucl₂_apply","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Coordinate access for eucl₂: (eucl₂ v) j = v j.","s":"∀ (v : Fin 2 → ℝ) (j : Fin 2), (Causalean.PartialID.RandomSet.eucl₂ v).ofLp j = v j","l":221,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestReject","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a measurable sample space with sampling measure, a measurable outcome space with outcome measure, a nonempty grid containing $k$ directions, an inner-product outcome space, an independent and identically distributed sample, a set-valued outcome function, its proposed center set, a grid of directions, a nonnegative sample size, and a real critical value, the rejection region is the set of sample points at which the finite-grid test statistic exceeds the critical value.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {k : ℕ} → [NeZero k] → {V : Type u_3} → [inst_3 : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → Causalean.Stat.IIDSample Ω X μ P → (X → Set V) → Set V → (Fin k → V) → ℕ → ℝ → Set Ω","l":95,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestStat","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a measurable sample space with sampling measure, a measurable outcome space with outcome measure, a nonempty grid containing $k$ directions, an inner-product outcome space, an independent and identically distributed sample, a set-valued outcome function, its proposed center set, and a grid of directions, the finite-grid test statistic, at every nonnegative sample size and sample point, is the largest absolute coordinate of the normalized centered support-process sum.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {k : ℕ} → [NeZero k] → {V : Type u_3} → [inst_3 : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → Causalean.Stat.IIDSample Ω X μ P → (X → Set V) → Set V → (Fin k → V) → ℕ → Ω → ℝ","l":75,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTestStat_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"The normalized finite-grid support-process statistic converges in distribution to the grid supremum of its Gaussian limit.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (Causalean.PartialID.RandomSet.gridTestStat S F EF p) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":120,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.gridTest_asymptotic_level","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.GridTest","a":"PO/ID/Partial/RandomSet","t":1,"d":"Asymptotic level of the finite-grid tail test. At any continuity point c of the Gaussian limit law of the grid test statistic — i.e. the limit law assigns zero mass to {c} — the tail (rejection) probability of the normalized finite-grid support-process statistic converges to the corresponding tail mass of the Gaussian limit law: μ (gridTestReject … c) → L(c, ∞).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ) → ∀ {c : ℝ}, (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) {c} = 0 → Filter.Tendsto (fun n => μ (Causalean.PartialID.RandomSet.gridTestReject S F EF p n c)) Filter.atTop (nhds ((MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) (Set.Ioi c)))","l":136,"f":"Causalean/PO/ID/Partial/RandomSet/GridTest.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"In a pseudo-metric space, for a set $A$ and a set $B$, the symmetric Hausdorff distance is the larger of the directed distance from $A$ to $B$ and the directed distance from $B$ to $A$.","s":"{α : Type u_1} → [PseudoMetricSpace α] → Set α → Set α → ℝ","l":68,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":1,"d":"The geometric keystone (Beresteanu–Molinari eq. (A.1), d = 1). For real numbers a ≤ b and c ≤ d forming two well-ordered closed intervals, the symmetric Hausdorff distance between [a,b] and [c,d] equals the larger of the two endpoint gaps: H([a,b], [c,d]) = max(|a − c|, |b − d|).","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc a b) (Set.Icc c d) = max |a - c| |b - d|","l":177,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.hausdorffDist_Icc_eq_supportFn","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":1,"d":"The d = 1 Hörmander identity (Beresteanu–Molinari eq. (A.1)). For real numbers a ≤ b and c ≤ d forming two well-ordered closed intervals, the Hausdorff distance between the intervals [a,b] and [c,d] equals the largest, over the two unit directions +1 and −1, of the absolute difference between their support functions in that direction.","s":"∀ {a b c d : ℝ}, a ≤ b → c ≤ d → Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc a b) (Set.Icc c d) = max |Causalean.PartialID.supportFn (Set.Icc a b) (-1) - Causalean.PartialID.supportFn (Set.Icc c d) (-1)| |Causalean.PartialID.supportFn (Set.Icc a b) 1 - Causalean.PartialID.supportFn (Set.Icc c d) 1|","l":84,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.infDist_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Hausdorff","a":"PO/ID/Partial/RandomSet","t":2,"d":"Distance from a real point to a closed interval. For c ≤ d, infDist x [c,d] = max 0 (max (c − x) (x − d)) — zero inside the interval, and the signed gap to the nearer endpoint outside it.","s":"∀ {c d : ℝ}, c ≤ d → ∀ (x : ℝ), Metric.infDist x (Set.Icc c d) = max 0 (max (c - x) (x - d))","l":77,"f":"Causalean/PO/ID/Partial/RandomSet/Hausdorff.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinMaxAbsKGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"For a measurable sample space equipped with a measure, a positive number of grid directions, and a vector-valued process on that space that is measurable and has an integrable squared norm under the measure, the law obtained by applying the grid supremum statistic to its Gaussian limit is a probability measure. This follows from taking the measurable pushforward of that Gaussian limit.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {k : ℕ} [inst_1 : NeZero k] {ψ : X → EuclideanSpace ℝ (Fin k)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar))","l":88,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinOfNatNatDirStatGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"For a measurable sample space equipped with a measure and a two-dimensional vector-valued process on that space that is measurable and has an integrable squared norm under the measure, the law obtained by applying the directed-Hausdorff functional to its Gaussian limit is a probability measure. This follows from taking the measurable pushforward of that Gaussian limit.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar))","l":201,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.instIsProbabilityMeasureRealMapEuclideanSpaceFinOfNatNatMaxAbsGaussianLimit","k":"instance","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"For a measurable sample space equipped with a measure and a two-dimensional vector-valued process on that space that is measurable and has an integrable squared norm under the measure, the law obtained by applying the maximum absolute endpoint-deviation statistic to its Gaussian limit is a probability measure. This follows from taking the measurable pushforward of that Gaussian limit.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar))","l":93,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.integral_le_integral_of_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"Monotonicity of the endpoint integrals (used to order the reported bounds).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → ∫ (ω : Ω), L ω ∂μ ≤ ∫ (ω : Ω), U ω ∂μ","l":139,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a measurable outcome space, a lower-endpoint outcome function, an upper-endpoint outcome function, and a measure on that outcome space, the centered interval-endpoint influence function maps each observation to its lower and upper endpoints less their respective population means, viewed as a two-dimensional Euclidean vector.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → ℝ) → (X → ℝ) → MeasureTheory.Measure X → X → EuclideanSpace ℝ (Fin 2)","l":145,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_hHmeas","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The scaled Hausdorff statistic is AEMeasurable for each n. Equals maxAbs ∘ (normalised sum) everywhere by maxAbs_normalizedSum_eq.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → Measurable yL → Measurable yU → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) μ","l":331,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"intervalIFVec is Bochner-integrable, from MemLp 2 ⇒ Integrable of the endpoints on a probability measure.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → MeasureTheory.Integrable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) P","l":267,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_mean_zero","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The endpoint influence function is centered: ∫ ψ dP = 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0","l":281,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_sum_aemeasurable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The vector normalised sum of intervalIFVec is AEMeasurable for each n, from measurability of the endpoints and of the sample coordinates S.Z i.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), Measurable yL → Measurable yU → ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ","l":316,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.intervalIFVec_var_integrable","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The variance term ‖ψ‖² ∈ L¹(P), from MemLp 2 of the two endpoints.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (yL yU : X → ℝ), MeasureTheory.MemLp yL 2 P → MeasureTheory.MemLp yU 2 P → MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P","l":249,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.interval_data_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"Beresteanu–Molinari Theorem 3.2 (scalar interval data). For an i.i.d. sample of interval data Yᵢ = [y_{iL}, y_{iU}] with the lower endpoint pointwise at most the upper endpoint and both endpoints integrable, assume the centered endpoint influence function is measurable, has finite second moment and is integrable, is centered, and its normalized partial sums and the resulting scaled Hausdorff statistic are almost-everywhere measurable at every sample size. Then the scaled Hausdorff distance between the sample-mean interval Ȳₙ and the population identified interval E[Y] = [E y_L, E y_U] converges in distribution to the max-abs of the bivariate Gaussian limit of the influence function:","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (hψ : Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2) P), MeasureTheory.Integrable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) P → ∫ (x : X), Causalean.PartialID.RandomSet.intervalIFVec yL yU P x ∂P = 0 → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n) μ) → ∀ (hHmeas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar)) μ hHmeas","l":347,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.interval_data_clt_of_memLp","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"Beresteanu–Molinari Theorem 3.2, self-contained MemLp 2 form. For an i.i.d. sample of interval data Yᵢ = [y_{iL}, y_{iU}] with the lower endpoint pointwise at most the upper endpoint, measurable endpoints, and finite second moments (MemLp 2) for both endpoints, the scaled Hausdorff distance between the sample-mean interval and the population identified interval E[Y] = [E y_L, E y_U] converges in distribution to the max-abs of the bivariate Gaussian limit of the endpoint influence function. All four multivariate-CLT hypotheses of interval_data_clt are discharged from these clean moment conditions on yL, yU.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] [inst_3 : MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ) (hLU : ∀ (z : X), yL z ≤ yU z) (hLmeas : Measurable yL) (hUmeas : Measurable yU) (hLsq : MeasureTheory.MemLp yL 2 P) (hUsq : MeasureTheory.MemLp yU 2 P), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit (Causalean.PartialID.RandomSet.measurable_intervalIFVec yL yU hLmeas hUmeas) (Causalean.PartialID.RandomSet.intervalIFVec_var_integrable yL yU hLsq hUsq))) μ (Causalean.PartialID.RandomSet.intervalIFVec_hHmeas S yL yU hLU hLmeas hUmeas (MeasureTheory.MemLp.integrable (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ENNReal Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ENNReal (Eq.refl 2)) (Eq.refl true)) hLsq) (MeasureTheory.MemLp.integrable (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ENNReal Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ENNReal (Eq.refl 2)) (Eq.refl true)) hUsq))","l":387,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.isBody_finsetSum","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"A finite Minkowski sum of bodies is a body.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.RandomSet.IsBody (∑ i ∈ s, F i)","l":87,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.isSelection_iff_exists_param","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"Measurable selection of an interval random set. A function f is a selection of [L, U] iff f = L + t·(U − L) for some measurable t : Ω → [0,1]. Elementary — no Kuratowski–Ryll-Nardzewski.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {L U : Ω → ℝ}, Measurable L → Measurable U → (∀ (ω : Ω), L ω ≤ U ω) → ∀ (f : Ω → ℝ), Causalean.PartialID.RandomSet.IsSelection L U f ↔ ∃ t, Measurable t ∧ (∀ (ω : Ω), t ω ∈ Set.Icc 0 1) ∧ ∀ (ω : Ω), f ω = L ω + t ω * (U ω - L ω)","l":85,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.isSelection_left","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":2,"d":"The lower endpoint is always a selection, so the random set has a measurable selection.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {L U : Ω → ℝ}, Measurable L → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.RandomSet.IsSelection L U L","l":79,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbs","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a two-dimensional endpoint-deviation vector, the maximum-absolute-deviation functional is $\\max\\{|w_0|,|w_1|\\}$. In the scalar random-set setting, this is the Hausdorff distance between intervals whose lower- and upper-endpoint gaps are the two coordinates.","s":"EuclideanSpace ℝ (Fin 2) → ℝ","l":73,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbsK","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a nonempty grid containing $k$ directions and a vector of support-process deviations over that grid, the grid supremum statistic is the largest absolute coordinate of the vector.","s":"{k : ℕ} → [NeZero k] → EuclideanSpace ℝ (Fin k) → ℝ","l":60,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.maxAbs_normalizedSum_eq","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"The Hausdorff bridge (Beresteanu–Molinari Theorem 3.2, statistic form). For an i.i.d. sample with interval endpoints yL, yU satisfying the lower endpoint pointwise at most the upper endpoint and both integrable, the max-abs functional applied to the centered endpoint normalised sum equals the scaled Hausdorff distance √n · H(Ȳₙ, E[Y]) between the sample-mean interval Ȳₙ = [ȳ_{nL}, ȳ_{nU}] and the population identified interval E[Y] = [E y_L, E y_U].","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → MeasureTheory.Integrable yL P → MeasureTheory.Integrable yU P → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.intervalIFVec yL yU P) (fun m => Finset.range m) n ω) = √↑n * Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc (Causalean.PartialID.RandomSet.sampleMean S yL n ω) (Causalean.PartialID.RandomSet.sampleMean S yU n ω)) (Set.Icc (∫ (x : X), yL x ∂P) (∫ (x : X), yU x ∂P))","l":197,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_dirStat","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"The directed-Hausdorff endpoint functional is measurable.","s":"Measurable Causalean.PartialID.RandomSet.dirStat","l":192,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_intervalIFVec","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"intervalIFVec is measurable from measurability of the two endpoint maps.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} (yL yU : X → ℝ), Measurable yL → Measurable yU → Measurable (Causalean.PartialID.RandomSet.intervalIFVec yL yU P)","l":237,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_maxAbs","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The max-absolute-value functional on endpoint deviations is measurable.","s":"Measurable Causalean.PartialID.RandomSet.maxAbs","l":84,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.measurable_maxAbsK","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"The grid supremum statistic is measurable.","s":"∀ {k : ℕ} [inst : NeZero k], Measurable Causalean.PartialID.RandomSet.maxAbsK","l":78,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.minkowskiMean","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"For an inner-product outcome space, an index population, a finite index set, and a family of subsets of that space, the empirical Minkowski average is the Minkowski sum of the selected sets, scaled by the reciprocal of the number of selected indices.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → {ι : Type u_2} → Finset ι → (ι → Set E) → Set E","l":120,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.norm_sq_intervalIFVec","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Pointwise squared norm of the endpoint influence function as the sum of the two squared centered endpoints.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} (yL yU : X → ℝ) (x : X), ‖Causalean.PartialID.RandomSet.intervalIFVec yL yU P x‖ ^ 2 = (yL x - ∫ (z : X), yL z ∂P) ^ 2 + (yU x - ∫ (z : X), yU z ∂P) ^ 2","l":225,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.normalizedSum_dirStat_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":2,"d":"Directed continuous-mapping CLT. dirStat of the vector normalised sum converges in distribution to (gaussianLimit ψ).map dirStat.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.dirStat (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_dirStat (hSum_meas n)","l":206,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.normalizedSum_maxAbs_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"Abstract continuous-mapping CLT. maxAbs of the vector normalised sum converges in distribution to the pushforward (gaussianLimit ψ).map maxAbs — the law of max(|z_L|, |z_U|) for the bivariate Gaussian limit. Immediate from the multivariate CLT (clt_normalizedSum_vec) and the continuous-mapping theorem (Tendsto_dist_vec.map_continuous).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {ψ : X → EuclideanSpace ℝ (Fin 2)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), MeasureTheory.Integrable ψ P → ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbs (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbs (hSum_meas n)","l":98,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.randomInterval","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a sample space, a lower endpoint function, and an upper endpoint function, the interval-valued random set assigns to every sample outcome the set of real numbers that are at least its lower-endpoint value and at most its upper-endpoint value.","s":"{Ω : Type u_1} → (Ω → ℝ) → (Ω → ℝ) → Ω → Set ℝ","l":64,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sInf_selectionExpectation","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the sharp lower endpoint of the identified set — the infimum of the selection (Aumann) expectation over all measurable selections of the interval-valued random set [L, U] — equals the expectation of the lower endpoint L: sInf (selectionExpectation L U μ) = ∫ L dμ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → sInf (Causalean.PartialID.RandomSet.selectionExpectation L U μ) = ∫ (ω : Ω), L ω ∂μ","l":188,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sSup_selectionExpectation","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the sharp upper endpoint of the identified set — the supremum of the selection (Aumann) expectation over all measurable selections of the interval-valued random set [L, U] — equals the expectation of the upper endpoint U: sSup (selectionExpectation L U μ) = ∫ U dμ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → sSup (Causalean.PartialID.RandomSet.selectionExpectation L U μ) = ∫ (ω : Ω), U ω ∂μ","l":200,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.sampleMean","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a measurable sample space with sampling measure, a measurable outcome space with outcome measure, an independent and identically distributed sample, a real-valued outcome function, a nonnegative sample size, and a realized sample point, the sample mean is the arithmetic average of the first $n$ observed outcome values, with the empty sum divided by zero interpreted by the library's real-number convention.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":135,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.sampleMean_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The sample-mean interval is well-ordered (lower ≤ upper) when y_L ≤ y_U.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (yL yU : X → ℝ), (∀ (z : X), yL z ≤ yU z) → ∀ (n : ℕ) (ω : Ω), Causalean.PartialID.RandomSet.sampleMean S yL n ω ≤ Causalean.PartialID.RandomSet.sampleMean S yU n ω","l":188,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.selectionExpectation","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a sample space equipped with a measurable structure, a lower endpoint function, an upper endpoint function, and a measure on the sample space, the selection, or Aumann, expectation is the set of real numbers for which there exists a function such that it is an everywhere measurable selection of the endpoint interval, it is integrable under the measure, and its integral under that measure equals the real number.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Set ℝ","l":128,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.selectionExpectation_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Interval","a":"PO/ID/Partial/RandomSet","t":1,"d":"Selection expectation equals [∫L, ∫U]. For measurable lower and upper endpoint functions L, U that are integrable and satisfy L pointwise at most U, the selection (Aumann) expectation of the interval-valued random set [L, U] — the set of integrals of its integrable measurable selections — equals the closed interval [∫L dμ, ∫U dμ]. The forward inclusion is integral monotonicity; the reverse inclusion realises every intermediate value with a constant mixing weight t ∈ [0,1], so no atomlessness is needed.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {L U : Ω → ℝ}, Measurable L → Measurable U → MeasureTheory.Integrable L μ → MeasureTheory.Integrable U μ → (∀ (ω : Ω), L ω ≤ U ω) → Causalean.PartialID.RandomSet.selectionExpectation L U μ = Set.Icc (∫ (ω : Ω), L ω ∂μ) (∫ (ω : Ω), U ω ∂μ)","l":144,"f":"Causalean/PO/ID/Partial/RandomSet/Interval.lean"},{"n":"Causalean.PartialID.RandomSet.setValued_supportProcess_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a set-valued random variable F with center EF, observed via an IID sample on a finite grid of directions p, if the centered support process is measurable, it has finite second moment, it is mean zero, E[ψ] = 0, and the normalized partial sums of the process are almost-everywhere measurable at every sample size, then the grid supremum statistic applied to those normalized sums converges in distribution to the corresponding supremum functional of the process's Gaussian limit.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {V : Type u_3} [inst_4 : NormedAddCommGroup V] [inst_5 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V) (hψ : Measurable (Causalean.PartialID.RandomSet.supportProcess F EF p)) (hvar : MeasureTheory.Integrable (fun x => ‖Causalean.PartialID.RandomSet.supportProcess F EF p x‖ ^ 2) P), ∫ (x : X), Causalean.PartialID.RandomSet.supportProcess F EF p x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":192,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.sqrt_inv_centered","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalCLT","a":"PO/ID/Partial/RandomSet","t":2,"d":"The centered-sum / √n identity (√n)⁻¹·(s − n·c) = √n·(s/n − c), the algebra turning a normalised centered sum into √n × (sample mean − population mean).","s":"∀ (n : ℕ) (s c : ℝ), (√↑n)⁻¹ * (s - ↑n * c) = √↑n * (s / ↑n - c)","l":121,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalCLT.lean"},{"n":"Causalean.PartialID.RandomSet.subset_dilate_iff_directedHausdorff_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Coverage characterization (Beresteanu–Molinari Proposition 2.7 core). For real numbers with μL ≤ μU and yl ≤ yu forming two closed intervals, and a nonnegative dilation radius r, the population identified interval [μL, μU] lies inside the dilated estimate [yl, yu] ⊕ [−r, r] if and only if the directed Hausdorff distance from [μL,μU] to [yl,yu] is at most r. This is the one-sided coverage event of the Beresteanu–Molinari confidence region.","s":"∀ {μL μU yl yu r : ℝ}, μL ≤ μU → yl ≤ yu → 0 ≤ r → (Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc yl yu) r ↔ Causalean.PartialID.RandomSet.directedHausdorff (Set.Icc μL μU) (Set.Icc yl yu) ≤ r)","l":95,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.subset_dilate_iff_hausdorff_le","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.IntervalInference","a":"PO/ID/Partial/RandomSet","t":1,"d":"Symmetric (two-sided) coverage characterization (Beresteanu–Molinari Uₙ / Theorem 2.4). For real numbers with μL ≤ μU and yl ≤ yu forming two closed intervals, and a nonnegative dilation radius r, each interval lies inside the other's dilation by r if and only if the symmetric Hausdorff distance between [μL,μU] and [yl,yu] is at most r. The mutual containment [μL,μU] ⊆ Uᵧ ∧ [yl,yu] ⊆ Uᵤ is exactly the two-sided event H ≤ r.","s":"∀ {μL μU yl yu r : ℝ}, μL ≤ μU → yl ≤ yu → 0 ≤ r → (Set.Icc μL μU ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc yl yu) r ∧ Set.Icc yl yu ⊆ Causalean.PartialID.RandomSet.dilate (Set.Icc μL μU) r ↔ Causalean.PartialID.RandomSet.hausdorffDist (Set.Icc μL μU) (Set.Icc yl yu) ≤ r)","l":113,"f":"Causalean/PO/ID/Partial/RandomSet/IntervalInference.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_Icc_neg_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function of a real interval at −1: the negated lower endpoint.","s":"∀ {a b : ℝ}, a ≤ b → Causalean.PartialID.supportFn (Set.Icc a b) (-1) = -a","l":76,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_Icc_one","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.Aumann","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function of a real interval at +1: the upper endpoint.","s":"∀ {a b : ℝ}, a ≤ b → Causalean.PartialID.supportFn (Set.Icc a b) 1 = b","l":69,"f":"Causalean/PO/ID/Partial/RandomSet/Aumann.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_finsetSum","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":2,"d":"Support function commutes with finite Minkowski sums: s(∑ᵢ Fᵢ, d) = ∑ᵢ s(Fᵢ, d).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E) (d : E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.supportFn (∑ i ∈ s, F i) d = ∑ i ∈ s, Causalean.PartialID.supportFn (F i) d","l":102,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.supportFn_minkowskiMean","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SetValued","a":"PO/ID/Partial/RandomSet","t":1,"d":"Keystone — Minkowski-mean support bridge. Given a finite index set s all of whose values F i are bodies — nonempty, compact, convex subsets, the support function of the empirical Minkowski average (1/|s|) · ∑ᵢ Fᵢ in a direction d equals the arithmetic average of the individual support functions: s(d, F̄ₙ) = (1/|s|) · ∑ᵢ s(d, Fᵢ). This is the identity that turns the support process into an ordinary sample mean, so the multivariate CLT applies.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {ι : Type u_2} (s : Finset ι) (F : ι → Set E) (d : E), (∀ i ∈ s, Causalean.PartialID.RandomSet.IsBody (F i)) → Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.minkowskiMean s F) d = (↑s.card)⁻¹ * ∑ i ∈ s, Causalean.PartialID.supportFn (F i) d","l":128,"f":"Causalean/PO/ID/Partial/RandomSet/SetValued.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess","k":"def","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For an observation space, a nonempty grid containing $k$ directions, an inner-product outcome space, a set-valued outcome function, a proposed center set, and a grid of directions, the centered support process maps each observation to the vector whose coordinate in each direction is the support value of its realized set minus the support value of the proposed center set.","s":"{X : Type u_2} → {k : ℕ} → {V : Type u_3} → [inst : NormedAddCommGroup V] → [InnerProductSpace ℝ V] → (X → Set V) → Set V → (Fin k → V) → X → EuclideanSpace ℝ (Fin k)","l":137,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_normalizedSum_apply","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"Each coordinate of the normalized support-process sum is the rescaled support deviation of the empirical Minkowski average.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {k : ℕ} {V : Type u_3} [inst_2 : NormedAddCommGroup V] [inst_3 : InnerProductSpace ℝ V] (S : Causalean.Stat.IIDSample Ω X μ P) (F : X → Set V) (EF : Set V) (p : Fin k → V), (∀ (x : X), Causalean.PartialID.RandomSet.IsBody (F x)) → ∀ (n : ℕ) (ω : Ω) (j : Fin k), (Causalean.Stat.IsAsymLinearVec.normalizedSum S (Causalean.PartialID.RandomSet.supportProcess F EF p) (fun m => Finset.range m) n ω).ofLp j = √↑n * (Causalean.PartialID.supportFn (Causalean.PartialID.RandomSet.minkowskiMean (Finset.range n) fun i => F (S.Z i ω)) (p j) - Causalean.PartialID.supportFn EF (p j))","l":162,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_ofLp","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":2,"d":"Each coordinate of the centered support process is the corresponding support deviation.","s":"∀ {X : Type u_2} {k : ℕ} {V : Type u_3} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (F : X → Set V) (EF : Set V) (p : Fin k → V) (x : X) (j : Fin k), (Causalean.PartialID.RandomSet.supportProcess F EF p x).ofLp j = Causalean.PartialID.supportFn (F x) (p j) - Causalean.PartialID.supportFn EF (p j)","l":155,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.RandomSet.supportProcess_sup_clt","k":"theorem","m":"Causalean.PO.ID.Partial.RandomSet.SupportProcess","a":"PO/ID/Partial/RandomSet","t":1,"d":"For a centered support process ψ on k fixed directions observed via an IID sample S, if ψ is mean zero, E[ψ] = 0, and the normalized partial sums built from S are almost-everywhere measurable at every sample size, then the grid supremum statistic maxAbsK applied to those normalized sums converges in distribution to maxAbsK applied to the Gaussian limit of ψ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {k : ℕ} [inst_3 : NeZero k] {ψ : X → EuclideanSpace ℝ (Fin k)} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ (fun m => Finset.range m) n ω)) (MeasureTheory.Measure.map Causalean.PartialID.RandomSet.maxAbsK (Causalean.Stat.gaussianLimit hψ hvar)) μ fun n => Measurable.comp_aemeasurable Causalean.PartialID.RandomSet.measurable_maxAbsK (hSum_meas n)","l":96,"f":"Causalean/PO/ID/Partial/RandomSet/SupportProcess.lean"},{"n":"Causalean.PartialID.affineBall","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product-space domain, a real inner-product-space codomain, a continuous linear map from the domain to the codomain, a target vector in the codomain, and a real bound, the affine-ball feasible set consists exactly of the domain vectors whose image equals the target vector and whose norm is at most the bound.","s":"{H : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : NormedAddCommGroup F] → [inst_3 : InnerProductSpace ℝ F] → (H →L[ℝ] F) → F → ℝ → Set H","l":67,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.affineBall_point_identified_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. Let A be a continuous linear map between real inner-product spaces with b in its range, B a radius, and c a target direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is strictly less than B (strict slack in the norm bound). Then the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} collapses to a point if and only if the orthogonal projection of c onto the kernel of A vanishes, i.e. c is orthogonal to that kernel.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ < B → (Causalean.PartialID.width (Causalean.PartialID.affineBall A b B) c = 0 ↔ (Causalean.PartialID.opKer A).orthogonalProjection c = 0)","l":218,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.frechet_lower","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Fréchet–Hoeffding lower bound. For a fixed joint law of two real random variables X, Y under a probability measure, if Y is measurable, then the joint probability P(X ≤ u, Y ≤ v) is at least the larger of zero and the sum of the two marginal CDF values at u and v minus one: max (P(X ≤ u) + P(Y ≤ v) - 1) 0 ≤ P(X ≤ u, Y ≤ v).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (u v : ℝ), max ((P {ω | X ω ≤ u}).toReal + (P {ω | Y ω ≤ v}).toReal - 1) 0 ≤ (P {ω | X ω ≤ u ∧ Y ω ≤ v}).toReal","l":83,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.frechet_upper","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Fréchet–Hoeffding upper bound. For any threshold u for X and any threshold v for Y, the joint probability P(X ≤ u, Y ≤ v) is at most the smaller of the two marginal probabilities P(X ≤ u) and P(Y ≤ v).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ} (u v : ℝ), (P {ω | X ω ≤ u ∧ Y ω ≤ v}).toReal ≤ min (P {ω | X ω ≤ u}).toReal (P {ω | Y ω ≤ v}).toReal","l":67,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.identifiedInterval_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"Sharp interval (order-connected form). For an abstract objective function over a feasible parameter set, suppose the objective is bounded below by L on every feasible parameter, bounded above by U on every feasible parameter, the value L itself is attained by some feasible parameter, the value U itself is attained by some feasible parameter, and the set of attainable objective values is order-connected — it contains every real number between any two of its members. Then the identified interval — the set of all objective values attainable over the feasible parameter set — equals the closed interval [L, U] exactly. Order-connectedness is the abstract substitute for \"no gaps\", supplied concretely by identifiedInterval_param_Icc through continuity + connectedness of a parameterization.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {L U : ℝ}, (∀ (x : α), feasible x → L ≤ obj x) → (∀ (x : α), feasible x → obj x ≤ U) → L ∈ Causalean.PartialID.IdentifiedInterval obj feasible → U ∈ Causalean.PartialID.IdentifiedInterval obj feasible → (Causalean.PartialID.IdentifiedInterval obj feasible).OrdConnected → Causalean.PartialID.IdentifiedInterval obj feasible = Set.Icc L U","l":67,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.identifiedInterval_eq_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Bridge to the scalar engine. For an objective functional obj and a feasible set C, the scalar identified interval of obj relative to membership in C equals exactly the image of obj on C. Specialised to obj = ⟪d, ·⟫, this identifies the scalar engine's output with the linear image studied here.","s":"∀ {α : Type u_2} (obj : α → ℝ) (C : Set α), (Causalean.PartialID.IdentifiedInterval obj fun x => x ∈ C) = obj '' C","l":170,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.identifiedInterval_param_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":1,"d":"Mixing-pattern constructor. Suppose the feasible parameter set is exactly the image of the unit interval [0, 1] under a path γ, the objective composed with γ is continuous on [0, 1], the objective value at the path's start equals L, the objective value at the path's end equals U, and the objective stays between L and U at every point along the path. Then the sharp identified interval is exactly [L, U]. This is the canonical partial-identification \"mixing\" shape: an unidentified nuisance ranging over a connected parameter set sweeps the objective continuously across the whole interval between its extreme values.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {γ : ℝ → α} {L U : ℝ}, (∀ (x : α), feasible x ↔ ∃ t ∈ Set.Icc 0 1, γ t = x) → ContinuousOn (fun t => obj (γ t)) (Set.Icc 0 1) → obj (γ 0) = L → obj (γ 1) = U → (∀ t ∈ Set.Icc 0 1, L ≤ obj (γ t) ∧ obj (γ t) ≤ U) → Causalean.PartialID.IdentifiedInterval obj feasible = Set.Icc L U","l":86,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.identifiedInterval_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Outer bound. If the objective is uniformly bounded below by L and above by U over the feasible set, the sharp identified interval is contained in [L, U].","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {L U : ℝ}, (∀ (x : α), feasible x → L ≤ obj x) → (∀ (x : α), feasible x → obj x ≤ U) → Causalean.PartialID.IdentifiedInterval obj feasible ⊆ Set.Icc L U","l":58,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.kerHasOrthogonalProjection","k":"instance","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":2,"d":"For a complete real inner-product space serving as the domain, a real inner-product space serving as the codomain, and a continuous linear map from the domain to the codomain, the map's kernel has an orthogonal projection. This follows because the kernel is closed and hence complete.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F), (Causalean.PartialID.opKer A).HasOrthogonalProjection","l":61,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.l2Ball","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space, a base direction in that space, and a real radius, the chi-square/L2 ambiguity set consists exactly of the vectors whose inner product with the base direction is one and whose distance from that direction is at most the radius.","s":"{H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → H → ℝ → Set H","l":175,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.l2Ball_eq_translate","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Reduction to the affine-ball engine. When the base direction has unit inner product with itself, ⟪e, e⟫ = 1, the χ²/L² ambiguity set l2Ball e ρ equals the translate by e of the affine ball of radius ρ for the linear functional ⟪e, ·⟫ centered at the origin — the substitution v = w - e turns the normalization constraint ⟪e, w⟫ = 1 into the linear constraint ⟪e, v⟫ = 0.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {e : H}, inner ℝ e e = 1 → ∀ (ρ : ℝ), Causalean.PartialID.l2Ball e ρ = (fun v => e + v) '' Causalean.PartialID.affineBall ((innerSL ℝ) e) 0 ρ","l":195,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.l2Ball_point_identified_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. For a unit-norm base direction e and a strictly positive radius ρ, the χ²/L² robust interval for the target c collapses to a single point exactly when ‖c‖² equals ⟪e, c⟫², i.e. when c is collinear with e — equivalently, equality holds in the Cauchy–Schwarz inequality, meaning the represented outcome is almost-surely constant.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 < ρ → (Causalean.PartialID.width (Causalean.PartialID.l2Ball e ρ) c = 0 ↔ ‖c‖ ^ 2 = inner ℝ e c ^ 2)","l":274,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.le_robustUpper","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Any admissible weight has a reweighted mean no larger than the upper robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c w : E}, w ∈ W → BddAbove ((fun x => inner ℝ c x) '' W) → inner ℝ c w ≤ Causalean.PartialID.robustUpper W c","l":116,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.le_supportFn","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Lower sandwich. For a feasible set C in a real inner-product space and a direction d, if a point x belongs to C and the linear functional ⟪d, ·⟫ is bounded above on C, then x's functional value is at most the support value: ⟪d, x⟫ ≤ supportFn C d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d x : E}, x ∈ C → BddAbove ((fun x => inner ℝ d x) '' C) → inner ℝ d x ≤ Causalean.PartialID.supportFn C d","l":71,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.linearImage_eq_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Sharp identified interval (attainment form). For a convex feasible set C in a real inner-product space and a linear target direction d, assuming the target functional is bounded above on C in the direction d and in the opposite direction -d, and that the upper support value is attained by some feasible point and the lower support value -supportFn C (-d) is likewise attained, then the identified set of the target functional over C is exactly the closed interval [-supportFn C (-d), supportFn C d].","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, Convex ℝ C → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → Causalean.PartialID.supportFn C d ∈ (fun x => inner ℝ d x) '' C → -Causalean.PartialID.supportFn C (-d) ∈ (fun x => inner ℝ d x) '' C → (fun x => inner ℝ d x) '' C = Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":86,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_eq_Icc_of_isCompact","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Sharp identified interval (compact convex form). For a linear target direction d and a feasible set that is compact, convex, and nonempty, the identified set of the target functional ⟪d, ·⟫ over C is exactly the closed interval [-supportFn C (-d), supportFn C d]: a continuous functional attains its supremum and infimum on a compact set and is automatically bounded there.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, IsCompact C → Convex ℝ C → C.Nonempty → (fun x => inner ℝ d x) '' C = Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":102,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_ordConnected","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Order-connectedness. The linear image of a convex set is convex in ℝ, hence order-connected: the identified set has \"no gaps\".","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, Convex ℝ C → ((fun x => inner ℝ d x) '' C).OrdConnected","l":77,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.linearImage_subset_Icc","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Outer bound. The identified set of the linear target lies inside the support interval [-supportFn C (-d), supportFn C d].","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → (fun x => inner ℝ d x) '' C ⊆ Set.Icc (-Causalean.PartialID.supportFn C (-d)) (Causalean.PartialID.supportFn C d)","l":68,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.makarov_lower_iSup","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Makarov lower bound (easy direction), sup-convolution envelope form. For a fixed joint law of two real random variables X, Y, if Y is measurable, then taking the supremum, over every reference point, of the per-threshold Makarov lower bound still lower-bounds the CDF of the difference X - Y at the given threshold.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (s : ℝ), ⨆ a, max ((P {ω | X ω ≤ a}).toReal - (P {ω | Y ω < a - s}).toReal) 0 ≤ (P {ω | X ω - Y ω ≤ s}).toReal","l":132,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.makarov_lower_param","k":"theorem","m":"Causalean.PO.ID.Partial.Frechet","a":"PO/ID/Partial","t":1,"d":"Makarov lower bound (easy direction), per-threshold form. For a fixed joint law of two real random variables X, Y, if Y is measurable, then for every reference point a and threshold s, the CDF of the difference X - Y at s is at least the larger of zero and the gap between the CDF of X at a and the CDF of Y just below a - s: max (P(X ≤ a) - P(Y < a - s)) 0 ≤ P(X - Y ≤ s).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] {X Y : Ω → ℝ}, Measurable Y → ∀ (s a : ℝ), max ((P {ω | X ω ≤ a}).toReal - (P {ω | Y ω < a - s}).toReal) 0 ≤ (P {ω | X ω - Y ω ≤ s}).toReal","l":98,"f":"Causalean/PO/ID/Partial/Frechet.lean"},{"n":"Causalean.PartialID.mem_Icc_csInf_csSup","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Worst/best case over a nuisance. If a set s ⊆ ℝ is bounded, every one of its members lies between sInf s and sSup s. Applied with s = range obj this is the engine form of \"the truth is bracketed by the extreme feasible values\".","s":"∀ {s : Set ℝ} {y : ℝ}, BddBelow s → BddAbove s → y ∈ s → y ∈ Set.Icc (sInf s) (sSup s)","l":47,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_Icc_of_sandwich","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"Sandwich → membership. The literal content of a two-sided bound L ≤ θ ≤ U: the target functional θ lies in the reported interval [L, U]. Names the step that turns the inequality pair every concrete bound produces into the Set.Icc vocabulary.","s":"∀ {θ L U : ℝ}, L ≤ θ → θ ≤ U → θ ∈ Set.Icc L U","l":39,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_affineBall","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Membership in the affine-ball fiber is exactly satisfying the equation and the norm bound.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F] {A : H →L[ℝ] F} {b : F} {B : ℝ} {h : H}, h ∈ Causalean.PartialID.affineBall A b B ↔ A h = b ∧ ‖h‖ ≤ B","l":74,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.mem_identifiedInterval","k":"theorem","m":"Causalean.PO.ID.Partial.Basic","a":"PO/ID/Partial","t":2,"d":"A feasible parameter's objective value belongs to the identified interval.","s":"∀ {α : Type u_1} {obj : α → ℝ} {feasible : α → Prop} {x : α}, feasible x → obj x ∈ Causalean.PartialID.IdentifiedInterval obj feasible","l":34,"f":"Causalean/PO/ID/Partial/Basic.lean"},{"n":"Causalean.PartialID.mem_l2Ball","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Membership in the chi-square or L2 ambiguity set is exactly normalization and the radius bound.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] {e : H} {ρ : ℝ} {w : H}, w ∈ Causalean.PartialID.l2Ball e ρ ↔ inner ℝ e w = 1 ∧ ‖w - e‖ ≤ ρ","l":181,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.neg_supportFn_neg_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Lower endpoint bound. If ⟪-d, ·⟫ is bounded above on C, then the lower endpoint -supportFn C (-d) is a lower bound for the functional ⟪d, ·⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d x : E}, x ∈ C → BddAbove ((fun x => inner ℝ (-d) x) '' C) → -Causalean.PartialID.supportFn C (-d) ≤ inner ℝ d x","l":59,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.norm_orthogonalProjection_opKer_innerSL","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The norm of the free-direction projection equals the standard deviation √(‖c‖² − ⟪e,c⟫²) under ‖e‖ = 1.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H), ‖↑((Causalean.PartialID.opKer ((innerSL ℝ) e)).orthogonalProjection c)‖ = √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":215,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.opKer","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space serving as the domain, a real inner-product space serving as the codomain, and a continuous linear map between them, the free-direction subspace is the kernel of that map: the vectors whose image is zero.","s":"{H : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup H] → [inst_1 : InnerProductSpace ℝ H] → [inst_2 : NormedAddCommGroup F] → [inst_3 : InnerProductSpace ℝ F] → (H →L[ℝ] F) → Submodule ℝ H","l":56,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.opKer_innerSL_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The free-direction kernel for normalization against the base direction is its orthogonal complement: ker (innerSL ℝ e) = (ℝ ∙ e)ᗮ.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] (e : H), Causalean.PartialID.opKer ((innerSL ℝ) e) = (ℝ ∙ e)ᗮ","l":187,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustInterval_eq_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"The robust interval is the identified set. When the ambiguity set W is compact, convex, and nonempty, the set of reweighted means ⟪c, w⟫ attained as w ranges over W is exactly the closed interval [robustLower W c, robustUpper W c] — the robust bounds are sharp.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c : E}, IsCompact W → Convex ℝ W → W.Nonempty → (fun x => inner ℝ c x) '' W = Set.Icc (Causalean.PartialID.robustLower W c) (Causalean.PartialID.robustUpper W c)","l":158,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space, a set of admissible reweighting vectors, and a target direction, the lower robust bound is the infimum, over the admissible vectors, of their inner product with the target direction.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":100,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_antitone","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotonicity in the budget (lower). A larger ambiguity set lowers the lower robust bound, so the robust interval widens.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W W' : Set E} {c : E}, W ⊆ W' → W.Nonempty → BddAbove ((fun x => inner ℝ (-c) x) '' W') → Causalean.PartialID.robustLower W' c ≤ Causalean.PartialID.robustLower W c","l":135,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Any admissible weight has a reweighted mean no smaller than the lower robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c w : E}, w ∈ W → BddAbove ((fun x => inner ℝ (-c) x) '' W) → Causalean.PartialID.robustLower W c ≤ inner ℝ c w","l":122,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_le_robustUpper","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The robust interval [robustLower, robustUpper] is well-ordered under the usual boundedness conditions: its width is width W c ≥ 0.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W : Set E} {c : E}, W.Nonempty → BddAbove ((fun x => inner ℝ c x) '' W) → BddAbove ((fun x => inner ℝ (-c) x) '' W) → Causalean.PartialID.robustLower W c ≤ Causalean.PartialID.robustUpper W c","l":105,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustLower_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"No ambiguity implies point identification (lower). When the ambiguity set collapses to the singleton {w₀}, the lower robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (c w₀ : E), Causalean.PartialID.robustLower {w₀} c = inner ℝ c w₀","l":151,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space, a set of admissible reweighting vectors, and a target direction, the upper robust bound is the supremum, over the admissible vectors, of their inner product with the target direction.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":95,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper_mono","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotonicity in the budget (upper). A larger ambiguity set raises the upper robust bound.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {W W' : Set E} {c : E}, W ⊆ W' → W.Nonempty → BddAbove ((fun x => inner ℝ c x) '' W') → Causalean.PartialID.robustUpper W c ≤ Causalean.PartialID.robustUpper W' c","l":128,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.robustUpper_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"No ambiguity implies point identification (upper). When the ambiguity set collapses to the singleton {w₀}, the upper robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (c w₀ : E), Causalean.PartialID.robustUpper {w₀} c = inner ℝ c w₀","l":144,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.supportFn","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space, a set of feasible vectors in that space, and a direction in the same space, the support function of the set in that direction is the supremum of the inner products of the direction with all feasible vectors.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":55,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_add_dir_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Subadditivity in the direction (sublinearity of the support function). For a feasible set C in a real inner-product space and directions d₁, d₂, assuming C is nonempty and the linear functionals ⟪d₁, ·⟫ and ⟪d₂, ·⟫ are each bounded above on C, then the support value in the combined direction is at most the sum of the support values in each direction: supportFn C (d₁ + d₂) ≤ supportFn C d₁ + supportFn C d₂.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d₁ d₂ : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d₁ x) '' C) → BddAbove ((fun x => inner ℝ d₂ x) '' C) → Causalean.PartialID.supportFn C (d₁ + d₂) ≤ Causalean.PartialID.supportFn C d₁ + Causalean.PartialID.supportFn C d₂","l":102,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_affineBall_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form support value over the affine-ball fiber. Let A be a continuous linear map between real inner-product spaces, b a target value, B a radius, and c a direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is at most B (h₀ fits the ball). Then the support value of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals ⟪c, h₀⟫ + √(B² − ‖h₀‖²)·‖P c‖, where P is the orthogonal projection onto the kernel of A.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ ≤ B → Causalean.PartialID.supportFn (Causalean.PartialID.affineBall A b B) c = inner ℝ c h₀ + √(B ^ 2 - ‖h₀‖ ^ 2) * ‖↑((Causalean.PartialID.opKer A).orthogonalProjection c)‖","l":78,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.supportFn_eq_iSup_image","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The support function is definitionally the supremum of the linear target over the set.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (C : Set E) (d : E), Causalean.PartialID.supportFn C d = sSup ((fun x => inner ℝ d x) '' C)","l":65,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_inter_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Intersection rule (one-sided). The support function of an intersection is at most the minimum of the parts.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, (C ∩ D).Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C ∩ D) d ≤ min (Causalean.PartialID.supportFn C d) (Causalean.PartialID.supportFn D d)","l":145,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_l2Ball_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form worst case (mean plus ρ·SD). For a unit-norm base direction e and a nonnegative radius ρ, the largest reweighted value ⟪c, w⟫ attains over the χ²/L² ambiguity set l2Ball e ρ equals ⟪e, c⟫ + ρ · √(‖c‖² − ⟪e, c⟫²) — the base-direction mean plus ρ standard deviations of c.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 ≤ ρ → Causalean.PartialID.supportFn (Causalean.PartialID.l2Ball e ρ) c = inner ℝ e c + ρ * √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":241,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.supportFn_le","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Basic","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Upper sandwich. A uniform upper bound b on the functional over a nonempty C bounds the support value: supportFn C d ≤ b.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {b : ℝ}, C.Nonempty → (∀ x ∈ C, inner ℝ d x ≤ b) → Causalean.PartialID.supportFn C d ≤ b","l":80,"f":"Causalean/PO/ID/Partial/SupportFunction/Basic.lean"},{"n":"Causalean.PartialID.supportFn_minkowski","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Minkowski additivity. The support function is additive over Minkowski sums: supportFn (C + D) d = supportFn C d + supportFn D d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C.Nonempty → D.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C + D) d = Causalean.PartialID.supportFn C d + Causalean.PartialID.supportFn D d","l":229,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_mono","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Monotone in the set. If C ⊆ D (with C nonempty and the functional bounded above on D), then supportFn C d ≤ supportFn D d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C ⊆ D → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn C d ≤ Causalean.PartialID.supportFn D d","l":121,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_nonneg_of_zero_mem","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"If 0 ∈ C then the support value is nonnegative (the functional value 0 at the origin is a lower bound for the sup).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, 0 ∈ C → BddAbove ((fun x => inner ℝ d x) '' C) → 0 ≤ Causalean.PartialID.supportFn C d","l":57,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_singleton","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Support function of a singleton is the functional value at the point.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (d x₀ : E), Causalean.PartialID.supportFn {x₀} d = inner ℝ d x₀","l":52,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_smul_dir","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Positive homogeneity in the direction. For t ≥ 0, supportFn C (t • d) = t * supportFn C d.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {t : ℝ}, 0 ≤ t → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn C (t • d) = t * Causalean.PartialID.supportFn C d","l":65,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_smul_set","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Positive scaling rule. Scaling the set by a ≥ 0 scales the support value by a.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} {a : ℝ}, 0 ≤ a → C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn (a • C) d = a * Causalean.PartialID.supportFn C d","l":182,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_translate","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Translation rule. Translating the set by x₀ shifts the support value by ⟪d, x₀⟫.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E} (x₀ : E), C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → Causalean.PartialID.supportFn ((fun x => x₀ + x) '' C) d = inner ℝ d x₀ + Causalean.PartialID.supportFn C d","l":155,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.supportFn_union","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Calculus","a":"PO/ID/Partial/SupportFunction","t":2,"d":"Union rule. The support function of a union is the maximum of the parts.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C D : Set E} {d : E}, C.Nonempty → D.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ d x) '' D) → Causalean.PartialID.supportFn (C ∪ D) d = max (Causalean.PartialID.supportFn C d) (Causalean.PartialID.supportFn D d)","l":129,"f":"Causalean/PO/ID/Partial/SupportFunction/Calculus.lean"},{"n":"Causalean.PartialID.width","k":"def","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"For a real inner-product space, a set of feasible vectors in that space, and a direction in the same space, the width of the associated identified interval is its upper endpoint minus its lower endpoint, equivalently the sum of the support values in the direction and its negative.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → Set E → E → ℝ","l":125,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.width_affineBall_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.AffineBall","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form width of the affine-ball identified set. Under the same hypotheses as supportFn_affineBall_eq — h₀ solves A h₀ = b, h₀ is orthogonal to the kernel of A, and h₀'s norm is at most B — the width of the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals 2·√(B² − ‖h₀‖²)·‖P c‖, twice the residual radius times the norm of c's projection onto the kernel of A.","s":"∀ {H : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [inst_2 : CompleteSpace H] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] (A : H →L[ℝ] F) {b : F} {B : ℝ} {c h₀ : H}, A h₀ = b → h₀ ∈ (Causalean.PartialID.opKer A)ᗮ → ‖h₀‖ ≤ B → Causalean.PartialID.width (Causalean.PartialID.affineBall A b B) c = 2 * √(B ^ 2 - ‖h₀‖ ^ 2) * ‖↑((Causalean.PartialID.opKer A).orthogonalProjection c)‖","l":198,"f":"Causalean/PO/ID/Partial/SupportFunction/AffineBall.lean"},{"n":"Causalean.PartialID.width_eq_zero_iff","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Point identification. For a feasible set C and direction d in a real inner-product space, assuming C is nonempty and the target functional ⟪d, ·⟫ is bounded above on C in the direction d and in the opposite direction -d, then the identified set collapses to a point (zero width) if and only if the target functional is constant on C, equal to its support value.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → (Causalean.PartialID.width C d = 0 ↔ ∀ x ∈ C, inner ℝ d x = Causalean.PartialID.supportFn C d)","l":142,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.PartialID.width_l2Ball_eq","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Sensitivity","a":"PO/ID/Partial/SupportFunction","t":1,"d":"Closed-form width. For a unit-norm base direction e and a nonnegative radius ρ, the width of the χ²/L² robust interval for the target c equals 2ρ · √(‖c‖² − ⟪e, c⟫²), twice the radius times the standard deviation of c.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] {e : H}, ‖e‖ = 1 → ∀ (c : H) {ρ : ℝ}, 0 ≤ ρ → Causalean.PartialID.width (Causalean.PartialID.l2Ball e ρ) c = 2 * ρ * √(‖c‖ ^ 2 - inner ℝ e c ^ 2)","l":263,"f":"Causalean/PO/ID/Partial/SupportFunction/Sensitivity.lean"},{"n":"Causalean.PartialID.width_nonneg","k":"theorem","m":"Causalean.PO.ID.Partial.SupportFunction.Interval","a":"PO/ID/Partial/SupportFunction","t":2,"d":"The identified width is nonnegative.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {C : Set E} {d : E}, C.Nonempty → BddAbove ((fun x => inner ℝ d x) '' C) → BddAbove ((fun x => inner ℝ (-d) x) '' C) → 0 ≤ Causalean.PartialID.width C d","l":131,"f":"Causalean/PO/ID/Partial/SupportFunction/Interval.lean"},{"n":"Causalean.SCM","k":"structure","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"A generalized structural causal model bundles a SWIG graph together with an edge-type assignment recording functional restrictions on each edge, the requirement that each fixed parameter and its corresponding random node share the same value space, a deterministic structural function for each observed node mapping its parents' values to its own, and an independent probability law on each latent root node.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type (max u_2 u_3)","l":41,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode","k":"inductive","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain node type consists of the treatment node and the outcome node.","s":"Type","l":61,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.dag","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain directed acyclic graph is the two-node graph whose only directed edge is from treatment $d$ to outcome $y$.","s":"Causalean.DAG Causalean.SCM.Assumptions.BoolChainNode","l":108,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.edge","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain edge relation holds for the ordered pair consisting of treatment $d$ and outcome $y$, and holds for no other ordered pair of Boolean-chain nodes.","s":"Causalean.SCM.Assumptions.BoolChainNode → Causalean.SCM.Assumptions.BoolChainNode → Prop","l":79,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.edgeDecidable","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"For every ordered pair of Boolean-chain nodes, a decision procedure for whether the pair is a directed edge is provided.","s":"DecidableRel Causalean.SCM.Assumptions.BoolChainNode.edge","l":85,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.instFintype","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"A finite enumeration of the Boolean-chain node type is provided by the two-element collection containing treatment and outcome together with the assertion that every Boolean-chain node belongs to that collection.","s":"Fintype Causalean.SCM.Assumptions.BoolChainNode","l":69,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.BoolChainNode.topo","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain topological ranking assigns rank zero to treatment $d$ and rank one to outcome $y$.","s":"Causalean.SCM.Assumptions.BoolChainNode → ℕ","l":94,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.MonotoneMechanism","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"For a finite population of nodes with measurable ordered value spaces, a child node, a parent node, and a structural causal model, the monotone-mechanism condition holds precisely when the child is observed, the parent is a parent of that child, and for every two parent-value assignments that agree at all other parent coordinates and are ordered at the designated parent coordinate, the child's structural-function value is ordered in the same direction.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [(n : N) → Preorder (Ω n)] → Causalean.SWIGNode N → Causalean.SWIGNode N → Causalean.SCM N Ω → Prop","l":38,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.antitoneBoolSCM","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The antitone Boolean structural causal model is the Boolean-chain model with no fixed or latent nodes, nonparametric edge labels, and the reversing structural equation.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ","l":250,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.antitoneBoolSCM_violates","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The reversing Boolean structural causal model violates monotonicity of the outcome mechanism in the designated parent coordinate.","s":"¬Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d) Causalean.SCM.Assumptions.antitoneBoolSCM","l":305,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainDParent","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The designated Boolean-chain outcome-parent coordinate is the random treatment node $d$, the sole parent of the random outcome node $y$.","s":"↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))","l":160,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainMeasurableSpace","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"For each Boolean-chain node, the measurable-space structure on its Boolean value space is the discrete measurable space, with the treatment case and the outcome case specified separately.","s":"(n : Causalean.SCM.Assumptions.BoolChainNode) → MeasurableSpace (Causalean.SCM.Assumptions.boolChainΩ n)","l":150,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainPreorder","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"For each Boolean-chain node, the preorder on its Boolean value space is the usual Boolean preorder, with the treatment case and the outcome case specified separately.","s":"(n : Causalean.SCM.Assumptions.BoolChainNode) → Preorder (Causalean.SCM.Assumptions.boolChainΩ n)","l":155,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainSWIG","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain single-world intervention graph has random observed treatment $d$ and outcome $y$, no fixed nodes, and no unobserved nodes, with the Boolean-chain directed graph.","s":"Causalean.SWIGGraph Causalean.SCM.Assumptions.BoolChainNode","l":119,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolChainΩ","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The Boolean-chain value-space assignment gives both treatment $d$ and outcome $y$ the two-point Boolean value space.","s":"Causalean.SCM.Assumptions.BoolChainNode → Type","l":146,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolParentAssignment","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"For a Boolean value, the Boolean-chain outcome-parent assignment assigns that value to the designated treatment-parent coordinate and false to every other parent coordinate.","s":"Bool → (w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w","l":268,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.boolParentAssignment_boolChainDParent","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"Evaluating the Boolean parent assignment at the designated parent returns the assigned value.","s":"∀ (b : Bool), Causalean.SCM.Assumptions.boolParentAssignment b Causalean.SCM.Assumptions.boolChainDParent = b","l":287,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.copyStructFun","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"For an observed node of the Boolean-chain graph, the Boolean copying structural equation returns false at treatment $d$ and, at outcome $y$, returns the value assigned to the treatment-parent coordinate.","s":"(v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed) → ((w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents ↑v)) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑v","l":166,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.copyStructFun_measurable","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"Each structural function of the copy mechanism on the Boolean chain example is measurable.","s":"∀ (v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed), Measurable (Causalean.SCM.Assumptions.copyStructFun v)","l":202,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.flipStructFun","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"For an observed node of the Boolean-chain graph, the Boolean reversing structural equation returns false at treatment $d$ and, at outcome $y$, returns the Boolean negation of the value assigned to the treatment-parent coordinate.","s":"(v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed) → ((w : ↥(Causalean.SCM.Assumptions.boolChainSWIG.dag.parents ↑v)) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑v","l":184,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.flipStructFun_measurable","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"Each structural function of the flip mechanism on the Boolean chain example is measurable.","s":"∀ (v : ↥Causalean.SCM.Assumptions.boolChainSWIG.observed), Measurable (Causalean.SCM.Assumptions.flipStructFun v)","l":216,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instDecidableEqBoolChainNode","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"DecidableEq Causalean.SCM.Assumptions.BoolChainNode","l":65,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instPreorderSwigΩ","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"For a finite collection of variables whose members can be compared for equality and that have measurable ordered value spaces and each random or fixed copy of a variable, an order on that copy's value space is provided by the order on the corresponding base-variable value space, with the random-copy case and the fixed-copy case treated identically.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [(n : N) → Preorder (Ω n)] → (s : Causalean.SWIGNode N) → Preorder (Causalean.swigΩ Ω s)","l":29,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.instReprBoolChainNode","k":"instance","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":2,"d":"","s":"Repr Causalean.SCM.Assumptions.BoolChainNode","l":65,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.monotoneBoolSCM","k":"def","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The monotone Boolean structural causal model is the Boolean-chain model with no fixed or latent nodes, nonparametric edge labels, and the copying structural equation.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ","l":232,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.Assumptions.monotoneBoolSCM_satisfies","k":"theorem","m":"Causalean.SCM.ID.Assumptions.Monotonicity","a":"SCM/ID/Assumptions","t":1,"d":"The copying Boolean structural causal model satisfies monotonicity of the outcome mechanism in the designated parent coordinate.","s":"Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d) Causalean.SCM.Assumptions.monotoneBoolSCM","l":293,"f":"Causalean/SCM/ID/Assumptions/Monotonicity.lean"},{"n":"Causalean.SCM.BackdoorPositivityAE","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, a finite treatment set, a finite adjustment-node set, the condition that every adjustment node is observed, the condition that every random treatment copy and adjustment node is observed, and fixed-node values, the backdoor product-positivity condition states that the product of the observational treatment marginal and observational adjustment marginal is absolutely continuous with respect to the observational joint law of treatment and adjustment coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (Z : Finset (Causalean.SWIGNode N)) → Z ⊆ M.observed → Finset.image Causalean.SWIGNode.random X ∪ Z ⊆ M.observed → M.FixedValues → Prop","l":70,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.DistMarkovEquiv","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"For a family of value spaces indexed by the base variables and two directed acyclic graphs on the same split-node set, distributional Markov equivalence means that, for every structural causal model with those value spaces and every finite measure on its random-value space, the measure is a global I-map of the first graph if and only if it is a global I-map of the second.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (Ω : N → Type uΩ) → [(n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → Causalean.DAG (Causalean.SWIGNode N) → Prop","l":64,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.DominatedObs","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, and a reference-measure family, observational domination holds exactly when, for every assignment of fixed-node values, the model's observational law is absolutely continuous with respect to the joint reference measure on its observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM.ReferenceMeasures Ω → Prop","l":68,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.Equiv","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and two structural causal models, structural equivalence holds exactly when their single-world intervention graphs are equivalent, their edge-type labels agree on every directed edge, and their structural functions and latent-root probability laws agree.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":279,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.heq_jointKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for jointKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → M₁.jointKernel ≍ M₂.jointKernel","l":245,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_latentProduct","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for latentProduct.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.unobserved = M₂.unobserved → M₁.latentDist ≍ M₂.latentDist → M₁.latentProduct ≍ M₂.latentProduct","l":216,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_obsCondKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"HEq transport for obsCondKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω} (hGraph : M₁.Equivalent M₂.toSWIGGraph), M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → ∀ (Y CC : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hCC₁ : CC ⊆ M₁.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))], have hY₂ := fun ⦃v⦄ hv => Eq.mpr (id (congrArg (fun _a => v ∈ _a) (Eq.symm hGraph.right.right.left))) (hY₁ hv); have hCC₂ := fun ⦃v⦄ hv => Eq.mpr (id (congrArg (fun _a => v ∈ _a) (Eq.symm hGraph.right.right.left))) (hCC₁ hv); M₁.obsCondKernel Y CC hY₁ hCC₁ ≍ M₂.obsCondKernel Y CC hY₂ hCC₂","l":378,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.heq_obsKernel","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":1,"d":"HEq transport for obsKernel. For two structural causal models M₁, M₂ that are structurally equivalent, have heterogeneously equal structural functions, and have heterogeneously equal latent-root distributions, then their observational kernels are heterogeneously equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → M₁.latentDist ≍ M₂.latentDist → M₁.obsKernel ≍ M₂.obsKernel","l":296,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.Equiv.refl","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Structural equivalence is reflexive.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.Equiv M","l":295,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.symm","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Structural equivalence is symmetric.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equiv M₂ → M₂.Equiv M₁","l":303,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Equiv.trans","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"Structural equivalence is transitive: if M₁ and M₂ are structurally equivalent and M₂ and M₃ are structurally equivalent, then M₁ and M₃ are structurally equivalent.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ M₃ : Causalean.SCM N Ω}, M₁.Equiv M₂ → M₂.Equiv M₃ → M₁.Equiv M₃","l":313,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.Examples.BackDoor.BDNode","k":"inductive","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor-example node type consists of the treatment vertex, the outcome vertex, the observed-confounder vertex, the latent root for the confounder, the latent root for treatment, and the latent root for the outcome.","s":"Type","l":70,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdDAG","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor-adjustment directed acyclic graph has the specified backdoor edge relation and the displayed topological ordering, and is therefore acyclic.","s":"Causalean.DAG Causalean.SCM.Examples.BackDoor.BDNode","l":151,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdEdge","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The backdoor-edge relation contains exactly the arrow from the observed confounder to treatment, the arrow from the observed confounder to outcome, the arrow from treatment to outcome, the arrow from the first latent root to the confounder, the arrow from the second latent root to treatment, and the arrow from the third latent root to outcome; all other ordered pairs have no edge.","s":"Causalean.SCM.Examples.BackDoor.BDNode → Causalean.SCM.Examples.BackDoor.BDNode → Prop","l":114,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdEdgeTypes","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The edge-type assignment for the backdoor graph classifies every graph edge as nonparametric.","s":"Causalean.EdgeTypeAssignment Causalean.SCM.Examples.BackDoor.bdDAG","l":373,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdSWIG","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The pre-intervention SWIG graph for the backdoor example has the backdoor directed acyclic graph, no fixed nodes, treatment, outcome, and confounder as observed random nodes, and the three latent roots as unobserved random nodes.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.BackDoor.BDNode","l":208,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdTopo","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"The topological-order label for the backdoor graph assigns label 0 to the first latent root, label 1 to the second latent root, label 2 to the third latent root, label 3 to the observed confounder, label 4 to treatment, and label 5 to outcome.","s":"Causalean.SCM.Examples.BackDoor.BDNode → ℕ","l":132,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.bdTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"Every edge of the backdoor-example graph connects a node with a strictly smaller assigned order label to one with a strictly larger label — the chosen ordering is a valid topological order for the graph.","s":"∀ (u v : Causalean.SCM.Examples.BackDoor.BDNode), Causalean.SCM.Examples.BackDoor.bdEdge u v → Causalean.SCM.Examples.BackDoor.bdTopo u < Causalean.SCM.Examples.BackDoor.bdTopo v","l":141,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instDecidableEqBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.BackDoor.BDNode","l":78,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instDecidableRelBDNodeBdEdge","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"For every ordered pair of backdoor-example vertices, a decision procedure for whether the pair is a directed edge is provided.","s":"DecidableRel Causalean.SCM.Examples.BackDoor.bdEdge","l":124,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instFintypeBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"A finite enumeration of the backdoor-example node type is provided by the collection of its six named vertices together with the assertion that every backdoor-example vertex belongs to that collection.","s":"Fintype Causalean.SCM.Examples.BackDoor.BDNode","l":105,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instReprBDNode","k":"instance","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":2,"d":"A textual representation structure for backdoor-example vertices is provided by rendering each vertex through its fully qualified constructor name.","s":"Repr Causalean.SCM.Examples.BackDoor.BDNode","l":101,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.BackDoor.instReprBDNode.repr","k":"def","m":"Causalean.SCM.Examples.BackDoor","a":"SCM/Examples","t":1,"d":"For each vertex of the backdoor-adjustment DAG and each natural-number precedence level, this method returns a formatted textual rendering of that vertex. There are no additional hypotheses or side conditions.","s":"Causalean.SCM.Examples.BackDoor.BDNode → ℕ → Std.Format","l":84,"f":"Causalean/SCM/Examples/BackDoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The node set of the continuous-backdoor example consists of three positions representing, in order, the confounder, treatment, and outcome.","s":"Type","l":78,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.CBΩ","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The value-space assignment for the continuous-backdoor example gives every node the real-valued state space.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Type","l":81,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Xidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The treatment node index is the second of the three node positions in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":86,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Yidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The outcome node index is the third of the three node positions in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":88,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.Zidx","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The confounder node index is the first of the three node positions in the continuous-backdoor example.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":84,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbDAG","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The directed acyclic graph of the continuous backdoor example has the stated three-node edge relation and topological ordering, and is acyclic.","s":"Causalean.DAG Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":117,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbEdge","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The edge relation of the continuous-backdoor graph holds exactly when the Boolean edge indicator is true, namely for arrows from confounder to treatment or outcome and from treatment to outcome.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Prop","l":102,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbEdgeBool","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The Boolean edge indicator is true exactly for an arrow from confounder to treatment, from confounder to outcome, or from treatment to outcome.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Causalean.SCM.Examples.ContinuousBackdoor.CBNode → Bool","l":95,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The pre-intervention SWIG graph for the continuous-backdoor example has no fixed or unobserved nodes and has the confounder, treatment, and outcome as observed random nodes.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.ContinuousBackdoor.CBNode","l":127,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbTopo","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The topological-order label of the continuous-backdoor graph is each node's position, so it places the confounder before treatment and treatment before outcome.","s":"Causalean.SCM.Examples.ContinuousBackdoor.CBNode → ℕ","l":109,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cbTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"Every edge in the continuous-backdoor graph points from an earlier to a later node in the chosen topological order.","s":"∀ (u v : Causalean.SCM.Examples.ContinuousBackdoor.CBNode), Causalean.SCM.Examples.ContinuousBackdoor.cbEdge u v → Causalean.SCM.Examples.ContinuousBackdoor.cbTopo u < Causalean.SCM.Examples.ContinuousBackdoor.cbTopo v","l":112,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment's fixed node is not already fixed in the computable continuous-backdoor SWIG graph.","s":"∀ D ∈ {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}, Causalean.SWIGNode.fixed D ∉ Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.fixed","l":217,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"Treatment random-image together with the adjustment set is observed.","s":"Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ∪ {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":286,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment random-image is observed.","s":"Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":278,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment's observed random node belongs to the computable continuous-backdoor SWIG graph.","s":"∀ D ∈ {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}, Causalean.SWIGNode.random D ∈ Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.observed","l":204,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"{random Y} is an observed node set.","s":"{Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":264,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"{random Z} (the adjustment set) is an observed node set.","s":"{Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} ⊆ Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.observed","l":271,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_backdoor_criterion","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The observed confounder satisfies the graphical backdoor criterion relative to the treatment and outcome nodes in the continuous-backdoor example graph.","s":"Causalean.SCM.Examples.ContinuousBackdoor.cbSWIGGraph.backdoorCriterion {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}","l":223,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_backdoor_identified","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"Backdoor adjustment identity on the continuous-backdoor example. Fix an assignment s0 of values to the model's fixed background variables. If for every post-intervention background assignment, the do(X)-intervened marginal law of Z is absolutely continuous with respect to its purely observational marginal law — the Rule-2 joint overlap condition, and the product of the observational marginal laws of the treatment's random image and of Z is absolutely continuous with respect to their joint observational law — the joint positivity condition, then almost everywhere under that product measure, the conditional law of the outcome Y given Z under the intervention that fixes the treatment equals the purely observational conditional law of Y given both the treatment and Z — the backdoor Rule-2 adjustment identity.","s":"∀ (s0 : Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.FixedValues), (∀ (s : (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSet {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed).FixedValues), Causalean.SCM.ID.Rule2JointOverlap Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs s) → (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ) × Causalean.ValuesOn {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Xr_obs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) (Causalean.swigΩ Causalean.SCM.Examples.ContinuousBackdoor.CBΩ)) (MeasureTheory.Measure.map (Causalean.valuesProjection Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsKernel s0))), ((Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSet {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed).obsCondKernel {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx} (Eq.symm (Causalean.SCM.fixSet_observed Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed) ▸ Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs) (Eq.symm (Causalean.SCM.fixSet_observed Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed) ▸ Causalean.SCM.Examples.ContinuousBackdoor.cb_Zobs)) (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.fixSetExtend {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} Causalean.SCM.Examples.ContinuousBackdoor.cb_Xrand_obs Causalean.SCM.Examples.ContinuousBackdoor.cb_Xfixed s0 p.1, p.2) = (Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM.obsCondKernel {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx} ∪ {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}) Causalean.SCM.Examples.ContinuousBackdoor.cb_Yobs Causalean.SCM.Examples.ContinuousBackdoor.cb_XrZ_obs) (s0, Causalean.valuesUnionMk p.1 p.2)","l":301,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_disj_XrZ","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The treatment random-image is disjoint from the adjustment set.","s":"Disjoint (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx}) {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Zidx}","l":296,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.cb_disj_YXr","k":"theorem","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"The outcome is disjoint from the treatment random-image.","s":"Disjoint {Causalean.SWIGNode.random Causalean.SCM.Examples.ContinuousBackdoor.Yidx} (Finset.image Causalean.SWIGNode.random {Causalean.SCM.Examples.ContinuousBackdoor.Xidx})","l":291,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.continuousBackdoorSCM","k":"def","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":1,"d":"The continuous-backdoor structural causal model has the specified real-valued pre-intervention graph, no latent variables, nonparametric edges, and constant-zero structural responses for every observed node.","s":"Causalean.SCM Causalean.SCM.Examples.ContinuousBackdoor.CBNode Causalean.SCM.Examples.ContinuousBackdoor.CBΩ","l":158,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.ContinuousBackdoor.instDecidableRelCBNodeCbEdge","k":"instance","m":"Causalean.SCM.Examples.ContinuousBackdoor","a":"SCM/Examples","t":2,"d":"For every ordered pair of continuous-backdoor vertices, a decision procedure for whether the pair is a directed edge is provided.","s":"DecidableRel Causalean.SCM.Examples.ContinuousBackdoor.cbEdge","l":105,"f":"Causalean/SCM/Examples/ContinuousBackdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.FDNode","k":"inductive","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor-example node type consists of the latent-confounder vertex, the treatment vertex, the mediator vertex, and the outcome vertex.","s":"Type","l":34,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdDAG","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor directed acyclic graph has the specified frontdoor edge relation and topological ordering, and is acyclic.","s":"Causalean.DAG Causalean.SCM.Examples.Frontdoor.FDNode","l":85,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdEdge","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The frontdoor-edge relation contains exactly the arrow from the latent confounder to treatment, the arrow from the latent confounder to outcome, the arrow from treatment to the mediator, and the arrow from the mediator to outcome; all other ordered pairs have no edge.","s":"Causalean.SCM.Examples.Frontdoor.FDNode → Causalean.SCM.Examples.Frontdoor.FDNode → Prop","l":53,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdSWIG","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The pre-intervention SWIG graph for the frontdoor example has the frontdoor directed acyclic graph, no fixed nodes, treatment, mediator, and outcome as observed random nodes, and the latent confounder as an unobserved random node.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.Frontdoor.FDNode","l":95,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdTopo","k":"def","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"The topological-order label for the frontdoor graph assigns label 0 to the latent confounder, label 1 to treatment, label 2 to the mediator, and label 3 to outcome.","s":"Causalean.SCM.Examples.Frontdoor.FDNode → ℕ","l":69,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.fdTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":1,"d":"Every edge of the frontdoor-example graph connects a node with a smaller assigned order label to one with a larger label, so the chosen ordering is a valid topological order.","s":"∀ (u v : Causalean.SCM.Examples.Frontdoor.FDNode), Causalean.SCM.Examples.Frontdoor.fdEdge u v → Causalean.SCM.Examples.Frontdoor.fdTopo u < Causalean.SCM.Examples.Frontdoor.fdTopo v","l":76,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instDecidableEqFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.Frontdoor.FDNode","l":40,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instDecidableRelFDNodeFdEdge","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"For every ordered pair of frontdoor-example vertices, a decision procedure for whether the pair is a directed edge is provided.","s":"DecidableRel Causalean.SCM.Examples.Frontdoor.fdEdge","l":61,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instFintypeFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"A finite enumeration of the frontdoor-example node type is provided by the collection of its four named vertices together with the assertion that every frontdoor-example vertex belongs to that collection.","s":"Fintype Causalean.SCM.Examples.Frontdoor.FDNode","l":44,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.Frontdoor.instReprFDNode","k":"instance","m":"Causalean.SCM.Examples.Frontdoor","a":"SCM/Examples","t":2,"d":"","s":"Repr Causalean.SCM.Examples.Frontdoor.FDNode","l":40,"f":"Causalean/SCM/Examples/Frontdoor.lean"},{"n":"Causalean.SCM.Examples.IV.IVNode","k":"inductive","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"For the instrumental-variable example, the set of node categories consists of an instrument, a treatment, an outcome, and an unobserved confounder.","s":"Type","l":47,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instDecidableEqIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"","s":"DecidableEq Causalean.SCM.Examples.IV.IVNode","l":53,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instDecidableRelIVNodeIvEdge","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"Decidability of the instrumental-variable edge relation determines, for every ordered pair of instrumental-variable nodes, whether that pair is an edge.","s":"DecidableRel Causalean.SCM.Examples.IV.ivEdge","l":95,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instFintypeIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"The finite enumeration of instrumental-variable nodes lists the instrument, treatment, outcome, and unobserved confounder, and establishes that every such node occurs in that list.","s":"Fintype Causalean.SCM.Examples.IV.IVNode","l":78,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instReprIVNode","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"A textual rendering of an instrumental-variable node assigns to each of the four nodes its fully qualified constructor name by the stated rendering rule.","s":"Repr Causalean.SCM.Examples.IV.IVNode","l":74,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.instReprIVNode.repr","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"For each vertex of the instrumental-variable DAG and each natural-number precedence level, this method returns a formatted textual rendering of that vertex. There are no additional hypotheses or side conditions.","s":"Causalean.SCM.Examples.IV.IVNode → ℕ → Std.Format","l":59,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivDAG","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable directed acyclic graph has the stated instrumental-variable edge relation and topological ordering, and is acyclic.","s":"Causalean.DAG Causalean.SCM.Examples.IV.IVNode","l":122,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivDoDGraph","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The intervention graph for treatment in the instrumental-variable example is the single-world intervention graph obtained by fixing treatment in the instrumental-variable directed acyclic graph.","s":"Causalean.DAG (Causalean.SWIGNode Causalean.SCM.Examples.IV.IVNode)","l":391,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivEdge","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The instrumental-variable edge relation contains exactly the arrow from instrument to treatment, the arrow from treatment to outcome, the arrow from the latent confounder to treatment, and the arrow from the latent confounder to outcome; all other ordered pairs have no edge.","s":"Causalean.SCM.Examples.IV.IVNode → Causalean.SCM.Examples.IV.IVNode → Prop","l":87,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivEdgeTypes","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The edge-type assignment for the instrumental-variable graph classifies the instrument-to-treatment edge as strictly increasing and classifies every other ordered pair as nonparametric.","s":"Causalean.EdgeTypeAssignment Causalean.SCM.Examples.IV.ivDAG","l":281,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivSCM","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The toy instrumental-variable structural causal model has the stated instrumental-variable graph, one-point node value spaces, constant structural functions, and a point-mass distribution for the latent root.","s":"Causalean.SCM Causalean.SCM.Examples.IV.IVNode Causalean.SCM.Examples.IV.ivΩ","l":329,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivSWIGGraph","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The pre-intervention SWIG graph for the instrumental-variable example has the instrumental-variable directed acyclic graph, no fixed nodes, instrument, treatment, and outcome as observed random nodes, and the latent confounder as an unobserved random node.","s":"Causalean.SWIGGraph Causalean.SCM.Examples.IV.IVNode","l":228,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivTopo","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The topological-order label for the instrumental-variable graph assigns label 0 to the latent confounder, label 1 to the instrument, label 2 to treatment, and label 3 to outcome.","s":"Causalean.SCM.Examples.IV.IVNode → ℕ","l":105,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivTopo_lt","k":"theorem","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"Every edge of the instrumental-variable example graph connects a node with a smaller assigned order label to one with a larger label, so the chosen ordering is a valid topological order.","s":"∀ (u v : Causalean.SCM.Examples.IV.IVNode), Causalean.SCM.Examples.IV.ivEdge u v → Causalean.SCM.Examples.IV.ivTopo u < Causalean.SCM.Examples.IV.ivTopo v","l":112,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivΩ","k":"def","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":1,"d":"The value-space assignment for the toy instrumental-variable model gives every node a one-point state space.","s":"Causalean.SCM.Examples.IV.IVNode → Type","l":320,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.IV.ivΩ_measurable","k":"instance","m":"Causalean.SCM.Examples.IV","a":"SCM/Examples","t":2,"d":"For each node of the toy instrumental-variable model, the measurable structure on its one-point value space is the trivial measurable structure.","s":"(n : Causalean.SCM.Examples.IV.IVNode) → MeasurableSpace (Causalean.SCM.Examples.IV.ivΩ n)","l":326,"f":"Causalean/SCM/Examples/IV.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"For a Boolean-chain structural causal model and a Boolean treatment value, the outcome-parent assignment gives each coordinate corresponding to a parent of the outcome that treatment value when it is the designated treatment parent and false otherwise.","s":"(M : Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ) → Bool → (w : ↥(M.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y))) → Causalean.swigΩ Causalean.SCM.Assumptions.boolChainΩ ↑w","l":38,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn_parent","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The designated parent coordinate receives the assigned Boolean value.","s":"∀ (M : Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ) (hparent : Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d ∈ M.dag.parents (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y)) (b : Bool), Causalean.SCM.Examples.MonotoneCounterfactualBound.boolParentAssignmentIn M b ⟨Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d, hparent⟩ = b","l":55,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.boolScore","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"For a Boolean value, its real-valued score is one when it is true and zero when it is false.","s":"Bool → ℝ","l":31,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"Structural monotonicity of the outcome mechanism in the treatment parent forces every response contrast in the compatible class to lie in the valid [0, 1] partial-identification interval.","s":"Causalean.SCM.PartialID.compatibleInterval Causalean.SCM.Assumptions.boolChainSWIG (Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d)) Causalean.SCM.Assumptions.monotoneBoolSCM Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast ⊆ Set.Icc 0 1","l":108,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound_assumption_satisfiable","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"The class of structural causal models compatible with the Boolean-chain graph and monotone in the outcome's treatment parent, evaluated against the copying model as reference, is nonempty.","s":"Causalean.SCM.PartialID.CompatibleSCM Causalean.SCM.Assumptions.boolChainSWIG (Causalean.SCM.Assumptions.MonotoneMechanism (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.y) (Causalean.SWIGNode.random Causalean.SCM.Assumptions.BoolChainNode.d)) Causalean.SCM.Assumptions.monotoneBoolSCM Causalean.SCM.Assumptions.monotoneBoolSCM","l":153,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneCounterfactualBound_fails_without_monotonicity","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"Without the monotonicity restriction, the unconstrained compatible class's response contrast need not lie in [0, 1].","s":"¬Causalean.SCM.PartialID.compatibleInterval Causalean.SCM.Assumptions.boolChainSWIG (fun x => True) Causalean.SCM.Assumptions.antitoneBoolSCM Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast ⊆ Set.Icc 0 1","l":189,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast","k":"def","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":1,"d":"For each Boolean-chain structural causal model, the monotone response contrast is the real-valued difference between the scored outcome response when the designated treatment parent is true and when it is false, and it is zero whenever the outcome is not observed or the designated treatment parent is absent.","s":"Causalean.SCM Causalean.SCM.Assumptions.BoolChainNode Causalean.SCM.Assumptions.boolChainΩ → ℝ","l":70,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast_antitoneBoolSCM","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The reversing Boolean SCM has response contrast minus one.","s":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast Causalean.SCM.Assumptions.antitoneBoolSCM = -1","l":177,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast_monotoneBoolSCM","k":"theorem","m":"Causalean.SCM.Examples.MonotoneCounterfactualBound","a":"SCM/Examples","t":2,"d":"The copying Boolean SCM has response contrast one.","s":"Causalean.SCM.Examples.MonotoneCounterfactualBound.monotoneResponseContrast Causalean.SCM.Assumptions.monotoneBoolSCM = 1","l":165,"f":"Causalean/SCM/Examples/MonotoneCounterfactualBound.lean"},{"n":"Causalean.SCM.FixedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the fixed-value assignments assign one value to every fixed intervention node of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":115,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.FullCondIndep","k":"def","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"For a population of nodes that is finite and has decidable equality, a value-space assignment to those nodes with a measurable structure on every assigned value space, a structural causal model whose full random state is a standard Borel space, three sets of random nodes that are each contained in the model's random-node set, and a finite measure on the full random state, full conditional independence asserts that the coordinate values of the first and second node sets are conditionally independent given the σ-algebra generated by the coordinate values of the third set.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ M.randomVars → Y ⊆ M.randomVars → Z ⊆ M.randomVars → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":33,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.FullCondIndep_iff","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Full conditional independence of two node sets given a third, under a measure on the model's random values, is exactly conditional independence of the two coordinate projections given the σ-algebra generated by the conditioning projection.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] (X Y Z : Finset (Causalean.SWIGNode N)) (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) (μ : MeasureTheory.Measure M.RandomValues) [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ ↔ ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ","l":56,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.ID.CFactorReachableRec","k":"inductive","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite collection of distinguishable node labels and a SWIG graph, the recursive c-factor reachability relation relates any source node set $T$ to any target node set $C$ when either the target is nonempty, is contained in the source, and its observed ancestral set in the source-induced graph is exactly the target, or the target is nonempty and contained in the source, that ancestral set is neither the target nor the source, and the target is recursively reachable from its containing c-component in the ancestral induced graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":80,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.CFactorReachableRec.target_subset","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The target set of a recursive c-factor reachability certificate is contained in the source set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {G : Causalean.SWIGGraph N} {T C : Finset (Causalean.SWIGNode N)}, Causalean.SCM.ID.CFactorReachableRec G T C → C ⊆ T","l":1073,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.CausalQuery","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a finite set of node labels, measurable value spaces assigned to those nodes, and a result space, a causal query maps each structural causal model on those nodes to one value in the result space.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type u_4 → Type (max (max u_4 u_3) u_2)","l":62,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.DiscreteID.DiscretePositive","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"For a finite node-label set, measurable node-value spaces, and a structural causal model, discrete positivity means that, at every assignment of its fixed variables, the model's observational probability measure assigns nonzero mass to every assignment of its observed variables.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":46,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"For a measure on a measurable space, positive point mass means that every point of that space has nonzero singleton mass.","s":"{α : Type u_3} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → Prop","l":33,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.eq_of_ae_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Under positive point mass at every point, almost-everywhere equality is pointwise equality.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α}, Causalean.SCM.ID.DiscreteID.PositiveMass μ → ∀ {f g : α → β}, f =ᵐ[μ] g → ∀ (x : α), f x = g x","l":38,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.map_of_surjective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Positive point mass is preserved by a measurable surjection.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] {μ : MeasureTheory.Measure α} {f : α → β}, Measurable f → Causalean.SCM.ID.DiscreteID.PositiveMass μ → Function.Surjective f → Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map f μ)","l":57,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.map_valuesProjection","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"If a measure assigns nonzero point mass to every value assignment on the full index set, then restricting to the values on a subset of coordinates reached by projection preserves this: the pushed-forward measure still assigns nonzero point mass to every value assignment on that subset.","s":"∀ {M : Type u_3} {I J : Finset M} {Ω' : M → Type u_4} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : M), Nonempty (Ω' m)] [MeasurableSingletonClass (Causalean.ValuesOn J Ω')] {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω')}, Causalean.SCM.ID.DiscreteID.PositiveMass μ → ∀ (hJI : J ⊆ I), Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map (Causalean.valuesProjection hJI) μ)","l":79,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.PositiveMass.of_map_injective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"Positive point mass can be pulled back across an injective measurable map when the pushed-forward measure is positive at every image value.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] {μ : MeasureTheory.Measure α} {f : α → β}, Measurable f → Function.Injective f → Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map f μ) → Causalean.SCM.ID.DiscreteID.PositiveMass μ","l":96,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.StandardDiscretePositive","k":"def","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":1,"d":"For a finite node-label set, measurable node-value spaces, and a structural causal model, standard discrete positivity means that the model is standard and has discrete positivity.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":129,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For a measure on a product of two measurable spaces and a value of the second coordinate, the conditional denominator is the singleton mass of that value under the measure's second-coordinate marginal.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure (α × β) → β → ENNReal","l":64,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator_apply","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure (α × β)) (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b = Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map Prod.snd μ) b","l":290,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalDenominator_ne_zero_of_positive_marginal","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Positive","a":"SCM/ID/DiscreteID","t":2,"d":"A positive marginal point mass is exactly the nonzero denominator needed by the finite conditional-mass ratio.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure (α × β)}, Causalean.SCM.ID.DiscreteID.PositiveMass (MeasureTheory.Measure.map Prod.snd μ) → ∀ (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ 0","l":120,"f":"Causalean/SCM/ID/DiscreteID/Positive.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For a measure on a product of two measurable spaces, a value of the first coordinate, and a value of the second coordinate, the conditional mass is the joint singleton mass at the two values divided by the singleton mass of the second value under the second-coordinate marginal.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure (α × β) → α → β → ENNReal","l":51,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.conditionalMass_mul_denominator","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Multiplying a discrete conditional mass by its actual denominator recovers the joint point mass, provided that denominator is nonzero and finite.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure (α × β)) (a : α) (b : β), Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ 0 → Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b ≠ ⊤ → Causalean.SCM.ID.DiscreteID.conditionalMass μ a b * Causalean.SCM.ID.DiscreteID.conditionalDenominator μ b = Causalean.SCM.ID.DiscreteID.singletonMass μ (a, b)","l":296,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.eq_of_ae_eq_of_forall_singletonMass_ne_zero","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"An almost-everywhere equality is pointwise when every singleton has nonzero mass.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → β}, (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x ≠ 0) → f =ᵐ[μ] g → ∀ (x : α), f x = g x","l":84,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.marginalMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For a measure, a map to a second measurable space, proof that the map is measurable, and a value in that second space, the marginal mass is the singleton mass of that value under the pushforward of the measure by the map.","s":"{α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → MeasureTheory.Measure α → (f : α → β) → Measurable f → β → ENNReal","l":43,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.measure_eq_of_singletonMass_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Measures on a countable space are equal when all singleton masses agree.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [Countable α] {μ ν : MeasureTheory.Measure α}, (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x = Causalean.SCM.ID.DiscreteID.singletonMass ν x) → μ = ν","l":77,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass","k":"def","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For a measure on a measurable space and a point in that space, the singleton mass is the mass that the measure assigns to the set containing that point alone.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → α → ENNReal","l":36,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_apply","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (x : α), Causalean.SCM.ID.DiscreteID.singletonMass μ x = μ {x}","l":72,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_of_const","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A finite kernel mixture has the common singleton mass of its slices when the finite singleton masses of the mixing measure sum to one.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (y : β) (c : ENNReal), ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = 1 → (∀ (x : α), Causalean.SCM.ID.DiscreteID.singletonMass (κ x) y = c) → Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = c","l":217,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_of_pointMass","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A finite kernel mixture has the singleton mass of one slice when the mixing measure is a point mass at that slice.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Finite α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (x0 : α) (y : β), Causalean.SCM.ID.DiscreteID.singletonMass μ x0 = 1 → (∀ (x : α), x ≠ x0 → Causalean.SCM.ID.DiscreteID.singletonMass μ x = 0) → Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = Causalean.SCM.ID.DiscreteID.singletonMass (κ x0) y","l":239,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_comp_eq_sum","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite source space, the singleton mass of a kernel mixture is the finite weighted sum of the singleton masses of the kernel slices.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (μ.bind ⇑κ) y = ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x * Causalean.SCM.ID.DiscreteID.singletonMass (κ x) y","l":204,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"The singleton mass of a mapped measure is the mass of the corresponding fibre.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = μ (f ⁻¹' {y})","l":105,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_one","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"A pointwise constant measurable map from a probability measure gives unit singleton mass at the constant value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 : β), (∀ (x : α), f x = y0) → μ Set.univ = 1 → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y0 = 1","l":141,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_univ","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"If a measurable map is pointwise constant, the pushed-forward measure has all mass at that constant value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 : β), (∀ (x : α), f x = y0) → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y0 = μ Set.univ","l":113,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_const_eq_zero_of_ne","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"If a measurable map is pointwise constant at y0, the pushed-forward measure has zero singleton mass at every different value.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y0 y : β), (∀ (x : α), f x = y0) → y ≠ y0 → Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = 0","l":126,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.singletonMass_map_eq_sum_fiber","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton space, the singleton mass of a pushed-forward measure is the finite sum of singleton masses over the fiber.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : Fintype α] [MeasurableSingletonClass α] [MeasurableSingletonClass β] [inst_5 : DecidableEq β] (μ : MeasureTheory.Measure α) {f : α → β}, Measurable f → ∀ (y : β), Causalean.SCM.ID.DiscreteID.singletonMass (MeasureTheory.Measure.map f μ) y = ∑ x, if f x = y then Causalean.SCM.ID.DiscreteID.singletonMass μ x else 0","l":152,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.sum_singletonMass_eq_one","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton probability space, singleton masses sum to one.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : Fintype α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ], ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = 1","l":195,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.sum_singletonMass_eq_univ","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"On a finite measurable-singleton space, the sum of all singleton masses is the total mass of the measure.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : Fintype α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α), ∑ x, Causalean.SCM.ID.DiscreteID.singletonMass μ x = μ Set.univ","l":179,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.valuesOn_measure_eq_of_singletonMass_eq","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":1,"d":"For two measures on a finite-coordinate product over countable value spaces, if every value assignment carries the same singleton point mass under both measures, then the two measures are equal.","s":"∀ {M : Type u_1} {Ω' : M → Type u_2} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : M), Countable (Ω' m)] (I : Finset M) {μ ν : MeasureTheory.Measure (Causalean.ValuesOn I Ω')}, (∀ (x : Causalean.ValuesOn I Ω'), Causalean.SCM.ID.DiscreteID.singletonMass μ x = Causalean.SCM.ID.DiscreteID.singletonMass ν x) → μ = ν","l":278,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DiscreteID.valuesProjection_surjective","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Mass","a":"SCM/ID/DiscreteID","t":2,"d":"Coordinate restriction is surjective when every omitted coordinate has at least one default value.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω' : M → Type u_2} [inst : (m : M) → MeasurableSpace (Ω' m)] [∀ (m : ↥I), Nonempty (Ω' ↑m)] (hJI : J ⊆ I), Function.Surjective (Causalean.valuesProjection hJI)","l":259,"f":"Causalean/SCM/ID/DiscreteID/Mass.lean"},{"n":"Causalean.SCM.ID.DistrictIdConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite node-label set, measurable node-value spaces, a structural causal model, a node set, and an intervention target set, the district-identification conclusion is the Q-factor identity conclusion obtained by taking the ancestral node set to be all observed nodes of the model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":645,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.ENNReal.div_mul_common","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Cancelling a common nonzero finite factor preserves a quotient of extended nonnegative reals.","s":"∀ {a b r : ENNReal}, r ≠ 0 → r ≠ ⊤ → a * r / (b * r) = a / b","l":385,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.ENNReal.prod_div_prod","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"A finite product of quotients is the quotient of the finite products when denominators are nonzero and finite.","s":"∀ {ι : Type u_3} [DecidableEq ι] (t : Finset ι) (f g : ι → ENNReal), (∀ i ∈ t, g i ≠ 0) → (∀ i ∈ t, g i ≠ ⊤) → ∏ i ∈ t, f i / g i = (∏ i ∈ t, f i) / ∏ i ∈ t, g i","l":700,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.Finset.prod_ne_top_of_ne_top","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"A finite product of finite extended nonnegative reals is finite.","s":"∀ {ι : Type u_3} [DecidableEq ι] (s : Finset ι) (f : ι → ENNReal), (∀ i ∈ s, f i ≠ ⊤) → ∏ i ∈ s, f i ≠ ⊤","l":397,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.Identifiable","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, and a causal query, identifiability means that every pair of structural causal models whose SWIG graphs both equal the given graph and whose observational probability kernels are equal has the same query value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":99,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.IdentifiableUnder","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, a functional-assumption predicate, a structural-assumption predicate, and a causal query, identifiability under the two assumptions means that every pair of models whose SWIG graphs equal the given graph, which both satisfy each predicate, and which have equal observational probability kernels, has the same query value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → (Causalean.SCM N Ω → Prop) → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":155,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.InterventionTargetSimpConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite node-label set, measurable node-value spaces, a structural causal model, a first intervention target set, and a second intervention target set, the intervention-target simplification conclusion asserts that the two target sets are disjoint, that each listed intervention is well formed, and that intervening first on the first set and then on the second produces a structurally equivalent model to intervening once on their union.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset N → Finset N → Prop","l":451,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.KernelGlobalMarkovOn","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, a finite coordinate set, and a finite measure on its coordinate-value space, the kernel global Markov property means that every three pairwise disjoint coordinate blocks contained in that set which are d-separated in the graph are conditionally independent under the measure, provided the coordinate-value space is standard Borel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":128,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.KernelObsCondIndepOn","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node-label set, measurable node-value spaces, a finite coordinate set, two coordinate blocks, a conditioning coordinate block, proofs that all three blocks are contained in the coordinate set, and a finite measure on the values of that coordinate set, kernel observational conditional independence means that the first two blocks are conditionally independent given the third under that measure, provided the coordinate-value space is standard Borel.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → (D X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ D → Y ⊆ D → Z ⊆ D → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":104,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.NonIdentifiable","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, and a causal query, non-identifiability means that the query is not identifiable from the observational distribution: it is not the case that every two models with that graph and equal observational probability kernels have equal query values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → {α : Type u_3} → Causalean.SWIGGraph N → Causalean.SCM.ID.CausalQuery N Ω α → Prop","l":121,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.QFactorIdentityConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite node-label set, measurable node-value spaces, a structural causal model, a node set, a second node set, and an intervention target set, the Q-factor identity conclusion asserts that, whenever the first node set is ancestrally closed, the second lies among the observed nodes of the induced model, and the targets are observed random nodes not already fixed with no descendants in the second set or its induced Q-factor parents after intervention, the indicated induced and post-intervention conditional kernels are equal at every fixed-value slice, provided their stated finite-measure, standard-Borel, nonemptiness, and countable-generation conditions hold. It uses the induced model, the post-intervention model, and the induced Q-factor parent set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":112,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.QFactorMarginalFixingConclusion","k":"def","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite node-label set, measurable node-value spaces, a structural causal model, a node set, a second node set, and an intervention target set, the Q-factor marginal-fixing conclusion asserts the following. Whenever the first node set is ancestrally closed, the second lies among the observed nodes of the induced model, the intervention targets are observed random nodes not already fixed and have no descendants in the second set after intervention, and a fixed-value slice is chosen, the two specified marginal observational measures on that second set are equal.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset N → Prop","l":69,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.Rule2JointOverlap","k":"def","m":"Causalean.SCM.ID.Overlap","a":"SCM/ID","t":1,"d":"For a finite node-label set, measurable node-value spaces, a structural causal model, an intervention target set, proof that each target is an observed random node not already fixed, an additional observed-node set, proof that the random copies of the targets together with that set are observed, and a fixed-value assignment after intervention, Rule 2 joint overlap means that the post-intervention observational marginal on those nodes is absolutely continuous with respect to the corresponding pre-intervention observational marginal at the projected fixed-value assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M' : Causalean.SCM N Ω) → (Z : Finset N) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) → (W : Finset (Causalean.SWIGNode N)) → Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed → (M'.fixSet Z hZ_obs hZ_fixed).FixedValues → Prop","l":48,"f":"Causalean/SCM/ID/Overlap.lean"},{"n":"Causalean.SCM.ID.cComponentDensityFactor_eq_mechCFactor","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"(A) Tian Lemma 1 (Eq. 37 = Eq. 36). For a standard causal model M, i.e. one with no fixed nodes, a reference family that is faithful — every coordinate value has nonzero reference mass, and an observational kernel at a fixed-value slice s with full point-mass support (every observed assignment has nonzero probability), fix a node set S contained in the observed coordinates that is a full c-component of the model's SWIG graph. Then the observational full-prefix c-component density cComponentDensityFactor on S equals the mechanism c-factor Q[S] (mechCFactor) — the do-complement marginal density of S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), M.isStandard → ∀ (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), S ∈ M.cComponentSet → Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ [inst_5 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_6 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_7 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.cComponentDensityFactor ref s S x = M.mechCFactor ref S hS s x","l":944,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.cComponentSet_nonempty","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"Every c-component in a graph's c-component set contains at least one node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → C.Nonempty","l":71,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachable","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables, a SWIG graph, a proposed containing node set, and a target node set, structural c-factor reachability holds exactly when the target is nonempty, is contained in the proposed set, and is a c-component of the graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":65,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.cFactorReachableRecB","k":"def","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"For a finite population of variables, a SWIG graph, a nonnegative fuel bound, a proposed containing node set, and a target node set, the computable reachability checker returns false at zero fuel, while at positive fuel it accepts exactly the stated one-step or recursively reachable cases.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → ℕ → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Bool","l":36,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachableRecB_sound","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"Soundness of the computable reachability check. If the fuel-bounded checker accepts, the inductive reachability certificate holds.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (fuel : ℕ) (T C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.cFactorReachableRecB G fuel T C = true → Causalean.SCM.ID.CFactorReachableRec G T C","l":81,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.cFactorReachable_base_toRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"The no-fixing certificate is the base case of the recursive one. For a SWIG graph G and a district S, if S is already reachable from its containing district under the plain no-fixing certificate, then S is recursively reachable from its containing district — which, since S is a full c-component, is S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (S : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.cFactorReachable G (Causalean.SCM.ID.containingCComponent G S) S → Causalean.SCM.ID.CFactorReachableRec G (Causalean.SCM.ID.containingCComponent G S) S","l":157,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.component_ref_atom_product_eq_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Multiplying the singleton reference masses for all observed variables in one c-component equals the singleton mass of their joint reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), S ∈ M.cComponentSet → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∏ i with M.cComponentOf ↑(M.observedAt i) = S, (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} = (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hS x}","l":881,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.component_ref_atom_product_eq_jointRef_prefixIn","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"For a subset of an ordered finite graph set, the product of coordinate reference-measure masses at selected values equals the reference measure's mass at their joint singleton outcome.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (S : Finset (Causalean.SWIGNode N)) (hSD : S ⊆ D) (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), ∏ i with ↑(H.nodesAt D i) ∈ S, (ref.μ ↑(H.nodesAt D i)) {xD (H.nodesAt D i)} = (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hSD xD}","l":1196,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.containingCComponent","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables, a SWIG graph, and a finite node set, the containing c-component is the c-component containing an arbitrary member of that set when it is nonempty, and is empty otherwise.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":54,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.containingCComponent_eq_of_mem_of_subset","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"The unique c-component containing a nonempty set S is containingCComponent. Bridges the checker's cComponentSet.any search to the reachability predicate's choice-based containingCComponent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {S C' : Finset (Causalean.SWIGNode N)}, S.Nonempty → C' ∈ G.cComponentSet → S ⊆ C' → Causalean.SCM.ID.containingCComponent G S = C'","l":58,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.containingCComponent_induce_subset","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"A containing c-component in an induced graph lies inside the inducing set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (A C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.containingCComponent (G.induce A) C ⊆ A","l":97,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.containingCComponent_of_mem_cComponentSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"When S is a full c-component of G, its containing district is S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ S ∈ G.cComponentSet, Causalean.SCM.ID.containingCComponent G S = S","l":129,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.containingCComponent_subset_observed","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"The containing c-component is always a set of observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (S : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.containingCComponent G S ⊆ G.observed","l":88,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.defaultInterventionalKernel","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a population of variables with nonempty measurable value spaces, an intervention set, and an outcome-node set, the default interventional kernel is the constant kernel concentrated at an arbitrary outcome assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → [inst_1 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (X : Finset N) → (Y : Finset (Causalean.SWIGNode N)) → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":45,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.district_id","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"Corollary (District identification). For a structural causal model M, a c-component T of M's full graph, and an intervention set Wn, the c-factor of T equals, almost everywhere, the observational conditional obtained by intervening on Wn.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.DistrictIdConclusion M T Wn","l":655,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"T2, abstract density recovery statement. Fix a standard structural causal model M and an intervention target set X for which every targeted node is currently a random observed node and none of its fixed copies is already fixed, an output set Y, and a reference-measure family faithful to the graph. For a district S of the truncated c-component set of the post-intervention ancestral graph and a c-component C of the base graph that is factor-reachable from S, assume every fixed-value assignment gives an observational kernel with everywhere-positive point masses, no intervention target's random form lies in Y, and that an extension map from ancestral assignments to full observed assignments restricts back to the identity and agrees with the intervention values sDo on the targeted coordinates. Then the district factor of S computed from the density of the do-law's ancestral marginal equals, almost everywhere, the full-graph c-component density factor of C evaluated at the extension of the ancestral assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.cFactorReachable M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) C (extend xD)","l":1160,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_core","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"T2 density-recovery core.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.cFactorReachable M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) C (extend xD)","l":1075,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_core_self","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Finite same-district recovery: the do-law's S-district density factor equals S's observational c-component density factor (after extension), for a district S that is already a full observational c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":1000,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doAncestralDistrictDensity_recovered_from_obs_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive district-density recovery from the observational kernel. Fix a standard structural causal model M and an intervention target set X for which every targeted node is currently a random observed node and none of its fixed copies is already fixed, an output set Y, and a reference-measure family faithful to the graph. For a district S of the truncated c-component set of the post-intervention ancestral graph and a c-component C of the base graph that is recursively factor-reachable from S, assume every fixed-value assignment gives an observational kernel with everywhere-positive point masses, no intervention target's random form lies in Y, and that an extension map from ancestral assignments to full observed assignments restricts back to the identity and agrees with the intervention values sDo on the targeted coordinates. Then the district factor of S computed from the density of the do-law's ancestral marginal equals, almost everywhere, the full-graph c-component density factor of C evaluated at the extension of the ancestral assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [inst_4 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (S C : Finset (Causalean.SWIGNode N)), S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y → Causalean.SCM.ID.CFactorReachableRec M.toSWIGGraph C S → C ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => Causalean.SCM.ID.recoveredFactorRec M ref (M.fixSetProj X hObs hFix sDo) C S (extend xD)","l":352,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_cfactor_decomposition","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":1,"d":"Valid-branch kernel equality under a successful no-fixing ID certificate. For two finite structural causal models M₁, M₂ that share the same SWIG graph G, are each dominated by a reference-measure family ref that is faithful to the graph, satisfy discrete positivity of their observational kernels, and have heterogeneously equal observational kernels, if the total interventional query on outcome set Y under intervention X is well formed in both models and X, Y admit a successful no-fixing ID certificate on G, then the two models' post-intervention outcome kernels for Y are heterogeneously equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), M₁.doKernelY X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left (Causalean.SCM.ID.standardFixedValues M₁ hvalid₁.right.right.right) = M₂.doKernelY X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left (Causalean.SCM.ID.standardFixedValues M₂ hvalid₂.right.right.right)","l":504,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_cfactor_decomposition_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Recursive valid-branch kernel equality. For two finite structural causal models M₁, M₂ that share the same SWIG graph G, are each dominated by a reference-measure family ref that is faithful to the graph, satisfy discrete positivity of their observational kernels, and have heterogeneously equal observational kernels, if the total interventional query on outcome set Y under intervention X is well formed in both models and X, Y admit a successful full recursive ID certificate on G, then the two models' post-intervention outcome kernels for Y are heterogeneously equal. This is the same transport as doKernelY_eq_cfactor_decomposition, with the recursive Y-marginal wrapper in place of the no-fixing one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), M₁.doKernelY X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left (Causalean.SCM.ID.standardFixedValues M₁ hvalid₁.right.right.right) = M₂.doKernelY X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left (Causalean.SCM.ID.standardFixedValues M₂ hvalid₂.right.right.right)","l":851,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doKernelY_eq_of_doObsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.DoLawTransport","a":"SCM/ID","t":1,"d":"Structural transport. Fix two causal models that share the same underlying SWIG graph, together with a treatment set X that is a valid intervention in the first model — every node of X is observed and not already fixed there and likewise valid in the second model, and an outcome set Y that is observed in the first model and observed in the second. If the two models' base fixed-value slices coincide and their Y-marginals of the do-observational law agree, up to the type-level identification the shared graph provides, then the resulting post-intervention Y-marginal kernels doKernelY agree at that shared base slice.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (s0₁ : M₁.FixedValues) (s0₂ : M₂.FixedValues), s0₁ ≍ s0₂ → Causalean.SCM.ID.doObsKernelYMarginal M₁ X hObs₁ hFix₁ Y hY₁ ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hObs₂ hFix₂ Y hY₂ → M₁.doKernelY X hObs₁ hFix₁ Y hY₁ s0₁ = M₂.doKernelY X hObs₂ hFix₂ Y hY₂ s0₂","l":61,"f":"Causalean/SCM/ID/DoLawTransport.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, and a query-node set, the post-intervention observed-ancestral marginal kernel maps fixed intervention values to the distribution of the observed ancestors of the query.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → ProbabilityTheory.Kernel (M.fixSet X hObs hFix).FixedValues (Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω))","l":439,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_dominated","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"A faithful finite product reference dominates the post-intervention observed-ancestral marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (s : (M.fixSet X hObs hFix).FixedValues), ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y))","l":133,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_globalMarkovOn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The do-law ancestral marginal is globally Markov with respect to the pure ancestral graph G_X[D]. This is the SCM-to-measure bridge for T1; it does not assert that D is an ancestrally closed SCM support.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (M.fixSet X hObs hFix).RandomValues] [StandardBorelSpace (M.fixSet X hObs hFix).ObservedValues] [∀ (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).jointKernel s)] [∀ (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).obsKernel s)] (s : (M.fixSet X hObs hFix).FixedValues) [inst_9 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)], have A := Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y; let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; have H := (M.fixSet X hObs hFix).induce A; Causalean.SCM.ID.KernelGlobalMarkovOn H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)","l":832,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational density have the same post-intervention law on the observed ancestors selected by a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hvalid₁.left hvalid₁.right.left Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hvalid₂.left hvalid₂.right.left Y","l":154,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_heq_of_obsDensity_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive Tian–Shpitser density core for the observed-ancestral do-law.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hvalid₁.left hvalid₁.right.left Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hvalid₂.left hvalid₂.right.left Y","l":565,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_positiveMass","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For an intervention set X whose random copies are observed and whose fixed copies are not already frozen, a positive observational kernel at every fixed-value assignment, and an outcome set Y disjoint from the random copies of X, the do(X)-law ancestral marginal kernel used in the identification density assembly also has everywhere-positive point mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)","l":1682,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.ID.doObsKernelAncestralMarginal_tian_cfactorization_density","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"ID-specific T1 wrapper. For an intervention target set X where every targeted node is currently a random observed node with no fixed copy already fixed, if the ancestral marginal of the do-law ν_M = (M.fixSet X).obsKernel.map π_D is absolutely continuous with respect to the product reference measure on the ancestral observed set, then its Radon–Nikodym density equals, almost everywhere, the product over the c-components of the induced post-intervention ancestral graph G_X[D] of their Tian district-density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : (M.fixSet X hObs hFix).FixedValues) [StandardBorelSpace (Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω))] [StandardBorelSpace (M.fixSet X hObs hFix).RandomValues] [StandardBorelSpace (M.fixSet X hObs hFix).ObservedValues] [∀ (s' : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).jointKernel s')] [∀ (s' : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((M.fixSet X hObs hFix).obsKernel s')] [inst_12 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)] [inst_13 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_14 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))], ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y)) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s).rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s) ref S x","l":1246,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal","k":"def","m":"Causalean.SCM.ID.DoLawTransport","a":"SCM/ID","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, an outcome-node set contained in the observed nodes, the do-observational outcome marginal kernel maps fixed intervention values to the induced distribution of those outcomes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hObs hFix).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":37,"f":"Causalean/SCM/ID/DoLawTransport.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_eq_ancestralMarginal_map","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"The outcome marginal of the do-law is the projection of the observed-ancestral do-law marginal to the outcome coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed), Causalean.SCM.ID.doObsKernelYMarginal M X hObs hFix Y hY = (Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y).map (Causalean.valuesProjection (Causalean.SCM.ID.subset_fixObservedAncestralSet M X hObs hFix Y hY))","l":63,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_ancestralMarginal_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"If two models with the same SWIG graph have equal observed-ancestral do-law marginals, then their outcome marginals agree.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed), Causalean.SCM.ID.doObsKernelAncestralMarginal M₁ X hObs₁ hFix₁ Y ≍ Causalean.SCM.ID.doObsKernelAncestralMarginal M₂ X hObs₂ hFix₂ Y → Causalean.SCM.ID.doObsKernelYMarginal M₁ X hObs₁ hFix₁ Y hY₁ ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hObs₂ hFix₂ Y hY₂","l":88,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational density have the same post-intervention outcome marginal under a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":426,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsDensity_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive density-to-Y-marginal wrapper. The projection from the observed-ancestral marginal to the query coordinates is identical to the no-fixing proof; only the ancestral density core changes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsDensity ref ≍ M₂.obsDensity ref → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":803,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two dominated models with the same SWIG graph and observational law have the same post-intervention outcome marginal under a successful ID certificate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":475,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.doObsKernelYMarginal_heq_of_obsKernel_heq_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Recursive observational-kernel wrapper. Equal observational kernels give equal observational densities, which feed the recursive density core.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → ∀ (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = G → M₂.toSWIGGraph = G → M₁.DominatedObs ref → M₂.DominatedObs ref → Causalean.SCM.ID.DiscreteID.DiscretePositive M₁ → Causalean.SCM.ID.DiscreteID.DiscretePositive M₂ → M₁.obsKernel ≍ M₂.obsKernel → ∀ (hvalid₁ : Causalean.SCM.ID.interventionalQueryValid X Y M₁) (hvalid₂ : Causalean.SCM.ID.interventionalQueryValid X Y M₂), Causalean.SCM.ID.doObsKernelYMarginal M₁ X hvalid₁.left hvalid₁.right.left Y hvalid₁.right.right.left ≍ Causalean.SCM.ID.doObsKernelYMarginal M₂ X hvalid₂.left hvalid₂.right.left Y hvalid₂.right.right.left","l":828,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extendTianPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a SWIG graph, a finite node set, and an index strictly below its cardinality, the prefix-extension map combines an assignment on the first indexed nodes with an assignment on the next node into an assignment on the one-node-longer prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → {n : ℕ} → (hn : n < D.card) → Causalean.ValuesOn (H.prefixIn D n) (Causalean.swigΩ Ω) × Causalean.ValuesOn {↑(H.nodesAt D ⟨n, hn⟩)} (Causalean.swigΩ Ω) → Causalean.ValuesOn (H.prefixIn D (n + 1)) (Causalean.swigΩ Ω)","l":169,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.extractDistrictObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a population of variables with finite value spaces, an observed-node set, a SWIG graph, an ancestral node set, a target district contained in the observed set, and a nonnegative mass function on observed assignments, the observed-set district-extraction functional is the product of the successive marginal-ratio factors indexed by the graph order of nodes in the target district.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (A : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → A ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":77,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extractDistrictObserved_eq_extractDistrict","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G' : Causalean.SWIGGraph N) (A C' : Finset (Causalean.SWIGNode N)) (hA : A ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.extractDistrictObserved M.observed G' A C' hA q = Causalean.SCM.extractDistrict M.observed G' A C' hA q","l":144,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.extractDistrict_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Extracting an induced district from the local mass on an ancestral set recovers that district's local q-mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (A C' : Finset (Causalean.SWIGNode N)) (hA : A ⊆ M.observed) (hC' : C' ∈ (M.induce A).cComponentSet), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.extractDistrict M.observed (M.induce A) A C' hA (M.qLocalMass s A hA) x = M.qLocalMass s C' (fun x hv => hA (have hHobs := of_eq_true (Eq.trans (congrFun' (congrArg Eq (congrArg Causalean.SWIGGraph.observed (Causalean.SWIGGraph.mk.congr_simp (M.inducedDag ({s ∈ M.fixed | Causalean.iotaMap s ∈ A ∩ M.observed} ∪ A ∩ M.observed ∪ {u ∈ M.unobserved | ∃ v ∈ A ∩ M.observed, M.dag.edge u v})) (M.inducedDag ({x ∈ M.fixed | Causalean.iotaMap x ∈ A} ∪ (A ∪ {x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v}))) (congrArg M.inducedDag (Eq.trans (congr (congrArg Union.union (congr (congrArg Union.union (Finset.filter_congr fun x a => Iff.of_eq (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) (Causalean.iotaMap x)))) (Finset.inter_eq_left.mpr hA))) (Finset.filter_congr fun x a => Iff.of_eq (congrArg Exists (funext fun v => congrFun' (congrArg And (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) v)) (M.dag.edge x v))))) (Finset.union_assoc ({x ∈ M.fixed | Causalean.iotaMap x ∈ A}) A ({x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v})))) ({s ∈ M.fixed | Causalean.iotaMap s ∈ A ∩ M.observed}) ({x ∈ M.fixed | Causalean.iotaMap x ∈ A}) (Finset.filter_congr fun x a => Iff.of_eq (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) (Causalean.iotaMap x))) (A ∩ M.observed) A (Finset.inter_eq_left.mpr hA) ({u ∈ M.unobserved | ∃ v ∈ A ∩ M.observed, M.dag.edge u v}) ({x ∈ M.unobserved | ∃ v ∈ A, M.dag.edge x v}) (Finset.filter_congr fun x a => Iff.of_eq (congrArg Exists (funext fun v => congrFun' (congrArg And (congrFun' (congrArg Membership.mem (Finset.inter_eq_left.mpr hA)) v)) (M.dag.edge x v)))) (Causalean.SWIGGraph.induce._proof_1 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_2 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_3 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_4 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_5 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_6 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_7 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_8 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_9 M.toSWIGGraph A) (Causalean.SWIGGraph.induce._proof_10 M.toSWIGGraph A)))) A) (eq_self A)); hHobs ▸ Causalean.SWIGGraph.cComponentSet_subset_observed (M.induce A) C' hC' hv)) x","l":964,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.fixAncestralSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, and a query-node set, the post-intervention ancestral set is the query set together with all of its ancestors in the intervened graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":113,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixObservedAncestralSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, and a query-node set, the observed post-intervention ancestral set is the intersection of its post-intervention ancestors with the observed nodes of the intervened model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":158,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixObservedAncestralSet_obsParent_closed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The observed post-intervention ancestral support is closed under observed parents in the do-model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (M.fixSet X hObs hFix).ObsParentClosed (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y)","l":420,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixSet_cComponentSet_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Intervening does not change the full c-component partition.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).cComponentSet = M.cComponentSet","l":400,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixSet_cComponentSet_mem","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Membership in the full c-component partition transports across fixSet.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (S : Finset (Causalean.SWIGNode N)), S ∈ (M.fixSet X hObs hFix).cComponentSet ↔ S ∈ M.cComponentSet","l":410,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.fixTruncCComponentSet","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, and a query-node set, the truncated c-component index set is the set of c-components of the intervened graph induced on the query's post-intervention ancestors.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Finset (Causalean.SWIGNode N) → Finset (Finset (Causalean.SWIGNode N))","l":131,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.idAlgorithm","k":"def","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"For a finite population of variables, a fuel bound, a SWIG graph, an intervention set, and an outcome-node set, the executable ID checker returns true exactly when the intervention is valid, the outcomes are observed and disjoint from intervention random nodes, and every post-intervention ancestral c-component passes the fuel-bounded reachability check.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → ℕ → Causalean.SWIGGraph N → Finset N → Finset (Causalean.SWIGNode N) → Bool","l":127,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idAlgorithm_sound","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":1,"d":"Soundness of the executable ID algorithm. When the runnable checker idAlgorithm returns true on the graph G, intervention set X, query Y, and the given fuel bound, the interventional query P(Y ∣ do(X)) is identified within the standard discrete positive model class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (fuel : ℕ) (G : Causalean.SWIGGraph N) (X : Finset N) (Y : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.idAlgorithm fuel G X Y = true → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":171,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idAlgorithm_success_toRec","k":"theorem","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"A successful run yields the graphical certificate. idAlgorithm … = true implies the declarative recursive success certificate idSucceedsRec.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (fuel : ℕ) (G : Causalean.SWIGGraph N) (X : Finset N) (Y : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.idAlgorithm fuel G X Y = true → Causalean.SCM.ID.idSucceedsRec X Y G","l":150,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.idSucceeds","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables, an intervention set, an outcome-node set, and a SWIG graph, the no-additional-fixing ID success certificate holds when the intervention is valid, the outcome nodes are observed, no intervention variable's random node is an outcome node, and every c-component of the post-intervention ancestral induced graph is structurally c-factor reachable from its containing original c-component.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N → Prop","l":81,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.idSucceedsRec","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite collection of distinguishable node labels, an intervention variable set, an outcome-node set, and a SWIG graph, the full recursive ID success certificate holds exactly when the intervention set is valid for the graph, the outcome nodes are observed, no random counterpart of an intervention variable is an outcome node, and every district of the post-intervention ancestral graph is recursively reachable from its containing district in the original graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N → Prop","l":105,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.idSucceeds_toRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"idSucceedsRec generalizes idSucceeds. For an intervention target set X, an outcome node set Y, and a SWIG graph G, if the plain no-fixing ID certificate succeeds for X, Y on G, then the full recursive ID certificate also succeeds for X, Y on G, so soundness proved for idSucceedsRec subsumes the no-fixing headline.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.idSucceedsRec X Y G","l":170,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.id_sound","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":1,"d":"Soundness of the no-fixing ID algorithm. Fix an intervention target set X, an outcome node set Y, a SWIG graph G, and a reference-measure family ref that is faithful to the graph. Then whenever the no-fixing ID certificate succeeds for X, Y on G, the interventional query mapping X to Y is identifiable within the class of models dominated by ref with discretely positive observational kernels: any two such models that share graph G and observational kernel agree on the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun M => M.DominatedObs ref ∧ Causalean.SCM.ID.DiscreteID.DiscretePositive M) (Causalean.SCM.ID.interventionalQuery X Y)","l":554,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.id_sound_discrete","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDSoundDiscrete","a":"SCM/ID/GraphicalThms","t":1,"d":"Discrete soundness of the ID assembly for the no-additional-fixing (full-district) fragment (on-contract). For treatment nodes X, outcome nodes Y, and a SWIG graph G, a successful idSucceeds certificate implies that the interventional query is identified from the observational distribution within the standard discrete positive model class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceeds X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":30,"f":"Causalean/SCM/ID/GraphicalThms/IDSoundDiscrete.lean"},{"n":"Causalean.SCM.ID.id_sound_rec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Soundness of the full recursive ID algorithm for finite discrete-positive models. Fix an intervention target set X, an outcome node set Y, a SWIG graph G, and a reference-measure family ref that is faithful to the graph. Then whenever the full recursive Tian–Shpitser IDENTIFY certificate succeeds for X, Y on G, the interventional query mapping X to Y is identifiable within the class of models dominated by ref with discretely positive observational kernels: any two such models that share graph G and observational kernel agree on the query. This generalizes id_sound from the no-fixing fragment to the full Tian-Shpitser IDENTIFY recursion; see the module docstring for the proof architecture.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → Causalean.SCM.ID.idSucceedsRec X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun M => M.DominatedObs ref ∧ Causalean.SCM.ID.DiscreteID.DiscretePositive M) (Causalean.SCM.ID.interventionalQuery X Y)","l":888,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.id_sound_rec_discrete","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"Discrete soundness of the full recursive ID algorithm (on-contract). For an intervention target set X, an outcome node set Y, and a SWIG graph G, if the full recursive Tian–Shpitser IDENTIFY certificate succeeds for X, Y on G, then the interventional query mapping X to Y is identifiable within the standard discrete positive model class. Obtained from id_sound_rec at the counting reference by collapsing DominatedObs · countingRef to StandardDiscretePositive, exactly as id_sound_discrete is obtained from id_sound. This subsumes id_sound_discrete (via idSucceeds_toRec).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] [inst_4 : ∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (G : Causalean.SWIGGraph N), Causalean.SCM.ID.idSucceedsRec X Y G → Causalean.SCM.ID.IdentifiableUnder G (fun x => True) Causalean.SCM.ID.DiscreteID.StandardDiscretePositive (Causalean.SCM.ID.interventionalQuery X Y)","l":925,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifiableUnder_mono","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"Monotonicity of identifiability under assumptions. Fix functional-assumption predicates Af₁, Af₂ and structural-assumption predicates As₁, As₂ on causal models sharing a graph G, together with a causal query Φ. If every model satisfying Af₂ also satisfies Af₁, every model satisfying As₂ also satisfies As₁, and Φ is identifiable under the assumption pair (Af₁, As₁), then Φ is identifiable under (Af₂, As₂): passing to the more restrictive assumption predicates Af₂, As₂ cannot destroy identifiability.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Af₁ Af₂ As₁ As₂ : Causalean.SCM N Ω → Prop) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), (∀ (M : Causalean.SCM N Ω), Af₂ M → Af₁ M) → (∀ (M : Causalean.SCM N Ω), As₂ M → As₁ M) → Causalean.SCM.ID.IdentifiableUnder G Af₁ As₁ Φ → Causalean.SCM.ID.IdentifiableUnder G Af₂ As₂ Φ","l":189,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.identifiable_eq_identifiableUnder_true","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":2,"d":"Identifiability without additional assumptions is the special case where both Af and As are trivially satisfied.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), Causalean.SCM.ID.Identifiable G Φ ↔ Causalean.SCM.ID.IdentifiableUnder G (fun x => True) (fun x => True) Φ","l":181,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.identifyMassRecObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a population of variables with finite value spaces, an observed-node set, a SWIG graph, a containing node set, a target district contained in the observed set, and a nonnegative mass function on observed assignments, the recursive observed-set mass identification functional first [forms the induced ancestral set and records that it is observed](step:1,step:2), then returns the appropriate marginal, original mass, or recursively extracted district mass according to its ancestral-set cases.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (T : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → T ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":97,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifyMassRecObserved_eq_identifyMassRec","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.identifyMassRecObserved M.observed G T C hT q = Causalean.SCM.identifyMassRec M.observed G T C hT q","l":154,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.identifyMassRec_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The obs-side IDENTIFY recursion recovers the local q-mass of the target district from the local q-mass of any recursively reachable source district.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (hReach : Causalean.SCM.ID.CFactorReachableRec M.toSWIGGraph T C) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.identifyMassRec M.observed M.toSWIGGraph T C hT (M.qLocalMass s T hT) x = M.qLocalMass s C (fun x hv => hT (Causalean.SCM.ID.CFactorReachableRec.target_subset hReach hv)) x","l":1092,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.inducedAncestral","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite collection of distinguishable node labels, a SWIG graph, a node set on which to induce a subgraph, and a target node set, the induced ancestral set is the observed part of the ancestors of the target set in the graph induced on the first set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":35,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_parent_closed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"The induced ancestral set is closed under observed parents inside the ambient district T.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {T C : Finset (Causalean.SWIGNode N)}, T ⊆ G.observed → ∀ v ∈ T, ∀ w ∈ Causalean.SCM.ID.inducedAncestral G T C, G.dag.edge v w → v ∈ Causalean.SCM.ID.inducedAncestral G T C","l":48,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_self_of_mem_cComponentSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaRec","a":"SCM/ID/GraphicalThms","t":2,"d":"The observed ancestors of a full c-component S within G_S are S itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ S ∈ G.cComponentSet, Causalean.SCM.ID.inducedAncestral G S S = S","l":142,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaRec.lean"},{"n":"Causalean.SCM.ID.inducedAncestral_subset_left","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"The induced ancestral set lies inside the ambient observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)), Causalean.SCM.ID.inducedAncestral G T C ⊆ T","l":67,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.instDecidableInterventionValid","k":"instance","m":"Causalean.SCM.ID.DiscreteID.Checker","a":"SCM/ID/DiscreteID","t":2,"d":"For a finite collection of distinguishable node labels, an intervention-variable set, and a SWIG graph, decidability of intervention validity determines whether splitting the graph at that intervention set is valid.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (X : Finset N) → (G : Causalean.SWIGGraph N) → Decidable (Causalean.SCM.ID.interventionValid X G)","l":118,"f":"Causalean/SCM/ID/DiscreteID/Checker.lean"},{"n":"Causalean.SCM.ID.instIsFiniteMeasure_doObsKernelAncestralMarginal","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"For a finite collection of distinguishable node labels, measurable node-value spaces, a structural causal model, an intervention set whose random intervention nodes are observed and whose fixed intervention nodes are not already fixed, a query-node set, and fixed intervention values, the observed-ancestral post-intervention marginal measure has finite total mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (s : (M.fixSet X hObs hFix).FixedValues), MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) s)","l":118,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.instIsMarkovKernel_doObsKernelAncestralMarginal","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"For a finite collection of distinguishable node labels, measurable node-value spaces, a structural causal model, an intervention set whose random intervention nodes are observed and whose fixed intervention nodes are not already fixed, and a query-node set, the observed-ancestral post-intervention marginal is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), ProbabilityTheory.IsMarkovKernel (Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y)","l":458,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.interventionValid","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables, an intervention set in a SWIG graph is valid exactly when every intervention variable's random node is observed and its fixed node is not already fixed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Finset N → Causalean.SWIGGraph N → Prop","l":45,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.intervention_target_simp","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Proposition (fact4 — intervention target simplification).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Dn Yn : Finset N), Causalean.SCM.ID.InterventionTargetSimpConclusion M Dn Yn","l":485,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.interventionalQuery","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a finite population of variables with nonempty measurable value spaces, an intervention set, and an outcome-node set, the interventional query maps each structural causal model to its post-intervention outcome kernel when the query is valid, and otherwise to the default constant kernel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (X : Finset N) → (Y : Finset (Causalean.SWIGNode N)) → Causalean.SCM.ID.CausalQuery N Ω (ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω)))","l":94,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQueryValid","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a finite population of variables with measurable value spaces, an intervention set, an outcome-node set, and a structural causal model, interventional-query validity holds exactly when all intervention random nodes are observed, their fixed nodes are absent, all outcomes are observed, and the model is standard.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Finset N → Finset (Causalean.SWIGNode N) → Causalean.SCM N Ω → Prop","l":79,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQueryValid_iff_of_toSWIGGraph_eq","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"Well-formedness invariance under matching SWIG graphs. For an intervention target set X and outcome set Y, if two structural causal models M₁, M₂ have the same SWIG graph, then they agree on whether the total interventional query for X, Y is well formed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → (Causalean.SCM.ID.interventionalQueryValid X Y M₁ ↔ Causalean.SCM.ID.interventionalQueryValid X Y M₂)","l":147,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQuery_eq_default_of_not_valid","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":2,"d":"Outside the well-formed branch, the total interventional query is the fixed fallback kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M : Causalean.SCM N Ω), ¬Causalean.SCM.ID.interventionalQueryValid X Y M → Causalean.SCM.ID.interventionalQuery X Y M = Causalean.SCM.ID.defaultInterventionalKernel X Y","l":136,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.interventionalQuery_eq_doKernelY_of_valid","k":"theorem","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For treatment nodes X and outcome nodes Y satisfying the well-formedness conditions for a valid interventional query in a model M, the total interventional query evaluated at M equals the post-intervention outcome-marginal kernel doKernelY.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (X : Finset N) (Y : Finset (Causalean.SWIGNode N)) (M : Causalean.SCM N Ω) (h : Causalean.SCM.ID.interventionalQueryValid X Y M), Causalean.SCM.ID.interventionalQuery X Y M = M.doKernelY X h.left h.right.left Y h.right.right.left (Causalean.SCM.ID.standardFixedValues M h.right.right.right)","l":122,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.jointRef_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor prefix extension carries the product of the old-prefix reference and the next singleton reference to the successor-prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) {k : ℕ} (hk : k < D.card), MeasureTheory.Measure.map (Causalean.SCM.ID.extendTianPrefix H D hk) ((Causalean.SCM.jointRef ref (H.prefixIn D k)).prod (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨k, hk⟩)})) = Causalean.SCM.jointRef ref (H.prefixIn D (k + 1))","l":208,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.jointRef_map_prefixIn_absolutelyContinuous","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The reference marginal of any D-prefix is absolutely continuous with respect to the corresponding product reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω) (k : ℕ), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) (Causalean.SCM.jointRef ref D)).AbsolutelyContinuous (Causalean.SCM.jointRef ref (H.prefixIn D k))","l":252,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.marginalizeOnObserved","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a population of variables with finite value spaces, an observed-node set, a node subset to be marginalized contained in that observed set, and a nonnegative mass function on observed assignments, the observed-set marginalization functional assigns to each observed assignment the sum of that mass function over all replacements on the specified subset.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O W : Finset (Causalean.SWIGNode N)) → W ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":63,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.marginalizeOnObserved_eq_marginalizeOn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.marginalizeOnObserved M.observed W hW q = Causalean.SCM.marginalizeOn M.observed W hW q","l":138,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.markov_tian_cfactorization_density","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"Density form of the Markov-to-c-factorization theorem. Let H be a pure SWIG graph, D a finite set of SWIG nodes, μ a finite (probability) measure on the assignments to D, and ref a family of reference measures. If D is exactly the observed-node set of H and μ is absolutely continuous with respect to the product reference measure on D, then the Radon–Nikodym density of μ against that product reference equals, almost everywhere, the product over the c-components of H of their Tian district-density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.observed = D → ∀ (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → μ.rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D μ ref S x","l":800,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measurable_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix extension is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), Measurable (Causalean.SCM.ID.extendTianPrefix H D hn)","l":185,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measure_map_prefixIn_absolutelyContinuous_jointRef","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Domination of a dominated law's prefix marginal by the prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω), μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → ∀ (k : ℕ), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) μ).AbsolutelyContinuous (Causalean.SCM.jointRef ref (H.prefixIn D k))","l":300,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.measure_prefixIn_rnDeriv_eq_tianPrefixDensityProductInPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix-level Radon--Nikodym chain rule for Tian's arbitrary-measure conditional density product.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → ∀ k ≤ D.card, (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D k)) μ).rnDeriv (Causalean.SCM.jointRef ref (H.prefixIn D k)) =ᵐ[Causalean.SCM.jointRef ref (H.prefixIn D k)] Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref k","l":520,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.mechCFactor_eq_qLocalMass_div_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), Causalean.SCM.ReferenceFaithful ref → (∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.mechCFactor ref S hS s x = M.qLocalMass s S hS x / (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection hS x}","l":735,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.mechCFactor_fixSet_invariant","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"Do(X)-invariance of the c-factor Q[S] (Tian Lemma 4). Let an intervention set X have random copies that are all observed and fixed copies that are not already fixed in the base model, giving the intervened model M.fixSet X. For a standard base model M, a reference family that is faithful, and a node set S contained in the observed coordinates whose random copies avoid every intervened node in X, if the base assignment x records, at each intervened node, the same value that the intervened model's fixed values sDo assign to the corresponding fixed coordinate, then the S-c-factor Q[S] of the intervened model M.fixSet X at sDo, x equals the S-c-factor of the base model M at the projected fixed values M.fixSetProj X sDo, x.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (X : Finset N), Causalean.SCM.ReferenceFaithful ref → M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ n ∈ X, Causalean.SWIGNode.random n ∉ S) → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (D : N) (hD : D ∈ X), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (M.fixSet X hObs hFix).mechCFactor ref S (id hS) sDo x = M.mechCFactor ref S hS (M.fixSetProj X hObs hFix sDo) x","l":1094,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.mem_cComponent_iff_cComponentOf_eq","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"For an observed node and a c-component listed in cComponentSet, membership in that component is the same as saying that the node's computed c-component is that listed component. This converts Tian's district factor, which filters by membership in a district, into the fiber form needed for finite-product regrouping.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N} {S : Finset (Causalean.SWIGNode N)}, v ∈ G.observed → S ∈ G.cComponentSet → (v ∈ S ↔ G.cComponentOf v = S)","l":237,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.mem_prefixIn_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Membership in a graph-ordered prefix of D is exactly index membership below the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ) (v : Causalean.SWIGNode N), v ∈ H.prefixIn D n ↔ ∃ (h : v ∈ D), ↑(H.nodeIndex D ⟨v, h⟩) < n","l":63,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_mem_prefixIn_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The node at index i belongs to the first n D-nodes iff i < n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ) (i : Fin D.card), ↑(H.nodesAt D i) ∈ H.prefixIn D n ↔ ↑i < n","l":87,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_not_mem_prefixIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The next D-node is not in the previous D-prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), ↑(H.nodesAt D ⟨n, hn⟩) ∉ H.prefixIn D n","l":126,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nodesAt_toSWIGGraph_observed_eq_observedAt","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The graph-level topological enumeration agrees definitionally with the SCM-level observed-node enumeration.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (i : Fin M.observed.card), M.nodesAt M.observed i = M.observedAt i","l":911,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.nonIdentifiable_iff","k":"theorem","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a SWIG graph G and a causal query Φ, the causal query is non-identifiable from the observational distribution if and only if there exist two causal models with SWIG graph G and the same observational law that disagree on the value of Φ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {α : Type u_3} (G : Causalean.SWIGGraph N) (Φ : Causalean.SCM.ID.CausalQuery N Ω α), Causalean.SCM.ID.NonIdentifiable G Φ ↔ ∃ M₁ M₂, M₁.toSWIGGraph = G ∧ M₂.toSWIGGraph = G ∧ Causalean.SCM.ID.obsEquiv M₁ M₂ ∧ Φ M₁ ≠ Φ M₂","l":133,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.obsCondKernel_slice_ae_eq_of_pairMeasure_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Equal pair laws give equal conditional-kernel slices a.e.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hCC₁ : CC ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hCC₂ : CC ⊆ M₂.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [∀ (s : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s)] [∀ (s : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s)] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [inst_8 : MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s₁ : M₁.FixedValues) (s₂ : M₂.FixedValues), (M₁.obsCondPairKernel Y CC hY₁ hCC₁) s₁ = (M₂.obsCondPairKernel Y CC hY₂ hCC₂) s₂ → (fun c => ((M₁.obsCondKernel Y CC hY₁ hCC₁).sectR s₁) c) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hCC₁) (M₁.obsKernel s₁)] fun c => ((M₂.obsCondKernel Y CC hY₂ hCC₂).sectR s₂) c","l":222,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.obsEquiv","k":"def","m":"Causalean.SCM.ID.Identifiable","a":"SCM/ID","t":1,"d":"For a finite node-label set, measurable node-value spaces, and two structural causal models on them, observational equivalence means that their derived observational probability kernels are equal.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":84,"f":"Causalean/SCM/ID/Identifiable.lean"},{"n":"Causalean.SCM.ID.pinnedExtend","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with nonempty measurable value spaces, a structural causal model, an intervention set whose random nodes are observed and whose fixed nodes are not already fixed, a query-node set, and a fixed-value intervention slice, the pinned extension maps each assignment on the observed post-intervention ancestors to an assignment on all original observed nodes, using the slice's fixed value at intervened random coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : N), Nonempty (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → (M.fixSet X hObs hFix).FixedValues → Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)","l":176,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.pinnedExtend_pin_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The pinned extension reads intervened random coordinates from the matching fixed coordinate of the do-slice.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (sDo : (M.fixSet X hObs hFix).FixedValues) (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), Causalean.SCM.ID.pinnedExtend M X hObs hFix Y sDo xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩","l":382,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.pinnedExtend_projection_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The pinned extension projects back to the ancestral assignment when natural intervened coordinates are excluded from the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : N), Nonempty (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (Causalean.SCM.ID.pinnedExtend M X hObs hFix Y sDo xD) = xD","l":349,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.prefixIn_card","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The full D-prefix is D.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.prefixIn D D.card = D","l":108,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_disjoint_singleton_next","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The previous D-prefix is disjoint from the singleton next node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), Disjoint (H.prefixIn D n) {↑(H.nodesAt D ⟨n, hn⟩)}","l":161,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_mono","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Prefix sets are monotone in the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {m k : ℕ}, m ≤ k → H.prefixIn D m ⊆ H.prefixIn D k","l":118,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_parent_closed_induce_observed","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"In an induced set of observed variables, every topological prefix contains all parents within that set of each variable it contains.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ∀ A ⊆ M.observed, ∀ (k : ℕ), ∀ v ∈ A, ∀ w ∈ (M.induce A).prefixIn A k, M.dag.edge v w → v ∈ (M.induce A).prefixIn A k","l":405,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_qProduct_ratio_eq_component_ratio_of_family","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"In a product over an abstract pairwise-disjoint family, the prefix-ratio step at a node of S cancels every factor except the S factor.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))), ∀ S ∈ 𝒞, (∀ C ∈ 𝒞, C ≠ S → Disjoint C S) → ∀ (i : Fin D.card) (hDobs : D ⊆ M.observed), ↑(H.nodesAt D i) ∈ S → Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x) / ∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x = M.qLocalMass s (S ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x / M.qLocalMass s (S ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x","l":571,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_qProduct_ratio_eq_component_ratio_of_family_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Nonzero-denominator variant of prefixIn_qProduct_ratio_eq_component_ratio_of_family.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))), ∀ S ∈ 𝒞, (∀ C ∈ 𝒞, C ≠ S → Disjoint C S) → ∀ (i : Fin D.card) (hDobs : D ⊆ M.observed), ↑(H.nodesAt D i) ∈ S → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∏ C ∈ 𝒞 \\ {S}, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x ≠ 0 → (∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x) / ∏ C ∈ 𝒞, M.qLocalMass s (C ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x = M.qLocalMass s (S ∩ H.prefixIn D (↑i + 1)) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1) (Finset.mem_of_mem_inter_right hv))) x / M.qLocalMass s (S ∩ H.prefixIn D ↑i) (fun x hv => hDobs (Causalean.SWIGGraph.prefixIn_subset H D (↑i) (Finset.mem_of_mem_inter_right hv))) x","l":710,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prefixIn_succ","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor D-prefix is obtained by adjoining the next D-node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card), H.prefixIn D (n + 1) = H.prefixIn D n ∪ {↑(H.nodesAt D ⟨n, hn⟩)}","l":133,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixIn_zero","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The D-prefix of length zero is empty.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.prefixIn D 0 = ∅","l":77,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.prefixNodes_obsParentClosed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The first k observed nodes in a structural causal model's topological order form a set closed under observed parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (k : ℕ), M.ObsParentClosed (M.prefixNodes k)","l":324,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.prefix_pair_singleton_mass_eq_succ_prefix_mass","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The mass of a realized prefix together with its next coordinate equals the mass of the same realization of the successor prefix under any measure on the ordered graph values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (i : Fin D.card) (x : Causalean.ValuesOn D (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) ω, Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) ω)) μ) {(Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x, Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) x)} = (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1))) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1)) x}","l":1250,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.prod_filter_div_telescope","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Pure ENNReal telescope for products of selected adjacent ratios.","s":"∀ (a : ℕ → ENNReal) (m : ℕ), ∀ T ⊆ Finset.range m, (∀ i ≤ m, a i ≠ 0) → (∀ i ≤ m, a i ≠ ⊤) → (∀ i < m, i ∉ T → a (i + 1) = a i) → ∏ i ∈ T, a (i + 1) / a i = a m / a 0","l":1794,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.ID.prod_tianDistrictDensity_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Tian's full prefix-chain density regroups exactly as the product of the Tian district factors over the graph c-components. This is pure finite-product algebra: each prefix index maps to the c-component of its node, and the district factor is precisely the product over the corresponding fiber.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)), H.observed = D → ∀ (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], (fun x => ∏ S ∈ H.cComponentSet, Causalean.SCM.ID.tianDistrictDensity H D μ ref S x) = fun x => Causalean.SCM.ID.tianDensityProduct H D μ ref x","l":256,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_fixSet_invariant","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Local q-mass is invariant under a fixSet intervention when no coordinate in S is one of the intervened random nodes and the full assignment pins each intervened random node to the corresponding fixed value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (sDo : (M.fixSet X hObs hFix).FixedValues) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ n ∈ X, Causalean.SWIGNode.random n ∉ S) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (D : N) (hD : D ∈ X), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (M.fixSet X hObs hFix).qLocalMass sDo S (id hS) x = M.qLocalMass (M.fixSetProj X hObs hFix sDo) S hS x","l":1045,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_ne_top","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every local q-mass associated with fixed values, an observed node set, and an observed assignment is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x ≠ ⊤","l":376,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.ID.qLocalMass_obsProof_irrel","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"The local q-mass of an observed variable set does not depend on which proof establishes that the set is observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT hT' : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x = M.qLocalMass s T hT' x","l":1082,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.q_factor_identity","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":1,"d":"Proposition 2.19 (Q-factor identity / Tian's lemma). For an ancestrally-closed node set R and a c-component T of the induced subgraph on R in a structural causal model M, after intervening on a node set Wn, the structurally-defined c-factor on T equals, almost everywhere, the conditional of T given its q-factor parents extracted from that intervention. See QFactorIdentityConclusion for the precise hypothesis frame.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.QFactorIdentityConclusion M R T Wn","l":290,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.q_factor_marginal_fixing","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.QFactorIdentity","a":"SCM/ID/GraphicalThms","t":2,"d":"Marginal fixing form of Tian's Q-factor identity.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R T : Finset (Causalean.SWIGNode N)) (Wn : Finset N), Causalean.SCM.ID.QFactorMarginalFixingConclusion M R T Wn","l":178,"f":"Causalean/SCM/ID/GraphicalThms/QFactorIdentity.lean"},{"n":"Causalean.SCM.ID.random_intervened_mem_fixAncestralSet_iff_mem_Y","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"For an intervened variable, its random copy is a post-intervention ancestor of the query exactly when it is explicitly queried.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) {d : N}, d ∈ X → (Causalean.SWIGNode.random d ∈ Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y ↔ Causalean.SWIGNode.random d ∈ Y)","l":323,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.recoveredFactorRec","k":"def","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model, reference measures, a fixed-value assignment, a containing c-component, and a target district, assuming finite value spaces, finite observational-kernel slices, standard-Borel and nonempty one-node observed value spaces, and countably generated prefix value spaces, the recursively recovered factor assigns to every observed-data realization the recursive identification mass for the target divided by its reference atom, and is zero when either named node set is not observed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → [hfin : (n : N) → Fintype (Ω n)] → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":191,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.recoveredFactorRec_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithmRec","a":"SCM/ID/GraphicalThms","t":2,"d":"Equal observational kernels transport the recursive recovered factor.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (C S : Finset (Causalean.SWIGNode N)), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ [hfin : (n : N) → Fintype (Ω n)] [inst_3 : ∀ (s' : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M₁.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M₁.observed.card), Nonempty (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn (M₁.prefixNodes k) (Causalean.swigΩ Ω))] [inst_7 : ∀ (s' : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < M₂.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < M₂.observed.card), Nonempty (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn (M₂.prefixNodes k) (Causalean.swigΩ Ω))], (fun s => Causalean.SCM.ID.recoveredFactorRec M₁ ref s C S) ≍ fun s => Causalean.SCM.ID.recoveredFactorRec M₂ ref s C S","l":237,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithmRec.lean"},{"n":"Causalean.SCM.ID.rnDeriv_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The density of any dominated finite law on a finite coordinate set factors as the product of its one-coordinate conditional densities along the chosen topological order. The conditional density at each coordinate is computed from Mathlib's regular conditional distribution given the preceding prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] [StandardBorelSpace (Causalean.ValuesOn D (Causalean.swigΩ Ω))] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref D) → μ.rnDeriv (Causalean.SCM.jointRef ref D) =ᵐ[Causalean.SCM.jointRef ref D] Causalean.SCM.ID.tianDensityProduct H D μ ref","l":724,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.splitMono_bidirectedReachable_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"A valid monolithic intervention split leaves bidirected reachability between any two SWIG nodes unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v w : Causalean.SWIGNode N), (G.splitMono X hObs hFix).bidirectedReachable v w ↔ G.bidirectedReachable v w","l":246,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_cComponentOf_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, the bidirected component containing any SWIG node is the same as it was before intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), (G.splitMono X hObs hFix).cComponentOf v = G.cComponentOf v","l":269,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_cComponentSet_eq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Splitting a graph under a valid monolithic intervention leaves its partition into bidirected connected components unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).cComponentSet = G.cComponentSet","l":289,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_directlyConfounded_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, two nodes are directly confounded exactly when they were directly confounded in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v₁ v₂ : Causalean.SWIGNode N), (G.splitMono X hObs hFix).directlyConfounded v₁ v₂ ↔ G.directlyConfounded v₁ v₂","l":227,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_edge_from_unobserved_iff","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, a directed edge from an unobserved node exists exactly when that edge existed in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {u v : Causalean.SWIGNode N}, u ∈ G.unobserved → ((G.splitMono X hObs hFix).dag.edge u v ↔ G.dag.edge u v)","l":212,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_no_edge_from_intervened_random","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"Under a valid monolithic intervention split, the random copy of an intervened variable has no outgoing directed edge.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {d : N}, d ∈ X → ∀ (v : Causalean.SWIGNode N), ¬(G.splitMono X hObs hFix).dag.edge (Causalean.SWIGNode.random d) v","l":299,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.splitMono_not_isAncestor_from_intervened_random","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"After a valid monolithic intervention split, the random copy of an intervened variable is not an ancestor of any node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) {d : N}, d ∈ X → ∀ (v : Causalean.SWIGNode N), ¬(G.splitMono X hObs hFix).dag.isAncestor (Causalean.SWIGNode.random d) v","l":310,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.standardFixedValues","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a finite population of variables with measurable value spaces, a structural causal model that is standard, the canonical fixed-value assignment is the unique assignment on its empty fixed-node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.isStandard → M.FixedValues","l":63,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.standardFixedValues_heq","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.IDAlgorithm","a":"SCM/ID/GraphicalThms","t":2,"d":"Two standard models with the same SWIG graph have the same canonical fixed-value assignment after type transport.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (hf : M₁.fixed = M₂.fixed) (h₁ : M₁.isStandard), Causalean.SCM.ID.standardFixedValues M₁ h₁ ≍ Causalean.SCM.ID.standardFixedValues M₂ (id (Eq.mpr (id (congrArg (fun _a => _a = ∅) (Eq.symm hf))) h₁))","l":46,"f":"Causalean/SCM/ID/GraphicalThms/IDAlgorithm.lean"},{"n":"Causalean.SCM.ID.subset_fixAncestralSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":2,"d":"The post-intervention ancestral set contains the query.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), Y ⊆ Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y","l":149,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.subset_fixObservedAncestralSet","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormula","a":"SCM/ID/GraphicalThms","t":1,"d":"For a valid intervention set X (observed, not already fixed) and an outcome set Y contained in the observed nodes, Y is contained in its own post-intervention observed-ancestral closure: it lies in the post-intervention ancestral set (subset_fixAncestralSet) and in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), ∀ Y ⊆ M.observed, Y ⊆ Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y","l":473,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormula.lean"},{"n":"Causalean.SCM.ID.subset_inducedAncestral","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"If C is observed and contained in T, then it is contained in its induced ancestral set inside T.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {T C : Finset (Causalean.SWIGNode N)}, C ⊆ T → C ⊆ G.observed → C ⊆ Causalean.SCM.ID.inducedAncestral G T C","l":76,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.ID.swigValueNonempty","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a population of variables with nonempty value spaces and a SWIG node, a witness that that node's value space is nonempty is provided.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [∀ (n : N), Nonempty (Ω n)] (w : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω w)","l":25,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.tianDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, a finite node set, a finite measure on its value assignments, and a family of reference measures, Tian's full prefix-chain density maps each node assignment to the product of the prefix-step densities for every node in the graph's ordering; every singleton node-value space in that ordering is required to be nonempty and standard Borel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":216,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, a finite node set, a finite measure on its value assignments, a family of reference measures, and a selected node set, Tian's district density maps each node assignment to the product of the prefix-step densities for exactly those ordered nodes that lie in the selected set; every singleton node-value space in the ordering is required to be nonempty and standard Borel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → Finset (Causalean.SWIGNode N) → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(G.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":188,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity_eq_mechCFactor_doModel","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"(B) Tian Lemma 1 for the do-model ancestral marginal. Consider a structural causal model M with an intervention set X whose random copies are observed and whose fixed copies are not already frozen, an outcome set Y disjoint from the random copies of X, under a faithful reference-measure family, a positive observational kernel at every fixed-value assignment, and the standing assumption that M is a standard model. For a set S that is simultaneously a district of the post-intervention ancestral graph and a full c-component of M, and an extension map that inverts the projection onto the ancestral observed coordinates, the Tian district density read off the do(X)-law ancestral marginal at S agrees with the mechanism c-factor of the post-intervention model at the extended point.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → M.isStandard → ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, ∀ (hSfull : S ∈ M.cComponentSet) [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianDistrictDensity ((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)) (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S xD = (M.fixSet X hObs hFix).mechCFactor ref S (have hSobs := Causalean.SWIGGraph.cComponentSet_subset_observed M.toSWIGGraph S hSfull; Eq.mpr (id ge_iff_le._simp_1) hSobs) sDo (extend xD)","l":1704,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianDistrictDensity_eq_qLocalMass_div_jointRef_district","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"Consider an intervention set X whose random copies are observed and whose fixed copies are not already frozen together with an outcome set Y disjoint from the random copies of X, under a faithful reference-measure family and a positive observational kernel at every fixed-value assignment. For any district S of the post-intervention ancestral graph and an extension map inverting the projection onto the ancestral observed coordinates, the Tian district density read off the do(X)-law ancestral marginal at S equals the do-model local q-mass on S divided by the reference atom mass of S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ d ∈ X, Causalean.SWIGNode.random d ∉ Y) → ∀ (S : Finset (Causalean.SWIGNode N)) (hS : S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y) [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), let MX := M.fixSet X hObs hFix; let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := MX.induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S xD = MX.qLocalMass sDo S (have this := fun ⦃v⦄ hv => have hSHobs := Causalean.SWIGGraph.cComponentSet_subset_observed H S (id hS); have hvD := id (hSHobs hv); (Finset.mem_inter.mp hvD).right; this) (extend xD) / (Causalean.SCM.jointRef ref S) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => have hSHobs := Causalean.SWIGGraph.cComponentSet_subset_observed H S (id hS); id (hSHobs hv); this) xD}","l":1975,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a SWIG graph, a finite node set, a finite measure on its joint value space, and reference measures, assuming every graph-ordered singleton node value space is standard Borel and nonempty, the Tian prefix density product at a prefix length and an assignment on that prefix is one at length zero and otherwise the preceding product times the next conditional-density factor, or one when that next index is outside the node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω) → ENNReal","l":312,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix_card_eq_tianDensityProduct","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"At the full D prefix, the recursive prefix density product is Tian's finite product over all D indices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (y : Causalean.ValuesOn (H.prefixIn D D.card) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref D.card y = Causalean.SCM.ID.tianDensityProduct H D μ ref ((Causalean.valuesEquivOfEq (Causalean.SCM.ID.prefixIn_card H D)) y)","l":470,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixDensityProductInPrefix_eq_range_product","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The recursive prefix density product is the range product of its one-step factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))], ∀ k ≤ D.card, ∀ (z : Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω)), Causalean.SCM.ID.tianPrefixDensityProductInPrefix H D μ ref k z = ∏ i ∈ Finset.range k, Causalean.SCM.ID.tianPrefixStepDensityInPrefix H D μ ref k z i","l":407,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensity","k":"def","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node-label set, measurable node-value spaces, a SWIG graph, a finite node set, a finite measure on its value assignments, a family of reference measures, and a position in the graph's ordering of that set, Tian's prefix-step density maps each assignment of the selected nodes to the Radon–Nikodym density of the selected node's conditional distribution given its preceding nodes, relative to its reference measure; the selected node's value space is required to be nonempty and standard Borel.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (G : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → (i : Fin D.card) → [MeasureTheory.IsFiniteMeasure μ] → [StandardBorelSpace (Causalean.ValuesOn {↑(G.nodesAt D i)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(G.nodesAt D i)} (Causalean.swigΩ Ω))] → Causalean.ValuesOn D (Causalean.swigΩ Ω) → ENNReal","l":149,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensityInPrefix","k":"def","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":1,"d":"For a finite population of variables with measurable value spaces, a SWIG graph, a finite node set, a finite measure on its joint value space, reference measures, assuming every graph-ordered singleton node value space is standard Borel and nonempty, a prefix length, an assignment on that prefix, and a node index, the one-step Tian density is the conditional-density factor for that indexed node when it lies in both the prefix and the node set, and is one otherwise.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (H : Causalean.SWIGGraph N) → (D : Finset (Causalean.SWIGNode N)) → (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) → Causalean.SCM.ReferenceMeasures Ω → [MeasureTheory.IsFiniteMeasure μ] → [∀ (i : ℕ) (hi : i < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ) (hi : i < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω) → ℕ → ENNReal","l":360,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensityInPrefix_eq_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.TianMassBridge","a":"SCM/ID/Density","t":1,"d":"Fix a step index i earlier than the prefix length k that also lies within the node domain D. Provided the conditional distribution of the i-th coordinate given the shorter prefix is absolutely continuous with respect to the reference measure on that coordinate, and the reference-measure mass of the singleton value at that coordinate is nonzero and finite, then the Tian prefix-step density at index i equals the prefix conditional singleton mass divided by the corresponding singleton mass of the product reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω) [inst_3 : MeasureTheory.IsFiniteMeasure μ] [inst_4 : ∀ (j : ℕ) (hj : j < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨j, hj⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (j : ℕ) (hj : j < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨j, hj⟩)} (Causalean.swigΩ Ω))] (k i : ℕ) (hi : i < k) (hcard : i < D.card) [MeasurableSingletonClass (Causalean.ValuesOn {↑(H.nodesAt D ⟨i, hcard⟩)} (Causalean.swigΩ Ω))] (z : Causalean.ValuesOn (H.prefixIn D k) (Causalean.swigΩ Ω)), ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D ⟨i, hcard⟩).property; this)) (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D i)) μ) (Causalean.valuesProjection (Causalean.SCM.ID.prefixIn_mono H D (Nat.le_of_lt hi)) z)).AbsolutelyContinuous (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) → (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} ≠ 0 → (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} ≠ ⊤ → Causalean.SCM.ID.tianPrefixStepDensityInPrefix H D μ ref k z i = ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D ⟨i, hcard⟩).property; this)) (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D i)) μ) (Causalean.valuesProjection (Causalean.SCM.ID.prefixIn_mono H D (Nat.le_of_lt hi)) z)) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z} / (Causalean.SCM.jointRef ref {↑(H.nodesAt D ⟨i, hcard⟩)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ Eq.mpr (id (congrArg (fun _a => _a) (propext (Causalean.SCM.ID.nodesAt_mem_prefixIn_iff H D k ⟨i, hcard⟩)))) hi; this) z}","l":35,"f":"Causalean/SCM/ID/Density/TianMassBridge.lean"},{"n":"Causalean.SCM.ID.tianPrefixStepDensity_eq_prefix_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"When the preceding prefix has nonzero singleton mass, the one-step Tian density is the ratio of the successive prefix singleton masses, divided by the singleton reference mass of the added variable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn D (Causalean.swigΩ Ω))) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ [inst_5 : MeasureTheory.IsFiniteMeasure μ] [inst_6 : ∀ (k : ℕ) (hk : k < D.card), StandardBorelSpace (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_7 : ∀ (k : ℕ) (hk : k < D.card), Nonempty (Causalean.ValuesOn {↑(H.nodesAt D ⟨k, hk⟩)} (Causalean.swigΩ Ω))] (i : Fin D.card) (x : Causalean.ValuesOn D (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i)) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x} ≠ 0 → Causalean.SCM.ID.tianPrefixStepDensity H D μ ref i x = (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1))) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D (↑i + 1)) x} / (MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i)) μ) {Causalean.valuesProjection (Causalean.SWIGGraph.prefixIn_subset H D ↑i) x} / (Causalean.SCM.jointRef ref {↑(H.nodesAt D i)}) {Causalean.valuesProjection (have this := fun ⦃v⦄ hv => Eq.symm (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_singleton)) hv) ▸ (H.nodesAt D i).property; this) x}","l":1350,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.tian_full_cComponent_density_recovery_core","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"Core Tian c-factor recovery for a full observational c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":918,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.ID.tian_full_cComponent_density_recovery_core_direct","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":1,"d":"District recovery (Tian Lemma 4 projection consistency). Consider a standard structural causal model M with an intervention set X whose random copies are observed and whose fixed copies are not already frozen, under a faithful reference-measure family and a positive observational kernel at every fixed-value assignment, for an outcome set Y disjoint from the random copies of X. For a set S that is simultaneously a district of the post-intervention ancestral graph and a full c-component of M, and an extension map that inverts the projection onto the ancestral observed coordinates and reproduces the intervention values sDo on the intervened coordinates, the Tian district density read off the do(X)-law ancestral marginal at S agrees, almost everywhere with respect to the product reference measure on the ancestral observed coordinates, with the observational c-component density factor at S pulled back through the extension.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Nonempty (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N), M.isStandard → ∀ (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (sDo : (M.fixSet X hObs hFix).FixedValues), ∀ S ∈ Causalean.SCM.ID.fixTruncCComponentSet M X hObs hFix Y, S ∈ M.cComponentSet → ∀ [inst_6 : MeasureTheory.IsFiniteMeasure ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo)] [inst_7 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), StandardBorelSpace (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y).card), Nonempty (Causalean.ValuesOn {↑(((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).nodesAt (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_11 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_12 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (∀ (s' : M.FixedValues), Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s')) → (∀ D ∈ X, Causalean.SWIGNode.random D ∉ Y) → ∀ (extend : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)), Causalean.valuesProjection (have this := Finset.inter_subset_right; this) (extend xD) = xD) → (∀ (xD : Causalean.ValuesOn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) (Causalean.swigΩ Ω)) (D : N) (hD : D ∈ X), extend xD ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sDo ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → let D := Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y; let H := (M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y); Causalean.SCM.ID.tianDistrictDensity H D ((Causalean.SCM.ID.doObsKernelAncestralMarginal M X hObs hFix Y) sDo) ref S =ᵐ[Causalean.SCM.jointRef ref D] fun xD => M.cComponentDensityFactor ref (M.fixSetProj X hObs hFix sDo) S (extend xD)","l":2248,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.valuesOnNonempty","k":"def","m":"Causalean.SCM.ID.Query","a":"SCM/ID","t":1,"d":"For a population of variables with nonempty value spaces and a finite SWIG-node set, a witness that the corresponding joint value space is nonempty is provided.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [∀ (n : N), Nonempty (Ω n)] (Y : Finset (Causalean.SWIGNode N)), Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":34,"f":"Causalean/SCM/ID/Query.lean"},{"n":"Causalean.SCM.ID.valuesProjection_extend_eq_of_subset","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"If an extension preserves all values on a larger observed set, then restricting the extension to any subset gives the same values as restricting the original assignment directly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {D S : Finset (Causalean.SWIGNode N)} (hDobs : D ⊆ M.observed) (hSD : S ⊆ D) (hSobs : S ⊆ M.observed) (extend : Causalean.ValuesOn D (Causalean.swigΩ Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (∀ (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), Causalean.valuesProjection hDobs (extend xD) = xD) → ∀ (xD : Causalean.ValuesOn D (Causalean.swigΩ Ω)), Causalean.valuesProjection hSobs (extend xD) = Causalean.valuesProjection hSD xD","l":1452,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SCM.ID.valuesUnionEquiv_extendTianPrefix","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.DoGFormulaTian","a":"SCM/ID/GraphicalThms","t":2,"d":"The successor-prefix extension is inverse to the union-equivalence view of the successor prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (H : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) {n : ℕ} (hn : n < D.card) (p : Causalean.ValuesOn (H.prefixIn D n) (Causalean.swigΩ Ω) × Causalean.ValuesOn {↑(H.nodesAt D ⟨n, hn⟩)} (Causalean.swigΩ Ω)), (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.ID.prefixIn_disjoint_singleton_next H D hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.ID.prefixIn_succ H D hn)) (Causalean.SCM.ID.extendTianPrefix H D hn p)) = p","l":194,"f":"Causalean/SCM/ID/GraphicalThms/DoGFormulaTian.lean"},{"n":"Causalean.SCM.IsFaithful","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"For a directed acyclic graph on the split nodes, a structural causal model, and a finite measure on that model's random-value space, the faithfulness property means that, for every three pairwise-disjoint finite sets of random nodes contained respectively in the model's random variables, their conditional independence under the measure implies that the graph d-separates the first and second sets given the third.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":53,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.IsGlobalIMap","k":"def","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"For a directed acyclic graph on the split nodes, a structural causal model, and a finite measure on that model's random-value space, the global I-map property means that, for every three finite sets of random nodes contained respectively in the model's random variables, d-separation of the first and second sets given the third in the graph implies their conditional independence under the measure.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.DAG (Causalean.SWIGNode N) → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.RandomValues] → (μ : MeasureTheory.Measure M.RandomValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":43,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.LatentValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the latent-value assignments assign one value to every unobserved latent root node of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":127,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.ObsCondIndep","k":"def","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, three sets of observed nodes, respectively the first, second, and conditioning sets, each of which is contained in the model's observed-node set, and a finite measure on the standard-Borel observed-value space, the observational conditional-independence relation asserts that the value vectors on the first and second node sets are conditionally independent given the value vector on the conditioning set under that measure.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [StandardBorelSpace M.ObservedValues] → (X Y Z : Finset (Causalean.SWIGNode N)) → X ⊆ M.observed → Y ⊆ M.observed → Z ⊆ M.observed → (μ : MeasureTheory.Measure M.ObservedValues) → [MeasureTheory.IsFiniteMeasure μ] → Prop","l":40,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.ObsParentClosed","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model and a finite node set, the observed-parent-closure condition holds exactly when every member of the set is observed and every observed parent of each member also belongs to the set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Prop","l":690,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.ObsStepFiberRN","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a family of $\\sigma$-finite reference measures for its node values, and a fixed-node assignment, assuming finite observational kernels, standard-Borel and nonempty one-node value spaces, and countably generated prefix value spaces, the stepwise fibre Radon--Nikodym condition requires that, at every observed position, the next observed node is selected, the product reference measure on its preceding prefix is formed, the observational law of that prefix is formed, the conditional kernel of that next node given the prefix is formed, and that kernel is almost surely dominated by its reference measure under the prefix law while its fibre derivative is jointly almost-everywhere measurable under the product reference measure.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Prop","l":149,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.ObservedPrefixValues","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a natural number $n$ no greater than its number of observed nodes, the observed-prefix value space is the one-point space of the empty assignment when $n=0$, and the product of the preceding prefix space and the value space of the $n$-th observed node when $n$ is positive.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → n ≤ M.observed.card → Type uΩ","l":32,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.ObservedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the observed-value assignments assign one value to every observed node of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":121,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.OrderedLatentPrefixValues","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Random values consisting of the latent tuple paired with an observed prefix. This is the state-space of the kernel at step n in the factored construction.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → n ≤ M.observed.card → Type (max (max uΩ uN) uΩ)","l":59,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.PartialID.CompatibleSCM","k":"def","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a SWIG graph, a predicate specifying structural assumptions, and a reference structural causal model, the compatible-model class is the collection of structural causal models that have the given graph, satisfy the given assumptions, and are observationally equivalent to the reference model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → Causalean.SCM N Ω → Prop","l":43,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.IsSharp","k":"def","m":"Causalean.SCM.PartialID.SharpnessCertificate","a":"SCM/PartialID","t":1,"d":"For a finite node set with measurable node-value spaces, a SWIG graph, a class of structural causal models, a baseline structural causal model, a real-valued query of such models, and two real numbers $L$ and $U$, the sharpness predicate holds exactly when the query's range over models compatible with the graph, class, and baseline model is the closed interval $[L,U]$.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → (Causalean.SCM N Ω → ℝ) → ℝ → ℝ → Prop","l":28,"f":"Causalean/SCM/PartialID/SharpnessCertificate.lean"},{"n":"Causalean.SCM.PartialID.compatibleInterval","k":"def","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a SWIG graph, a predicate specifying structural assumptions, a reference structural causal model, and a real-valued causal query, the compatible interval is the set of all query values attained by structural causal models compatible with that graph, assumptions, and reference model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SWIGGraph N → (Causalean.SCM N Ω → Prop) → Causalean.SCM N Ω → (Causalean.SCM N Ω → ℝ) → Set ℝ","l":75,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.compatibleSCM_mono","k":"theorem","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"If the structural-assumption predicate As' is stronger than As, i.e. every model satisfying As' also satisfies As, then every model compatible with the reference model M₀ under the stricter assumptions As' is also compatible under the weaker assumptions As — strengthening the structural assumptions can only shrink the compatible class.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {G : Causalean.SWIGGraph N} {As As' : Causalean.SCM N Ω → Prop} {M₀ : Causalean.SCM N Ω}, (∀ (M : Causalean.SCM N Ω), As' M → As M) → ∀ (M : Causalean.SCM N Ω), Causalean.SCM.PartialID.CompatibleSCM G As' M₀ M → Causalean.SCM.PartialID.CompatibleSCM G As M₀ M","l":66,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.compatibleSCM_self","k":"theorem","m":"Causalean.SCM.PartialID.CanonicalModel","a":"SCM/PartialID","t":1,"d":"For any graph G, structural-assumption predicate As, and reference model M₀, if M₀'s own SWIG graph is G and M₀ satisfies the structural assumptions As, then M₀ belongs to its own compatible class CompatibleSCM G As M₀.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (As : Causalean.SCM N Ω → Prop) (M₀ : Causalean.SCM N Ω), M₀.toSWIGGraph = G → As M₀ → Causalean.SCM.PartialID.CompatibleSCM G As M₀ M₀","l":58,"f":"Causalean/SCM/PartialID/CanonicalModel.lean"},{"n":"Causalean.SCM.PartialID.isSharp_of_attaining","k":"theorem","m":"Causalean.SCM.PartialID.SharpnessCertificate","a":"SCM/PartialID","t":1,"d":"Sharpness certificate. For a real-valued query obj over the compatible class of (G, As, M₀), if the identified set is contained in the interval [L, U] (soundness), the identified set is order-connected, some compatible model attains the value L, and some compatible model attains the value U, then the identified set equals [L, U], i.e. the bound is sharp.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (As : Causalean.SCM N Ω → Prop) (M₀ : Causalean.SCM N Ω) (obj : Causalean.SCM N Ω → ℝ) (L U : ℝ), Causalean.SCM.PartialID.compatibleInterval G As M₀ obj ⊆ Set.Icc L U → (Causalean.SCM.PartialID.compatibleInterval G As M₀ obj).OrdConnected → (∃ M, Causalean.SCM.PartialID.CompatibleSCM G As M₀ M ∧ obj M = L) → (∃ M, Causalean.SCM.PartialID.CompatibleSCM G As M₀ M ∧ obj M = U) → Causalean.SCM.PartialID.IsSharp G As M₀ obj L U","l":41,"f":"Causalean/SCM/PartialID/SharpnessCertificate.lean"},{"n":"Causalean.SCM.QmechMeasure","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node set with measurable node-value spaces, a structural causal model, a set of SWIG nodes contained in its observed-node set, and an assignment to the fixed nodes of the model obtained by intervening on its mechanism complement, the mechanism $Q[S]$ measure is the intervened model's observational law projected onto the given SWIG-node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (S : Finset (Causalean.SWIGNode N)) → S ⊆ M.observed → (M.fixSet (M.mechComplementNames S) (Causalean.SCM.mechComplementNames_random_mem_observed M S) (Causalean.SCM.mechComplementNames_fixed_not_mem M S)).FixedValues → MeasureTheory.Measure (Causalean.ValuesOn S (Causalean.swigΩ Ω))","l":150,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.QmechMeasure_singleton_eq_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The do-complement Q[S] atom equals the local q-mass on S.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)) (hS : S ⊆ M.observed), (∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v) → ∀ (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (M.QmechMeasure S hS (M.mechDoValues S s x)) {Causalean.valuesProjection hS x} = M.qLocalMass s S hS x","l":195,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.RandomValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the random-value assignments assign one value to every observed or unobserved latent node of the model.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":151,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.ReferenceFaithful","k":"def","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"For any node set with measurable node-value spaces and a family of reference measures, reference faithfulness means that every value at every random or fixed node has nonzero mass under that node's reference measure.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM.ReferenceMeasures Ω → Prop","l":70,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.ReferenceMeasures","k":"structure","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"A reference measure family assigns a measure on every random or fixed node's value space, required to be σ-finite.","s":"{N : Type u_1} → (Ω : N → Type u_3) → [(n : N) → MeasurableSpace (Ω n)] → Type (max u_1 u_3)","l":35,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.UnobservedValues","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the unobserved-value assignments are exactly the model's latent-value assignments.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Type (max uΩ uN)","l":133,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.absolutelyContinuous_jointRef_of_faithful","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"Faithful references dominate. On a finite coordinate product indexed by a node set I, if the reference family ref is faithful — every single coordinate value carries nonzero reference mass, then every measure μ on that product is absolutely continuous with respect to the joint reference measure.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (I : Finset (Causalean.SWIGNode N)) (μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))), μ.AbsolutelyContinuous (Causalean.SCM.jointRef ref I)","l":120,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.absolutelyContinuous_of_map_measurableEmbedding","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Pull absolute continuity back through a measurable embedding.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {f : α → β}, MeasurableEmbedding f → ∀ {μ ν : MeasureTheory.Measure α}, (MeasureTheory.Measure.map f μ).AbsolutelyContinuous (MeasureTheory.Measure.map f ν) → μ.AbsolutelyContinuous ν","l":468,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.absolutelyContinuous_of_singleton_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"Any measure is absolutely continuous with respect to a measure that gives every singleton nonzero mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α), (∀ (x : α), ν {x} ≠ 0) → μ.AbsolutelyContinuous ν","l":80,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.adjustmentKernelY","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, a finite treatment set, the condition that every random treatment copy is observed, the condition that no fixed treatment copy is already fixed, a finite observed outcome-node set, a finite observed adjustment-node set, and fixed-node values, the treatment-indexed backdoor-adjustment outcome kernel maps each treatment value to the backdoor-adjustment law of the outcome coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":116,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.adjustmentKernelY_empty_eq","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"With an empty adjustment set, the outcome adjustment kernel equals the observed conditional kernel of the outcome variables given the treated variables at the same fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wr : Finset (Causalean.SWIGNode N)) (hWr : Wr ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.adjustmentKernelY X hObs hFix Wr ∅ hWr (Finset.empty_subset M.observed) s0) t = (M.obsCondKernel Wr (Finset.image Causalean.SWIGNode.random X) hWr hXr) (s0, t)","l":285,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.aemeasurable_fiber_rnDeriv_of_finite","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":1,"d":"For finite index types α and β, a joint measure μ on α × β, a measure ρ on β, and a Markov kernel κ from α to β, the fibre Radon-Nikodym derivative selector (κ p.1).rnDeriv ρ p.2 is almost-everywhere measurable with respect to μ.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Finite α] [Finite β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure (α × β)) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β), AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) μ","l":131,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.ancestralFactorization","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"Ancestral factorization (Lemma lem:scm-ancestral-factor). Fix a structural causal model M, a target node set T, two fixed-value assignments s, s', two latent assignments ℓ, ℓ', and a node v with v belonging to T and v an observed node. If s and s' agree on every fixed node that equals or is an ancestor of some node of T and ℓ and ℓ' agree on every unobserved node that equals or is an ancestor of some node of T, then the evaluation of M at v with (s, ℓ) equals the evaluation with (s', ℓ').","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {s s' : M.FixedValues} {ℓ ℓ' : M.LatentValues}, (∀ (d : Causalean.SWIGNode N) (hd : d ∈ M.fixed), (∃ v ∈ T, d = v ∨ M.dag.isAncestor d v) → s ⟨d, hd⟩ = s' ⟨d, hd⟩) → (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ v ∈ T, u = v ∨ M.dag.isAncestor u v) → ℓ ⟨u, hu⟩ = ℓ' ⟨u, hu⟩) → ∀ {v : Causalean.SWIGNode N}, v ∈ T → ∀ (hv_obs : v ∈ M.observed), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv_obs⟩ = M.evalMap s' ℓ' ⟨v, Finset.mem_union_left M.unobserved hv_obs⟩","l":541,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.backdoorAdjustment","k":"def","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"For a finite node set with measurable node-value spaces in which every random or fixed node value space is standard Borel and nonempty, a structural causal model, a treatment node set whose random copies are observed, whose fixed copies are not already fixed in the model, and outcome and adjustment SWIG-node sets, each contained in the observed-node set, the backdoor adjustment functional is the kernel from post-intervention fixed-node assignments to distributions of the outcome-node assignment. It first forms the observational marginal distribution of the adjustment nodes, then that marginal indexed by the post-intervention fixed-node assignment, and then the conditional outcome distribution given the treatment and adjustment-node assignments.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hX_obs hX_fixed).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":69,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.backdoorAdjustment_invariant","k":"theorem","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Cross-SCM invariance of backdoorAdjustment. Fix two causal models M₁ and M₂ that share the same SWIG graph and the same observational kernel, and a treatment set X, an outcome set Y, and an adjustment set Z that are valid intervention data and observed variables in both models. Then the backdoor-adjustment kernel for Y given Z after intervening on X, computed within M₁, is the same as the one computed within M₂.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ (X : Finset N) (hX_obs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hX_fixed₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hX_obs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hX_fixed₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.backdoorAdjustment X hX_obs₁ hX_fixed₁ Y Z hY₁ hZ₁ ≍ M₂.backdoorAdjustment X hX_obs₂ hX_fixed₂ Y Z hY₂ hZ₂","l":283,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.backdoorCriterion_W_nonDesc","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Backdoor criterion (i), post-intervention non-descendance form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), M.backdoorCriterion X hObs hFix Y Z → ∀ x ∈ X, ∀ v ∈ Z, ¬(M.fixSet X hObs hFix).dag.isAncestor (Causalean.SWIGNode.fixed x) v","l":107,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoorCriterion_W_nonDescM1","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Backdoor criterion (i), base-graph non-descendance form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), M.backdoorCriterion X hObs hFix Y Z → ∀ D ∈ X, ∀ w ∈ Z, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w","l":124,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoorCriterion_dSep_fixSet","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":1,"d":"Backdoor criterion (ii), extended to the post-intervention fixed nodes. Fix a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, an outcome set Y and a conditioning set Z with Y contained in the observed nodes and the randomized image of X contained in the observed nodes. If M's DAG satisfies the backdoor criterion for X, Y given Z, then in the post-intervention graph obtained by fixing X, Y is d-separated from the randomized image of X given Z together with the post-intervention fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)), Y ⊆ M.observed → Finset.image Causalean.SWIGNode.random X ⊆ M.observed → M.backdoorCriterion X hObs hFix Y Z → (M.fixSet X hObs hFix).dag.dSep Y (Finset.image Causalean.SWIGNode.random X) (Z ∪ (M.fixSet X hObs hFix).fixed)","l":55,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoor_completeness_ae","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Backdoor completeness, a.e. in the treatment value. Fix a causal model M and a treatment set X whose random copy is observed and whose fixed copy is not already held fixed in M, and let Y, Z be observed outcome and adjustment sets, neither of which overlaps the treatment's random nodes, satisfying the backdoor criterion relative to (X,Y). Given a base configuration s0 at which overlap holds along every post-intervention slice and the product of the observational treatment and adjustment-set marginals is absolutely continuous with respect to their joint observational law, then for almost every treatment value t under the observational treatment marginal, the post-intervention outcome distribution at t equals the backdoor-adjustment functional evaluated at t.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.backdoorCriterion X hObs hFix Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s) → M.BackdoorPositivityAE X Z hZ (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s0 → ∀ᵐ (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) ∂M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0, (M.doKernelY X hObs hFix Y hY s0) t = (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0) t","l":380,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_completeness_ae_compProd","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Joint (compProd) completeness — primary form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.backdoorCriterion X hObs hFix Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s) → M.BackdoorPositivityAE X Z hZ (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ) s0 → (M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0).compProd (M.doKernelY X hObs hFix Y hY s0) = (M.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs) hZ)) s0).compProd (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0)","l":216,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_identifiable_ae","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"Backdoor identifiability, a.e. in the treatment value (cross-SCM corollary). Fix two causal models M₁, M₂ that share the same SWIG graph and the same observational kernel, with base configurations that agree under this shared identification. Fix a treatment set X valid in both models and disjoint outcome/adjustment sets Y, Z that are observed in both models and disjoint from the treatment's random nodes, such that Z satisfies the backdoor criterion for (X,Y) in each model separately, with overlap holding along every post-intervention slice in each model and product positivity of the observational treatment and adjustment-set marginals holding in each model. Then for almost every treatment value t under the shared observational treatment marginal, the post-intervention outcome distribution computed in M₁ at t equals the one computed in M₂ at t, since each model individually satisfies backdoor completeness and the two share the same graph and observational kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → ∀ (X : Finset N) (Y Z : Finset (Causalean.SWIGNode N)) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.backdoorCriterion X hObs₁ hFix₁ Y Z → M₂.backdoorCriterion X hObs₂ hFix₂ Y Z → Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → ∀ (s0₁ : M₁.FixedValues) (s0₂ : M₂.FixedValues), (∀ (s : (M₁.fixSet X hObs₁ hFix₁).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M₁ X hObs₁ hFix₁ Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁) s) → (∀ (s : (M₂.fixSet X hObs₂ hFix₂).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M₂ X hObs₂ hFix₂ Z (Finset.union_subset (Finset.image_subset_iff.mpr hObs₂) hZ₂) s) → M₁.BackdoorPositivityAE X Z hZ₁ (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁) s0₁ → M₂.BackdoorPositivityAE X Z hZ₂ (Finset.union_subset (Finset.image_subset_iff.mpr hObs₂) hZ₂) s0₂ → M₁.obsKernel ≍ M₂.obsKernel → s0₁ ≍ s0₂ → ⇑(M₁.doKernelY X hObs₁ hFix₁ Y hY₁ s0₁) =ᵐ[M₁.treatmentMarginal X (LE.le.trans Finset.subset_union_left (Finset.union_subset (Finset.image_subset_iff.mpr hObs₁) hZ₁)) s0₁] ⇑(M₂.doKernelY X hObs₂ hFix₂ Y hY₂ s0₂)","l":424,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.backdoor_rule2_ae","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Rule-2 applicator (backdoor form).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed) (hXrZ : Finset.image Causalean.SWIGNode.random X ∪ Z ⊆ M.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random X) → Disjoint (Finset.image Causalean.SWIGNode.random X) Z → M.backdoorCriterion X hObs hFix Y Z → ∀ (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix Z hXrZ s) → (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hXr) (M.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hXrZ) (M.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) × Causalean.ValuesOn Z (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hXr) (M.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0))), ((M.fixSet X hObs hFix).obsCondKernel Y Z (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hZ)) (M.fixSetExtend X hObs hFix s0 p.1, p.2) = (M.obsCondKernel Y (Finset.image Causalean.SWIGNode.random X ∪ Z) hY hXrZ) (s0, Causalean.valuesUnionMk p.1 p.2)","l":138,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.backdoor_rule3_Z_marginal","k":"theorem","m":"Causalean.SCM.ID.BackdoorCriterion","a":"SCM/ID","t":1,"d":"Backdoor Rule-3 leg. Fix a causal model M and a treatment set X whose random copy is observed and whose fixed copy is not already held fixed in M, and a set Z that is observed in M such that no node of Z is a descendant of any treatment random node — the non-descendant clause of the backdoor criterion. Then, at any post-intervention configuration s_post, the Z-marginal of the post-intervention observational law at s_post equals the Z-marginal of the original observational law at the pre-intervention configuration underlying s_post.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (hZ : Z ⊆ M.observed), (∀ z ∈ Z, ∀ D ∈ X, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) z) → ∀ (s_post : (M.fixSet X hX_obs hX_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M X hX_obs hX_fixed) ▸ hZ)) ((M.fixSet X hX_obs hX_fixed).obsKernel s_post) = MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel (M.fixSetProj X hX_obs hX_fixed s_post))","l":109,"f":"Causalean/SCM/ID/BackdoorCriterion.lean"},{"n":"Causalean.SCM.bbZAncestors_fixSet_transport","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A node in the Bayes-ball ancestor set of a target set and the model's fixed nodes remains in the corresponding ancestor set after intervention, once the fixed treatment copies are added to the target set, provided no treatment's random copy is an ancestor of a target node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {m : Causalean.SWIGNode N}, m ∈ M.dag.bbZAncestors (W ∪ M.fixed) → m ∈ (M.fixSet Z hZ_obs hZ_fixed).dag.bbZAncestors (W ∪ M.fixed ∪ Finset.image Causalean.SWIGNode.fixed Z)","l":256,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cComponentDensityFactor","k":"def","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":1,"d":"For a finite node set with measurable node-value spaces, a structural causal model, a family of reference measures, an assignment to the model's fixed nodes, and a set of SWIG nodes, assuming every observational law is finite, every singleton observed-node value space is standard Borel and nonempty, and every fixed-node space together with every observed-node prefix space satisfies the stated countability condition, the per-c-component density factor maps a full observed-node assignment to the product of its one-node conditional density factors over precisely those observed nodes whose c-component is the given set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Finset (Causalean.SWIGNode N) → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":31,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.cComponentDensityFactor_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":2,"d":"Cross-model c-component density-factor transport. If two structural causal models share the same SWIG graph and have heterogeneously-equal observational kernels, then every c-component density factor recovered from the observational chain-rule density is heterogeneously equal across the two models.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (C : Finset (Causalean.SWIGNode N)), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ [inst_3 : ∀ (s' : M₁.FixedValues), MeasureTheory.IsFiniteMeasure (M₁.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M₁.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M₁.observed.card), Nonempty (Causalean.ValuesOn {↑(M₁.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₁.FixedValues (Causalean.ValuesOn (M₁.prefixNodes k) (Causalean.swigΩ Ω))] [inst_7 : ∀ (s' : M₂.FixedValues), MeasureTheory.IsFiniteMeasure (M₂.obsKernel s')] [inst_8 : ∀ (k : ℕ) (hk : k < M₂.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_9 : ∀ (k : ℕ) (hk : k < M₂.observed.card), Nonempty (Causalean.ValuesOn {↑(M₂.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_10 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M₂.FixedValues (Causalean.ValuesOn (M₂.prefixNodes k) (Causalean.swigΩ Ω))], (fun s => M₁.cComponentDensityFactor ref s C) ≍ fun s => M₂.cComponentDensityFactor ref s C","l":95,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.c_component_factorization","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"Theorem (Tian 2002, per-node chain-rule factorization). For a structural causal model M, at a fixed assignment s, its observational kernel equals the full chain-rule product of one-node conditional kernels along the topological order of observed nodes. Tian's Pa⁺(V) is interpreted as the full observed history before V, so this statement is the ordinary iterated-disintegration factorization and does not use graphical Markov or do-calculus reasoning.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.obsKernel s = M.qFactorProduct s","l":137,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SCM.compProd_deterministic_apply","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Composing a kernel with a deterministic second kernel gives the distribution obtained by drawing from the first kernel and appending the deterministic output to that draw. This map-valued identity is useful when constructing factored kernels recursively.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] {f : α × β → γ} (hf : Measurable f) (a : α), (κ.compProd (ProbabilityTheory.Kernel.deterministic f hf)) a = MeasureTheory.Measure.map (fun b => (b, f (a, b))) (κ a)","l":150,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.compProd_singleton_mass","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"The singleton mass of a measure-kernel composition product factors as the conditioning singleton mass times the fibre singleton mass.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (b : α) (y : β), (μ.compProd κ) {(b, y)} = μ {b} * (κ b) {y}","l":100,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condDistrib_eq_of_map_prod_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Two conditional distributions with the same joint law are equal.","s":"∀ {α : Type u_3} {α' : Type u_4} {β : Type u_5} {γ : Type u_6} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace α'] [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] [inst_4 : StandardBorelSpace β] [inst_5 : Nonempty β] {X : α → γ} {Y : α → β} {X' : α' → γ} {Y' : α' → β} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α'} [inst_6 : MeasureTheory.IsFiniteMeasure μ] [inst_7 : MeasureTheory.IsFiniteMeasure ν], MeasureTheory.Measure.map (fun a => (X a, Y a)) μ = MeasureTheory.Measure.map (fun a => (X' a, Y' a)) ν → ProbabilityTheory.condDistrib Y X μ = ProbabilityTheory.condDistrib Y' X' ν","l":52,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.condDistrib_fixSet_cross_SCM_bridge","k":"theorem","m":"Causalean.SCM.Do.Rule2AE","a":"SCM/Do","t":2,"d":"Joint-μ_C-a.e. cross-SCM condDistrib bridge (isolated hard lemma).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) (Causalean.valuesUnionMk p.1 p.2)","l":41,"f":"Causalean/SCM/Do/Rule2AE.lean"},{"n":"Causalean.SCM.condDistrib_intervention_ancestral_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":1,"d":"Rule 3 core — intervention on non-ancestors of T is irrelevant (Option B in the design; covers the simplified Z_Y = Z case of the tex's Rule 3, sufficient for backdoor / frontdoor demos). Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and a target block T of observed variables. If none of the fixed copies of Z's nodes is an ancestor, in the intervention SWIG graph, of any node in T, then the T-marginal law of the intervened model equals the T-marginal law of the base model evaluated at the corresponding fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M'.observed), (∀ z ∈ Z, ∀ v ∈ T, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hT)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection hT) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":338,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.condDistrib_singleton_mass_eq_conditionalMass","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"For a measurable map Y and a measurable map Z out of a finite measure space, at a conditioning value z with nonzero pushforward mass under Z, the singleton mass that Mathlib's conditional distribution condDistrib Y Z μ assigns to a value y at z equals the discrete conditional-mass ratio conditionalMass, computed from the joint pushforward law of (Y, Z), evaluated at (y, z).","s":"∀ {α : Type u_3} {β : Type u_4} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] [MeasurableSingletonClass β] [MeasurableSingletonClass γ] {μ : MeasureTheory.Measure α} [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Y : α → γ} {Z : α → β}, Measurable Y → Measurable Z → ∀ (z : β) (y : γ), (MeasureTheory.Measure.map Z μ) {z} ≠ 0 → ((ProbabilityTheory.condDistrib Y Z μ) z) {y} = Causalean.SCM.ID.DiscreteID.conditionalMass (MeasureTheory.Measure.map (fun ω => (Y ω, Z ω)) μ) y z","l":159,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condDistrib_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"Mathlib's measure-level conditional distribution has singleton mass equal to the corresponding joint singleton mass divided by the conditioning mass.","s":"∀ {α : Type u_3} {β : Type u_4} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : StandardBorelSpace γ] [inst_4 : Nonempty γ] [MeasurableSingletonClass β] [MeasurableSingletonClass γ] {μ : MeasureTheory.Measure α} [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Y : α → γ} {Z : α → β}, Measurable Y → ∀ (z : β) (y : γ), (MeasureTheory.Measure.map Z μ) {z} ≠ 0 → ((ProbabilityTheory.condDistrib Y Z μ) z) {y} = (MeasureTheory.Measure.map (fun ω => (Z ω, Y ω)) μ) {(z, y)} / (MeasureTheory.Measure.map Z μ) {z}","l":140,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.condIndepFun_of_map","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":2,"d":"Pushforward bridge for CondIndepFun.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : StandardBorelSpace α] {β : Type u_2} [inst_2 : MeasurableSpace β] [inst_3 : StandardBorelSpace β] {γ : Type u_3} [inst_4 : MeasurableSpace γ] [StandardBorelSpace γ] [Nonempty γ] {δ : Type u_4} [inst_7 : MeasurableSpace δ] [StandardBorelSpace δ] [Nonempty δ] {ε : Type u_5} [inst_10 : MeasurableSpace ε] {φ : α → β} (hφ : Measurable φ) {X : β → γ}, Measurable X → ∀ {Y : β → δ}, Measurable Y → ∀ {Z : β → ε} (hZ : Measurable Z) {ν : MeasureTheory.Measure α} [inst_11 : MeasureTheory.IsFiniteMeasure ν] [inst_12 : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.map φ ν)], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Z ∘ φ) inferInstance) (Measurable.comap_le (Measurable.comp hZ hφ)) (X ∘ φ) (Y ∘ φ) ν → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap Z inferInstance) (Measurable.comap_le hZ) X Y (MeasureTheory.Measure.map φ ν)","l":54,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.condIndepFun_of_shared_base_valuesProjection_pi","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"Under an SCM's latent product measure, two outcomes that depend on a shared latent block and otherwise on disjoint latent blocks are conditionally independent after conditioning on any measurable summary of the shared block.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {U Rx Ry : Finset (Causalean.SWIGNode N)} [inst_3 : StandardBorelSpace M.LatentValues] {β : Type u_1} {γ : Type u_2} {δ : Type u_3} [inst_4 : MeasurableSpace β] [inst_5 : MeasurableSpace γ] [inst_6 : MeasurableSpace δ] (hU : U ⊆ M.unobserved) (hRx : Rx ⊆ M.unobserved) (hRy : Ry ⊆ M.unobserved) {baseMap : Causalean.ValuesOn U (Causalean.swigΩ Ω) → β} (hbaseMap : Measurable baseMap) {leftMap : β × Causalean.ValuesOn Rx (Causalean.swigΩ Ω) → γ}, Measurable leftMap → ∀ {rightMap : β × Causalean.ValuesOn Ry (Causalean.swigΩ Ω) → δ}, Measurable rightMap → Disjoint U Rx → Disjoint U Ry → Disjoint Rx Ry → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (baseMap ∘ Causalean.valuesProjection hU) inferInstance) (Measurable.comap_le (Measurable.comp hbaseMap (Causalean.measurable_valuesProjection hU))) (fun ω => leftMap (baseMap (Causalean.valuesProjection hU ω), Causalean.valuesProjection hRx ω)) (fun ω => rightMap (baseMap (Causalean.valuesProjection hU ω), Causalean.valuesProjection hRy ω)) M.latentProduct","l":376,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_contraction_axiom","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Contraction for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hW : W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ hW)) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ hW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (Finset.union_subset hY hW)) μ","l":184,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_decomposition","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Decomposition for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":135,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_subset_right","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Subset-right for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} {Ω : N → Type uΩ} [inst : (n : N) → MeasurableSpace (Ω n)] {I X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hY' : Y' ⊆ I) (hZ : Z ⊆ I), Y' ⊆ Y → ∀ [inst_1 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_2 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY') μ","l":111,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_symm","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Symmetry for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} {Ω : N → Type uΩ} [inst : (n : N) → MeasurableSpace (Ω n)] {I X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hY : Y ⊆ I) (hZ : Z ⊆ I) [inst_1 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_2 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hY) (Causalean.valuesProjection hX) μ","l":89,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condIndep_valuesProjection_weak_union_axiom","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Weak union for CondIndepFun when coordinates are given by valuesProjection.","s":"∀ {N : Type uN} [inst : DecidableEq N] {Ω : N → Type uΩ} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZ : Z ⊆ I) [inst_2 : StandardBorelSpace (Causalean.ValuesOn I (Causalean.swigΩ Ω))] {μ : MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW))) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW))) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":158,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.condKernel_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"A disintegration conditional kernel at a positive conditioning atom is the joint singleton mass divided by the conditioning singleton mass.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace β] [inst_3 : Nonempty β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] (ρ : MeasureTheory.Measure (α × β)) [inst_6 : MeasureTheory.IsFiniteMeasure ρ] (b : α) (y : β), ρ.fst {b} ≠ 0 → (ρ.condKernel b) {y} = ρ {(b, y)} / ρ.fst {b}","l":124,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.countingRef","k":"def","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"For any node set with measurable node-value spaces, provided every random or fixed node has a countable value space and every base-node singleton is measurable, the counting reference family assigns counting measure to every random and fixed node coordinate and records that each such measure is σ-finite.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → [∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] → [∀ (n : N), MeasurableSingletonClass (Ω n)] → Causalean.SCM.ReferenceMeasures Ω","l":26,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.cutsetLatent","k":"def","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"For a structural causal model, a target set of graph nodes, and an override block of graph nodes, the latent cutset is the set of unobserved nodes that either are themselves targets or have a directed path to some target whose strictly interior nodes avoid the override block.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_3} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":173,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.cutsetLatent_dSep_of_dSep","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":1,"d":"Concatenation d-separation for the latent cutset. Suppose W consists of observed nodes and F consists of fixed nodes, and that the target set Y is d-separated from the randomised do-block Zr given W together with F. Then the latent cutset cutsetLatent Y (Zr ∪ W) — the latent roots reaching Y along a directed path whose interior avoids Zr ∪ W — is likewise d-separated from Zr given the same W ∪ F.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y Zr W F : Finset (Causalean.SWIGNode N)), W ⊆ M.observed → F ⊆ M.fixed → M.dag.dSep Y Zr (W ∪ F) → M.dag.dSep (M.cutsetLatent Y (Zr ∪ W)) Zr (W ∪ F)","l":44,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cutsetLatent_dSep_of_fixSet_dSep","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":1,"d":"Cross-model concatenation d-separation for the latent cutset. Suppose that, in the intervened model M.fixSet Z _ _, the target set Y is d-separated from the randomised do-block Zr = Z.image .random given the adjustment set W together with the post-intervention fixed block, where W consists of observed nodes and no w ∈ W is, in the base graph M.dag, a descendant of any treatment's random copy .random D (D ∈ Z) — backdoor criterion (i). Then, in the base model M, the latent cutset cutsetLatent Y (Zr ∪ W) is d-separated from Zr given W ∪ M.fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y W : Finset (Causalean.SWIGNode N)), W ⊆ M.observed → (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → (M.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M.fixSet Z hZ_obs hZ_fixed).fixed) → M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed)","l":768,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.cutsetLatent_subset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"The latent cutset is contained in the model's unobserved node set.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y C : Finset (Causalean.SWIGNode N)), M.cutsetLatent Y C ⊆ M.unobserved","l":202,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.cutset_condIndep_condDistrib","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cut-set conditional independence, in condDistrib-pair form.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M.observed) (hW : W ⊆ M.observed), M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues), (fun p => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct) p) =ᵐ[MeasureTheory.Measure.map (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct] fun p => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) p.2","l":367,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.cutset_factor_pointwise","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cut-set factorization, pointwise, at the realized override.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [∀ (n : N), Nonempty (Ω n)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) (hY : Y ⊆ M.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M.observed) (s : M.FixedValues), ∃ h, (Measurable fun p => h p.1.1 p.1.2 p.2) ∧ (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection (fun x hz => hZrW (Finset.mem_union_left W hz)) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) ∧ ∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M.evalMap_overrideC hY hZrW s (Causalean.valuesUnionMk zr w') ℓ","l":210,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.disjoint_base_of_disjoint_random_image","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"Disjoint random-node images have disjoint underlying base-variable sets.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [Fintype N] (X W : Finset N), Disjoint (Finset.image Causalean.SWIGNode.random W) (Finset.image Causalean.SWIGNode.random X) → Disjoint W X","l":41,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.disjoint_fixed_observed","k":"theorem","m":"Causalean.SCM.ID.Toolkit.Derivation","a":"SCM/ID/Toolkit","t":2,"d":"Fixed and observed node sets of an SCM are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω), Disjoint M.fixed M.observed","l":44,"f":"Causalean/SCM/ID/Toolkit/Derivation.lean"},{"n":"Causalean.SCM.distMarkovEquiv_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"Easy half. For two DAGs G₁, G₂ on the same node set, if they are Markov equivalent — they declare exactly the same d-separations, then they are distributionally Markov equivalent: a distribution is a global I-map of one exactly when it is a global I-map of the other.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {G₁ G₂ : Causalean.DAG (Causalean.SWIGNode N)}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SCM.DistMarkovEquiv Ω G₁ G₂","l":86,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.doKernelY","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, a finite treatment set, the condition that every random treatment copy is observed, the condition that no fixed treatment copy is already fixed, a finite observed outcome-node set, and fixed-node values, the treatment-indexed post-intervention outcome kernel maps each treatment value to the post-intervention observational law of the outcome coordinates.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":97,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doKernelY_disintegrate","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Treatment-indexed chain rule. At treatment value t, the post-do(X) Y-marginal kernel disintegrates through Z: it is the conditional Y ∣ Z (under do(X)) composed with the post-do(X) Z-marginal kernel. The doKernelY-level form of obsKernel_map_eq_obsCondKernel_comp.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.doKernelY X hObs hFix Y hY s0) t = ((M.doKernelY X hObs hFix Z hZ s0) t).bind ⇑(((M.fixSet X hObs hFix).obsCondKernel Y Z (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M X hObs hFix) ▸ hZ)).sectR (M.fixSetExtend X hObs hFix s0 t))","l":161,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doKernelY_marginal_const","k":"theorem","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"Treatment-indexed Rule 3 (marginal invariance). If no X-copy is an ancestor of any Z-node, the post-do(X) Z-marginal kernel is constant in the treatment value and equals the observational Z-marginal at base s₀. The doKernelY-level form of backdoor_rule3_Z_marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (hZ : Z ⊆ M.observed), (∀ z ∈ Z, ∀ D ∈ X, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) z) → ∀ (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), (M.doKernelY X hObs hFix Z hZ s0) t = MeasureTheory.Measure.map (Causalean.valuesProjection hZ) (M.obsKernel s0)","l":189,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.doSide_M2_condDistrib_eq_M1_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Do-side analytic core: the M2 W-conditional equals the M1 witness kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M', so that intervening on Z is well-formed; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed and Y disjoint from the treatments' pre-intervention nodes. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value and additionally, for every candidate treatment/W pair, agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, the measure-level conditional law of Y given W equals, for almost every w under the W-marginal of the original model's observational kernel at s0, the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M'.LatentValues), Causalean.valuesProjection hY (M'.randomToObserved (M'.evalMap s0 ℓ)) = h (Causalean.valuesProjection hZr (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → (fun w => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t))) w) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0)] fun w => MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":1052,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.doSide_M2_pullback_eq_M1_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"Do-side cross-SCM core: the M2-pullback W-conditional on M2.latentProduct equals the M1 posterior witness kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → (fun w => (ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ∘ fun ℓ => (M'.fixSet Z hZ_obs hZ_fixed).randomToObserved ((M'.fixSet Z hZ_obs hZ_fixed).evalMap (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t) ℓ)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) ∘ fun ℓ => (M'.fixSet Z hZ_obs hZ_fixed).randomToObserved ((M'.fixSet Z hZ_obs hZ_fixed).evalMap (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t) ℓ)) (M'.fixSet Z hZ_obs hZ_fixed).latentProduct) w) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M'.obsKernel s0)] fun w => MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection (fun x hw => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hw)) (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":828,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.doSide_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Do-side per-slice witness identity (M1-witness-kernel form). Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value, and additionally for every candidate treatment/W pair agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, for almost every w under the W-marginal of the original model's observational kernel at s0, the intervened model's conditional-probability kernel for Y given W at (s', w) equals the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace M'.RandomValues] [inst_4 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_14 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M'.LatentValues), Causalean.valuesProjection hY (M'.randomToObserved (M'.evalMap s0 ℓ)) = h (Causalean.valuesProjection hZr (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (∀ (zr : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (w' : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : M'.LatentValues), h zr w' (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ) = M'.evalMap_overrideC hY hZrW s0 (Causalean.valuesUnionMk zr w') ℓ) → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 t, w) = MeasureTheory.Measure.map (h t w) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M' Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M'.randomToObserved (M'.evalMap s0 ℓ))) M'.latentProduct) w)","l":1205,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.do_rule1","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 1: Insertion/deletion of observations (single-SCM form). On any structural causal model M', suppose Y, Z, and W are all observed nodes of M', and that Y is d-separated from Z given W together with the fixed nodes M'.fixed, in the split graph M'.dag. Then, at any fixed-value point s, Y and Z are conditionally independent given W under the observational kernel M'.obsKernel s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (Y Z W : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Z (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Z (Causalean.swigΩ Ω))] (hY : Y ⊆ M'.observed) (hZ : Z ⊆ M'.observed) (hW : W ⊆ M'.observed), M'.dag.dSep Y Z (W ∪ M'.fixed) → ∀ (s : M'.FixedValues), M'.ObsCondIndep Y Z W hY hZ hW (M'.obsKernel s)","l":57,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule2_kernel","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 2: Action/observation exchange (single-SCM form, kernel-native). Fix a structural causal model M' and a treatment set Z for which each member's random copy is already observed in M' and each member's fixed copy is not yet among M''s fixed nodes, with outcome and conditioning sets Y and W, and with the random copies of Z and their union with W all observed in M'. Suppose that, in the model intervened on Z, Y is d-separated from the random copies of Z given W together with the post-intervention fixed nodes, that no fixed copy of a Z-variable is a post-intervention ancestor of any node in W, and that no random copy of a Z-variable is an M'-ancestor of any node in W, so W is not downstream of the intervention in either graph. At a fixed-value point s0, assume the law obtained by independently pairing a treatment value drawn from the observational marginal of Z's random copies with a conditioning value drawn from the observational marginal of W is absolutely continuous with respect to the actual observational joint law of Z's random copies and W. Then for almost every such independently-paired pair (t, w), the Y-given-W conditional kernel of the model intervened at t, evaluated at the fixed value extended by t and at w, equals the Y-given-(Z ∪ W) conditional kernel of M' evaluated at s0 and the point filled by combining t and w.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":95,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule3","k":"theorem","m":"Causalean.SCM.Do.DoCalculus","a":"SCM/Do","t":1,"d":"Rule 3: insertion/deletion of actions (simplified joint-marginal form). Fix a structural causal model M' and a treatment set Z for which each member's random copy is already observed in M' and each member's fixed copy is not yet among M''s fixed nodes, together with outcome and conditioning sets Y and W, both observed in M'. Suppose no node of Y ∪ W is, in the model intervened on Z, a descendant of the fixed copy of any variable in Z. Then the joint law of (Y, W) under the intervened model, at any post-intervention fixed value, equals the joint law of (Y, W) under the base model M' at the corresponding pre-intervention fixed value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed), (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ Finset.union_subset hY hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.union_subset hY hW)) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":181,"f":"Causalean/SCM/Do/DoCalculus.lean"},{"n":"Causalean.SCM.do_rule3_conditional","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":1,"d":"Conditional Rule 3 (headline, a.e. obsCondKernel form). Pearl's Rule 3 for the deletion of actions, stated against the project's jointly-measurable conditional kernel. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on, an outcome block Y and a conditioning block W of observed variables. If none of the fixed copies of Z's nodes is an ancestor, in the intervention SWIG graph, of any node in Y ∪ W, then for almost every value w of W under the intervened model's W-marginal, the Y-given-W conditional kernel of the model intervened at do(Z) equals the Y-given-W conditional kernel of the base model, both evaluated at the corresponding fixed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))], (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (fun w => ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s', w)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))] fun w => (M'.obsCondKernel Y W hY hW) (M'.fixSetProj Z hZ_obs hZ_fixed s', w)","l":213,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.do_rule3_conditional_condDistrib","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Conditional Rule 3 (literal condDistrib form).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))], (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), ProbabilityTheory.condDistrib (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY)) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":180,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.dominatedObs_countingRef","k":"theorem","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"For every structural causal model M, each of its observational laws is absolutely continuous with respect to the counting reference family's product measure, because a faithful reference dominates every measure on a countable coordinate product. Note this holds for *all* M, with no positivity or graph hypothesis.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] [inst_4 : ∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω), M.DominatedObs Causalean.SCM.countingRef","l":51,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.double_edge_to_doW_of_not_fixedX","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"For two disjoint intervention blocks, an edge remaining after first fixing the X block and then the W block also remains after fixing W alone when its source is not a fixed copy from X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (hDisj : Disjoint X W) {u v : Causalean.SWIGNode N}, (∀ D ∈ X, u ≠ Causalean.SWIGNode.fixed D) → ((M.fixSet X hX_obs hX_fixed).fixSet W (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M W X hX_obs hX_fixed hW_fixed hDisj)).dag.edge u v → (M.fixSet W hW_obs hW_fixed).dag.edge u v","l":173,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.double_edge_to_doX_of_not_fixedW","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"For two disjoint intervention blocks, an edge remaining after first fixing the W block and then the X block also remains after fixing X alone when its source is not a fixed copy from W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (hDisj : Disjoint W X) {u v : Causalean.SWIGNode N}, (∀ D ∈ W, u ≠ Causalean.SWIGNode.fixed D) → ((M.fixSet W hW_obs hW_fixed).fixSet X (fun D hD => id (hX_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M X W hW_obs hW_fixed hX_fixed hDisj)).dag.edge u v → (M.fixSet X hX_obs hX_fixed).dag.edge u v","l":60,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.edge_fixSet_of_edge","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A directed edge in the original graph remains an edge after fixing treatments when its source is not the random copy of any treatment being fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, M.dag.edge u v → (∀ D ∈ Z, u ≠ Causalean.SWIGNode.random D) → (M.fixSet Z hZ_obs hZ_fixed).dag.edge u v","l":165,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.edge_of_edge_fixSet","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Every directed edge after fixing treatments is either an original edge or the redirected outgoing edge from the fixed copy of a treatment being fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, (M.fixSet Z hZ_obs hZ_fixed).dag.edge u v → M.dag.edge u v ∨ ∃ D ∈ Z, u = Causalean.SWIGNode.fixed D ∧ M.dag.edge (Causalean.SWIGNode.random D) v","l":190,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.emptyPrefixValues","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model, the empty-prefix assignment is the unique assignment of values to its empty initial observed-node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.ValuesOn (M.prefixNodes 0) (Causalean.swigΩ Ω)","l":392,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.evalMap","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"For a structural causal model, the evaluation map maps each fixed-value assignment and latent realization to the resulting assignment of all random-node values, using the recursive structural evaluation for observed nodes and the supplied latent realization for unobserved nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → M.RandomValues","l":170,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_agree_iff_localConsistent","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"On a set P of observed nodes closed under taking observed parents (every observed parent of a member of P is itself in P), agreement of the structural evaluation evalMap with the recorded observed assignment x at every node of P is equivalent to pointwise local consistency of x against a fixed-value slice s and latent realization ℓ at every node of P.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved (hP.left hv)⟩ = x ⟨v, hP.left hv⟩) ↔ ∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.localConsistent s x v (hP.left hv) ℓ","l":699,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_eq_iff_localConsistent_of_observed_parent_agree","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"When all observed parents agree with their assigned values, evaluation at an observed node equals its assignment exactly when the latent values are locally consistent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), (∀ (w : Causalean.SWIGNode N) (hw : w ∈ M.observed), M.dag.edge w v → M.evalMap s ℓ ⟨w, Finset.mem_union_left M.unobserved hw⟩ = x ⟨w, hw⟩) → (M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = x ⟨v, hv⟩ ↔ M.localConsistent s x v hv ℓ)","l":430,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_eq_of_equiv","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":1,"d":"Cross-SCM pointwise equality of evalMap under SCM.Equiv. For two structural causal models M₁, M₂ that are structurally equivalent — the same graph up to topological reordering, matching edge types and have heterogeneously equal structural functions, fixed-value assignments s₁, s₂ and latent assignments ℓ₁, ℓ₂ on the two models, and a shared node w that is a random-variable node of M₁ and a random-variable node of M₂: if s₁ and s₂ agree coordinatewise on every fixed node shared by both models and ℓ₁ and ℓ₂ agree coordinatewise on every unobserved node shared by both models, then the two models' evaluation maps agree at w: M₁'s evaluation with (s₁, ℓ₁) equals M₂'s evaluation with (s₂, ℓ₂).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → ∀ (s₁ : M₁.FixedValues) (ℓ₁ : M₁.LatentValues) (s₂ : M₂.FixedValues) (ℓ₂ : M₂.LatentValues), (∀ {d : Causalean.SWIGNode N} (hd₁ : d ∈ M₁.fixed) (hd₂ : d ∈ M₂.fixed), s₁ ⟨d, hd₁⟩ = s₂ ⟨d, hd₂⟩) → (∀ {u : Causalean.SWIGNode N} (hu₁ : u ∈ M₁.unobserved) (hu₂ : u ∈ M₂.unobserved), ℓ₁ ⟨u, hu₁⟩ = ℓ₂ ⟨u, hu₂⟩) → ∀ {w : Causalean.SWIGNode N} (hw₁ : w ∈ M₁.randomVars) (hw₂ : w ∈ M₂.randomVars), M₁.evalMap s₁ ℓ₁ ⟨w, hw₁⟩ = M₂.evalMap s₂ ℓ₂ ⟨w, hw₂⟩","l":105,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.evalMap_eq_x_of_observedAt_eq","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"If structural evaluation at an indexed observed node equals its recorded value, then the same equality holds after replacing that node by any equal observed node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (j : Fin M.observed.card) {v : Causalean.SWIGNode N} (hv : v ∈ M.observed), ↑(M.observedAt j) = v → M.evalMap s ℓ ⟨↑(M.observedAt j), Finset.mem_union_left M.unobserved (M.observedAt j).property⟩ = x (M.observedAt j) → M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = x ⟨v, hv⟩","l":488,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.evalMap_factors_excluding_latent","k":"theorem","m":"Causalean.SCM.Model.EvalLatent","a":"SCM/Model","t":1,"d":"Latent-restricted factorization away from a chosen latent root. Fix a structural causal model M, a fixed-value assignment s, and a latent root node a. For a set T of random-variable nodes such that no node of T is a descendant of a, and none equals a, then the projection of the evaluation evalMap s to T factors through a measurable function of the latent coordinates other than a's — i.e. it does not depend on the latent value at a.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (a : Causalean.SWIGNode N) (ha : a ∈ M.unobserved) (T : Finset (Causalean.SWIGNode N)) (hT_sub : T ⊆ M.randomVars), (∀ v ∈ T, ¬M.dag.isAncestor a v ∧ v ≠ a) → ∃ g, Measurable g ∧ ∀ (ℓ : M.LatentValues), Causalean.valuesProjection hT_sub (M.evalMap s ℓ) = g fun i => ℓ ↑i","l":49,"f":"Causalean/SCM/Model/EvalLatent.lean"},{"n":"Causalean.SCM.evalMap_factors_through_ancestors","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"Ancestor factorization (existence form). Fix a structural causal model M and a node v with v an observed node. Then there is a measurable function g of the fixed-ancestor projection and the latent-ancestor projection of v such that, for every fixed-value assignment s and latent assignment ℓ, the evaluated value at v equals g applied to those two projections of s and ℓ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [∀ (n : N), Nonempty (Ω n)] (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), ∃ g, Measurable (Function.uncurry g) ∧ ∀ (s : M.FixedValues) (ℓ : M.LatentValues), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = g (Causalean.valuesProjection (Causalean.SCM.fixedAncestorsOfNode_subset M v) s) (Causalean.valuesProjection (Causalean.SCM.latentAncestorsOfNode_subset M v) ℓ)","l":210,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.evalMap_factors_through_parents","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"Parent factorization of evalMap at observed nodes. Fix a structural causal model M, a fixed-value assignment s, and a node v with v an observed node. Then there is a measurable function g of the assignment restricted to v's random parent coordinates such that, for every latent assignment ℓ, the evaluated value at v equals g applied to that restriction of the joint evaluation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed), ∃ g, Measurable g ∧ ∀ (ℓ : M.LatentValues), M.evalMap s ℓ ⟨v, Finset.mem_union_left M.unobserved hv⟩ = g (Causalean.valuesProjection Finset.inter_subset_right (M.evalMap s ℓ))","l":58,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.evalMap_fixSet_factual_eq","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":1,"d":"prop:scm-cf-consistency (Basic Concepts.tex L475–480). Fix a structural causal model M, an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, a base fixed-value assignment s, a latent assignment ℓ, and an intervened fixed-value assignment sx for the model obtained by fixing X. If sx agrees with s on the model's original fixed coordinates and for every targeted node the base evaluation at its random form already equals sx's value at its fixed form — the factual consistency condition, then for every observed node v, evaluating the intervened model with sx and ℓ at v agrees with evaluating the base model with s and ℓ at v.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s : M.FixedValues) (ℓ : M.LatentValues) (sx : (M.fixSet X hObs hFix).FixedValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), sx ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed X) hv⟩ = s ⟨v, hv⟩) → (∀ (D : N) (hD : D ∈ X), M.evalMap s ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left M.unobserved (hObs D hD)⟩ = sx ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → ∀ (v : ↥M.observed), (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved v.property⟩ = M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩","l":110,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_fixSet_observed_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"For one recursive evaluation step after an intervention, the value of an observed variable is the original structural equation evaluated at the parent values where intervened parents are pinned to their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (sx : (M.fixSet X hObs hFix).FixedValues) (ℓ : M.LatentValues) (v : ↥M.observed), (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved v.property⟩ = M.structFun v (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix ↑v fun w => if huo : ↑w ∈ (M.fixSet X hObs hFix).unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ (M.fixSet X hObs hFix).fixed then sx ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents (M.fixSet X hObs hFix).dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp ((M.fixSet X hObs hFix).dag_edges_classified (↑w) (↑v) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; (M.fixSet X hObs hFix).evalMap sx ℓ ⟨↑w, Finset.mem_union_left (M.fixSet X hObs hFix).unobserved hobs⟩)","l":73,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_fixSet_union_eq","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":1,"d":"Composition of a combined intervention with a single-stage intervention. Fix a structural causal model M, a latent assignment ℓ, and two intervention target sets X₁, X₂ such that every node of X₁ is currently a random observed node with no fixed copy already fixed, and likewise every node of X₁ ∪ X₂, with an X₁-only intervened fixed-value assignment sx₁ and a combined-intervention fixed-value assignment sxU for X₁ ∪ X₂. If sxU agrees with sx₁ on the original fixed coordinates and on the X₁ intervention coordinates and the X₁-intervened model's value at each X₂ node already equals sxU's assignment there — the intermediate condition, then for every observed node v, evaluating the combined-intervention model with sxU and ℓ at v agrees with evaluating the X₁-only intervened model with sx₁ and ℓ at v.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X₁ X₂ : Finset N) (hObs₁ : ∀ D ∈ X₁, Causalean.SWIGNode.random D ∈ M.observed) (hFix₁ : ∀ D ∈ X₁, Causalean.SWIGNode.fixed D ∉ M.fixed) (hObsU : ∀ D ∈ X₁ ∪ X₂, Causalean.SWIGNode.random D ∈ M.observed) (hFixU : ∀ D ∈ X₁ ∪ X₂, Causalean.SWIGNode.fixed D ∉ M.fixed) (ℓ : M.LatentValues) (sx₁ : (M.fixSet X₁ hObs₁ hFix₁).FixedValues) (sxU : (M.fixSet (X₁ ∪ X₂) hObsU hFixU).FixedValues), (∀ (v : Causalean.SWIGNode N) (hv : v ∈ M.fixed), sxU ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed (X₁ ∪ X₂)) hv⟩ = sx₁ ⟨v, Finset.mem_union_left (Finset.image Causalean.SWIGNode.fixed X₁) hv⟩) → (∀ (D : N) (hD : D ∈ X₁), sxU ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨Finset.mem_union_left X₂ hD, rfl⟩))⟩ = sx₁ ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → (∀ (D : N) (hD : D ∈ X₂), (M.fixSet X₁ hObs₁ hFix₁).evalMap sx₁ ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left (M.fixSet X₁ hObs₁ hFix₁).unobserved (hObsU D (Finset.mem_union_right X₁ hD))⟩ = sxU ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨Finset.mem_union_right X₁ hD, rfl⟩))⟩) → ∀ (v : ↥M.observed), (M.fixSet (X₁ ∪ X₂) hObsU hFixU).evalMap sxU ℓ ⟨↑v, Finset.mem_union_left (M.fixSet (X₁ ∪ X₂) hObsU hFixU).unobserved v.property⟩ = (M.fixSet X₁ hObs₁ hFix₁).evalMap sx₁ ℓ ⟨↑v, Finset.mem_union_left (M.fixSet X₁ hObs₁ hFix₁).unobserved v.property⟩","l":293,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.evalMap_measurable","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The evaluation map is jointly measurable in the fixed-value assignment and latent realization.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable (Function.uncurry M.evalMap)","l":415,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_observed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"On an observed node, the evaluation map is the auxiliary topological-order evaluation transported to that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (w : ↥M.randomVars) (hobs : ↑w ∈ M.observed), M.evalMap s ℓ w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑w, hobs⟩)) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":197,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_observed_unfold","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"At an observed node, the evaluation map unfolds to the structural function applied to fixed, latent, or recursively evaluated parent values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (v : ↥M.observed), M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved v.property⟩ = M.structFun v fun w => if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) (↑v) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; M.evalMap s ℓ ⟨↑w, Finset.mem_union_left M.unobserved hobs⟩","l":297,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalMap_overrideC","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"For a structural causal model, a target set contained in its observed nodes, an override block contained in its observed nodes, an assignment of fixed values, an assignment on the override block, and a latent realization, the overridden evaluation map returns the values of every target node, holding every node in the override block to its assigned override value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {Y C : Finset (Causalean.SWIGNode N)} → Y ⊆ M.observed → C ⊆ M.observed → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → Causalean.ValuesOn Y (Causalean.swigΩ Ω)","l":216,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_agree_anc","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Two assignments that override observed variables give the same evaluated values at every target variable when they agree on which relevant ancestors are overridden and on the values assigned there.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C₁ C₂ : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC₁ : C₁ ⊆ M.observed) (hC₂ : C₂ ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (c₁ : Causalean.ValuesOn C₁ (Causalean.swigΩ Ω)) (c₂ : Causalean.ValuesOn C₂ (Causalean.swigΩ Ω)), (∀ (x : Causalean.SWIGNode N), (∃ y ∈ Y, x = y ∨ M.dag.isAncestor x y) → (x ∈ C₁ ↔ x ∈ C₂) ∧ ∀ (h₁ : x ∈ C₁) (h₂ : x ∈ C₂), c₁ ⟨x, h₁⟩ = c₂ ⟨x, h₂⟩) → M.evalMap_overrideC hY hC₁ s c₁ ℓ = M.evalMap_overrideC hY hC₂ s c₂ ℓ","l":192,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalMap_overrideC_agree_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"The overridden evaluation on the target set depends on latent values only through the latent cutset.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ₁ ℓ₂ : M.LatentValues), Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ₁ = Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ₂ → M.evalMap_overrideC hY hC s c ℓ₁ = M.evalMap_overrideC hY hC s c ℓ₂","l":296,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.evalMap_overrideC_apply_of_mem_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At a target node inside the override block, the overridden evaluation returns the assigned override value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y) (hvC : ↑v ∈ C), M.evalMap_overrideC hY hC s c ℓ v = c ⟨↑v, hvC⟩","l":285,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_apply_of_not_mem_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At a target node outside the override block, the overridden evaluation applies the structural function to overridden parent values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y), ↑v ∉ C → M.evalMap_overrideC hY hC s c ℓ v = Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.structFun (M.observedAt (M.observedIndex ⟨↑v, hY v.property⟩)) fun w => M.parentMapOverride s c ℓ (M.observedIndex ⟨↑v, hY v.property⟩).isLt (fun m x hm_card => M.evalObservedAuxOverride hC s c ℓ m hm_card) w","l":315,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_at_self","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"Cornerstone: overriding by the model's own values changes nothing. Fix a structural causal model M, a target set Y and an override set C with Y contained in the observed nodes and C contained in the observed nodes, a fixed-value assignment s, and a latent assignment ℓ. Then overriding C with the values that evalMap s ℓ would itself have produced there leaves the resulting evaluation on Y equal to the plain evaluation evalMap s ℓ on Y. No d-separation hypothesis is needed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues), M.evalMap_overrideC hY hC s (fun v' => Causalean.SCM.observedAt_observedIndex M ⟨↑v', hC v'.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v', hC v'.property⟩)) (M.observedIndex ⟨↑v', hC v'.property⟩).isLt) ℓ = fun v => Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v, hY v.property⟩)) (M.observedIndex ⟨↑v, hY v.property⟩).isLt","l":446,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_dropZr_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Filled intervention coordinates do not affect the post-intervention override on Y.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY_M2 : Y ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed), Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → ∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)), (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC hY_M2 hZrW s (M'.fillZrW Z hZ_obs hZ_fixed W s w) ℓ = (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC hY_M2 (LE.le.trans Finset.subset_union_right hZrW) s w ℓ","l":217,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalMap_overrideC_eq","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At any target node, the overridden evaluation map is the transported override auxiliary value at that node's topological index.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥Y), M.evalMap_overrideC hY hC s c ℓ v = Causalean.SCM.observedAt_observedIndex M ⟨↑v, hY v.property⟩ ▸ M.evalObservedAuxOverride hC s c ℓ (↑(M.observedIndex ⟨↑v, hY v.property⟩)) (M.observedIndex ⟨↑v, hY v.property⟩).isLt","l":231,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalMap_overrideC_fixSet_compat_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructCrossSCM","a":"SCM/Do/Rule2Kernel/Structural","t":1,"d":"Original and post-intervention override evaluations agree at fillZrW. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with the union of the random copies of Z and a set W contained in the intervened model's observed variables, fix an intervened fixed assignment s, a conditioning value w on W, and a latent draw ℓ. Then at any observed node v of the intervened model, the base model's override-evaluation at the projected fixed assignment and the filled point built from s and w, applied to the reindexed latent draw, equals the intervened model's override-evaluation at s, the same filled point, and ℓ, evaluated at v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW_M2 : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ (M'.fixSet Z hZ_obs hZ_fixed).observed) (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), M'.evalMap_overrideC (Finset.Subset.refl M'.observed) (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed ▸ hZrW_M2) (M'.fixSetProj Z hZ_obs hZ_fixed s) (M'.fillZrW Z hZ_obs hZ_fixed W s w) (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, hv⟩ = (M'.fixSet Z hZ_obs hZ_fixed).evalMap_overrideC (Finset.Subset.refl (M'.fixSet Z hZ_obs hZ_fixed).observed) hZrW_M2 s (M'.fillZrW Z hZ_obs hZ_fixed W s w) ℓ ⟨v, hv⟩","l":178,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructCrossSCM.lean"},{"n":"Causalean.SCM.evalMap_overrideC_observed_unfold","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructCrossSCM","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"Cast-free observed unfold for evalMap_overrideC (Y = M.observed case).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (v : ↥M.observed), M.evalMap_overrideC (Finset.Subset.refl M.observed) hC s c ℓ ⟨↑v, v.property⟩ = if hvC : ↑v ∈ C then c ⟨↑v, hvC⟩ else M.structFun v (Causalean.SCM.parentDispatchOverride✝ M hC s c ℓ v)","l":129,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructCrossSCM.lean"},{"n":"Causalean.SCM.evalMap_unobserved","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"On a latent node, the evaluation map is direct projection from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (w : ↥M.randomVars) (huo : ↑w ∈ M.unobserved), M.evalMap s ℓ w = ℓ ⟨↑w, huo⟩","l":209,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"For a structural causal model, an assignment of its fixed values, a realization of its latent values, and an observed-node index, the auxiliary observed-node evaluator assigns, for every proof that the index is valid, the value of that observed node by recursively applying its structural function to its parent values in topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → (n : ℕ) → (hn : n < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":136,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"For a structural causal model, an override block contained in its observed nodes, an assignment of fixed values, an assignment on the override block, a latent realization, and an observed-node index, the override auxiliary evaluator gives, for every proof that the index is valid, the node's override value when it lies in the override block and otherwise its structural-function value computed recursively from overridden parent values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {C : Finset (Causalean.SWIGNode N)} → C ⊆ M.observed → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → (n : ℕ) → (hn : n < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":176,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_agree_anc","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"If two overrides use the same values on every observed ancestor of a target set, their recursive structural evaluations agree at every such ancestor and target.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {C₁ C₂ : Finset (Causalean.SWIGNode N)} (hC₁ : C₁ ⊆ M.observed) (hC₂ : C₂ ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (c₁ : Causalean.ValuesOn C₁ (Causalean.swigΩ Ω)) (c₂ : Causalean.ValuesOn C₂ (Causalean.swigΩ Ω)), (∀ (x : Causalean.SWIGNode N), (∃ t ∈ T, x = t ∨ M.dag.isAncestor x t) → (x ∈ C₁ ↔ x ∈ C₂) ∧ ∀ (h₁ : x ∈ C₁) (h₂ : x ∈ C₂), c₁ ⟨x, h₁⟩ = c₂ ⟨x, h₂⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ t ∈ T, ↑(M.observedAt ⟨n, hn⟩) = t ∨ M.dag.isAncestor (↑(M.observedAt ⟨n, hn⟩)) t) → M.evalObservedAuxOverride hC₁ s c₁ ℓ n hn = M.evalObservedAuxOverride hC₂ s c₂ ℓ n hn","l":109,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_agree_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"If two latent assignments agree at every unobserved node that is a target or reaches a target along a directed path whose interior avoids the override block, their recursive override evaluations agree at every observed node that is a target or reaches one along such a path.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ₁ ℓ₂ : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ t ∈ T, u = t ∨ M.dag.isAncestorAvoiding C u t) → ℓ₁ ⟨u, hu⟩ = ℓ₂ ⟨u, hu⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ t ∈ T, ↑(M.observedAt ⟨n, hn⟩) = t ∨ M.dag.isAncestorAvoiding C (↑(M.observedAt ⟨n, hn⟩)) t) → M.evalObservedAuxOverride hC s c ℓ₁ n hn = M.evalObservedAuxOverride hC s c ℓ₂ n hn","l":213,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_eq","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override auxiliary evaluator unfolds to either the override value or the structural function applied to overridden parents.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAuxOverride hC s c ℓ n hn = if hcSelf : ↑(M.observedAt ⟨n, hn⟩) ∈ C then c ⟨↑(M.observedAt ⟨n, hn⟩), hcSelf⟩ else M.structFun (M.observedAt ⟨n, hn⟩) fun w => M.parentMapOverride s c ℓ hn (fun m x hm_card => M.evalObservedAuxOverride hC s c ℓ m hm_card) w","l":196,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_eq_evalObservedAux_at_self","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"When an observed-node override uses the model's own recursively evaluated values, the overridden recursive evaluator agrees with the original evaluator at every topological position.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAuxOverride hC s (fun v' => Causalean.SCM.observedAt_observedIndex M ⟨↑v', hC v'.property⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑v', hC v'.property⟩)) (M.observedIndex ⟨↑v', hC v'.property⟩).isLt) ℓ n hn = M.evalObservedAux s ℓ n hn","l":383,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAuxOverride_measurable","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"At every position in an SCM's topological order, its observed-variable evaluator with specified observed values overridden is jointly measurable in fixed, override, and latent inputs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)} (hC : C ⊆ M.observed) (n : ℕ) (hn : n < M.observed.card), Measurable fun p => M.evalObservedAuxOverride hC p.1.1 p.1.2 p.2 n hn","l":486,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAux_agree_anc","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"An observed variable has the same recursively evaluated value under two inputs when those inputs agree on every fixed and latent cause that can affect the target variables. This expresses the local ancestral dependence of a structural causal model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (T : Finset (Causalean.SWIGNode N)) {s s' : M.FixedValues} {ℓ ℓ' : M.LatentValues}, (∀ (d : Causalean.SWIGNode N) (hd : d ∈ M.fixed), (∃ v ∈ T, d = v ∨ M.dag.isAncestor d v) → s ⟨d, hd⟩ = s' ⟨d, hd⟩) → (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.unobserved), (∃ v ∈ T, u = v ∨ M.dag.isAncestor u v) → ℓ ⟨u, hu⟩ = ℓ' ⟨u, hu⟩) → ∀ (n : ℕ) (hn : n < M.observed.card), (∃ v ∈ T, ↑(M.observedAt ⟨n, hn⟩) = v ∨ M.dag.isAncestor (↑(M.observedAt ⟨n, hn⟩)) v) → M.evalObservedAux s ℓ n hn = M.evalObservedAux s' ℓ' n hn","l":474,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux_cast_collapse_at_observedAt","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"For an observed node, transporting its recursively evaluated value through the round-trip topological-index lookup leaves that value unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (j : Fin M.observed.card) (hvObs : ↑(M.observedAt j) ∈ M.observed), Causalean.SCM.observedAt_observedIndex M ⟨↑(M.observedAt j), hvObs⟩ ▸ M.evalObservedAux s ℓ (↑(M.observedIndex ⟨↑(M.observedAt j), hvObs⟩)) (M.observedIndex ⟨↑(M.observedAt j), hvObs⟩).isLt = M.evalObservedAux s ℓ (↑j) j.isLt","l":345,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.evalObservedAux_eq","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The auxiliary evaluator unfolds to its structural function applied to the parent tuple.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n < M.observed.card), M.evalObservedAux s ℓ n hn = M.structFun (M.observedAt ⟨n, hn⟩) fun w => M.parentMap s ℓ hn (fun m x hm_card => M.evalObservedAux s ℓ m hm_card) w","l":152,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.evalObservedAux_measurable","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"At every position in a causal model's topological ordering of observed variables, the recursively evaluated observed value is measurable as a function of the model's fixed and latent inputs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n < M.observed.card), Measurable fun p => M.evalObservedAux p.1 p.2 n hn","l":348,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.exists_evalMap_overrideC_factors_cutset","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":1,"d":"Cutset factorization of an override evaluation. Fix a structural causal model M, a target node set Y and an override node set C with Y contained in the observed nodes and C contained in the observed nodes, a fixed-value assignment s, and an override assignment c on C. Then there is a measurable map from assignments on the latent cutset of Y relative to C to assignments on Y such that, for every latent assignment, the overridden evaluation at Y equals this map applied to the latent assignment's projection onto the cutset.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)), ∃ h, Measurable h ∧ ∀ (ℓ : M.LatentValues), M.evalMap_overrideC hY hC s c ℓ = h (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y C) ℓ)","l":334,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.extendObsPrefix","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model and an index strictly below its number of observed nodes, the extended prefix assignment maps an assignment on the first $n$ observed nodes together with a value for the next node to the assignment on the first $n+1$ observed nodes that retains the prefix values and appends that value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n < M.observed.card) → Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩) → Causalean.ValuesOn (M.prefixNodes (n + 1)) (Causalean.swigΩ Ω)","l":414,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.extendOrderedLatentPrefix","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a natural number $n$, and evidence that the model has at least $n+1$ observed nodes, the ordered-latent prefix extension map takes a latent assignment, an assignment to the first $n$ observed nodes, and the value of the next observed node, and returns the same latent assignment paired with the resulting length-$n+1$ observed prefix.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩) → M.OrderedLatentPrefixValues (n + 1) hn","l":117,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.extractDistrict","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For an ambient node set, a SWIG graph, a node set, a district node set, the condition that the node set is contained in the ambient set, and a nonnegative extended-real mass function on ambient assignments, the district factor maps each ambient assignment to the product, over the nodes of the district in graph topological order, of the ratio of the two adjacent prefix marginals obtained by summing out the remaining nodes of $A$.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (A : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → A ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":116,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.fillZrW","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For a structural causal model, a set of treatment-variable names, the condition that each corresponding random copy is observed, the condition that none of their fixed copies is already fixed, a set of free conditioning nodes, and a fixed-value assignment for the model after fixing those treatments, the Rule 2 filler maps each free conditioning assignment to an assignment on the union of the treatment random copies and free nodes, inserting the treatment values from the fixed-value assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M' : Causalean.SCM N Ω) → (Z : Finset N) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) → (W : Finset (Causalean.SWIGNode N)) → (M'.fixSet Z hZ_obs hZ_fixed).FixedValues → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω)","l":80,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fillZrW_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, filling the conditioning assignment z against the extended post-intervention fixed assignment built from the base assignment s0 and treatment value t equals directly combining t and z into one assignment on their union.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Z : Finset (Causalean.SWIGNode N)) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (z : Causalean.ValuesOn Z (Causalean.swigΩ Ω)), M.fillZrW X hObs hFix Z (M.fixSetExtend X hObs hFix s0 t) z = Causalean.valuesUnionMk t z","l":304,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fillZrW_image_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"The image of a W-event under the filled-assignment map fillZrW Z hZ_obs hZ_fixed W s is exactly the set of assignments on Z.image .random ∪ W whose W-projection lies in the event and whose Z.image .random projection equals the fixed intervention value zFixedAsRandom (valuesProjection (fixSet_image_fixed_subset ...) s).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))), M'.fillZrW Z hZ_obs hZ_fixed W s' '' A = (fun c => Causalean.valuesProjection Finset.subset_union_right c) ⁻¹' A ∩ (fun c => Causalean.valuesProjection Finset.subset_union_left c) ⁻¹' {Causalean.SCM.zFixedAsRandom (Causalean.valuesProjection (Causalean.SCM.fixSet_image_fixed_subset M' Z hZ_obs hZ_fixed) s')}","l":47,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.fillZrW_injective","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"fillZrW Z _ _ W s' is injective when Z.image .random and W are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), Function.Injective (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":112,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.fillZrW_random_eq_fixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"The filled assignment pins intervention values on the random copies of Z.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) {D : N} (hD : D ∈ Z) (hRD : Causalean.SWIGNode.random D ∈ Finset.image Causalean.SWIGNode.random Z ∪ W), M'.fillZrW Z hZ_obs hZ_fixed W s w ⟨Causalean.SWIGNode.random D, hRD⟩ = s ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩","l":289,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixMono","k":"def","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, and a finite set of intervention targets whose random copies are observed and whose fixed copies are not already fixed, the monolithic intervened structural causal model simultaneously splits every target, first forming the split graph and then assigning each split edge the corresponding original edge type. It retains the original latent laws and structural mechanisms after reindexing their parent-value inputs.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Causalean.SCM N Ω","l":206,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap","k":"def","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"For a finite node population with node-value spaces, a SWIG graph, a finite set of intervention targets whose random copies are observed and whose fixed copies are not already fixed, a node, and an assignment of values to that node's parents after the simultaneous split, the monolithic parent reindexing map returns the corresponding assignment on the node's parents before the split. It reads a targeted random parent from its new fixed-copy coordinate and otherwise preserves the parent coordinate; these are respectively the random-parent clause and the fixed-parent clause.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → (G : Causalean.SWIGGraph N) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → (v : Causalean.SWIGNode N) → ((w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') → (w : ↥(G.dag.parents v)) → Causalean.swigΩ Ω ↑w","l":44,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At a fixed-coordinate parent, the monolithic parent reindexer reads the same fixed coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (d : N) (hwVal : Causalean.SWIGNode.fixed d ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.fixed d, hwVal⟩ = ξ ⟨Causalean.SWIGNode.fixed d, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.fixed d)).mpr (Or.inl ⟨hwVal, fun x x_1 heq => Eq.casesOn (motive := fun a t => Causalean.SWIGNode.random x = a → heq ≍ t → False) heq (fun h => False.elim (noConfusion_of_Nat Causalean.SWIGNode.ctorIdx h)) (Eq.refl (Causalean.SWIGNode.random x)) (HEq.refl heq)⟩)⟩","l":81,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_random","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At a treated random-coordinate parent, the monolithic parent reindexer reads the corresponding fixed coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (D : N) (hD : D ∈ X) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (hD_parent : Causalean.SWIGNode.random D ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.random D, hD_parent⟩ = ξ ⟨Causalean.SWIGNode.fixed D, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.fixed D)).mpr (Or.inr (Exists.intro D ⟨hD, ⟨rfl, hD_parent⟩⟩))⟩","l":126,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMonoParentMap_apply_random_notMem","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"At an untreated random-coordinate parent, the monolithic parent reindexer reads the same random coordinate from the split graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N) (ξ : (w' : ↥((G.splitMono X hObs hFix).dag.parents v)) → Causalean.swigΩ Ω ↑w') (u : N) (hu : u ∉ X) (hwVal : Causalean.SWIGNode.random u ∈ G.dag.parents v), Causalean.SCM.fixMonoParentMap G X hObs hFix v ξ ⟨Causalean.SWIGNode.random u, hwVal⟩ = ξ ⟨Causalean.SWIGNode.random u, (Causalean.SWIGGraph.splitMono_parents_char G X hObs hFix v (Causalean.SWIGNode.random u)).mpr (Or.inl ⟨hwVal, fun x hD heq => hu (Eq.symm (Causalean.SWIGNode.random.inj heq) ▸ hD)⟩)⟩","l":102,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Fixed-node set of the monolithic intervention. For a SWIG graph G and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, the monolithic intervention graph's fixed node set equals G's fixed node set together with the fixed copies of the targeted nodes in X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).fixed = G.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":293,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The original fixed node set is contained in the fixed node set after the monolithic intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), G.fixed ⊆ (G.splitMono X hObs hFix).fixed","l":315,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_image_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The fixed copies of the intervention targets are contained in the fixed node set after the monolithic intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), Finset.image Causalean.SWIGNode.fixed X ⊆ (G.splitMono X hObs hFix).fixed","l":325,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_latentDist","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Latent-distribution invariance of the monolithic intervention. For a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, and any latent-root node u of the monolithically intervened model, the intervened model's latent distribution at u equals M's original latent distribution at u.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : ↥(M.fixMono X hObs hFix).unobserved), (M.fixMono X hObs hFix).latentDist u = M.latentDist u","l":303,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_observed","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":1,"d":"Observed-node invariance of the monolithic intervention. For a SWIG graph G and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed, the monolithic intervention graph obtained by fixing X has the same observed node set as G.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).observed = G.observed","l":278,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"If no fixed copy of a target is a parent of a vertex after intervention, that vertex has the same parents as before.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {v : Causalean.SWIGNode N}, (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixMono X hObs hFix).dag.parents v) → (M.fixMono X hObs hFix).dag.parents v = M.dag.parents v","l":335,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixMono_structFun_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"After an intervention on a set of observed variables, each observed variable is still computed by the original structural equation, with intervened parents replaced by their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (v : ↥(M.fixMono X hObs hFix).observed) (ξ : (w : ↥((M.fixMono X hObs hFix).dag.parents ↑v)) → Causalean.swigΩ Ω ↑w), (M.fixMono X hObs hFix).structFun v ξ = M.structFun ⟨↑v, v.property⟩ (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix (↑v) ξ)","l":45,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.fixMono_unobserved","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The monolithic intervention preserves the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).unobserved = G.unobserved","l":287,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.fixSet","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, and a finite set of intervention targets whose random copies are observed and whose fixed copies are not already fixed, the standard multi-target intervention is the model obtained by simultaneously splitting all targets. It is defined as the monolithic intervention.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Causalean.SCM N Ω","l":41,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSetExtend","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For a structural causal model, a set of treatment-variable names, the condition that each corresponding random copy is observed, the condition that none of their fixed copies is already fixed, an assignment on the model's original fixed nodes, and an assignment to the treatment random copies, the extended fixed-value assignment is the fixed-value assignment for the model after those treatments are fixed, retaining original fixed values and using the matching treatment values for newly fixed copies.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → M.FixedValues → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) → (M.fixSet X hObs hFix).FixedValues","l":221,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetExtend_apply_new_fixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"A newly fixed coordinate in fixSetExtend reads the matching observed intervention coordinate.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) {D : N} (hD : D ∈ X), M.fixSetExtend X hObs hFix s0 t ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M X hObs hFix hD⟩ = t ⟨Causalean.SWIGNode.random D, Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩)⟩","l":236,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetProj","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, a finite intervention-target set whose random copies are observed and whose fixed copies are not already fixed, the projection from post-intervention fixed-value assignments to original fixed-value assignments restricts an assignment to the model's original fixed coordinates. It is defined by coordinate projection.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (M.fixSet X hObs hFix).FixedValues → M.FixedValues","l":727,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSetProj_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, projecting the extended post-intervention fixed assignment — built from the base assignment s0 and treatment value t — back onto the original fixed coordinates recovers s0.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), M.fixSetProj X hObs hFix (M.fixSetExtend X hObs hFix s0 t) = s0","l":269,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.fixSetZSlice","k":"def","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, an initial intervention-target set whose random copies are observed and whose fixed copies are not already fixed, and a second intervention-target set whose random copies are observed after the first intervention and whose fixed copies are not already fixed after the first intervention, the second-intervention fixed-slice extractor maps every fixed-value assignment after both interventions to its coordinates for the fixed copies of the second targets. It is defined by coordinate projection.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X Z : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) → (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hX_obs hX_fixed).fixed) → ((M.fixSet X hX_obs hX_fixed).fixSet Z hZ_obs hZ_fixed).FixedValues → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed Z) (Causalean.swigΩ Ω)","l":756,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_fixed_mem_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"For a variable included in the intervention set, the edge label from its fixed-coordinate node equals the original label from its random-coordinate node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (d : N) (v : Causalean.SWIGNode N), d ∈ X → (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v = M.edgeTypes.edgeType (Causalean.SWIGNode.random d) v","l":467,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_fixed_notMem_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"For a variable outside the intervention set, the edge label from its fixed-coordinate node is unchanged by the intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (d : N) (v : Causalean.SWIGNode N), d ∉ X → (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v = M.edgeTypes.edgeType (Causalean.SWIGNode.fixed d) v","l":486,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edgeType_random_eq","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"After intervening on a set of variables, the label of every edge leaving a random-coordinate node is unchanged from the original causal model.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : N) (v : Causalean.SWIGNode N), (M.fixSet X hObs hFix).edgeTypes.edgeType (Causalean.SWIGNode.random u) v = M.edgeTypes.edgeType (Causalean.SWIGNode.random u) v","l":450,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_edge_iff","k":"theorem","m":"Causalean.SCM.Model.InterventionAncestry","a":"SCM/Model","t":2,"d":"An edge in a causal model after intervention is exactly the corresponding edge produced by splitting the intervened variables in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u v : Causalean.SWIGNode N), (M.fixSet X hX_obs hX_fixed).dag.edge u v ↔ Causalean.SWIGGraph.splitMonoEdgeRel M.dag.edge X u v","l":30,"f":"Causalean/SCM/Model/InterventionAncestry.lean"},{"n":"Causalean.SCM.fixSet_empty_edge","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Edges of fixSet ∅ coincide with the base.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (u v : Causalean.SWIGNode N), (G.splitMono ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.edge u v ↔ G.dag.edge u v","l":159,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_empty_equiv","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"For a structural causal model M, intervening on the empty set of nodes yields a structural causal model that is equivalent to M.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), (M.fixSet ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).Equiv M","l":176,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_empty_parents","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Parent set of fixSet ∅ coincides with the base.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), (M.fixSet ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.parents v = M.dag.parents v","l":145,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_equiv_congr","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":1,"d":"fixSet preserves SCM.Equiv. Given a structural equivalence h between M₁ and M₂ and a finite set of nodes X whose members are, in M₁, observed in their random coordinate and not already held fixed, intervening on X in both models produces two structurally equivalent post-intervention models — the analogous observed/not-fixed conditions transfer automatically to M₂ via h.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω} (h : M₁.Equiv M₂) (X : Finset N) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed), have hObs₂ := fun D hD => Eq.mpr (id (congrArg (fun _a => Causalean.SWIGNode.random D ∈ _a) (Eq.symm h.left.right.right.left))) (hObs₁ D hD); have hFix₂ := fun D hD hDfix => hFix₁ D hD (Eq.mpr (id (congrArg (fun _a => Causalean.SWIGNode.fixed D ∈ _a) h.left.right.left)) hDfix); (M₁.fixSet X hObs₁ hFix₁).Equiv (M₂.fixSet X hObs₂ hFix₂)","l":270,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_evalMap_levelset_compat","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.LevelsetCompat","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM evalMap level-set bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose on the \"level set\" picked out by ℓ, the base model's evaluation map at the projected fixed assignment already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues), (∀ (D : N) (hD : D ∈ Z), M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨Causalean.SWIGNode.random D, Finset.mem_union_left M'.unobserved (hZ_obs D hD)⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":33,"f":"Causalean/SCM/Do/Rule2Kernel/LevelsetCompat.lean"},{"n":"Causalean.SCM.fixSet_evalMap_levelset_compat_M2","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.LevelsetCompat","a":"SCM/Do/Rule2Kernel","t":1,"d":"M2-direction level-set evalMap bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose the intervened model's own evaluation map at (s', ℓ) already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues), (∀ (D : N) (hD : D ∈ Z), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨Causalean.SWIGNode.random D, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved (hZ_obs D hD)⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":273,"f":"Causalean/SCM/Do/Rule2Kernel/LevelsetCompat.lean"},{"n":"Causalean.SCM.fixSet_evalMap_nonAnc_compat","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"Cross-SCM evalMap bridge for Rule 3.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (ℓ : (M'.fixSet Z hZ_obs hZ_fixed).LatentValues) {v : Causalean.SWIGNode N} (hv : v ∈ (M'.fixSet Z hZ_obs hZ_fixed).observed), (∀ z ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (M'.fixSet Z hZ_obs hZ_fixed).evalMap s' ℓ ⟨v, Finset.mem_union_left (M'.fixSet Z hZ_obs hZ_fixed).unobserved hv⟩ = M'.evalMap (M'.fixSetProj Z hZ_obs hZ_fixed s') (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed))) ℓ) ⟨v, Finset.mem_union_left M'.unobserved (le_of_eq (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) hv)⟩","l":91,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.fixSet_fixed","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet enlarges fixed by exactly X.image SWIGNode.fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).fixed = M.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":66,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_fixed_not_mem_of_disjoint","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"Fixing one intervention block does not add fixed copies from a disjoint block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hW_obs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed), (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → Disjoint W X → ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixSet W hW_obs hW_fixed).fixed","l":47,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.fixSet_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet only enlarges the fixed set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), M.fixed ⊆ (M.fixSet X hObs hFix).fixed","l":80,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_image_fixed_subset","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"X.image SWIGNode.fixed ⊆ (M.fixSet X _ _).fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), Finset.image Causalean.SWIGNode.fixed X ⊆ (M.fixSet X hObs hFix).fixed","l":87,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_isAncestor_fixed_forward","k":"theorem","m":"Causalean.SCM.Model.InterventionAncestry","a":"SCM/Model","t":1,"d":"Forward direction: .fixed D-ancestry in fixSet X lifts to .random D-ancestry in the base graph. Fix a structural causal model M and an intervention target set X such that every targeted node is currently a random observed node and none of its fixed copies is already fixed. For a targeted node D and a node v, if the fixed copy of D is a proper ancestor of v in the post-intervention graph obtained by fixing X, then the random copy of D is a proper ancestor of v in the original base graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ X → ∀ {v : Causalean.SWIGNode N}, (M.fixSet X hX_obs hX_fixed).dag.isAncestor (Causalean.SWIGNode.fixed D) v → M.dag.isAncestor (Causalean.SWIGNode.random D) v","l":42,"f":"Causalean/SCM/Model/InterventionAncestry.lean"},{"n":"Causalean.SCM.fixSet_latentDist","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet inherits latentDist verbatim. Both sides have the same type ((M.fixSet X _ _).unobserved = M.unobserved by rfl) and equal body.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (u : ↥(M.fixSet X hObs hFix).unobserved), (M.fixSet X hObs hFix).latentDist u = M.latentDist u","l":72,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_latentProduct_compat","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"Cross-SCM latent-product transport for Rule 3.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed), MeasureTheory.Measure.map (Causalean.valuesProjection (le_of_eq (Eq.symm (Causalean.SCM.fixSet_unobserved M' Z hZ_obs hZ_fixed)))) (M'.fixSet Z hZ_obs hZ_fixed).latentProduct = M'.latentProduct","l":30,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.fixSet_observed","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet preserves observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).observed = M.observed","l":54,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"SCM-level parent-set coincidence at non-.fixed-targeted vertices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {v : Causalean.SWIGNode N}, (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hObs hFix).dag.parents v) → (M.fixSet X hObs hFix).dag.parents v = M.dag.parents v","l":103,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixSet_random_no_children","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":1,"d":"In the intervened model, consider a name D belonging to the intervention set Z, where random copies of names in Z are observed in the base model and their fixed copies are not yet part of the base model's fixed coordinates. Then the random copy of D has no outgoing edge, in the post-intervention DAG, to any node v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ Z → ∀ (v : Causalean.SWIGNode N), ¬(M.fixSet Z hZ_obs hZ_fixed).dag.edge (Causalean.SWIGNode.random D) v","l":70,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixSet_random_not_isAncestor","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Structural.StructPointwise","a":"SCM/Do/Rule2Kernel/Structural","t":2,"d":"In M.fixSet Z _ _, every .random D (D ∈ Z) is not a proper ancestor of any node. Direct consequence of fixSet_random_no_children via isAncestor_child.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ Z → ∀ (v : Causalean.SWIGNode N), ¬(M.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.random D) v","l":91,"f":"Causalean/SCM/Do/Rule2Kernel/Structural/StructPointwise.lean"},{"n":"Causalean.SCM.fixSet_structFun_apply","k":"theorem","m":"Causalean.SCM.Model.CounterfactualLemmas","a":"SCM/Model","t":2,"d":"After an intervention on a set of observed variables, each observed variable is still computed by the original structural equation, with intervened parents replaced by their assigned intervention values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (v : ↥(M.fixSet X hObs hFix).observed) (ξ : (w : ↥((M.fixSet X hObs hFix).dag.parents ↑v)) → Causalean.swigΩ Ω ↑w), (M.fixSet X hObs hFix).structFun v ξ = M.structFun ⟨↑v, v.property⟩ (Causalean.SCM.fixMonoParentMap M.toSWIGGraph X hObs hFix (↑v) ξ)","l":59,"f":"Causalean/SCM/Model/CounterfactualLemmas.lean"},{"n":"Causalean.SCM.fixSet_unobserved","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSet preserves unobserved.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (M.fixSet X hObs hFix).unobserved = M.unobserved","l":60,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.fixedAncestorsOfNode","k":"def","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, and a node, the set of fixed ancestors of that node consists exactly of the model's fixed nodes that are either the node itself or ancestors of it in the directed graph. It is formed by filtering the fixed nodes by this condition.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":169,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.fixedAncestorsOfNode_subset","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"The fixed-ancestor set is contained in the fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), M.fixedAncestorsOfNode v ⊆ M.fixed","l":198,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.fixed_mem_fixSet","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Every SWIGNode.fixed D with D ∈ X is in the fixed set of fixSet X.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) {D : N}, D ∈ X → Causalean.SWIGNode.fixed D ∈ (M.fixSet X hObs hFix).fixed","l":94,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.frontdoorAdjustment","k":"def","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"For a finite node set with measurable node-value spaces in which every random or fixed node value space is standard Borel and nonempty, a structural causal model, a treatment node set whose random copies are observed, whose fixed copies are not already fixed in the model, and outcome and mediator SWIG-node sets, each contained in the observed-node set, the frontdoor adjustment functional is the kernel from post-intervention fixed-node assignments to distributions of the outcome-node assignment. It first forms the intervention treatment assignment, the mediator law conditional on that assignment, the observational treatment marginal, the conditional outcome law given observational treatment and mediator assignments, and the outcome law obtained by averaging over observational treatment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y Z : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → Z ⊆ M.observed → ProbabilityTheory.Kernel (M.fixSet X hX_obs hX_fixed).FixedValues (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":152,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.frontdoorAdjustment_invariant","k":"theorem","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":1,"d":"Cross-SCM invariance of frontdoorAdjustment. Fix two causal models M₁ and M₂ that share the same SWIG graph and the same observational kernel, and a treatment set X, an outcome set Y, and a mediator set Z that are valid intervention data and observed variables in both models. Then the frontdoor-adjustment kernel for Y through mediator Z after intervening on X, computed within M₁, is the same as the one computed within M₂.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M₁ M₂ : Causalean.SCM N Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → ∀ (X : Finset N) (hX_obs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₁.observed) (hX_fixed₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₁.fixed) (hX_obs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M₂.observed) (hX_fixed₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M₂.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hZ₁ : Z ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (hZ₂ : Z ⊆ M₂.observed), M₁.frontdoorAdjustment X hX_obs₁ hX_fixed₁ Y Z hY₁ hZ₁ ≍ M₂.frontdoorAdjustment X hX_obs₂ hX_fixed₂ Y Z hY₂ hZ₂","l":352,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.frontdoorKernelY","k":"def","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"For a finite node population with measurable, nonempty standard-Borel node-value spaces, a structural causal model, a treatment-variable set whose random copies are observed and whose fixed copies are not already fixed, an outcome-node set, a mediator-node set, the requirement that every outcome node is observed, the requirement that every mediator node is observed, and an assignment to the model's original fixed nodes, the frontdoor-adjustment conditional law of outcomes indexed by treatment values is the frontdoor-adjustment law evaluated after extending the fixed-node assignment by that treatment value. It is defined by reindexing the graph-level frontdoor-adjustment law along this extension.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] → [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) → (Y W : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → W ⊆ M.observed → M.FixedValues → ProbabilityTheory.Kernel (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":134,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.frontdoor_completeness_ae_compProd","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"Frontdoor completeness — joint (compProd), version-safe primary form. Let X be a valid intervention set — observed and not already fixed, and Wbase likewise a valid intervention set of mediators, with the outcome set Y observed, the random copies of Wbase observed, and the random copies of X observed. Suppose the frontdoor criterion holds for (X, Wbase, Y), that Y is disjoint from the random copies of Wbase and the random copies of Wbase are disjoint from those of X, and that each of the three legs of the frontdoor decomposition — the do(X) leg, the do(Wbase) leg adjusting for X, and the nested do(X)-then-do(Wbase) leg — satisfies the matching backdoor overlap and positivity conditions. Then the joint law of the treatment marginal with the post-intervention Y-marginal equals the joint law of the treatment marginal with the frontdoor-adjustment functional.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wbase : Finset N) (hWobs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hWfix : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hWr : Finset.image Causalean.SWIGNode.random Wbase ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed), M.frontdoorCriterion X hObs hFix Wbase hWobs hWfix Y → Disjoint Y (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WrXr : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)) (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s) → M.BackdoorPositivityAE X ∅ (Finset.empty_subset M.observed) (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s0 → (∀ (s : (M.fixSet Wbase hWobs hWfix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M Wbase hWobs hWfix (Finset.image Causalean.SWIGNode.random X) (Finset.union_subset hWr hXr) s) → M.BackdoorPositivityAE Wbase (Finset.image Causalean.SWIGNode.random X) hXr (Finset.union_subset hWr hXr) s0 → (∀ (s : ((M.fixSet X hObs hFix).fixSet Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr)))).FixedValues), Causalean.SCM.ID.Rule2JointOverlap (M.fixSet X hObs hFix) Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr))) ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr) s) → (∀ (s : (M.fixSet X hObs hFix).FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection (id hWr)) ((M.fixSet X hObs hFix).obsKernel s)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Wbase) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.empty_subset (M.fixSet X hObs hFix).observed)) ((M.fixSet X hObs hFix).obsKernel s))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr)) ((M.fixSet X hObs hFix).obsKernel s))) → (M.treatmentMarginal X hXr s0).compProd (M.doKernelY X hObs hFix Y hY s0) = (M.treatmentMarginal X hXr s0).compProd (M.frontdoorKernelY X hObs hFix Y (Finset.image Causalean.SWIGNode.random Wbase) hY hWr s0)","l":1091,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.frontdoor_fd1_rule3_nonDesc","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"G2: corrected-FD1 Rule-3 non-ancestry premise.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Wbase : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)), (M.fixSet X hX_obs hX_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.fixed X) (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WX : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)), ∀ v ∈ Y, ∀ d ∈ X, ¬((M.fixSet Wbase hW_obs hW_fixed).fixSet X (fun D hD => id (hX_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M X Wbase hW_obs hW_fixed hX_fixed (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX))).dag.isAncestor (Causalean.SWIGNode.fixed d) v","l":217,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.frontdoor_fd3_rule2_dSep","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":1,"d":"Rule-2 d-separation premise for the frontdoor third condition. Fix a structural causal model M and two intervention target sets X, Wbase such that every node of X is currently a random observed node with no fixed copy already fixed, and likewise every node of Wbase, and an outcome set Y with Y contained in the observed nodes, such that the base graph satisfies the backdoor criterion for Wbase, Y given the randomized image of X and the randomized images of Wbase and X are disjoint. Then in the graph obtained by first fixing X and then fixing Wbase, Y is d-separated from the randomized image of Wbase given exactly that double-intervention graph's fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Wbase : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hW_obs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hW_fixed : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed), ∀ Y ⊆ M.observed, M.backdoorCriterion Wbase hW_obs hW_fixed Y (Finset.image Causalean.SWIGNode.random X) → ∀ (hDisj_WX : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)), ((M.fixSet X hX_obs hX_fixed).fixSet Wbase (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hX_obs hX_fixed hW_fixed (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX)))).dag.dSep Y (Finset.image Causalean.SWIGNode.random Wbase) (∅ ∪ ((M.fixSet X hX_obs hX_fixed).fixSet Wbase (fun D hD => id (hW_obs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hX_obs hX_fixed hW_fixed (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WX)))).fixed)","l":316,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.frontdoor_identifiable_ae","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"Frontdoor identification, a.e. in the treatment value. Let X be a valid intervention set — observed and not already fixed, and Wbase likewise a valid intervention set of mediators, with the outcome set Y observed, the random copies of Wbase observed, and the random copies of X observed. Suppose the frontdoor criterion holds for (X, Wbase, Y), that Y is disjoint from the random copies of Wbase and the random copies of Wbase are disjoint from those of X, and that each of the three legs of the frontdoor decomposition — the do(X) leg, the do(Wbase) leg adjusting for X, and the nested do(X)-then-do(Wbase) leg — satisfies the matching backdoor overlap and positivity conditions. Then for treatmentMarginal-almost-every treatment value t, the post-intervention Y-marginal doKernelY at t equals the frontdoor-adjustment functional frontdoorKernelY at t.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Wbase : Finset N) (hWobs : ∀ D ∈ Wbase, Causalean.SWIGNode.random D ∈ M.observed) (hWfix : ∀ D ∈ Wbase, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hWr : Finset.image Causalean.SWIGNode.random Wbase ⊆ M.observed) (hXr : Finset.image Causalean.SWIGNode.random X ⊆ M.observed), M.frontdoorCriterion X hObs hFix Wbase hWobs hWfix Y → Disjoint Y (Finset.image Causalean.SWIGNode.random Wbase) → ∀ (hDisj_WrXr : Disjoint (Finset.image Causalean.SWIGNode.random Wbase) (Finset.image Causalean.SWIGNode.random X)) (s0 : M.FixedValues), (∀ (s : (M.fixSet X hObs hFix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M X hObs hFix ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s) → M.BackdoorPositivityAE X ∅ (Finset.empty_subset M.observed) (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random X))) M.observed)) hXr) s0 → (∀ (s : (M.fixSet Wbase hWobs hWfix).FixedValues), Causalean.SCM.ID.Rule2JointOverlap M Wbase hWobs hWfix (Finset.image Causalean.SWIGNode.random X) (Finset.union_subset hWr hXr) s) → M.BackdoorPositivityAE Wbase (Finset.image Causalean.SWIGNode.random X) hXr (Finset.union_subset hWr hXr) s0 → (∀ (s : ((M.fixSet X hObs hFix).fixSet Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr)))).FixedValues), Causalean.SCM.ID.Rule2JointOverlap (M.fixSet X hObs hFix) Wbase (fun D hD => id (hWobs D hD)) (Causalean.SCM.fixSet_fixed_not_mem_of_disjoint M Wbase X hObs hFix hWfix (Disjoint.symm (Causalean.SCM.disjoint_base_of_disjoint_random_image X Wbase hDisj_WrXr))) ∅ (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr) s) → (∀ (s : (M.fixSet X hObs hFix).FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection (id hWr)) ((M.fixSet X hObs hFix).obsKernel s)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Wbase) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection (Finset.empty_subset (M.fixSet X hObs hFix).observed)) ((M.fixSet X hObs hFix).obsKernel s))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.mpr (id (congrFun' (congrArg LE.le (Finset.union_empty (Finset.image Causalean.SWIGNode.random Wbase))) M.observed)) hWr)) ((M.fixSet X hObs hFix).obsKernel s))) → ∀ᵐ (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)) ∂M.treatmentMarginal X hXr s0, (M.doKernelY X hObs hFix Y hY s0) t = (M.frontdoorKernelY X hObs hFix Y (Finset.image Causalean.SWIGNode.random Wbase) hY hWr s0) t","l":1169,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.fullCondIndep_congr_left","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Transport FullCondIndep along a Finset equality in the first argument. Used in the Verma–Pearl induction to convert {a} ∪ A' to insert a A'.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X X' Y Z : Finset (Causalean.SWIGNode N)} (heq : X = X') {hX : X ⊆ M.randomVars} {hY : Y ⊆ M.randomVars} {hZ : Z ⊆ M.randomVars} {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X' Y Z (heq ▸ hX) hY hZ μ","l":160,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_congr_right","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Transport FullCondIndep along a Finset equality in the conditioning set. Used to align an empty conditioning set with parents v ∩ randomVars when v is a latent root.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Z Z' : Finset (Causalean.SWIGNode N)} (heq : Z = Z') {hX : X ⊆ M.randomVars} {hY : Y ⊆ M.randomVars} {hZ : Z ⊆ M.randomVars} {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X Y Z' hX hY (heq ▸ hZ) μ","l":172,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_const_left","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"The empty source set is conditionally independent of anything: with X = ∅, the projection valuesProjection ∅ is constant, so FullCondIndep holds trivially. Interprets the nil constructor of OrderedLocalSG.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {Y Z : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep ∅ Y Z (Finset.empty_subset M.randomVars) hY hZ μ","l":185,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_contraction","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"Contraction for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state. Suppose X, Y, W, and Z are all sets of M's random variables, and that, under μ, X is conditionally independent of Y given Z ∪ W and X is conditionally independent of W given Z. Then X is conditionally independent of Y ∪ W given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hW : W ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y (Z ∪ W) hX hY (Finset.union_subset hZ hW) μ → M.FullCondIndep X W Z hX hW hZ μ → M.FullCondIndep X (Y ∪ W) Z hX (Finset.union_subset hY hW) hZ μ","l":136,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_decomposition","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Decomposition for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hYW : Y ∪ W ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.FullCondIndep X Y Z hX (LE.le.trans Finset.subset_union_left hYW) hZ μ","l":104,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_of_orderedLocalSG","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"A conditional-independence conclusion derived from the graph's ordered local Markov statements and semi-graphoid rules also holds in the structural causal model's full joint distribution.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues) {X Y Z : Finset (Causalean.SWIGNode N)}, M.dag.OrderedLocalSG M.randomVars X Y Z → ∀ (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), M.FullCondIndep X Y Z hX hY hZ (M.jointKernel s)","l":540,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_ordered_local","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"For a random node, any set of random non-descendants that contains its random parents is conditionally independent of the remaining nodes in that set given those parents, under the model's joint kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] [inst_5 : ∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [∀ (v : Causalean.SWIGNode N), Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues) (v : Causalean.SWIGNode N) (hv : v ∈ M.randomVars) (P : Finset (Causalean.SWIGNode N)) (hP : P ⊆ M.randomVars), P ⊆ M.dag.nonDescendants v → M.dag.parents v ∩ M.randomVars ⊆ P → M.FullCondIndep {v} (P \\ (M.dag.parents v ∩ M.randomVars)) (M.dag.parents v ∩ M.randomVars) (Finset.singleton_subset_iff.mpr hv) (Finset.Subset.trans Finset.sdiff_subset hP) Finset.inter_subset_right (M.jointKernel s)","l":482,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_singleton_of_dSep_with_fixed","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"D-separation of one random node from a target set implies conditional independence after conditioning on the random nodes, even when the graph's conditioning set also contains fixed intervention nodes. The fixed nodes affect the graphical separation but do not appear among the random values being conditioned on.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (a : Causalean.SWIGNode N) (Y W_rand W_fix : Finset (Causalean.SWIGNode N)) (ha : a ∈ M.randomVars) (hY : Y ⊆ M.randomVars) (hW_rand : W_rand ⊆ M.randomVars), W_fix ⊆ M.fixed → M.dag.dSep {a} Y (W_rand ∪ W_fix) → ∀ (s : M.FixedValues), M.FullCondIndep {a} Y W_rand (Finset.singleton_subset_iff.mpr ha) hY hW_rand (M.jointKernel s)","l":611,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.fullCondIndep_subset_right","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Subset right for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hY' : Y' ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), Y' ⊆ Y → ∀ {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep X Y' Z hX hY' hZ μ","l":91,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_symm","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":2,"d":"Symmetry for FullCondIndep.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X Y Z hX hY hZ μ → M.FullCondIndep Y X Z hY hX hZ μ","l":80,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.fullCondIndep_weak_union","k":"theorem","m":"Causalean.SCM.Do.FullCondIndep","a":"SCM/Do","t":1,"d":"Weak union for full conditional independence. Let M be a structural causal model and let μ be a finite measure on the full random state (observed together with latent variables). Suppose X, Y ∪ W, and Z ∪ W are all sets of M's random variables, and that, under μ, X is conditionally independent of Y ∪ W given Z. Then X is conditionally independent of Y given Z ∪ W.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.randomVars) (hYW : Y ∪ W ⊆ M.randomVars) (hZW : Z ∪ W ⊆ M.randomVars) {μ : MeasureTheory.Measure M.RandomValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.FullCondIndep X (Y ∪ W) Z hX hYW (LE.le.trans Finset.subset_union_left hZW) μ → M.FullCondIndep X Y (Z ∪ W) hX (LE.le.trans Finset.subset_union_left hYW) hZW μ","l":117,"f":"Causalean/SCM/Do/FullCondIndep.lean"},{"n":"Causalean.SCM.full_globalMarkov","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"Full Global Markov Property. If X, Y, and Z are sets of nodes drawn from the model's random (observed and latent) nodes and X is d-separated from Y by Z in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given the Z-coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z : Finset (Causalean.SWIGNode N)) (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ : Z ⊆ M.randomVars), M.dag.dSep X Y Z → ∀ (s : M.FixedValues), M.FullCondIndep X Y Z hX hY hZ (M.jointKernel s)","l":759,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.full_globalMarkov_with_fixed","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"Full Global Markov with fixed-node conditioning shadow. If X, Y, and Z_rand are sets of nodes drawn from the model's random (observed and latent) nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_rand ∪ Z_fix in the model's causal graph, then under the joint distribution over all random coordinates at fixed value s, the X-coordinates and the Y-coordinates are conditionally independent given only the Z_rand-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z_rand Z_fix : Finset (Causalean.SWIGNode N)) (hX : X ⊆ M.randomVars) (hY : Y ⊆ M.randomVars) (hZ_rand : Z_rand ⊆ M.randomVars), Z_fix ⊆ M.fixed → M.dag.dSep X Y (Z_rand ∪ Z_fix) → ∀ (s : M.FixedValues), M.FullCondIndep X Y Z_rand hX hY hZ_rand (M.jointKernel s)","l":646,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.full_local_markov","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":1,"d":"Full Local Markov Property. If v is an observed node of the model, then under the joint distribution over all random (observed and latent) coordinates at fixed value s, the v-coordinate is conditionally independent of its non-descendants — restricted to random nodes — given all of its parents, including any latent parents, likewise restricted to random nodes.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) [StandardBorelSpace (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn {v} (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M.dag.nonDescendants v ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues), M.FullCondIndep {v} (M.dag.nonDescendants v ∩ M.randomVars) (M.dag.parents v ∩ M.randomVars) (Finset.singleton_subset_iff.mpr (Causalean.SCM.observed_subset_randomVars M hv)) Finset.inter_subset_right Finset.inter_subset_right (M.jointKernel s)","l":108,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.full_local_markov_latent","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":1,"d":"Latent Local Markov Property. If a is a latent (unobserved) node of the model, then under the joint distribution over all random coordinates at fixed value s, the a-coordinate is unconditionally independent of its non-descendants — restricted to random nodes — i.e. conditionally independent given the empty conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [StandardBorelSpace M.LatentValues] (a : Causalean.SWIGNode N) (ha : a ∈ M.unobserved) [StandardBorelSpace (Causalean.ValuesOn {a} (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn {a} (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M.dag.nonDescendants a ∩ M.randomVars) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M.dag.nonDescendants a ∩ M.randomVars) (Causalean.swigΩ Ω))] (s : M.FixedValues), M.FullCondIndep {a} (M.dag.nonDescendants a ∩ M.randomVars) ∅ (Finset.singleton_subset_iff.mpr (Finset.mem_union_right M.observed ha)) Finset.inter_subset_right (Finset.empty_subset M.randomVars) (M.jointKernel s)","l":238,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.get_appendTail","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Appending the tail of one nonempty list to another nonempty list preserves the first list's entries and then reads the remaining entries from the second list with a one-place shift.","s":"∀ {V : Type u_3} (pa q : List V), pa ≠ [] → ∀ (hq : q ≠ []), (pa ++ q.tail).length = pa.length + q.length - 1 ∧ (∀ (j : ℕ) (hj : j < pa.length), (pa ++ q.tail).get ⟨j, Eq.mpr (id (congrArg (fun _a => j < _a) List.length_append)) (have this := List.length_tail; Decidable.byContradiction fun a => Causalean.SCM.get_appendTail._proof_1 pa q j hj this a)⟩ = pa.get ⟨j, hj⟩) ∧ ∀ (j : ℕ) (hjL : pa.length ≤ j) (hj : j < (pa ++ q.tail).length), (pa ++ q.tail).get ⟨j, hj⟩ = q.get ⟨j - pa.length + 1, have this := List.length_pos_iff.mpr hq; Decidable.byContradiction fun a => Causalean.SCM.get_appendTail._proof_2 pa q j hjL (Eq.mp (congrArg (fun _a => j < pa.length + _a) List.length_tail) (Eq.mp (congrArg (fun _a => j < _a) List.length_append) hj)) this a⟩","l":446,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.globalMarkov","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":1,"d":"Global Markov Property. If X, Y, and Z are sets of observed nodes and X is d-separated from Y by Z in the model's full causal graph, which also includes any latent nodes, then under the observational distribution — the law of the observed coordinates alone, at fixed value s — the X-coordinates and Y-coordinates are conditionally independent given the Z-coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), M.dag.dSep X Y Z → ∀ (s : M.FixedValues), M.ObsCondIndep X Y Z hX hY hZ (M.obsKernel s)","l":96,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.globalMarkov_with_fixed","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":1,"d":"Global Markov with fixed-node conditioning. If X, Y, and Z_obs are sets of observed nodes and Z_fix is a set of the model's fixed (intervened) nodes, and X is d-separated from Y by the union Z_obs ∪ Z_fix in the model's full causal graph, then under the observational distribution at fixed value s, the X-coordinates and Y-coordinates are conditionally independent given only the Z_obs-coordinates — the fixed nodes contribute to the graphical separation but, since their values are already pinned by s, drop out of the probabilistic conditioning set.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (X Y Z_obs Z_fix : Finset (Causalean.SWIGNode N)) [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ_obs : Z_obs ⊆ M.observed), Z_fix ⊆ M.fixed → M.dag.dSep X Y (Z_obs ∪ Z_fix) → ∀ (s : M.FixedValues), M.ObsCondIndep X Y Z_obs hX hY hZ_obs (M.obsKernel s)","l":125,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.hNoDesc_descend_to_parent","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"If none of the fixed intervention nodes is an ancestor of a node, then none is an ancestor of any parent of that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M2 : Causalean.SCM N Ω} {Z : Finset N} {v w : Causalean.SWIGNode N}, w ∈ M2.dag.parents v → (∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) w","l":80,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.hNoDesc_implies_no_fixed_parent","k":"theorem","m":"Causalean.SCM.Do.Rule3","a":"SCM/Do","t":2,"d":"If none of the fixed intervention nodes is an ancestor of a node, none is a parent of that node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M2 : Causalean.SCM N Ω} {Z : Finset N} {v : Causalean.SWIGNode N}, (∀ z ∈ Z, ¬M2.dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ z ∈ Z, Causalean.SWIGNode.fixed z ∉ M2.dag.parents v","l":72,"f":"Causalean/SCM/Do/Rule3.lean"},{"n":"Causalean.SCM.identifyMassRec","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For an ambient node set and a SWIG graph, the mass-level IDENTIFY recursion maps every target node set, district node set, proof that the target set is contained in the ambient set, and nonnegative extended-real mass function on ambient assignments to a mass function on ambient assignments. It first forms the induced ancestral set and its containment proof, then returns the marginal eliminating the target coordinates outside the district when that ancestral set equals the district, returns the original mass function when it equals the target set, and otherwise recurses after extracting the containing district factor from the appropriate marginal.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O : Finset (Causalean.SWIGNode N)) → Causalean.SWIGGraph N → (T : Finset (Causalean.SWIGNode N)) → Finset (Causalean.SWIGNode N) → T ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":129,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_base","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C already equals C, then the mass-level IDENTIFY recursion identifyMassRec on T stops immediately and returns the marginal of the input mass function q obtained by summing out the coordinates in T \\ C.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C = C → Causalean.SCM.identifyMassRec M.observed G T C hT q = Causalean.SCM.marginalizeOn M.observed (T \\ C) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) q","l":164,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_hedge","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C does not equal C but it equals T itself — the hedge case, reached only after the base case has failed — then the recursion identifyMassRec on T returns the input mass function q unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C ≠ C → Causalean.SCM.ID.inducedAncestral G T C = T → Causalean.SCM.identifyMassRec M.observed G T C hT q = q","l":180,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.identifyMassRec_step","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a target set T contained in the observed coordinates, if the graph-induced ancestral set of T relative to C does not equal C and does not equal T either — i.e. neither the base nor the hedge case applies — then the recursion identifyMassRec on T unfolds one step: it extracts, from the mass function q marginalized onto the induced ancestral set, the district factor of the c-component of C inside that induced ancestral graph, and recurses on that district.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (G : Causalean.SWIGGraph N) (T C : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal), Causalean.SCM.ID.inducedAncestral G T C ≠ C → Causalean.SCM.ID.inducedAncestral G T C ≠ T → Causalean.SCM.identifyMassRec M.observed G T C hT q = let A := Causalean.SCM.ID.inducedAncestral G T C; have hA := fun x hv => hT (Causalean.SCM.ID.inducedAncestral_subset_left G T C hv); let C₁ := Causalean.SCM.ID.containingCComponent (G.induce A) C; have hC₁ := fun x hv => hT (Causalean.SCM.ID.inducedAncestral_subset_left G T C (Causalean.SCM.ID.containingCComponent_induce_subset G A C hv)); Causalean.SCM.identifyMassRec M.observed G C₁ C hC₁ (Causalean.SCM.extractDistrict M.observed (G.induce A) A C₁ hA (Causalean.SCM.marginalizeOn M.observed (T \\ A) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) q))","l":198,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.indepFun_valuesProjection_latentProduct","k":"theorem","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":2,"d":"Under M.latentProduct = ⊗_{u ∈ 𝐋} ℙ(L_u), the coordinate-tuple projections at two disjoint latent blocks A, B ⊆ M.unobserved are independent. The latent product is a Measure.pi over the subtype of unobserved nodes, so this is indepFun_pi_of_disjoint transported along the reindexSubtypeProj relabelling.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {A B : Finset (Causalean.SWIGNode N)} (hA : A ⊆ M.unobserved) (hB : B ⊆ M.unobserved), Disjoint A B → ProbabilityTheory.IndepFun (Causalean.valuesProjection hA) (Causalean.valuesProjection hB) M.latentProduct","l":338,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.induce","k":"def","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, a set of graph nodes, and the condition that this set is ancestrally closed in the SCM sense, the induced structural causal submodel retains the selected observed and fixed nodes and precisely the original latent roots feeding retained observed nodes. It is defined from the induced SWIG graph, inheriting the relevant value spaces, structural functions, and latent distributions.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (R : Finset (Causalean.SWIGNode N)) → M.isAncestrallyClosedSCM R → Causalean.SCM N Ω","l":148,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_evalMap_compat","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"Evaluation-map bridge for the induced sub-SCM. Fix a structural causal model M and a node set R that is ancestrally closed in the SCM sense. Then at every random node v retained by the induced model M.induce R hR, the induced evaluation map at the restriction of sTilde to R and the projected latent assignment computes the same value as the original evaluation map at sTilde.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R) (sTilde : M.FixedValues) (ℓ : M.LatentValues) {v : Causalean.SWIGNode N} (hvI : v ∈ (M.induce R hR).randomVars), have hvM := Or.casesOn (Finset.mem_union.mp hvI) (fun hvIObs => Finset.mem_union_left M.unobserved (Finset.inter_subset_right hvIObs)) fun hvIUo => Finset.mem_union_right M.observed (Causalean.SCM.induce_unobserved_subset M R hR hvIUo); (M.induce R hR).evalMap (Causalean.valuesProjection (Finset.filter_subset (fun s => Causalean.iotaMap s ∈ R ∩ M.observed) M.fixed) sTilde) (Causalean.valuesProjection (Causalean.SCM.induce_unobserved_subset M R hR) ℓ) ⟨v, hvI⟩ = M.evalMap sTilde ℓ ⟨v, hvM⟩","l":277,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_latentProduct_eq_map","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"The latent product of an induced sub-SCM is the marginal of the original latent product.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R), (M.induce R hR).latentProduct = MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SCM.induce_unobserved_subset M R hR)) M.latentProduct","l":253,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_marginal_compat","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"Marginal compatibility of the induced sub-SCM (prop:scm-induced-marginal). Fix a structural causal model M, an ancestrally closed node set R, and a fixed-value assignment sTilde on M. Then the observational kernel of the induced sub-SCM at the restriction of sTilde to R equals the pushforward of M's observational kernel at sTilde onto the induced observed coordinates R ∩ M.observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R) (sTilde : M.FixedValues), (M.induce R hR).obsKernel (Causalean.valuesProjection (Finset.filter_subset (fun s => Causalean.iotaMap s ∈ R ∩ M.observed) M.fixed) sTilde) = MeasureTheory.Measure.map (Causalean.valuesProjection (have this := Finset.inter_subset_right; this)) (M.obsKernel sTilde)","l":454,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_parents_eq_of_ancClosed","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"An ancestrally closed retained observed set keeps every parent needed to evaluate each retained observed node, so its induced graph has exactly the same parent set at those nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)), M.isAncestrallyClosedSCM R → ∀ {v : Causalean.SWIGNode N}, v ∈ (M.induce R).observed → (M.induce R).dag.parents v = M.dag.parents v","l":89,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.induce_unobserved_subset","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"The latent roots of an induced sub-SCM are original latent roots.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) (hR : M.isAncestrallyClosedSCM R), (M.induce R hR).unobserved ⊆ M.unobserved","l":189,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.instFintypeSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"For a collection of node labels, a family of finite base-node value spaces, and a random or fixed SWIG node, the finite enumeration of that node's value space is inherited from the finite enumeration of its corresponding base-node value space by the random-node case or by the fixed-node case.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (sn : Causalean.SWIGNode N) → Fintype (Causalean.swigΩ Ω sn)","l":26,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instFintypeValuesOnSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"For a distinguishable node population with measurable, finite-valued node spaces and a finite set of SWIG nodes, the finite enumeration structure on assignments of values to those nodes exists.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (I : Finset (Causalean.SWIGNode N)) → Fintype (Causalean.ValuesOn I (Causalean.swigΩ Ω))","l":32,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.instIsFiniteKernelAdjustmentKernelY","k":"instance","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable, standard-Borel, nonempty node-value spaces and a structural causal model, a finite treatment set, the requirement that every corresponding random treatment node is observed, the requirement that no corresponding fixed treatment node is already fixed, finite observed outcome and adjustment-node sets, and a fixed-node assignment, the treatment-indexed backdoor-adjustment outcome kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.adjustmentKernelY X hObs hFix Y Z hY hZ s0)","l":142,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.instIsFiniteKernelBackdoorAdjustment","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable, standard-Borel, nonempty node-value spaces and a structural causal model, a finite treatment set, the requirement that every corresponding random treatment node is observed, the requirement that no corresponding fixed treatment node is already fixed, and finite observed outcome and adjustment-node sets, the backdoor-adjustment functional is a finite kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), ProbabilityTheory.IsFiniteKernel (M.backdoorAdjustment X hX_obs hX_fixed Y Z hY hZ)","l":142,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsFiniteKernelDoKernelY","k":"instance","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable node-value spaces and a structural causal model, a finite treatment set, the requirement that every corresponding random treatment node is observed, the requirement that no corresponding fixed treatment node is already fixed, a finite observed outcome-node set, and a fixed-node assignment, the treatment-indexed post-intervention outcome kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.doKernelY X hObs hFix Y hY s0)","l":132,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.instIsFiniteKernelFrontdoorAdjustment","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable, standard-Borel, nonempty node-value spaces and a structural causal model, a finite treatment set, the requirement that every corresponding random treatment node is observed, the requirement that no corresponding fixed treatment node is already fixed, and finite observed outcome and mediator-node sets, the frontdoor-adjustment functional is a finite kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y Z : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed), ProbabilityTheory.IsFiniteKernel (M.frontdoorAdjustment X hX_obs hX_fixed Y Z hY hZ)","l":273,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsFiniteKernelFrontdoorKernelY","k":"instance","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable, standard-Borel, nonempty node-value spaces and a structural causal model, a finite treatment set, the requirement that every corresponding random treatment node is observed, the requirement that no corresponding fixed treatment node is already fixed, finite observed outcome and mediator-node sets, and a fixed-node assignment, the treatment-indexed frontdoor-adjustment outcome kernel is finite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) (s0 : M.FixedValues), ProbabilityTheory.IsFiniteKernel (M.frontdoorKernelY X hObs hFix Y W hY hW s0)","l":151,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.instIsFiniteMeasure_jointRef","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"For a collection of node labels, finite measurable base-node value spaces with measurable singletons, a family of reference measures, and a finite set of random or fixed SWIG nodes, the corresponding product reference measure has finite total mass.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Finite (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)), MeasureTheory.IsFiniteMeasure (Causalean.SCM.jointRef ref I)","l":62,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instIsFiniteMeasure_refMu","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"For a collection of node labels, finite measurable base-node value spaces with measurable singletons, a family of reference measures, and a random or fixed SWIG node, the reference measure at that node has finite total mass.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), Finite (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (v : Causalean.SWIGNode N), MeasureTheory.IsFiniteMeasure (ref.μ v)","l":52,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instIsMarkovKernelJointKernel","k":"instance","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"For a finite, distinguishable node population with measurable node-value spaces and a structural causal model, the joint conditional law of random-node values given fixed-node values is a Markov kernel: every conditional law is a probability measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.jointKernel","l":158,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.instIsMarkovKernelObsCondKernel","k":"instance","m":"Causalean.SCM.ID.Adjustment","a":"SCM/ID","t":2,"d":"For a finite, distinguishable node population with measurable, standard-Borel, nonempty node-value spaces and a structural causal model, a finite observed outcome-node set, and a finite observed conditioning-node set, the observational conditional kernel from the conditioning values to the outcome values is a Markov kernel: each of its values is a probability measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed), ProbabilityTheory.IsMarkovKernel (M.obsCondKernel Y CC hY hCC)","l":61,"f":"Causalean/SCM/ID/Adjustment.lean"},{"n":"Causalean.SCM.instIsMarkovKernelObsKernel","k":"instance","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"For a finite, distinguishable node population with measurable node-value spaces and a structural causal model, the conditional law of observed-node values given fixed-node values is a Markov kernel: every conditional law is a probability measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.obsKernel","l":166,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.instMeasurableSingletonClassSwigΩ","k":"instance","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"For a collection of node labels, a family of base-node value spaces with measurable singletons, and a random or fixed SWIG node, the measurable-singleton structure on that node's value space is inherited from its corresponding base-node value space in the random-node case or in the fixed-node case.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (sn : Causalean.SWIGNode N), MeasurableSingletonClass (Causalean.swigΩ Ω sn)","l":32,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.instMeasurableSpaceObservedPrefixValues","k":"instance","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":2,"d":"For a finite, distinguishable node population with measurable node-value spaces, a structural causal model, and any natural number no greater than the number of its observed nodes, the measurable-space structure on the corresponding observed-prefix value space is provided by the one-point measurable space for a zero-length prefix and by the product measurable space for a positive-length prefix.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n ≤ M.observed.card) → MeasurableSpace (M.ObservedPrefixValues n hn)","l":46,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.instProbabilityLatentProduct","k":"instance","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"For a finite node population with measurable value spaces and a structural causal model, the probability-measure structure for the model's latent product measure asserts that this measure assigns total mass one to the joint space of latent-root values.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), MeasureTheory.IsProbabilityMeasure M.latentProduct","l":197,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.instSetoidSCM","k":"instance","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"For a finite node population with measurable value spaces, the setoid structure on structural causal models uses structural equivalence as its equivalence relation and certifies that this relation is reflexive, symmetric, and transitive.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Setoid (Causalean.SCM N Ω)","l":327,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.instSigmaFiniteJointRef","k":"instance","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":2,"d":"For a node population with measurable value spaces, a reference-measure family, and a finite set of nodes, the σ-finiteness structure for their joint reference measure asserts that the finite product of the selected coordinate reference measures is σ-finite.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)), MeasureTheory.SigmaFinite (Causalean.SCM.jointRef ref I)","l":59,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.isAncestorAvoiding_of_sinks","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"","s":"∀ {V : Type u_3} [inst : DecidableEq V] [inst_1 : Fintype V] (G : Causalean.DAG V) (C : Finset V), (∀ c ∈ C, ∀ (v : V), ¬G.edge c v) → ∀ {u v : V}, G.isAncestor u v → G.isAncestorAvoiding C u v","l":102,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.isAncestor_fixSet_of_isAncestor","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A directed ancestry path in a causal graph remains after intervening on a set when every possible edge source along that path is not a random copy of an intervened variable. Thus the original ancestor remains an ancestor in the intervened graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) {u v : Causalean.SWIGNode N}, M.dag.isAncestor u v → (∀ (s : Causalean.SWIGNode N), s = u ∨ M.dag.isAncestor u s → M.dag.isAncestor s v → ∀ D ∈ Z, s ≠ Causalean.SWIGNode.random D) → (M.fixSet Z hZ_obs hZ_fixed).dag.isAncestor u v","l":211,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.isAncestrallyClosedSCM","k":"def","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, and a set of graph nodes, the SCM ancestral-closure condition holds exactly when every observed parent of every retained node is retained and the random counterpart of every fixed parent of every retained node is retained.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Prop","l":67,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.isFiniteMeasure_of_finite_measurableSingleton","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"A σ-finite measure on a finite measurable-singleton space is finite.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [Finite α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ], MeasureTheory.IsFiniteMeasure μ","l":38,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.isGlobalIMap_dag_self","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Distributional","a":"Graph/MarkovEquiv","t":1,"d":"The bridge, restated. For any structural causal model M and any point s of its fixed values, the joint distribution of M's random values under s is a global I-map of M's own DAG — this is exactly the global Markov property full_globalMarkov.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_6 : ∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (s : M.FixedValues), Causalean.SCM.IsGlobalIMap M.dag M (M.jointKernel s)","l":73,"f":"Causalean/Graph/MarkovEquiv/Distributional.lean"},{"n":"Causalean.SCM.isMarkov_jointKernelPrefix","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"For a finite collection of nodes with measurable value spaces, a structural causal model, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the Markov-kernel structure for the corresponding prefix kernel asserts that the kernel generating all latent values and the first specified observed values is a Markov kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.jointKernelPrefix n hn)","l":125,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_jointKernelPrefixZero","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"For a finite collection of nodes with measurable value spaces and a structural causal model, the Markov-kernel structure for the zero-step prefix kernel asserts that the kernel generating the latent values and the empty observed prefix is a Markov kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.jointKernelPrefixZero","l":89,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_latentKernelOnFixed","k":"instance","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":2,"d":"For a finite collection of nodes with measurable value spaces and a structural causal model, the Markov-kernel structure for the constant latent-value kernel asserts that the kernel assigning the latent product distribution to every fixed-value assignment is a Markov kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.latentKernelOnFixed","l":62,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsChainKernel","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"For a finite node population with measurable value spaces, a structural causal model, standard Borel and nonempty value spaces for every observed node, and countably generated conditioning σ-algebras for every observed prefix, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the Markov-kernel structure for the corresponding observational chain kernel asserts that the sequential conditional distribution of the first specified observed values is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] (n : ℕ) (hn : n ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.obsChainKernel n hn)","l":564,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsChainKernelZero","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"For a finite node population with measurable value spaces and a structural causal model, the Markov-kernel structure for the zero-step observational chain kernel asserts that the kernel assigning unit probability to the unique empty-prefix assignment is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), ProbabilityTheory.IsMarkovKernel M.obsChainKernelZero","l":405,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_obsStepCondKernel","k":"instance","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"For a finite node population with measurable value spaces, a structural causal model, an index strictly below its number of observed nodes, a standard Borel and nonempty value space for the observed node at that index, and a countably generated conditioning σ-algebra for the fixed values and preceding observed values, the Markov-kernel structure for the one-step observational conditional kernel asserts that this conditional distribution is a Markov kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))], ProbabilityTheory.IsMarkovKernel (M.obsStepCondKernel hn)","l":274,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.isMarkov_stepKernel","k":"instance","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":2,"d":"For a finite node population with measurable value spaces, a structural causal model, a nonnegative integer, and proof that adding one to this integer does not exceed the number of observed nodes, the Markov-kernel structure for the corresponding deterministic step kernel asserts that the kernel assigning the next observed value from its structural equation is a Markov kernel.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), ProbabilityTheory.IsMarkovKernel (M.stepKernel hn)","l":95,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.isStandard","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, standardness holds exactly when the model has no fixed intervention nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Prop","l":141,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.jointKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces and a structural causal model, the joint conditional law of all random-node values given fixed-node values assigns each fixed-value assignment the distribution obtained by evaluating the model under its product distribution of latent variables. It is defined by combining that latent distribution with the deterministic evaluation rule and projecting to evaluated random values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.RandomValues","l":71,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointKernelPrefix","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the prefix kernel maps each fixed-node assignment to the joint distribution of all latent-node values and the values of the first specified number of observed nodes. At zero observed nodes it is the zero-step prefix kernel; at each positive prefix length it extends the preceding prefix distribution by the next observed node's deterministic structural equation.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → (hn : n ≤ M.observed.card) → ProbabilityTheory.Kernel M.FixedValues (M.OrderedLatentPrefixValues n hn)","l":102,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.jointKernelPrefixZero","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the zero-step prefix kernel maps every fixed-node assignment to the joint distribution of all latent-node values and the unique empty observed-node prefix at prefix length zero.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues (M.OrderedLatentPrefixValues 0 (Nat.zero_le M.observed.card))","l":74,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.jointKernelPrefix_apply_eq","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":1,"d":"Main correspondence. For a structural causal model M, its length-n prefix kernel evaluated at a fixed assignment s equals the pushforward of the latent-value product measure through the deterministic partial evaluation map at s.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (n : ℕ) (hn : n ≤ M.observed.card), (M.jointKernelPrefix n hn) s = MeasureTheory.Measure.map (fun ℓ => M.partialEvalMap n hn s ℓ) M.latentProduct","l":326,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.jointKernel_apply_eq","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a structural causal model M and a fixed-node assignment s, the joint kernel evaluated at s equals the pushforward of the latent-variable product measure through the evaluation map at s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.jointKernel s = MeasureTheory.Measure.map (fun ℓ => M.evalMap s ℓ) M.latentProduct","l":105,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointKernel_eq_factored_kernel","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"Kernel-level factorization. For a structural causal model M, its joint kernel equals the prefix kernel at full length, pushed through the reindexing map identifying the full prefix state with the random coordinates. Follows from the pointwise form via kernel extensionality.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.jointKernel = (M.jointKernelPrefix M.observed.card (le_refl M.observed.card)).map M.orderedLatentPrefixFullToRandom","l":215,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.jointKernel_factored","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"Main factorization theorem (pointwise). For a structural causal model M, at each fixed assignment s, the joint kernel equals the pushforward of the full-length prefix kernel through the reindexing map identifying the full prefix state with the random coordinates.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.jointKernel s = MeasureTheory.Measure.map M.orderedLatentPrefixFullToRandom ((M.jointKernelPrefix M.observed.card (le_refl M.observed.card)) s)","l":173,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.jointKernel_map_commute","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The observational kernel can be written as one pushforward that evaluates the model and then projects to observed coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.obsKernel = ((ProbabilityTheory.Kernel.const M.FixedValues M.latentProduct).compProd (ProbabilityTheory.Kernel.deterministic (Function.uncurry M.evalMap) (Causalean.SCM.evalMap_measurable M))).map (M.randomToObserved ∘ Prod.snd)","l":185,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.jointRef","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"For a node population with measurable value spaces, a reference-measure family, and a finite node set, the joint reference measure is the finite product of the reference measures assigned to the nodes in that set.","s":"{N : Type u_1} → {Ω : N → Type u_2} → [inst : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM.ReferenceMeasures Ω → (I : Finset (Causalean.SWIGNode N)) → MeasureTheory.Measure (Causalean.ValuesOn I (Causalean.swigΩ Ω))","l":49,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.jointRef_extendObsPrefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The successor prefix extension carries the product of the old-prefix reference and the next-node reference to the successor-prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) {k : ℕ} (hk : k < M.observed.card), MeasureTheory.Measure.map (M.extendObsPrefix hk) ((Causalean.SCM.jointRef ref (M.prefixNodes k)).prod (ref.μ ↑(M.observedAt ⟨k, hk⟩))) = Causalean.SCM.jointRef ref (M.prefixNodes (k + 1))","l":255,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.jointRef_prefix_card_map","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Transport the full-prefix reference measure to the observed-value reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed","l":241,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.jointRef_singleton_eq_prod","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"The singleton mass of a finite product reference is the product of the coordinate singleton masses.","s":"∀ {N : Type u_1} [DecidableEq N] [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (I : Finset (Causalean.SWIGNode N)) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)), (Causalean.SCM.jointRef ref I) {x} = ∏ i, (ref.μ ↑i) {x i}","l":51,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.jointRef_singleton_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.FiniteReference","a":"SCM/ID/Density","t":2,"d":"A faithful reference family gives every point in a finite coordinate product nonzero joint reference mass.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω), Causalean.SCM.ReferenceFaithful ref → ∀ (I : Finset (Causalean.SWIGNode N)) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)), (Causalean.SCM.jointRef ref I) {x} ≠ 0","l":106,"f":"Causalean/SCM/ID/Density/FiniteReference.lean"},{"n":"Causalean.SCM.kernel_apply_univ","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"Every slice of a Markov kernel has total mass one.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ] (a : α), (κ a) Set.univ = 1","l":174,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.latentAncestorsOfNode","k":"def","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, and a node, the set of latent ancestors of that node consists exactly of the model's unobserved nodes that are either the node itself or ancestors of it in the directed graph. It is formed by filtering the unobserved nodes by this condition.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":151,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.latentAncestorsOfNode_subset","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"The latent-ancestor set is contained in the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : Causalean.SWIGNode N), M.latentAncestorsOfNode v ⊆ M.unobserved","l":204,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.latentAncestorsOfSet","k":"def","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"For a finite collection of nodes, a structural causal model, and a finite set of nodes in its single-world intervention graph, the latent ancestors of that set are exactly the unobserved nodes that either belong to the set or are ancestors of at least one node in the set.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":38,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.latentBlock","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model and a candidate node set, the latent block is the set of all unobserved nodes having a directed edge into at least one member of that set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":40,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"If C is a full c-component of the causal model's SWIG graph and C' is likewise a full c-component, and C and C' are distinct, then their latent-parent blocks — the unobserved nodes with an edge into the component — are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ M.cComponentSet → C' ∈ M.cComponentSet → C ≠ C' → Disjoint (M.latentBlock C) (M.latentBlock C')","l":49,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint_fixSet_induce_components","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Do-model specialization of induced-component latent-block disjointness.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (R : Finset (Causalean.SWIGNode N)) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ ((M.fixSet X hObs hFix).induce R).cComponentSet → C' ∈ ((M.fixSet X hObs hFix).induce R).cComponentSet → C ≠ C' → Disjoint ((M.fixSet X hObs hFix).latentBlock C) ((M.fixSet X hObs hFix).latentBlock C')","l":105,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentBlock_pairwise_disjoint_induce_components","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Distinct c-components of an induced SWIG have disjoint latent-parent blocks in the ambient SCM.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (R : Finset (Causalean.SWIGNode N)) {C C' : Finset (Causalean.SWIGNode N)}, C ∈ (M.induce R).cComponentSet → C' ∈ (M.induce R).cComponentSet → C ≠ C' → Disjoint (M.latentBlock C) (M.latentBlock C')","l":71,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.latentKernelOnFixed","k":"def","m":"Causalean.SCM.Factored.PrefixKernel","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the kernel giving the distribution of the latent-node values conditional on the fixed-node values assigns the same latent product distribution to every fixed-node assignment.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.LatentValues","l":50,"f":"Causalean/SCM/Factored/PrefixKernel.lean"},{"n":"Causalean.SCM.latentProduct","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the latent product measure is the product of the model's probability laws over all unobserved latent root nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → MeasureTheory.Measure M.LatentValues","l":183,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.latentProduct_singleton_eq_prod","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"The latent-product mass of a singleton latent assignment equals the product of the singleton masses assigned by the latent distributions at every unobserved node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (ℓ₀ : M.LatentValues), M.latentProduct {ℓ₀} = ∏ u, (M.latentDist u) {ℓ₀ u}","l":655,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a fixed-node assignment, an observed-node assignment, an observed node with evidence that it is observed, and a latent-node assignment, local consistency at that node holds exactly when its structural function, is indexed by that node's canonical observed-order position and, fed the corresponding fixed, latent, and earlier observed parent values, equals its assigned observed value.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → (v : Causalean.SWIGNode N) → v ∈ M.observed → M.LatentValues → Prop","l":120,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_depends_only_on_block","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Local consistency at v depends only on the latent block of v's c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) (ℓ ℓ' : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.latentBlock (M.cComponentOf v)), ℓ ⟨u, (Finset.mem_filter.mp hu).left⟩ = ℓ' ⟨u, (Finset.mem_filter.mp hu).left⟩) → (M.localConsistent s x v hv ℓ ↔ M.localConsistent s x v hv ℓ')","l":138,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_depends_only_on_latentBlock_of_mem","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local consistency at an observed node in a node set is unchanged when two latent assignments agree on every unobserved parent of a node in that set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) {U : Finset (Causalean.SWIGNode N)} {v : Causalean.SWIGNode N}, v ∈ U → ∀ (hv : v ∈ M.observed) (ℓ ℓ' : M.LatentValues), (∀ (u : Causalean.SWIGNode N) (hu : u ∈ M.latentBlock U), ℓ ⟨u, (Finset.mem_filter.mp hu).left⟩ = ℓ' ⟨u, (Finset.mem_filter.mp hu).left⟩) → (M.localConsistent s x v hv ℓ ↔ M.localConsistent s x v hv ℓ')","l":894,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_event_eq_component_biInter","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"For an observed-parent-closed node set, local consistency at all of its nodes is equivalent to local consistency within each of its confounded components.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ P), M.localConsistent s x v (hP.left hv) ℓ} = ⋂ C ∈ M.cComponentSet, {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ C ∩ P), M.localConsistent s x v (hP.left (Finset.mem_of_mem_inter_right hv)) ℓ}","l":1063,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_event_eq_family_biInter","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"For a finite family of observed node sets, local consistency over their union is equivalent to local consistency over every member of the family.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed) (hSup : 𝒞.sup id ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ 𝒞.sup id), M.localConsistent s x v (hSup hv) ℓ} = ⋂ U ∈ 𝒞, if hU : U ∈ 𝒞 then {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ U), M.localConsistent s x v (h𝒞obs U hU hv) ℓ} else Set.univ","l":1096,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.localConsistent_fixSet_iff","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"Consider an intervention on a node set W whose random copies are all observed and whose fixed copies are not already fixed in the base model, giving the intervened model M.fixSet W. Fix a node v that is not itself one of the intervened random copies and that is observed both in the intervened model and in the base model. If the intervened model's observed assignment x' agrees with the base assignment x on every base-observed coordinate, the base assignment records, at each intervened node, the same value that the intervened model's fixed values assign to the corresponding fixed coordinate, and the intervened model's fixed values sW project, via fixSetProj, onto the base fixed values s, then local consistency of the structural evaluation at v in the intervened model, under sW, x', and a latent realization ℓ, is equivalent to local consistency at v in the base model, under s, x, and the same ℓ.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (W : Finset N) (hObs : ∀ D ∈ W, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) (sW : (M.fixSet W hObs hFix).FixedValues) (s : M.FixedValues) (x' : Causalean.ValuesOn (M.fixSet W hObs hFix).observed (Causalean.swigΩ Ω)) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv' : v ∈ (M.fixSet W hObs hFix).observed) (hv : v ∈ M.observed), v ∉ Finset.image Causalean.SWIGNode.random W → (∀ (w : Causalean.SWIGNode N) (hw : w ∈ M.observed), x' ⟨w, id hw⟩ = x ⟨w, hw⟩) → (∀ (D : N) (hD : D ∈ W), x ⟨Causalean.SWIGNode.random D, hObs D hD⟩ = sW ⟨Causalean.SWIGNode.fixed D, Finset.mem_union_right M.fixed (Finset.mem_image.mpr (Exists.intro D ⟨hD, rfl⟩))⟩) → M.fixSetProj W hObs hFix sW = s → ∀ (ℓ : M.LatentValues), (M.fixSet W hObs hFix).localConsistent sW x' v hv' ℓ ↔ M.localConsistent s x v hv ℓ","l":511,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.localConsistent_iff_structFun_dispatch","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (v : Causalean.SWIGNode N) (hv : v ∈ M.observed) (ℓ : M.LatentValues), M.localConsistent s x v hv ℓ ↔ (M.structFun ⟨v, hv⟩ fun w => if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) v hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; x ⟨↑w, hobs⟩) = x ⟨v, hv⟩","l":382,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.map_pi_valuesEquivOfEq","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Measure.pi reindexes along valuesEquivOfEq, with no probability-measure hypothesis (the index equality reduces the equiv to the identity). This is the σ-finite-friendly companion of measurePreserving_valuesEquivOfEq.","s":"∀ {M' : Type u_3} [DecidableEq M'] {I J : Finset M'} {Ω' : M' → Type u_4} [inst : (n : M') → MeasurableSpace (Ω' n)] (h : I = J) (μ : (i : ↥I) → MeasureTheory.Measure (Ω' ↑i)), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq h)) (MeasureTheory.Measure.pi μ) = MeasureTheory.Measure.pi fun j => μ ⟨↑j, (fun h_1 => Eq.ndrec (motive := fun {J} => ∀ (j : ↥J), ↑j ∈ J → ↑j ∈ I) (fun j h => h) h j h_1) j.property⟩","l":197,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.marginalizeOn","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For an ambient set of graph nodes, a subset of coordinates to eliminate that is contained in the ambient set, and a nonnegative extended-real mass function on assignments to the ambient set, the marginalized mass function maps each ambient assignment to the sum of the mass function over all assignments on the eliminated coordinates, replacing those coordinates in the evaluation point.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → [(n : N) → Fintype (Ω n)] → (O W : Finset (Causalean.SWIGNode N)) → W ⊆ O → (Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal) → Causalean.ValuesOn O (Causalean.swigΩ Ω) → ENNReal","l":109,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SCM.marginalizeOn_empty","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Marginalizing a mass function over an empty set of observed coordinates leaves its value at every observed assignment unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] (M : Causalean.SCM N Ω) (q : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.marginalizeOn M.observed ∅ (of_eq_true (Finset.empty_subset._simp_1 M.observed)) q x = q x","l":179,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.measurableEmbedding_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"fillZrW Z _ _ W s' is a MeasurableEmbedding when Z.image .random and W are disjoint.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasurableEmbedding (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":206,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.measurableSet_comap_piFinset_of_depends","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"A set of finite product outcomes whose membership depends only on a specified finite set of coordinates is measurable with respect to the σ-algebra on those coordinates.","s":"∀ {ι : Type u_3} [Fintype ι] {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] [(i : ι) → Fintype (α i)] [∀ (i : ι), MeasurableSingletonClass (α i)] (S : Finset ι) (A : Set ((i : ι) → α i)), (∀ (ξ ξ' : (i : ι) → α i), (∀ i ∈ S, ξ i = ξ' i) → (ξ ∈ A ↔ ξ' ∈ A)) → MeasurableSet A","l":677,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.measurableSet_fillZrW_image","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":2,"d":"A measurable W-event has a measurable image under fillZrW.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet A → MeasurableSet (M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":135,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.measurableSet_measure_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The agreement set of two measurable measure-valued maps is measurable.","s":"∀ {γ : Type u_3} {β : Type u_4} [inst : MeasurableSpace γ] [inst_1 : MeasurableSpace β] [MeasurableSpace.CountablyGenerated β] {f g : γ → MeasureTheory.Measure β}, Measurable f → Measurable g → (∀ (p : γ), MeasureTheory.IsFiniteMeasure (f p)) → MeasurableSet {p | f p = g p}","l":88,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.measurable_cast_family","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Transporting a value along an equality of indices is measurable.","s":"∀ {I : Type u_1} {X : I → Type u_2} [inst : (i : I) → MeasurableSpace (X i)] {a b : I} (hab : a = b), Measurable (cast (congrArg X hab))","l":143,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SCM.measurable_evalMap_apply","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"For each fixed assignment to the intervened variables, the evaluation map is measurable as a function of the latent realization alone.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), Measurable (M.evalMap s)","l":464,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.measurable_evalMap_overrideC","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The overridden evaluation map is jointly measurable in fixed values, override values, and the latent realization.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (hC : C ⊆ M.observed), Measurable fun p => M.evalMap_overrideC hY hC p.1.1 p.1.2 p.2","l":578,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.measurable_evalMap_prod","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The evaluation map, read as a function of the pair (s, ℓ) rather than as a curried map, is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable fun p => M.evalMap p.1 p.2","l":454,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.measurable_extendObsPrefix","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix extension is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), Measurable (M.extendObsPrefix hn)","l":425,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_extendOrderedLatentPrefix","k":"theorem","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names the next node to be appended in the canonical topological order of observed nodes. Then the map that appends the freshly generated value of that node to a length-n prefix of previously observed values, together with the latent assignment, producing a length-(n + 1) prefix, is measurable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.extendOrderedLatentPrefix hn)","l":134,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.measurable_family_cast","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"A measurable function remains measurable after transporting its codomain index.","s":"∀ {I : Type u_1} {γ : Type u_2} {X : I → Type u_3} [inst : (i : I) → MeasurableSpace (X i)] [inst_1 : MeasurableSpace γ] {v w : I} (h : v = w) {f : γ → X v}, Measurable f → Measurable fun x => h ▸ f x","l":151,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SCM.measurable_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"The Rule 2 filler is measurable as a function of the free conditioning assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), Measurable (M'.fillZrW Z hZ_obs hZ_fixed W s')","l":155,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fillZrW_prod","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"The Rule 2 filler is jointly measurable in the post-intervention fixed slice and the free conditioning coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)), Measurable fun p => M'.fillZrW Z hZ_obs hZ_fixed W p.1 p.2","l":169,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fixMonoParentMap","k":"theorem","m":"Causalean.SCM.Model.InterventionMono","a":"SCM/Model","t":2,"d":"The parent values used by a monolithic intervention depend measurably on the original parent values, so this reindexing can be used safely when constructing intervened structural equations and probability kernels.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), Measurable (Causalean.SCM.fixMonoParentMap G X hObs hFix v)","l":149,"f":"Causalean/SCM/Model/InterventionMono.lean"},{"n":"Causalean.SCM.measurable_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Extending a base fixed-value assignment is measurable in the treatment value when the base assignment is held fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues), Measurable (M.fixSetExtend X hObs hFix s0)","l":249,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_fixSetProj","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSetProj is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), Measurable (M.fixSetProj X hObs hFix)","l":744,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.measurable_fixSetZSlice","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"fixSetZSlice is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X Z : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ (M.fixSet X hX_obs hX_fixed).fixed), Measurable (M.fixSetZSlice X Z hX_obs hX_fixed hZ_obs hZ_fixed)","l":776,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.measurable_observedPrefixValue","k":"theorem","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":2,"d":"observedPrefixValue is measurable in its prefix-state argument.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n ≤ M.observed.card) (i : Fin n), Measurable fun ξ => M.observedPrefixValue hn ξ i","l":89,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.measurable_orderedLatentPrefixFullToRandom","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":2,"d":"orderedLatentPrefixFullToRandom is measurable. Case-split mirrors the definition; the observed branch composes observedPrefixValue with a cast.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable M.orderedLatentPrefixFullToRandom","l":83,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.measurable_parentValuesFromPrefix","k":"theorem","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names a valid position in the canonical topological order of observed nodes. Then the map parentValuesFromPrefix that reads off the parent values of the n-th observed node from a fixed-value assignment, a latent assignment, and the already-generated length-n prefix of observed values is jointly measurable in these three arguments.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.parentValuesFromPrefix hn)","l":123,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.measurable_partialEvalMap","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"partialEvalMap is jointly measurable in (s, ℓ). Proved by induction on n: the base case is a product of projections, and the step case composes the measurable stepFun, extendOrderedLatentPrefix, and the inductive hypothesis.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (hn : n ≤ M.observed.card), Measurable fun sℓ => M.partialEvalMap n hn sℓ.1 sℓ.2","l":118,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.measurable_randomToObserved","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The observed-coordinate projection is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), Measurable M.randomToObserved","l":96,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.measurable_singletonValue","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Reading a singleton value is measurable.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] {v : ι}, Measurable Causalean.SCM.singletonValue","l":215,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Building a singleton tuple is measurable.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} [inst : (i : ι) → MeasurableSpace (α i)] {v : ι}, Measurable Causalean.SCM.singletonValues","l":224,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.measurable_stepFun","k":"theorem","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"Fix a structural causal model M and a step index n such that there are at least n + 1 observed nodes, so n names a valid position in the canonical topological order of observed nodes. Then the deterministic map stepFun, which produces the value of the n-th observed node by assembling its parent tuple from the fixed values, latent values, and previously generated observed prefix and applying the node's structural equation, is measurable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n + 1 ≤ M.observed.card), Measurable (M.stepFun hn)","l":62,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.measurable_xRandomAsFixed","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Reading observed treatment values as intervention-coordinate values is measurable.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {X : Finset N}, Measurable Causalean.SCM.xRandomAsFixed","l":212,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurable_zFixedAsRandom","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Reading intervention values as observed random-variable values is a measurable operation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {Z : Finset N}, Measurable Causalean.SCM.zFixedAsRandom","l":67,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.measurePreserving_valuesUnionEquiv","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"Measure.pi splits over a disjoint union. Given a family of σ-finite reference measures μ, one per coordinate, together with two finite index sets A and B assumed disjoint, the canonical equivalence carrying value assignments on A ∪ B to pairs of assignments on A and on B is measure-preserving from the product reference measure on A ∪ B to the product of the component reference measures on A and on B.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {A B : Finset (Causalean.SWIGNode N)} (hDisj : Disjoint A B) (μ : (v : Causalean.SWIGNode N) → MeasureTheory.Measure (Causalean.swigΩ Ω v)) [∀ (v : Causalean.SWIGNode N), MeasureTheory.SigmaFinite (μ v)], MeasureTheory.MeasurePreserving (⇑(Causalean.SCM.valuesUnionEquiv hDisj)) (MeasureTheory.Measure.pi fun i => μ ↑i) ((MeasureTheory.Measure.pi fun a => μ ↑a).prod (MeasureTheory.Measure.pi fun b => μ ↑b))","l":111,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.measure_pi_map_valuesProjection","k":"theorem","m":"Causalean.SCM.Model.Induced","a":"SCM/Model","t":2,"d":"Product measures marginalize under coordinate restriction.","s":"∀ {N : Type u_1} [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] {I J : Finset (Causalean.SWIGNode N)} (hJI : J ⊆ I) (μ : (i : ↥I) → MeasureTheory.Measure (Causalean.swigΩ Ω ↑i)) [∀ (i : ↥I), MeasureTheory.IsProbabilityMeasure (μ i)], MeasureTheory.Measure.map (Causalean.valuesProjection hJI) (MeasureTheory.Measure.pi μ) = MeasureTheory.Measure.pi fun j => μ ⟨↑j, hJI j.property⟩","l":197,"f":"Causalean/SCM/Model/Induced.lean"},{"n":"Causalean.SCM.mechCFactor","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node set with measurable node-value spaces, a structural causal model, a family of reference measures, a set of SWIG nodes contained in the model's observed-node set, and an assignment to the model's fixed nodes, the mechanism c-factor density maps every full observed-node assignment to the Radon--Nikodym derivative of the corresponding mechanism $Q[S]$ measure with respect to the product reference measure on the given node set, evaluated at that assignment's restriction to the set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → (S : Finset (Causalean.SWIGNode N)) → S ⊆ M.observed → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":175,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node set with measurable node-value spaces, a structural causal model and a set of SWIG nodes, the mechanism-complement node set consists exactly of the base nodes whose random copy is observed but absent from the given set and whose fixed copy is not already fixed in the model.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N) → Finset N","l":52,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_fixed_not_mem","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every mechanism-complement variable has a fixed copy that is not already fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), ∀ n ∈ M.mechComplementNames S, Causalean.SWIGNode.fixed n ∉ M.fixed","l":73,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_image_random_subset_observed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"The random copies of the mechanism-complement variables form a subset of the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), Finset.image Causalean.SWIGNode.random (M.mechComplementNames S) ⊆ M.observed","l":80,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechComplementNames_random_mem_observed","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"Every mechanism-complement variable has its random copy among the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (S : Finset (Causalean.SWIGNode N)), ∀ n ∈ M.mechComplementNames S, Causalean.SWIGNode.random n ∈ M.observed","l":66,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mechDoValues","k":"def","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":1,"d":"For a finite node set with measurable node-value spaces, a structural causal model, a set of SWIG nodes, an assignment to the model's already fixed nodes, and a full assignment to its observed nodes, the do-complement fixed-node assignment extends the original fixed-node assignment by assigning each mechanism-complement node the value of its random copy in the observed assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (S : Finset (Causalean.SWIGNode N)) → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → (M.fixSet (M.mechComplementNames S) (Causalean.SCM.mechComplementNames_random_mem_observed M S) (Causalean.SCM.mechComplementNames_fixed_not_mem M S)).FixedValues","l":130,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.mem_cutsetLatent","k":"theorem","m":"Causalean.SCM.Model.CutsetLatent","a":"SCM/Model","t":2,"d":"Membership in the latent cutset means being unobserved and reaching the target set along an avoiding path.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_3} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y C : Finset (Causalean.SWIGNode N)} {u : Causalean.SWIGNode N}, u ∈ M.cutsetLatent Y C ↔ u ∈ M.unobserved ∧ ∃ y ∈ Y, u = y ∨ M.dag.isAncestorAvoiding C u y","l":189,"f":"Causalean/SCM/Model/CutsetLatent.lean"},{"n":"Causalean.SCM.mem_fixedAncestorsOfNode","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"Membership in the fixed-ancestor set means being fixed and equal to or ancestral to the node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v d : Causalean.SWIGNode N}, d ∈ M.fixedAncestorsOfNode v ↔ d ∈ M.fixed ∧ (d = v ∨ M.dag.isAncestor d v)","l":189,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.mem_latentAncestorsOfNode","k":"theorem","m":"Causalean.SCM.Model.EvalFactorization","a":"SCM/Model","t":2,"d":"Membership in the latent-ancestor set means being unobserved and equal to or ancestral to the node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v u : Causalean.SWIGNode N}, u ∈ M.latentAncestorsOfNode v ↔ u ∈ M.unobserved ∧ (u = v ∨ M.dag.isAncestor u v)","l":179,"f":"Causalean/SCM/Model/EvalFactorization.lean"},{"n":"Causalean.SCM.mem_prefixNodes_iff","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Membership in prefixNodes is exactly having observed index below n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (v : Causalean.SWIGNode N), v ∈ M.prefixNodes n ↔ ∃ (h : v ∈ M.observed), ↑(M.observedIndex ⟨v, h⟩) < n","l":61,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.mu_C_comap_F_eq_nu_C_comap_F","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Cross-SCM pullback equality along the filled assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))) = MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s))","l":113,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.not_fixedTreatment_of_uadj","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"A node incident to a directed edge in the base graph cannot be the fixed copy of a treatment variable that is newly fixed by the intervention.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N), (∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) → ∀ {a v : Causalean.SWIGNode N}, M.dag.UAdj a v → ∀ {D : N}, D ∈ Z → v ≠ Causalean.SWIGNode.fixed D","l":278,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.not_fixedW_of_incoming_doX","k":"theorem","m":"Causalean.SCM.ID.Toolkit.FrontdoorGraph","a":"SCM/ID/Toolkit","t":2,"d":"If fixed copies of the W block are absent in the original model, no edge after fixing X can end at a fixed copy from W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X W : Finset N) (hX_obs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hX_fixed : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed), (∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ M.fixed) → ∀ {u w : Causalean.SWIGNode N}, (M.fixSet X hX_obs hX_fixed).dag.edge u w → ∀ D ∈ W, w ≠ Causalean.SWIGNode.fixed D","l":116,"f":"Causalean/SCM/ID/Toolkit/FrontdoorGraph.lean"},{"n":"Causalean.SCM.not_fixed_of_obs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An observed node cannot also be a fixed node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.observed → n ∉ G.fixed","l":167,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.not_obs_of_unobs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An unobserved node cannot also be an observed node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.unobserved → n ∉ G.observed","l":162,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.not_unobs_of_obs","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"An observed node cannot also be an unobserved node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {n : Causalean.SWIGNode N}, n ∈ G.observed → n ∉ G.unobserved","l":157,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.obsChainKernel","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model, standard Borel and nonempty value spaces for every observed node, and countably generated conditioning σ-algebras for every observed prefix, the observational chain kernel assigns to every nonnegative integer not exceeding the number of observed nodes the kernel obtained at zero by the point mass on the empty prefix and at each positive index by composing the preceding chain kernel with the next one-node conditional kernel and extending the prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] → (n : ℕ) → n ≤ M.observed.card → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))","l":534,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsChainKernelZero","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model, the zero-step observational chain kernel assigns, at every fixed-value setting, unit probability to the unique assignment on the empty observed prefix.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn (M.prefixNodes 0) (Causalean.swigΩ Ω))","l":400,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsChainKernel_absolutelyContinuous_jointRef_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Domination of the recursive observed-prefix chain by the prefix reference.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (k : ℕ) (hk : k ≤ M.observed.card), ((M.obsChainKernel k hk) s).AbsolutelyContinuous (Causalean.SCM.jointRef ref (M.prefixNodes k))","l":539,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_eq_qFactorDensityProduct_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Prefix-level analytic chain rule at the full observed prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":861,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_eq_qFactorDensityProduct_prefix_induction","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The analytic prefix induction for the density chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":827,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_pullback_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Push the full-prefix a.e. density identity forward to observed coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → (fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":934,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_card_rnDeriv_pullback_eq_qFactorDensityProduct_of_prefix","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Push the full-prefix a.e. density identity forward, assuming the reference transport and prefix-level chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed → ∀ [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)] fun y => M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)) → (fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":890,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsChainKernel_rnDeriv_eq_prefixDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"General prefix-level RN derivative for the recursive observational chain.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → ∀ (k : ℕ) (hk : k ≤ M.observed.card), ((M.obsChainKernel k hk) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes k)) =ᵐ[Causalean.SCM.jointRef ref (M.prefixNodes k)] M.prefixDensityProduct ref s k","l":604,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsCondIndep_contraction","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Contraction. Fix subsets X, Y, W, Z of the node set of the structural causal model M, with X, Y, W, and Z each contained in the observed nodes, and let μ be a finite measure on the observed values. If X is conditionally independent of Y given the union of Z and W and X is conditionally independent of W given Z, then X is conditionally independent of the union of Y and W given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y (Z ∪ W) hX hY (Finset.union_subset hZ hW) μ → M.ObsCondIndep X W Z hX hW hZ μ → M.ObsCondIndep X (Y ∪ W) Z hX (Finset.union_subset hY hW) hZ μ","l":261,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_decomposition","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Decomposition. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hYW : Y ∪ W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.ObsCondIndep X Y Z hX (LE.le.trans Finset.subset_union_left hYW) hZ μ","l":227,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_subset_right","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Subset right. If X ⊥ Y | Z and Y' ⊆ Y, then X ⊥ Y' | Z.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y Y' Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hY' : Y' ⊆ M.observed) (hZ : Z ⊆ M.observed), Y' ⊆ Y → ∀ {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y Z hX hY hZ μ → M.ObsCondIndep X Y' Z hX hY' hZ μ","l":214,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_symm","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":2,"d":"Symmetry. Conditional independence is symmetric in X and Y.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hW : W ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X Y W hX hY hW μ → M.ObsCondIndep Y X W hY hX hW μ","l":78,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondIndep_weak_union","k":"theorem","m":"Causalean.SCM.Do.SemiGraphoid","a":"SCM/Do","t":1,"d":"Weak union. Let X, Y, W, Z be finite node sets of the structural causal model M, with X and the union of Y and W contained in the observed nodes, and Z contained in the observed nodes, under a finite measure μ on the observed values. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given the union of Z and W.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.ObservedValues] {X Y W Z : Finset (Causalean.SWIGNode N)} (hX : X ⊆ M.observed) (hYW : Y ∪ W ⊆ M.observed) (hZ : Z ⊆ M.observed) {μ : MeasureTheory.Measure M.ObservedValues} [inst_4 : MeasureTheory.IsFiniteMeasure μ], M.ObsCondIndep X (Y ∪ W) Z hX hYW hZ μ → M.ObsCondIndep X Y (Z ∪ W) hX (LE.le.trans Finset.subset_union_left hYW) (Finset.union_subset hZ (LE.le.trans Finset.subset_union_right hYW)) μ","l":244,"f":"Causalean/SCM/Do/SemiGraphoid.lean"},{"n":"Causalean.SCM.obsCondKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, a target-node set, a conditioning-node set, the requirement that every target node is observed, the requirement that every conditioning node is observed, a nonempty standard-Borel target-value space, and a conditioning-value space satisfying the stipulated countable-generation condition, the jointly measurable conditional law of target coordinates given fixed-node and conditioning-coordinate values is obtained as a conditional law from the paired observational distribution.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (Y CC : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → CC ⊆ M.observed → [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (M.FixedValues × Causalean.ValuesOn CC (Causalean.swigΩ Ω)) (Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":230,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondKernel_ae_eq_condDistrib","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"At a fixed value s of the intervention-free (fixed) coordinates, provided the target coordinates Y are observed and the conditioning coordinates CC are observed, the jointly-measurable conditional kernel obsCondKernel built from Y and CC, evaluated at s, agrees almost everywhere — under the pushforward of the observational kernel onto CC — with the measure-theoretic conditional distribution of Y given CC under the observational kernel at s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), (fun c => (M.obsCondKernel Y CC hY hCC) (s, c)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)] ⇑(ProbabilityTheory.condDistrib (Causalean.valuesProjection hY) (Causalean.valuesProjection hCC) (M.obsKernel s))","l":257,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondKernel_cross_SCM_ae_eq_on_fillZrW","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.RectIdentity","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM conditional kernels agree along the fillZrW filled assignment. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, the union of the random copies of Z and W contained in the observed variables with the random copies of Z disjoint from W, fix an intervened fixed assignment s; then for a measurable outcome set B and almost every conditioning value w, taken with respect to the pullback of the base model's law on Z.random ∪ W under the filled-assignment map, the base model's conditional distribution of Y given Z.random ∪ W, evaluated at the filled point built from w, and the intervened model's corresponding conditional distribution, evaluated at the same filled point, assign B the same probability.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed), W ⊆ M'.observed → ∀ (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))), ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) (s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B","l":319,"f":"Causalean/SCM/Do/Rule2Kernel/RectIdentity.lean"},{"n":"Causalean.SCM.obsCondKernel_cross_eq_ae_of_discrete","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Discrete-treatment cross-SCM conditional-kernel equality.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed) [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)], (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [inst_7 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.obsKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).jointKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s)] [inst_14 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_16 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))).AbsolutelyContinuous (MeasureTheory.Measure.comap (M'.fillZrW Z hZ_obs hZ_fixed W s) (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s)))) → ∀ {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → ∀ᵐ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s), ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s, w)) B","l":176,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.obsCondKernel_dSep_collapse_ae","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.RectIdentity","a":"SCM/Do/Rule2Kernel","t":1,"d":"D-separation collapse for the post-intervention conditional kernel. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, suppose in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set. Then, fixing an intervened fixed assignment s and a measurable outcome set B, for almost every conditioning value on Z.random ∪ W, the intervened model's conditional distribution of Y given Z.random ∪ W assigns B the same probability as its conditional distribution of Y given W alone, evaluated at the W-projection of that conditioning value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)], (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → ∀ [inst_5 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_6 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet B → have hZr := fun ⦃v⦄ hv => Exists.casesOn (Finset.mem_image.mp hv) fun D h => And.casesOn h fun hD right => Eq.ndrec (motive := fun ⦃v⦄ => v ∈ Finset.image Causalean.SWIGNode.random Z → v ∈ M'.observed) (fun hv => hZ_obs D hD) right hv; have hZrW := Finset.union_subset hZr hW; ∀ᵐ (c : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω)) ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s), (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW)) (s, c)) B = (((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (s, Causalean.valuesProjection Finset.subset_union_right c)) B","l":47,"f":"Causalean/SCM/Do/Rule2Kernel/RectIdentity.lean"},{"n":"Causalean.SCM.obsCondKernel_empty_eq_marginal","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":2,"d":"Empty-conditioning collapse. Conditioning obsCondKernel Y on the empty set collapses to the plain Y-marginal (M.obsKernel s).map π_Y. This is the reusable step every do-calculus factorization performs when a conditioning block becomes empty (e.g. the first node of a fixing sequence); it is graph-agnostic in M, so it applies verbatim to any stacked-intervention model M.fixSet ….","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (s : M.FixedValues) (c : Causalean.ValuesOn ∅ (Causalean.swigΩ Ω)), (M.obsCondKernel Y ∅ hY (Finset.empty_subset M.observed)) (s, c) = MeasureTheory.Measure.map (Causalean.valuesProjection hY) (M.obsKernel s)","l":155,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsCondKernel_fixSet_M1_eq_ae_product","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Product-form cross-SCM bridge for the do-side conditional kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose that, in the intervened model's graph, no node of W is a descendant of any treatment's post-intervention node, that, in the original model's graph, no node of W is a descendant of any treatment's pre-intervention node, and that in the intervened model's graph, Y is d-separated from the treatments' pre-intervention nodes given W together with the intervened model's fixed nodes. Assume also, at a baseline assignment s0 to the original model's fixed nodes, an overlap condition: the pushforward, under combining a treatment value with a W value, of the product of the treatments' pre-intervention marginal law and the W-marginal law (both taken from the original model's observational kernel at s0) is absolutely continuous with respect to the original model's marginal law on the treatments' pre-intervention nodes together with W, again at s0. Then for almost every pair (t, w) drawn from that product law, the intervened model's conditional-probability kernel for Y given W, evaluated at treatment value t and conditioning value w, equals the original model's conditional-probability kernel for Y given the union of the treatments' pre-intervention nodes and W, evaluated at the combined value (t, w).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → Disjoint Y (Finset.image Causalean.SWIGNode.random Z) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → ∀ [StandardBorelSpace M'.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_6 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_7 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (M'.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).RandomValues] [StandardBorelSpace (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues] [StandardBorelSpace (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω))] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.jointKernel s)] [∀ (s : M'.FixedValues), MeasureTheory.IsFiniteMeasure (M'.obsKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).jointKernel s)] [∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.IsFiniteMeasure ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s)] [inst_18 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_20 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":1359,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsCondKernel_fixSet_eq_ae_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2AE","a":"SCM/Do","t":1,"d":"Rule 2, a.e. in the treatment value (product form + positivity). Consider intervening on the finite set of names Z, where each name's random copy is already observed in the base model and its fixed copy is not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, together with the random copies of Z and their union with W. Assume in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set, no fixed copy of a name in Z is an ancestor of any node of W in the post-intervention DAG, no random copy of a name in Z is an ancestor of any node of W in the original DAG, and positivity: the pushforward of the product of the treatment and conditioning marginals under the fill map is absolutely continuous with respect to the base model's law on Z.random ∪ W. Then for almost every pair (t, w) under that product measure, the W-conditional kernel of the model intervened at treatment value t, evaluated together with w, restricted to Y, equals the base model's conditional distribution of Y given Z.random ∪ W evaluated at the combined point (t, w).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), (M'.fixSet Z hZ_obs hZ_fixed).dag.dSep Y (Finset.image Causalean.SWIGNode.random Z) (W ∪ (M'.fixSet Z hZ_obs hZ_fixed).fixed) → (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → (∀ D ∈ Z, ∀ w ∈ W, ¬M'.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ [inst_3 : ∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [inst_4 : ∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated (M'.fixSet Z hZ_obs hZ_fixed).FixedValues (Causalean.ValuesOn W (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s0 : M'.FixedValues), (MeasureTheory.Measure.map (fun p => Causalean.valuesUnionMk p.1 p.2) ((MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))))).AbsolutelyContinuous (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel s0)) → ∀ᵐ (p : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) × Causalean.ValuesOn W (Causalean.swigΩ Ω)) ∂(MeasureTheory.Measure.map (Causalean.valuesProjection hZr) (M'.obsKernel s0)).compProd (ProbabilityTheory.Kernel.const (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)) (MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel s0))), ((M'.fixSet Z hZ_obs hZ_fixed).obsCondKernel Y W (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) (M'.fixSetExtend Z hZ_obs hZ_fixed s0 p.1, p.2) = (M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s0, Causalean.valuesUnionMk p.1 p.2)","l":139,"f":"Causalean/SCM/Do/Rule2AE.lean"},{"n":"Causalean.SCM.obsCondKernel_singleton_mass_of_ne_zero","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"For a query coordinate set Y contained in the observed nodes and a conditioning coordinate set CC contained in the observed nodes, at a conditioning value c with nonzero pushforward mass of the observational kernel under projection onto CC, the observational conditional kernel's singleton mass at a value y equals the observational kernel's joint singleton mass at the pair (c, y) divided by its singleton mass at c.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [MeasurableSingletonClass (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_6 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s)] [inst_8 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues) (c : Causalean.ValuesOn CC (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)) {c} ≠ 0 → ((M.obsCondKernel Y CC hY hCC) (s, c)) {y} = (MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection hCC ω, Causalean.valuesProjection hY ω)) (M.obsKernel s)) {(c, y)} / (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)) {c}","l":190,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsCondKernel_union_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The (Zr∪W)-conditional packaged as the witness kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_9 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_10 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_11 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (hY : Y ⊆ M.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hZrW) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_right hZrW) (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (fun c => (M.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (s, c)) =ᵐ[MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M.obsKernel s)] fun c => MeasureTheory.Measure.map (h (Causalean.valuesProjection Finset.subset_union_left c) (Causalean.valuesProjection Finset.subset_union_right c)) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection (LE.le.trans Finset.subset_union_right hZrW) (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) (Causalean.valuesProjection Finset.subset_union_right c))","l":690,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsCondPairKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces, a structural causal model, a target-node set, a conditioning-node set, the requirement that every target node is observed, and the requirement that every conditioning node is observed, the joint conditional law of the conditioning and target coordinates given fixed-node values maps each observational law to the ordered pair of its conditioning-coordinate and target-coordinate restrictions. It is defined by that paired coordinate projection.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (Y CC : Finset (Causalean.SWIGNode N)) → Y ⊆ M.observed → CC ⊆ M.observed → ProbabilityTheory.Kernel M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω) × Causalean.ValuesOn Y (Causalean.swigΩ Ω))","l":207,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsCondPairKernel_apply_eq_compProd","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Slice-level disintegration for the pair kernel defining obsCondKernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), (M.obsCondPairKernel Y CC hY hCC) s = (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)).compProd ((M.obsCondKernel Y CC hY hCC).sectR s)","l":489,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsCondPairKernel_eq","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"The conditional-pair kernel is the observational kernel pushed forward along the map that records the conditioning coordinates paired with the target coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed), M.obsCondPairKernel Y CC hY hCC = M.obsKernel.map fun ω => (Causalean.valuesProjection hCC ω, Causalean.valuesProjection hY ω)","l":219,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsDensity","k":"def","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, a reference-measure family, and fixed-node values, the joint observational density is the Radon--Nikodym derivative of the model's observational law at those fixed-node values with respect to the joint reference measure on the observed nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":78,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.obsDensity_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"Observational density chain rule. In a structural causal model whose observational law is absolutely continuous with respect to the joint reference measure on the observed nodes, if in addition the stepwise fibre Radon--Nikodym condition holds along the observed topological order, then the joint observational density agrees, almost everywhere with respect to that joint reference measure, with the product of the one-node conditional density factors taken in observed topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → M.obsDensity ref s =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":995,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsDensity_heq_of_obsKernel_heq","k":"theorem","m":"Causalean.SCM.ID.Density.DoLawMarginal","a":"SCM/ID/Density","t":1,"d":"Cross-model density transport. If two structural causal models M₁ and M₂ share the same SWIG graph and have heterogeneously-equal observational kernels, then their observational densities are heterogeneously equal. After unifying the SWIG-graph data the observed-value types coincide, the observational kernels become literally equal, and the density is the Radon–Nikodym derivative of that kernel against a fixed reference measure.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.obsKernel ≍ M₂.obsKernel → M₁.obsDensity ref ≍ M₂.obsDensity ref","l":28,"f":"Causalean/SCM/ID/Density/DoLawMarginal.lean"},{"n":"Causalean.SCM.obsKernel","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces and a structural causal model, the conditional law of observed-node values given fixed-node values is obtained by mapping the joint conditional law to its observed coordinates. It is defined by that observed-coordinate projection.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ProbabilityTheory.Kernel M.FixedValues M.ObservedValues","l":145,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsKernel_apply_univ","k":"theorem","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":2,"d":"Each observational-kernel slice has total mass one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), (M.obsKernel s) Set.univ = 1","l":180,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.obsKernel_disintegrate_rect","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":2,"d":"Base-model disintegration over condition coordinates.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues) {D : Set (Causalean.ValuesOn CC (Causalean.swigΩ Ω))} {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet D → MeasurableSet B → (M.obsKernel s) (Causalean.valuesProjection hCC ⁻¹' D ∩ Causalean.valuesProjection hY ⁻¹' B) = ∫⁻ (c : Causalean.ValuesOn CC (Causalean.swigΩ Ω)) in D, ((M.obsCondKernel Y CC hY hCC) (s, c)) B ∂MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)","l":664,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_empty_projection_eq_dirac","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":2,"d":"Empty-projection collapse. Projecting the observational kernel onto the empty node set yields a Dirac measure: ValuesOn ∅ is a subsingleton, so the projected Markov kernel is the point mass at its unique value. The base case of any conditioning set shrinking to ∅ in a fixing/adjustment step.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (hEmpty : ∅ ⊆ M.observed) (s : M.FixedValues) (c : Causalean.ValuesOn ∅ (Causalean.swigΩ Ω)), MeasureTheory.Measure.map (Causalean.valuesProjection hEmpty) (M.obsKernel s) = MeasureTheory.Measure.dirac c","l":124,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsKernel_eq_latentProduct_map","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"obsKernel s is the latent product pushed through randomToObserved ∘ evalMap s.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues), M.obsKernel s = MeasureTheory.Measure.map (fun ℓ => M.randomToObserved (M.evalMap s ℓ)) M.latentProduct","l":178,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsKernel_eq_of_obsDensity_ae_eq","k":"theorem","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"Within a model whose observational law is dominated by the joint reference measure, if the joint observational densities at two fixed-value slices agree almost everywhere with respect to that reference measure, then the two slices induce the same observational law.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s₁ s₂ : M.FixedValues), M.obsDensity ref s₁ =ᵐ[Causalean.SCM.jointRef ref M.observed] M.obsDensity ref s₂ → M.obsKernel s₁ = M.obsKernel s₂","l":95,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.obsKernel_eq_qFactorProduct","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model M, at a fixed assignment s, its observational kernel equals the full chain-rule product of one-node conditional kernels along the observed topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))], M.obsKernel s = M.qFactorProduct s","l":756,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_marginal_eq_M1_marginal","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.DiscreteZHelpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"Rule 3 W-marginal equality for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, and a conditioning set W contained in the observed variables, if no fixed copy of a name in Z is an ancestor, in the post-intervention DAG, of any node in W, then the W-marginal of the intervened model's observational kernel at a fixed assignment s equals the W-marginal of the base model's observational kernel at the corresponding projected assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M'.observed), (∀ z ∈ Z, ∀ v ∈ W, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed z) v) → ∀ (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) = MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s))","l":71,"f":"Causalean/SCM/Do/Rule2Kernel/DiscreteZHelpers.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_marginal_pushforward_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":1,"d":"W-marginal pushforward identity. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and an observation block W such that the union of Z's random copies and W is observed and disjoint from Z's random copies. If, under the intervened model's law of the observed variables given the fixed values s', the post-intervention random copies of Z almost surely equal their assigned intervention values, then for every measurable W-event A, the intervened model's marginal probability of A on W equals the base model's probability of the pullback of A through the map that fills in the pinned Z-random-copy coordinates alongside W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (∀ᵐ (ω : (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues) ∂(M'.fixSet Z hZ_obs hZ_fixed).obsKernel s', ∀ (D : N) (hD : D ∈ Z), ω ⟨Causalean.SWIGNode.random D, hZ_obs D hD⟩ = s' ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet A → (MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ fun ⦃v⦄ hv => hZrW (Finset.mem_union_right (Finset.image Causalean.SWIGNode.random Z) hv))) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s')) A = (MeasureTheory.Measure.map (Causalean.valuesProjection hZrW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))) (M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":223,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.obsKernel_fixSet_W_rect_integral_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WMarginal","a":"SCM/Do/Rule2Kernel","t":1,"d":"Cross-SCM rectangle bridge for the W-marginal. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on, together with an outcome block Y and a conditioning block W of observed variables such that the union of Z's random copies and W is observed and disjoint from Z's random copies. Assume that, under the intervened model's law given the fixed values s, the post-intervention random copies of Z almost surely equal their assigned intervention values. Then for every measurable W-event A and every measurable Y-event B, integrating, over A and with respect to the intervened model's W-marginal, the base model's conditional probability of B given the outcome of filling in Z's pinned random-copy values alongside each w reconstructs the intervened model's probability that both the W-event A and the Y-event B occur.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M'.FixedValues (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), (∀ᵐ (ω : (M'.fixSet Z hZ_obs hZ_fixed).ObservedValues) ∂(M'.fixSet Z hZ_obs hZ_fixed).obsKernel s, ∀ (D : N) (hD : D ∈ Z), ω ⟨Causalean.SWIGNode.random D, hZ_obs D hD⟩ = s ⟨Causalean.SWIGNode.fixed D, Causalean.SCM.fixed_mem_fixSet M' Z hZ_obs hZ_fixed hD⟩) → ∀ {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))} {B : Set (Causalean.ValuesOn Y (Causalean.swigΩ Ω))}, MeasurableSet A → MeasurableSet B → ∫⁻ (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) in A, ((M'.obsCondKernel Y (Finset.image Causalean.SWIGNode.random Z ∪ W) hY hZrW) (M'.fixSetProj Z hZ_obs hZ_fixed s, M'.fillZrW Z hZ_obs hZ_fixed W s w)) B ∂MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) = ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s) (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW) ⁻¹' A ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ⁻¹' B)","l":384,"f":"Causalean/SCM/Do/Rule2Kernel/WMarginal.lean"},{"n":"Causalean.SCM.obsKernel_heq_of_obsDensity_heq","k":"theorem","m":"Causalean.SCM.ID.Density.DoLawMarginal","a":"SCM/ID/Density","t":1,"d":"Cross-model law transport (converse). If two structural causal models M₁ and M₂ are each dominated by the same reference measure, share the same SWIG graph, and have heterogeneously-equal observational densities, then their observational kernels are heterogeneously equal. After unifying the SWIG-graph data the observed-value types coincide and the densities become literally equal; weighting the common joint reference by that density recovers each observational law (withDensity_obsDensity_eq), so the two laws agree. This is the converse of obsDensity_heq_of_obsKernel_heq: under dominance, equal density and equal law are interchangeable, letting the kernel-level identification tools be driven from a density hypothesis.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M₁ M₂ : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M₁.toSWIGGraph = M₂.toSWIGGraph → M₁.DominatedObs ref → M₂.DominatedObs ref → M₁.obsDensity ref ≍ M₂.obsDensity ref → M₁.obsKernel ≍ M₂.obsKernel","l":53,"f":"Causalean/SCM/ID/Density/DoLawMarginal.lean"},{"n":"Causalean.SCM.obsKernel_inter_Wset_Zrand_levelset_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":1,"d":"For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with the union of the random copies of Z and a conditioning set W contained in the observed variables and the random copies of Z disjoint from W, fix an intervened fixed assignment s'; for measurable subsets S of the observed-value space and measurable subsets A of the values on W, the intervened model's observational kernel at s', evaluated on S intersected with the preimage under the Z.random ∪ W-projection of the image of A under the filled-assignment map, equals the base model's observational kernel at the projected fixed assignment, evaluated on the analogous set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed), Disjoint (Finset.image Causalean.SWIGNode.random Z) W → ∀ [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) {S : Set M'.ObservedValues} {A : Set (Causalean.ValuesOn W (Causalean.swigΩ Ω))}, MeasurableSet S → MeasurableSet A → ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') (S ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW) ⁻¹' M'.fillZrW Z hZ_obs hZ_fixed W s' '' A) = (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s')) (S ∩ Causalean.valuesProjection hZrW ⁻¹' M'.fillZrW Z hZ_obs hZ_fixed W s' '' A)","l":316,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_inter_singleton_Zrand_eq","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.InterSingleton","a":"SCM/Do/Rule2Kernel","t":2,"d":"Joint kernel agreement on the Z.random-level-set event.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hZrW : Finset.image Causalean.SWIGNode.random Z ∪ W ⊆ M'.observed) [MeasurableSingletonClass (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z ∪ W) (Causalean.swigΩ Ω))] (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues) (w : Causalean.ValuesOn W (Causalean.swigΩ Ω)) {S : Set M'.ObservedValues}, MeasurableSet S → ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') (S ∩ Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hZrW) ⁻¹' {M'.fillZrW Z hZ_obs hZ_fixed W s' w}) = (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s')) (S ∩ Causalean.valuesProjection hZrW ⁻¹' {M'.fillZrW Z hZ_obs hZ_fixed W s' w})","l":39,"f":"Causalean/SCM/Do/Rule2Kernel/InterSingleton.lean"},{"n":"Causalean.SCM.obsKernel_map_W_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Rule 3\\* on the W-marginal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (W : Finset (Causalean.SWIGNode N)) (hW : W ⊆ M'.observed), (∀ v ∈ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (Causalean.valuesProjection hW) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":154,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.obsKernel_map_eq_obsCondKernel_comp","k":"theorem","m":"Causalean.SCM.ID.Toolkit.ObsChainRule","a":"SCM/ID/Toolkit","t":1,"d":"Observational chain rule. Fix a structural causal model M, a target node set Y and a conditioning node set CC with Y contained in the observed nodes and CC contained in the observed nodes. Then, at a fixed parameter slice s, the observational distribution of Y equals the observational conditional law of Y given CC composed with the observational marginal law of CC.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Y CC : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M.observed) (hCC : CC ⊆ M.observed) [inst_3 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn CC (Causalean.swigΩ Ω))] (s : M.FixedValues), MeasureTheory.Measure.map (Causalean.valuesProjection hY) (M.obsKernel s) = (MeasureTheory.Measure.map (Causalean.valuesProjection hCC) (M.obsKernel s)).bind ⇑((M.obsCondKernel Y CC hY hCC).sectR s)","l":40,"f":"Causalean/SCM/ID/Toolkit/ObsChainRule.lean"},{"n":"Causalean.SCM.obsKernel_map_prefixNodes","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix form of the observational chain rule.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [inst_3 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] (n : ℕ) (hn : n ≤ M.observed.card), MeasureTheory.Measure.map (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M n)) (M.obsKernel s) = (M.obsChainKernel n hn) s","l":623,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsKernel_map_prodWY_eq","k":"theorem","m":"Causalean.SCM.Do.Rule3Conditional","a":"SCM/Do","t":2,"d":"Rule 3\\* on the (W, Y) joint pushforward.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M' : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M'.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M'.fixed) (Y W : Finset (Causalean.SWIGNode N)) (hY : Y ⊆ M'.observed) (hW : W ⊆ M'.observed), (∀ v ∈ Y ∪ W, ∀ d ∈ Z, ¬(M'.fixSet Z hZ_obs hZ_fixed).dag.isAncestor (Causalean.SWIGNode.fixed d) v) → ∀ (s' : (M'.fixSet Z hZ_obs hZ_fixed).FixedValues), MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hW) ω, Causalean.valuesProjection (Eq.symm (Causalean.SCM.fixSet_observed M' Z hZ_obs hZ_fixed) ▸ hY) ω)) ((M'.fixSet Z hZ_obs hZ_fixed).obsKernel s') = MeasureTheory.Measure.map (fun ω => (Causalean.valuesProjection hW ω, Causalean.valuesProjection hY ω)) (M'.obsKernel (M'.fixSetProj Z hZ_obs hZ_fixed s'))","l":76,"f":"Causalean/SCM/Do/Rule3Conditional.lean"},{"n":"Causalean.SCM.obsKernel_map_proj_eq_of_equiv","k":"theorem","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":2,"d":"Structurally equivalent structural causal models assign the same observational marginal law to a shared set of observed variables when their fixed-variable assignments correspond.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equiv M₂ → ∀ (Y : Finset (Causalean.SWIGNode N)) (hY₁ : Y ⊆ M₁.observed) (hY₂ : Y ⊆ M₂.observed) (s₁ : M₁.FixedValues) (s₂ : M₂.FixedValues), s₁ ≍ s₂ → MeasureTheory.Measure.map (Causalean.valuesProjection hY₁) (M₁.obsKernel s₁) = MeasureTheory.Measure.map (Causalean.valuesProjection hY₂) (M₂.obsKernel s₂)","l":236,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SCM.obsKernel_marginal_singleton_eq_latentProduct_agree","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"Mapping an observational kernel to a subcollection of observed coordinates turns a singleton mass into the latent-product mass of the corresponding agreement event.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) {P : Finset (Causalean.SWIGNode N)} (hP : P ⊆ M.observed) [MeasurableSingletonClass (Causalean.ValuesOn P (Causalean.swigΩ Ω))] (x : M.ObservedValues), (MeasureTheory.Measure.map (Causalean.valuesProjection hP) (M.obsKernel s)) {Causalean.valuesProjection hP x} = M.latentProduct {ℓ | ∀ (v : ↥P), M.evalMap s ℓ ⟨↑v, Finset.mem_union_left M.unobserved (hP v.property)⟩ = x ⟨↑v, hP v.property⟩}","l":61,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsKernel_marginal_singleton_eq_prod_qLocalMass","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a set of observed nodes P that is closed under observed parents, the singleton mass of the projection of the observational law onto P equals the product, over the full c-components C of the graph, of the local q-mass on C ∩ P.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hP : M.ObsParentClosed P) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), (MeasureTheory.Measure.map (Causalean.valuesProjection hP.left) (M.obsKernel s)) {Causalean.valuesProjection hP.left x} = ∏ C ∈ M.cComponentSet, M.qLocalMass s (C ∩ P) (fun x hv => hP.left (Finset.mem_of_mem_inter_right hv)) x","l":1591,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.obsSide_eq_witness","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":1,"d":"Obs-side witness-kernel identity (pair-conditioned, on the latent space). Let M be a structural causal model with node sets Y, W and a set of treatment names Z, where Y is observed, the pre-intervention nodes of Z are observed, and W is observed. Suppose the latent cut-set for Y relative to Z's pre-intervention nodes together with W is d-separated, in M's causal graph, from Z's pre-intervention nodes given W and the fixed nodes, and let h be a jointly measurable map such that, at a fixed baseline assignment s, the realized value of Y factors pointwise as h applied to the realized pre-intervention-Z value, the realized W value, and the cut-set's latent value. Then, under M's latent product measure, the conditional law of Y given the pair of realized pre-intervention-Z and W values equals, for almost every such pair, the pushforward under h applied to that pair of the conditional law of the latent cut-set given the realized W value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [StandardBorelSpace M.RandomValues] [∀ (n : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω n)] [∀ (n : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω n)] [∀ (s : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.jointKernel s)] (Y W : Finset (Causalean.SWIGNode N)) (Z : Finset N) [inst_7 : StandardBorelSpace (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_8 : Nonempty (Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω))] [inst_9 : StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [inst_10 : Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hY : Y ⊆ M.observed) (hZr : Finset.image Causalean.SWIGNode.random Z ⊆ M.observed) (hW : W ⊆ M.observed), M.dag.dSep (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Finset.image Causalean.SWIGNode.random Z) (W ∪ M.fixed) → ∀ (s : M.FixedValues) (h : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn (M.cutsetLatent Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) (Causalean.swigΩ Ω) → Causalean.ValuesOn Y (Causalean.swigΩ Ω)), (Measurable fun p => h p.1.1 p.1.2 p.2) → (∀ (ℓ : M.LatentValues), Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ)) = h (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W)) ℓ)) → (fun p => (ProbabilityTheory.condDistrib (fun ℓ => Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ))) (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct) p) =ᵐ[MeasureTheory.Measure.map (fun ℓ => (Causalean.valuesProjection hZr (M.randomToObserved (M.evalMap s ℓ)), Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ)))) M.latentProduct] fun p => MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib (Causalean.valuesProjection (Causalean.SCM.cutsetLatent_subset M Y (Finset.image Causalean.SWIGNode.random Z ∪ W))) (fun ℓ => Causalean.valuesProjection hW (M.randomToObserved (M.evalMap s ℓ))) M.latentProduct) p.2)","l":597,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.obsStepCondDensity","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a family of $\\sigma$-finite reference measures for its node values, a fixed-node assignment, an observed-node position, and assuming finite observational kernels, standard-Borel and nonempty one-node value spaces, and countably generated prefix value spaces, the one-node conditional density factor maps each full observed assignment to the Radon--Nikodym derivative of the conditional law of the node at that position, given its preceding observed values, with respect to that node's reference measure.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → (i : Fin M.observed.card) → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑i) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":39,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.obsStepCondDensity_eq_mass_ratio","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":1,"d":"At the i-th observed coordinate, if the one-step observational conditional kernel is absolutely continuous with respect to the reference measure on that coordinate, and the reference measure's singleton mass at the recorded value is nonzero and finite, then the one-node observational step density obsStepCondDensity equals the conditional kernel's singleton mass at the recorded value divided by the reference measure's singleton mass there.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) (i : Fin M.observed.card) [MeasurableSingletonClass (Causalean.swigΩ Ω ↑(M.observedAt i))] [inst_4 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_5 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] [inst_6 : Nonempty (Causalean.ValuesOn {↑(M.observedAt i)} (Causalean.swigΩ Ω))] [inst_7 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑i) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ((M.obsStepCondKernel i.isLt) (s, Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M ↑i) x)).AbsolutelyContinuous (ref.μ ↑(M.observedAt i)) → (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} ≠ 0 → (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)} ≠ ⊤ → M.obsStepCondDensity ref s i x = ((M.obsStepCondKernel i.isLt) (s, Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M ↑i) x)) {x (M.observedAt i)} / (ref.μ ↑(M.observedAt i)) {x (M.observedAt i)}","l":250,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.obsStepCondKernel","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model, an index strictly below its number of observed nodes, a standard Borel and nonempty value space for the observed node at that index, and a countably generated conditioning σ-algebra for the fixed values and preceding observed values, the one-step observational conditional kernel gives the conditional distribution of that node's value given the fixed values and all earlier observed values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n < M.observed.card) → [StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (M.FixedValues × Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω)) (Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩))","l":253,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsStepCondKernel_map_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Mapping the scalar step kernel back to the singleton tuple recovers the conditional kernel it was built from.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))], (M.obsStepCondKernel hn).map Causalean.SCM.singletonValues = M.obsCondKernel {↑(M.observedAt ⟨n, hn⟩)} (M.prefixNodes n) (fun ⦃v⦄ hv => have hv_eq := Eq.mp Finset.mem_singleton._simp_1 hv; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hv_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) (Causalean.SCM.prefixNodes_subset_observed M n)","l":311,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obsStepCondKernel_sectR_map_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Slice form of obsStepCondKernel_map_singletonValues.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) [inst_3 : StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_4 : Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨n, hn⟩)} (Causalean.swigΩ Ω))] [inst_5 : MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω))] (s : M.FixedValues), ((M.obsStepCondKernel hn).sectR s).map Causalean.SCM.singletonValues = (M.obsCondKernel {↑(M.observedAt ⟨n, hn⟩)} (M.prefixNodes n) (fun ⦃v⦄ hv => have hv_eq := Eq.mp Finset.mem_singleton._simp_1 hv; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hv_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) (Causalean.SCM.prefixNodes_subset_observed M n)).sectR s","l":349,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.obs_condIndep_of_full","k":"theorem","m":"Causalean.SCM.Do.ObsMarkov","a":"SCM/Do","t":2,"d":"Projection lemma. For X, Y, Z ⊆ V (all observed), conditional independence at the full distribution level (jointKernel) implies conditional independence at the observational level (obsKernel).","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) [inst_3 : StandardBorelSpace M.RandomValues] [inst_4 : StandardBorelSpace M.ObservedValues] {X Y Z : Finset (Causalean.SWIGNode N)} [StandardBorelSpace (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn X (Causalean.swigΩ Ω))] [StandardBorelSpace (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] [Nonempty (Causalean.ValuesOn Y (Causalean.swigΩ Ω))] (hX : X ⊆ M.observed) (hY : Y ⊆ M.observed) (hZ : Z ⊆ M.observed) (s : M.FixedValues), M.FullCondIndep X Y Z (LE.le.trans hX (Causalean.SCM.observed_subset_randomVars M)) (LE.le.trans hY (Causalean.SCM.observed_subset_randomVars M)) (LE.le.trans hZ (Causalean.SCM.observed_subset_randomVars M)) (M.jointKernel s) → M.ObsCondIndep X Y Z hX hY hZ (M.obsKernel s)","l":45,"f":"Causalean/SCM/Do/ObsMarkov.lean"},{"n":"Causalean.SCM.observedAt","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, and a valid position among its observed nodes, the observed-node enumeration returns the observed node at that position in canonical topological order.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Fin M.observed.card → ↥M.observed","l":217,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedAt_mem_prefixNodes_iff","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"An observed node at index i belongs to the first n nodes iff i < n.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ) (i : Fin M.observed.card), ↑(M.observedAt i) ∈ M.prefixNodes n ↔ ↑i < n","l":89,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedAt_not_mem_prefixNodes","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The next observed node is not in the previous prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), ↑(M.observedAt ⟨n, hn⟩) ∉ M.prefixNodes n","l":118,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedAt_observedIndex","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Looking up an observed node by its canonical index recovers the same node.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (v : ↥M.observed), ↑(M.observedAt (M.observedIndex v)) = ↑v","l":235,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedAt_observedIndex_subtype","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"Taking an observed node to its topological index and back recovers the same observed-node subtype value.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {v : Causalean.SWIGNode N} (hv : v ∈ M.observed), M.observedAt (M.observedIndex ⟨v, hv⟩) = ⟨v, hv⟩","l":202,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.observedIndex","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, and an observed node, the observed-node index is that node's position in canonical topological order.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → ↥M.observed → Fin M.observed.card","l":226,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedIndex_observedAt","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":2,"d":"Looking up the canonical index of the observed node at a position recovers that position.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (k : Fin M.observed.card), M.observedIndex (M.observedAt k) = k","l":243,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observedPredecessors_observedAt","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"For the node at index n, Tian's full-history predecessor set is exactly the first n observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), M.observedPredecessors ↑(M.observedAt ⟨n, hn⟩) = M.prefixNodes n","l":159,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.observedPrefixValue","k":"def","m":"Causalean.SCM.Factored.PrefixState","a":"SCM/Factored","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a prefix length no greater than the number of its observed nodes, an assignment on that prefix, and a position within the prefix, the observed-prefix coordinate reader has no value when the prefix is empty, and otherwise returns the assigned value at that position, reading recursively from the preceding prefix or directly from its final coordinate.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n ≤ M.observed.card) → M.ObservedPrefixValues n hn → (i : Fin n) → Causalean.swigΩ Ω ↑(M.observedAt ⟨↑i, Nat.lt_of_lt_of_le i.isLt hn⟩)","l":69,"f":"Causalean/SCM/Factored/PrefixState.lean"},{"n":"Causalean.SCM.observedPrefixValue_succ_castSucc","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Appending an observed coordinate to a prefix leaves every earlier coordinate of the prefix unchanged.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {k : ℕ} (hn : k + 1 ≤ M.observed.card) (ξ : M.ObservedPrefixValues k (Nat.le_of_succ_le hn)) (y : Causalean.swigΩ Ω ↑(M.observedAt ⟨k, hn⟩)) (j : Fin k), M.observedPrefixValue hn (ξ, y) j.castSucc = M.observedPrefixValue (Nat.le_of_succ_le hn) ξ j","l":191,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.observedPrefixValue_succ_last","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Appending an observed coordinate to a prefix makes the final coordinate of the expanded prefix equal to the appended value.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {k : ℕ} (hn : k + 1 ≤ M.observed.card) (ξ : M.ObservedPrefixValues k (Nat.le_of_succ_le hn)) (y : Causalean.swigΩ Ω ↑(M.observedAt ⟨k, hn⟩)), M.observedPrefixValue hn (ξ, y) (Fin.last k) = y","l":181,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.observed_parent_index_lt","k":"theorem","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a structural causal model M, fix a valid position n among the observed nodes, and let p be a node such that there is an edge from p to the n-th node in the canonical observed order and p itself is observed; then the canonical index of p among the observed nodes is strictly less than n — an observed parent always precedes its child in the canonical observed topological order.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) {p : Causalean.SWIGNode N}, M.dag.edge p ↑(M.observedAt ⟨n, hn⟩) → ∀ (hobs : p ∈ M.observed), M.observedIndex ⟨p, hobs⟩ < ⟨n, hn⟩","l":254,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.observed_subset_randomVars","k":"theorem","m":"Causalean.SCM.Do.LocalMarkov","a":"SCM/Do","t":2,"d":"Observed nodes lie in randomVars.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.observed ⊆ M.randomVars","l":45,"f":"Causalean/SCM/Do/LocalMarkov.lean"},{"n":"Causalean.SCM.orderedLatentPrefixFullToRandom","k":"def","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the map from a completed latent-and-observed prefix state to an assignment of all random nodes assigns each observed node its completed prefix value and each unobserved node its latent value.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.OrderedLatentPrefixValues M.observed.card (le_refl M.observed.card) → M.RandomValues","l":51,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.overrideOn","k":"def","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"For a collection of nodes with a measurable outcome space for each node, a finite set of target coordinates, a finite set of replacement coordinates, an assignment on the target coordinates, and an assignment on the replacement coordinates, the overridden target-coordinate assignment uses the replacement assignment at every target coordinate that is also a replacement coordinate, and otherwise uses the original target-coordinate assignment.","s":"{N : Type u_1} → [DecidableEq N] → {Ω : N → Type u_2} → {I W : Finset (Causalean.SWIGNode N)} → Causalean.ValuesOn I (Causalean.swigΩ Ω) → Causalean.ValuesOn W (Causalean.swigΩ Ω) → Causalean.ValuesOn I (Causalean.swigΩ Ω)","l":41,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_mem","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"On overridden coordinates, overrideOn reads from the replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (i : ↥I) (hiW : ↑i ∈ W), Causalean.SCM.overrideOn x y i = y ⟨↑i, hiW⟩","l":55,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_notMem","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Away from overridden coordinates, overrideOn keeps the original assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)) (i : ↥I), ↑i ∉ W → Causalean.SCM.overrideOn x y i = x i","l":63,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.overrideOn_overrideOn","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Re-overriding the same coordinate set keeps the last replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [(n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y z : Causalean.ValuesOn W (Causalean.swigΩ Ω)), Causalean.SCM.overrideOn (Causalean.SCM.overrideOn x y) z = Causalean.SCM.overrideOn x z","l":71,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.parentMap","k":"def","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":1,"d":"For a structural causal model, an assignment of its fixed values, a realization of its latent values, an index strictly below the number of observed nodes, values already supplied for every earlier observed index, and a parent of the observed node at that index, the parent-value assignment gives that parent's value: it reads an unobserved parent from the latent realization, a fixed parent from the fixed assignment, and any observed parent from the earlier supplied values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → M.LatentValues → {n : ℕ} → (hn : n < M.observed.card) → ((m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":45,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMapOverride","k":"def","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":1,"d":"For a structural causal model, an override block, an assignment of fixed values, an override assignment on that block, a latent realization, an index strictly below the number of observed nodes, values supplied for every earlier observed index, and a parent of the observed node at that index, the override parent-value assignment gives that parent's value, reading an overridden observed parent from the override assignment and otherwise following the fixed, latent, or earlier-observed source appropriate to that parent.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {C : Finset (Causalean.SWIGNode N)} → M.FixedValues → Causalean.ValuesOn C (Causalean.swigΩ Ω) → M.LatentValues → {n : ℕ} → (hn : n < M.observed.card) → ((m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":70,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_C","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads an overridden observed parent directly from the override assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hc : ↑w ∈ C), M.parentMapOverride s c ℓ hn prev w = c ⟨↑w, hc⟩","l":136,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_fixed","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a fixed parent directly from the fixed-value assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hfix : ↑w ∈ M.fixed), M.parentMapOverride s c ℓ hn prev w = s ⟨↑w, hfix⟩","l":116,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_observed","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a non-overridden observed parent from the previous recursive values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hobs : ↑w ∈ M.observed), ↑w ∉ C → M.parentMapOverride s c ℓ hn prev w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ prev (↑(M.observedIndex ⟨↑w, hobs⟩)) (Causalean.SCM.observed_parent_index_lt M hn ((Causalean.DAG.mem_parents M.dag).mp w.property) hobs) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":152,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMapOverride_unobserved","k":"theorem","m":"Causalean.SCM.Model.EvalOverrideC","a":"SCM/Model","t":2,"d":"The override parent-value tuple reads a latent parent directly from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {C : Finset (Causalean.SWIGNode N)}, C ⊆ M.observed → ∀ (s : M.FixedValues) (c : Causalean.ValuesOn C (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (huo : ↑w ∈ M.unobserved), M.parentMapOverride s c ℓ hn prev w = ℓ ⟨↑w, huo⟩","l":103,"f":"Causalean/SCM/Model/EvalOverrideC.lean"},{"n":"Causalean.SCM.parentMap_fixed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads a fixed parent directly from the fixed-value assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hfix : ↑w ∈ M.fixed), M.parentMap s ℓ hn prev w = s ⟨↑w, hfix⟩","l":95,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMap_observed","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads an observed parent from the previously computed observed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (hobs : ↑w ∈ M.observed), M.parentMap s ℓ hn prev w = Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ prev (↑(M.observedIndex ⟨↑w, hobs⟩)) (Causalean.SCM.observed_parent_index_lt M hn ((Causalean.DAG.mem_parents M.dag).mp w.property) hobs) (M.observedIndex ⟨↑w, hobs⟩).isLt","l":113,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentMap_prevFromObservedValues_eq_dispatch","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) (ℓ : M.LatentValues) (j : Fin M.observed.card) (w : ↥(M.dag.parents ↑(M.observedAt j))), M.parentMap s ℓ j.isLt (M.prevFromObservedValues x) w = if huo : ↑w ∈ M.unobserved then ℓ ⟨↑w, huo⟩ else if hfix : ↑w ∈ M.fixed then s ⟨↑w, hfix⟩ else have hedge := (Causalean.DAG.mem_parents M.dag).mp w.property; have hobs := Or.casesOn (Finset.mem_union.mp (M.dag_edges_classified (↑w) (↑(M.observedAt j)) hedge).left) (fun h1 => Or.casesOn (Finset.mem_union.mp h1) (fun hfx => absurd hfx hfix) fun hob => hob) fun h2 => absurd h2 huo; x ⟨↑w, hobs⟩","l":230,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.parentMap_unobserved","k":"theorem","m":"Causalean.SCM.Model.Evaluation","a":"SCM/Model","t":2,"d":"The parent-value tuple reads a latent parent directly from the latent assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {n : ℕ} (hn : n < M.observed.card) (prev : (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)) (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) (huo : ↑w ∈ M.unobserved), M.parentMap s ℓ hn prev w = ℓ ⟨↑w, huo⟩","l":83,"f":"Causalean/SCM/Model/Evaluation.lean"},{"n":"Causalean.SCM.parentValuesFromPrefix","k":"def","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that the next position exists among the observed nodes, the map producing the values of all parents of the next observed node takes fixed-node values, latent-node values, and a prefix of the preceding observed-node values, and returns the corresponding value for every parent of that next observed node.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) → (w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))) → Causalean.swigΩ Ω ↑w","l":77,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.parent_unobserved_of_not_fixed_not_observed","k":"theorem","m":"Causalean.SCM.Factored.ParentLookup","a":"SCM/Factored","t":2,"d":"A parent that is neither fixed nor observed must be an unobserved node. This lets an evaluator classify a parent's value source without depending on where the target appears in an observation order.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.dag.edge u v → u ∉ G.fixed → u ∉ G.observed → u ∈ G.unobserved","l":57,"f":"Causalean/SCM/Factored/ParentLookup.lean"},{"n":"Causalean.SCM.partialEvalMap","k":"def","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":1,"d":"For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the deterministic prefix-state map takes fixed-node values and latent-node values and returns the latent-node values together with the values generated for the first specified number of observed nodes. At zero observed nodes it returns the latent-node values and the unique empty prefix; at each positive prefix length it first forms the preceding prefix and then appends the value given by the next node's structural equation.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (n : ℕ) → (hn : n ≤ M.observed.card) → M.FixedValues → M.LatentValues → M.OrderedLatentPrefixValues n hn","l":61,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_full_eq","k":"theorem","m":"Causalean.SCM.Factored.Factorization","a":"SCM/Factored","t":2,"d":"Bridge lemma: reindexing the deterministic full-prefix value built from partialEvalMap at length observed.card yields exactly evalMap s ℓ.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues), M.orderedLatentPrefixFullToRandom (M.partialEvalMap M.observed.card (le_refl M.observed.card) s ℓ) = M.evalMap s ℓ","l":115,"f":"Causalean/SCM/Factored/Factorization.lean"},{"n":"Causalean.SCM.partialEvalMap_latent","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"The first component of partialEvalMap is always the input latent tuple: the recursion only writes to the ObservedPrefixValues factor.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n ≤ M.observed.card), (M.partialEvalMap n hn s ℓ).1 = ℓ","l":89,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_observedPrefixValue","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Bridges partialEvalMap (kernel-side) to evalObservedAux (existing evaluator). Proven by induction on n mirroring the definitions: at each step, the newly-appended coordinate (index n) is stepFun hn (s, prev) = structFun v_n (parentValuesFromPrefix hn (s, prev)), which equals evalObservedAux M s ℓ n _ once one shows the parent lookups agree. Earlier indices are handled by the inductive hypothesis through observedPrefixValue of the extension.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) (n : ℕ) (hn : n ≤ M.observed.card) (i : Fin n), M.observedPrefixValue hn (M.partialEvalMap n hn s ℓ).2 i = M.evalObservedAux s ℓ (↑i) (Nat.lt_of_lt_of_le i.isLt hn)","l":226,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_succ_snd_fst","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Extending a deterministic evaluation prefix by one observed variable leaves the previously computed observed prefix unchanged.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {k : ℕ} (hn : k + 1 ≤ M.observed.card), (M.partialEvalMap (k + 1) hn s ℓ).2.1 = (M.partialEvalMap k (Nat.le_of_succ_le hn) s ℓ).2","l":202,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.partialEvalMap_succ_snd_snd","k":"theorem","m":"Causalean.SCM.Factored.EvalMapCorrespond","a":"SCM/Factored","t":2,"d":"Extending a deterministic evaluation prefix appends the value determined for the newly added observed variable.","s":"∀ {N : Type uN} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type uΩ} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (ℓ : M.LatentValues) {k : ℕ} (hn : k + 1 ≤ M.observed.card), (M.partialEvalMap (k + 1) hn s ℓ).2.2 = M.stepFun hn (s, M.partialEvalMap k (Nat.le_of_succ_le hn) s ℓ)","l":212,"f":"Causalean/SCM/Factored/EvalMapCorrespond.lean"},{"n":"Causalean.SCM.path_fixSet_active","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"An active path in the original structural causal model remains active after fixing the treatment set, when each of its directed edges has a non-treatment source.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (hZ_obs : ∀ D ∈ Z, Causalean.SWIGNode.random D ∈ M.observed) (hZ_fixed : ∀ D ∈ Z, Causalean.SWIGNode.fixed D ∉ M.fixed) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {P : List (Causalean.SWIGNode N)}, M.dag.IsActivePath (W ∪ M.fixed) P → (∀ (i : ℕ) (hi : i + 1 < P.length), (M.dag.edge (P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩) (P.get ⟨i + 1, hi⟩) → ∀ D ∈ Z, P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩ ≠ Causalean.SWIGNode.random D) ∧ (M.dag.edge (P.get ⟨i + 1, hi⟩) (P.get ⟨i, Decidable.byContradiction fun a => Causalean.SCM.path_fixSet_active._proof_1 i hi a⟩) → ∀ D ∈ Z, P.get ⟨i + 1, hi⟩ ≠ Causalean.SWIGNode.random D)) → (M.fixSet Z hZ_obs hZ_fixed).dag.IsActivePath (W ∪ M.fixed ∪ Finset.image Causalean.SWIGNode.fixed Z) P","l":294,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.prefixDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a family of $\\sigma$-finite reference measures for its node values, and a fixed-node assignment, assuming finite observational kernels, standard-Borel and nonempty one-node value spaces, and countably generated prefix value spaces, the recursive prefix-density product maps every prefix length and assignment on that prefix to the value $1$ for the empty prefix, and otherwise the preceding prefix density multiplied by the conditional density of the newly appended node, or by $1$ when that position is not observed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω) → ENNReal","l":106,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixDensityProduct_card_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"At the full observed prefix, the recursive prefix density product is the existing finite product over observed indices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (y : Causalean.ValuesOn (M.prefixNodes M.observed.card) (Causalean.swigΩ Ω)), M.prefixDensityProduct ref s M.observed.card y = M.qFactorDensityProduct ref s ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))) y)","l":401,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixDensityProduct_eq_range_product","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The recursive prefix density product is the range product of its one-step factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], ∀ k ≤ M.observed.card, ∀ (z : Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω)), M.prefixDensityProduct ref s k z = ∏ i ∈ Finset.range k, M.prefixStepDensityInPrefix ref s k z i","l":357,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixNodes","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a finite collection of distinguishable base-variable labels with a measurable value space attached to each label, a structural causal model, and a nonnegative integer, the prefix node set consists of precisely the first $n$ observed nodes in its canonical topological order; if $n$ is at least the number of observed nodes, it is the full observed-node set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → ℕ → Finset (Causalean.SWIGNode N)","l":55,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_card","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Every prefix at least as long as the observed-node list is the full observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ), M.observed.card ≤ n → M.prefixNodes n = M.observed","l":107,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_disjoint_singleton_next","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The previous prefix is disjoint from the singleton next node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), Disjoint (M.prefixNodes n) {↑(M.observedAt ⟨n, hn⟩)}","l":152,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_mono","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Prefix node sets are monotone in the prefix length.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {m k : ℕ}, m ≤ k → M.prefixNodes m ⊆ M.prefixNodes k","l":99,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixNodes_subset_observed","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Prefix nodes are observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (n : ℕ), M.prefixNodes n ⊆ M.observed","l":73,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_succ","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The prefix successor is obtained by adjoining the next observed node.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), M.prefixNodes (n + 1) = M.prefixNodes n ∪ {↑(M.observedAt ⟨n, hn⟩)}","l":125,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixNodes_zero","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The empty prefix has no nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.prefixNodes 0 = ∅","l":79,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prefixStepDensityInPrefix","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a family of $\\sigma$-finite reference measures for its node values, a fixed-node assignment, a prefix length $k$, an assignment on its prefix, and a natural-number position $i$, assuming finite observational kernels, standard-Borel and nonempty one-node value spaces, and countably generated prefix value spaces, the prefix-read one-step density factor is the conditional density factor at position $i$ when $i<k$ and that position is observed, the value $1$ when $i<k$ but the position is not observed, and the value $1$ when $i\\ge k$.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (i : ℕ) (hi : i < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ) (hi : i < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨i, hi⟩)} (Causalean.swigΩ Ω))] → [∀ (i : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes i) (Causalean.swigΩ Ω))] → (k : ℕ) → Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω) → ℕ → ENNReal","l":320,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.prefixSucc_projection_pair","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Projecting the successor prefix through the union equivalence gives the previous-prefix block and the singleton next-node block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card), (fun ω => (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.prefixNodes_disjoint_singleton_next M hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_succ M hn)) (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M (n + 1)) ω))) = fun ω => (Causalean.valuesProjection (Causalean.SCM.prefixNodes_subset_observed M n) ω, Causalean.valuesProjection (fun ⦃w⦄ hw => have hw_eq := Eq.mp Finset.mem_singleton._simp_1 hw; of_eq_true (Eq.trans (congrArg (Membership.mem M.observed) hw_eq) (eq_true (M.observedAt ⟨n, hn⟩).property))) ω)","l":436,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.prevFromObservedValues","k":"def","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a full assignment of its observed node values, a natural-number bound $n$, an index $m<n$, and evidence that this index is within the observed topological order, the previous-value reader returns from the assignment the value of the $m$-th observed node in that order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → {n : ℕ} → (m : ℕ) → m < n → (hm : m < M.observed.card) → Causalean.swigΩ Ω ↑(M.observedAt ⟨m, hm⟩)","l":27,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.prevFromObservedValues_apply_observed","k":"theorem","m":"Causalean.SCM.ID.Density.LatentBlocks","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)) {n : ℕ} {hn : n < M.observed.card} {w : ↥(M.dag.parents ↑(M.observedAt ⟨n, hn⟩))} (hobs : ↑w ∈ M.observed) (hlt : ↑(M.observedIndex ⟨↑w, hobs⟩) < n), Causalean.SCM.observedAt_observedIndex M ⟨↑w, hobs⟩ ▸ M.prevFromObservedValues x (↑(M.observedIndex ⟨↑w, hobs⟩)) hlt (M.observedIndex ⟨↑w, hobs⟩).isLt = x ⟨↑w, hobs⟩","l":209,"f":"Causalean/SCM/ID/Density/LatentBlocks.lean"},{"n":"Causalean.SCM.qFactor","k":"def","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"For a structural causal model, a finite set of observed SWIG vertices, and a fixed-value assignment, provided that the vertex set is contained in the observed-node set and that the value space of that vertex set is standard Borel and nonempty, every observational distribution at a fixed-value assignment is finite, and either the fixed-value space is countable or the conditioning-value space has a countably generated σ-algebra, the c-component conditional kernel is the conditional law of the values on the vertex set given the values on its conditioning-parent set under the model's observational law at that fixed-value assignment.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (C : Finset (Causalean.SWIGNode N)) → C ⊆ M.observed → [StandardBorelSpace (Causalean.ValuesOn C (Causalean.swigΩ Ω))] → [Nonempty (Causalean.ValuesOn C (Causalean.swigΩ Ω))] → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.qFactorParents C) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel (Causalean.ValuesOn (M.qFactorParents C) (Causalean.swigΩ Ω)) (Causalean.ValuesOn C (Causalean.swigΩ Ω))","l":104,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SCM.qFactorDensityProduct","k":"def","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a family of $\\sigma$-finite reference measures for its node values, and a fixed-node assignment, assuming finite observational kernels, standard-Borel and nonempty one-node value spaces, and countably generated prefix value spaces, the chain-rule density product maps each full observed assignment to the product of all one-node conditional density factors in the canonical observed topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → Causalean.SCM.ReferenceMeasures Ω → M.FixedValues → [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":71,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorDensityProduct_eq_prod_cComponentFactor","k":"theorem","m":"Causalean.SCM.ID.Density.CComponentDensity","a":"SCM/ID/Density","t":1,"d":"c-component regrouping of the chain-rule density product. For a structural causal model M, a reference measure family ref, a fixed assignment s, and a value assignment x to the observed coordinates, the product of all one-node conditional density factors equals the product, over the c-components of the graph, of the per-component density factors.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))] (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qFactorDensityProduct ref s x = ∏ C ∈ M.cComponentSet, M.cComponentDensityFactor ref s C x","l":63,"f":"Causalean/SCM/ID/Density/CComponentDensity.lean"},{"n":"Causalean.SCM.qFactorProduct","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model, standard Borel and nonempty value spaces for every observed node, and countably generated conditioning σ-algebras for every observed prefix, the full observational chain-rule product is the full-length observational chain kernel transported from the complete prefix to the observed-value space.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → [∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] → [∀ (k : Fin M.observed.card), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes ↑k) (Causalean.swigΩ Ω))] → ProbabilityTheory.Kernel M.FixedValues M.ObservedValues","l":607,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_obsChainKernel_card_pullback","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Peel the final qFactorProduct map back to the full prefix chain kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues) [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)","l":800,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_obsChainKernel_card_pullback_of_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"RN-derivative transport for qFactorProduct, assuming the final reference transport has already been identified.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω) (s : M.FixedValues), MeasureTheory.Measure.map (⇑(Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card)))) (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) = Causalean.SCM.jointRef ref M.observed → ∀ [∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] fun x => ((M.obsChainKernel M.observed.card (le_refl M.observed.card)) s).rnDeriv (Causalean.SCM.jointRef ref (M.prefixNodes M.observed.card)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_card M M.observed.card (le_refl M.observed.card))).symm x)","l":760,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qFactorProduct_rnDeriv_eq_qFactorDensityProduct","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Analytic chain rule for the mapped observational product kernel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues) [inst_3 : ∀ (s' : M.FixedValues), MeasureTheory.IsFiniteMeasure (M.obsKernel s')] [inst_4 : ∀ (k : ℕ) (hk : k < M.observed.card), StandardBorelSpace (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_5 : ∀ (k : ℕ) (hk : k < M.observed.card), Nonempty (Causalean.ValuesOn {↑(M.observedAt ⟨k, hk⟩)} (Causalean.swigΩ Ω))] [inst_6 : ∀ (k : ℕ), MeasurableSpace.CountableOrCountablyGenerated M.FixedValues (Causalean.ValuesOn (M.prefixNodes k) (Causalean.swigΩ Ω))], M.ObsStepFiberRN ref s → (M.qFactorProduct s).rnDeriv (Causalean.SCM.jointRef ref M.observed) =ᵐ[Causalean.SCM.jointRef ref M.observed] M.qFactorDensityProduct ref s","l":961,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.qLocalMass","k":"def","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a structural causal model, a fixed-value assignment, a finite set of observed SWIG vertices, contained in the model's observed-node set, and an assignment of values to all observed SWIG vertices, the local q-mass is the latent-product mass of latent-variable assignments for which every vertex in the specified set satisfies its local structural consistency condition at those fixed and observed values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.FixedValues → (T : Finset (Causalean.SWIGNode N)) → T ⊆ M.observed → Causalean.ValuesOn M.observed (Causalean.swigΩ Ω) → ENNReal","l":40,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_anti","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local q-mass is antitone in the constrained observed set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) {T T' : Finset (Causalean.SWIGNode N)}, T ⊆ T' → ∀ (hT : T ⊆ M.observed) (hT' : T' ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T' hT' x ≤ M.qLocalMass s T hT x","l":73,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_empty","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"The empty local q-mass is one.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s ∅ (of_eq_true (Finset.empty_subset._simp_1 M.observed)) x = 1","l":66,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_eq","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"The local q-mass of an observed subset is the latent-product mass of the set of latent draws that make every node of that subset locally consistent with the given fixed values and observed values.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x = M.latentProduct {ℓ | ∀ (v : Causalean.SWIGNode N) (hv : v ∈ T), M.localConsistent s x v (hT hv) ℓ}","l":53,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_marginalize_ancestralClosed","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a set of observed nodes T and a subset W of T such that every parent, within T, of a node in W also lies in W, marginalizing the local q-mass on T over the coordinates in T \\ W yields the local q-mass on W.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T W : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (hWT : W ⊆ T), (∀ v ∈ T, ∀ w ∈ W, M.dag.edge v w → v ∈ W) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), Causalean.SCM.marginalizeOn M.observed (T \\ W) (fun x hv => hT (Finset.mem_sdiff.mp hv).left) (M.qLocalMass s T hT) x = M.qLocalMass s W (fun x hv => hT (hWT hv)) x","l":534,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_pos_of_positiveObs","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Positive observational mass implies nonzero local q-mass.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) [MeasurableSingletonClass (Causalean.ValuesOn M.observed (Causalean.swigΩ Ω))], Causalean.SCM.ID.DiscreteID.PositiveMass (M.obsKernel s) → ∀ (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s T hT x ≠ 0","l":1623,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_prod_inter_of_latentBlock_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Local q-mass on a covered set factors over an abstract family after intersecting each family member with the covered set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (P : Finset (Causalean.SWIGNode N)) (hPobs : P ⊆ M.observed) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed), P ⊆ 𝒞.sup id → ((↑𝒞).Pairwise fun U U' => Disjoint (M.latentBlock U) (M.latentBlock U')) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s P hPobs x = ∏ U ∈ 𝒞, if hU : U ∈ 𝒞 then M.qLocalMass s (U ∩ P) (fun x hv => h𝒞obs U hU (Finset.mem_of_mem_inter_left hv)) x else 1","l":1356,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_prod_of_latentBlock_disjoint","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":1,"d":"For a finite family 𝒞 of observed node sets whose latent parent blocks are pairwise disjoint, the local q-mass on the union of the family equals the product, over the members U of 𝒞, of the local q-mass on U.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [(n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (𝒞 : Finset (Finset (Causalean.SWIGNode N))) (h𝒞obs : ∀ U ∈ 𝒞, U ⊆ M.observed), ((↑𝒞).Pairwise fun U U' => Disjoint (M.latentBlock U) (M.latentBlock U')) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), M.qLocalMass s (𝒞.sup id) (fun v hv => Exists.casesOn (Eq.mp (congrArg (fun _a => _a) (propext Finset.mem_sup)) hv) fun U h => And.casesOn h fun hU hvU => h𝒞obs U hU hvU) x = ∏ U ∈ 𝒞, if hU : U ∈ 𝒞 then M.qLocalMass s U (h𝒞obs U hU) x else 1","l":1229,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.qLocalMass_sum_point_eliminate","k":"theorem","m":"Causalean.SCM.ID.Density.QMass","a":"SCM/ID/Density","t":2,"d":"Summing a local q-mass over one childless observed coordinate removes that coordinate from the constrained set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] [inst_3 : (n : N) → Fintype (Ω n)] [∀ (n : N), MeasurableSingletonClass (Ω n)] (M : Causalean.SCM N Ω) (s : M.FixedValues) (T : Finset (Causalean.SWIGNode N)) (hT : T ⊆ M.observed) {v : Causalean.SWIGNode N}, v ∈ T → (∀ w ∈ T, ¬M.dag.edge v w) → ∀ (x : Causalean.ValuesOn M.observed (Causalean.swigΩ Ω)), ∑ ω, M.qLocalMass s T hT (Causalean.SCM.overrideOn x (Causalean.SCM.singletonValuePt✝ v ω)) = M.qLocalMass s (T.erase v) (fun x hv => hT (Finset.mem_of_mem_erase hv)) x","l":439,"f":"Causalean/SCM/ID/Density/QMass.lean"},{"n":"Causalean.SCM.randomToObserved","k":"def","m":"Causalean.SCM.Model.Kernel","a":"SCM/Model","t":1,"d":"For a finite node population with measurable node-value spaces and a structural causal model, the projection from all random-node values to observed-node values retains exactly the coordinates belonging to observed nodes. It is defined coordinate by coordinate.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → M.RandomValues → M.ObservedValues","l":91,"f":"Causalean/SCM/Model/Kernel.lean"},{"n":"Causalean.SCM.randomVars","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, its random-node set is the union of its observed nodes and its unobserved latent nodes.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → Finset (Causalean.SWIGNode N)","l":145,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.referenceFaithful_countingRef","k":"theorem","m":"Causalean.SCM.ID.Density.CountingReference","a":"SCM/ID/Density","t":1,"d":"The counting reference family is faithful: every singleton coordinate value has counting mass one, in particular nonzero.","s":"∀ {N : Type u_1} {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] [inst_1 : ∀ (sn : Causalean.SWIGNode N), Countable (Causalean.swigΩ Ω sn)] [inst_2 : ∀ (n : N), MeasurableSingletonClass (Ω n)], Causalean.SCM.ReferenceFaithful Causalean.SCM.countingRef","l":41,"f":"Causalean/SCM/ID/Density/CountingReference.lean"},{"n":"Causalean.SCM.reindexSubtypeProj","k":"def","m":"Causalean.SCM.Do.GlobalMarkov","a":"SCM/Do","t":1,"d":"For a finite coordinate population with a measurable outcome space for each coordinate, a finite coordinate subset, a finite set of coordinates, and the condition that this set is contained in the subset, the measurable equivalence between the two coordinate tuples relabels an assignment indexed first by membership in the set and then by membership in the subset as the same assignment indexed directly by membership in the set.","s":"{M' : Type u_1} → [inst : DecidableEq M'] → [Fintype M'] → {Ω' : M' → Type u_2} → [inst_2 : (n : M') → MeasurableSpace (Ω' n)] → {P : Finset M'} → (S : Finset M') → S ⊆ P → ((i : ↥(Finset.subtype (fun x => x ∈ P) S)) → Ω' ↑↑i) ≃ᵐ ((j : ↥S) → Ω' ↑j)","l":314,"f":"Causalean/SCM/Do/GlobalMarkov.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_prod_sigmaFinite_of_fiber_ac","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"σ-finite product-reference RN derivative for a finite kernel, under explicit fibre domination and fibre-density measurability.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) (f : α → ENNReal) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsFiniteKernel κ], μ.AbsolutelyContinuous ν → (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (ν.prod ρ) → μ.rnDeriv ν =ᵐ[ν] f → (μ.compProd κ).rnDeriv (ν.prod ρ) =ᵐ[ν.prod ρ] fun p => f p.1 * (κ p.1).rnDeriv ρ p.2","l":516,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_same_left","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Conditional factor of the composition-product Radon–Nikodym derivative.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] (μ : MeasureTheory.Measure α) (κ η : ProbabilityTheory.Kernel α β) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ (a : α), (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":437,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_compProd_same_left_ae","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"Same-left composition-product RN derivative under a.e. fibre absolute continuity. This is the a.e. variant needed after extracting fibre domination from product domination.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace.CountableOrCountablyGenerated α β] (μ : MeasureTheory.Measure α) (κ η : ProbabilityTheory.Kernel α β) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) → (μ.compProd κ).rnDeriv (μ.compProd η) =ᵐ[μ.compProd η] fun p => κ.rnDeriv η p.1 p.2","l":482,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.rnDeriv_mul_measure_singleton","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"On a measurable singleton, the RN derivative multiplied by the base mass recovers the numerator singleton mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : α), ν {x} * μ.rnDeriv ν x = μ {x}","l":27,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.rnDeriv_singleton_eq_div","k":"theorem","m":"Causalean.SCM.ID.Density.MassBridge","a":"SCM/ID/Density","t":2,"d":"On a positive finite singleton of the reference measure, the RN derivative is the ratio of numerator mass to reference mass.","s":"∀ {α : Type u_3} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : α), ν {x} ≠ 0 → ν {x} ≠ ⊤ → μ.rnDeriv ν x = μ {x} / ν {x}","l":41,"f":"Causalean/SCM/ID/Density/MassBridge.lean"},{"n":"Causalean.SCM.singletonValue","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For an index population, a singleton index in a family of value spaces, and an assignment on that singleton, the singleton-coordinate value is the assignment's value at that index.","s":"{ι : Type u_3} → {α : ι → Type u_4} → {v : ι} → Causalean.ValuesOn {v} α → α v","l":201,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValue_singletonValues","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Reading the tuple built from a singleton value returns that value.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} {v : ι} (x : α v), Causalean.SCM.singletonValue (Causalean.SCM.singletonValues x) = x","l":236,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValues","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For an index population, a singleton index in a family of value spaces, and a value at that index, the singleton assignment is the assignment on the singleton set whose sole coordinate equals that value.","s":"{ι : Type u_3} → {α : ι → Type u_4} → {v : ι} → α v → Causalean.ValuesOn {v} α","l":207,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.singletonValues_map_ref_eq_jointRef","k":"theorem","m":"Causalean.SCM.ID.Density.ChainRuleDensity","a":"SCM/ID/Density","t":2,"d":"The scalar reference on one node maps to the singleton product reference.","s":"∀ {N : Type u_1} [DecidableEq N] [Fintype N] {Ω : N → Type u_2} [inst : (n : N) → MeasurableSpace (Ω n)] (ref : Causalean.SCM.ReferenceMeasures Ω) (v : Causalean.SWIGNode N), MeasureTheory.Measure.map Causalean.SCM.singletonValues (ref.μ v) = Causalean.SCM.jointRef ref {v}","l":211,"f":"Causalean/SCM/ID/Density/ChainRuleDensity.lean"},{"n":"Causalean.SCM.singletonValues_singletonValue","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Building a singleton tuple from its only coordinate returns the tuple.","s":"∀ {ι : Type u_3} {α : ι → Type u_4} {v : ι} (x : Causalean.ValuesOn {v} α), Causalean.SCM.singletonValues (Causalean.SCM.singletonValue x) = x","l":243,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.standard_fixed_random_edgeless","k":"theorem","m":"Causalean.SCM.ID.Density.MechCFactor","a":"SCM/ID/Density","t":2,"d":"","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω), M.isStandard → ∀ (n : N), Causalean.SWIGNode.fixed n ∈ M.fixed → ∀ (v : Causalean.SWIGNode N), ¬M.dag.edge (Causalean.SWIGNode.random n) v","l":284,"f":"Causalean/SCM/ID/Density/MechCFactor.lean"},{"n":"Causalean.SCM.stepFun","k":"def","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model and an index $n$ for which at least $n+1$ observed vertices exist, the deterministic step function maps a fixed-value assignment together with the latent and already generated observed-prefix values to the value of the $n$-th observed vertex in the model's canonical topological order, by applying that vertex's structural function to the values of its parents.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn) → Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)","l":50,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.stepKernel","k":"def","m":"Causalean.SCM.Factored.StepKernel","a":"SCM/Factored","t":1,"d":"For a structural causal model and an index $n$ for which at least $n+1$ observed vertices exist, the step kernel is the probability kernel that assigns unit mass to the value of the $n$-th observed vertex produced by its structural function from the fixed values, latent values, and already generated observed-prefix values.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → {n : ℕ} → (hn : n + 1 ≤ M.observed.card) → ProbabilityTheory.Kernel (M.FixedValues × M.OrderedLatentPrefixValues n (Nat.le_of_succ_le hn)) (Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩))","l":80,"f":"Causalean/SCM/Factored/StepKernel.lean"},{"n":"Causalean.SCM.structFun_apply_eq_of_equiv","k":"theorem","m":"Causalean.SCM.Model.EquivKernel","a":"SCM/Model","t":2,"d":"Equivalent structural graphs with matching structural functions produce the same output at an observed node whenever corresponding parent values agree.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] {M₁ M₂ : Causalean.SCM N Ω}, M₁.Equivalent M₂.toSWIGGraph → M₁.structFun ≍ M₂.structFun → ∀ (v : Causalean.SWIGNode N) (hv_obs₁ : v ∈ M₁.observed) (hv_obs₂ : v ∈ M₂.observed) {ξ₁ : (w : ↥(M₁.dag.parents v)) → Causalean.swigΩ Ω ↑w} {ξ₂ : (w : ↥(M₂.dag.parents v)) → Causalean.swigΩ Ω ↑w}, (∀ (p : Causalean.SWIGNode N) (hp₁ : p ∈ M₁.dag.parents v) (hp₂ : p ∈ M₂.dag.parents v), ξ₁ ⟨p, hp₁⟩ = ξ₂ ⟨p, hp₂⟩) → M₁.structFun ⟨v, hv_obs₁⟩ ξ₁ = M₂.structFun ⟨v, hv_obs₂⟩ ξ₂","l":40,"f":"Causalean/SCM/Model/EquivKernel.lean"},{"n":"Causalean.SCM.swigInterventionSet_insert_equiv","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Insert form of monolithic intervention composition.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (y : N) (hyX : y ∉ X) (hInsert_obs : ∀ D ∈ insert y X, Causalean.SWIGNode.random D ∈ M.observed) (hInsert_fixed : ∀ D ∈ insert y X, Causalean.SWIGNode.fixed D ∉ M.fixed), have hX_obs := fun D hD => hInsert_obs D (Finset.mem_insert_of_mem hD); have hX_fixed := fun D hD => hInsert_fixed D (Finset.mem_insert_of_mem hD); have hy_obs := fun D hD => have hDy := Finset.mem_singleton.mp hD; Eq.ndrec (motive := fun D => D ∈ {y} → Causalean.SWIGNode.random D ∈ (M.fixSet X hX_obs hX_fixed).observed) (fun hD => id (hInsert_obs y (Finset.mem_insert_self y X))) (Eq.symm hDy) hD; have hy_fixed := fun D hD hDfixed => have hDy := Finset.mem_singleton.mp hD; Eq.ndrec (motive := fun D => D ∈ {y} → Causalean.SWIGNode.fixed D ∈ (M.fixSet X hX_obs hX_fixed).fixed → False) (fun hD hDfixed => Or.casesOn (Finset.mem_union.mp (Eq.mp (congrArg (fun _a => Causalean.SWIGNode.fixed y ∈ _a) (Causalean.SCM.fixSet_fixed M X hX_obs hX_fixed)) hDfixed)) (fun hyfixed => hInsert_fixed y (Finset.mem_insert_self y X) hyfixed) fun hyimage => Exists.casesOn (Finset.mem_image.mp hyimage) fun z h => And.casesOn h fun hz hzy => hyX (Eq.symm (Eq.symm (Causalean.SWIGNode.fixed.inj hzy)) ▸ hz)) (Eq.symm hDy) hD hDfixed; ((M.fixSet X hX_obs hX_fixed).fixSet {y} hy_obs hy_fixed).Equiv (M.fixSet (insert y X) hInsert_obs hInsert_fixed)","l":683,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SCM.topoLinearOrder","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a finite node population with measurable value spaces and a structural causal model, the canonical linear order on graph nodes ranks nodes by the model graph's topological ordering.","s":"{N : Type uN} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type uΩ} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → Causalean.SCM N Ω → LinearOrder (Causalean.SWIGNode N)","l":211,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SCM.treatmentMarginal","k":"def","m":"Causalean.SCM.ID.Backdoor","a":"SCM/ID","t":1,"d":"For a finite node population with measurable value spaces, a structural causal model, a finite treatment set, the condition that its random treatment copies are observed, and fixed-node values, the observational treatment marginal is the observational law of the random treatment coordinates at those fixed-node values.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → {Ω : N → Type u_2} → [inst_2 : (n : N) → MeasurableSpace (Ω n)] → (M : Causalean.SCM N Ω) → (X : Finset N) → Finset.image Causalean.SWIGNode.random X ⊆ M.observed → M.FixedValues → MeasureTheory.Measure (Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω))","l":60,"f":"Causalean/SCM/ID/Backdoor.lean"},{"n":"Causalean.SCM.treatment_not_isAncestor_cond","k":"theorem","m":"Causalean.SCM.Do.CutsetDSep","a":"SCM/Do","t":2,"d":"Under the backdoor non-descendancy condition, a treatment node cannot be a proper ancestor of any node in the observed-or-fixed conditioning block.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (Z : Finset N) (W : Finset (Causalean.SWIGNode N)), (∀ D ∈ Z, ∀ w ∈ W, ¬M.dag.isAncestor (Causalean.SWIGNode.random D) w) → ∀ {D : N}, D ∈ Z → ∀ {c : Causalean.SWIGNode N}, c ∈ W ∪ M.fixed → M.dag.isAncestor (Causalean.SWIGNode.random D) c → False","l":236,"f":"Causalean/SCM/Do/CutsetDSep.lean"},{"n":"Causalean.SCM.unionSumEquiv","k":"def","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":1,"d":"For a collection of nodes, two finite sets of nodes, and their disjointness, the index equivalence between their tagged sum and their union maps each tagged node to that same node regarded as a member of the union.","s":"{N : Type u_1} → [inst : DecidableEq N] → {A B : Finset (Causalean.SWIGNode N)} → Disjoint A B → ↥A ⊕ ↥B ≃ ↥(A ∪ B)","l":89,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.valuesEquivOfEq_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"A value block is HEq to its reindexing along a Finset equality. The bridge that lets the equiv-mediated Eq laws above discharge legacy HEq goals in one line.","s":"∀ {M : Type u_3} {Ω : M → Type u_4} [inst : (m : M) → MeasurableSpace (Ω m)] {I J : Finset M} (h : I = J) (x : Causalean.ValuesOn I Ω), (Causalean.valuesEquivOfEq h) x ≍ x","l":61,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesOn_heq_of_coord","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":1,"d":"HEq of value assignments from coordinatewise agreement. Given two node sets I and J that are equal as sets and value assignments f on I and g on J, if f and g agree at every node common to both index sets, then f and g are heterogeneously equal (HEq).","s":"∀ {M : Type u_3} {Ω : M → Type u_4} [(m : M) → MeasurableSpace (Ω m)] {I J : Finset M}, I = J → ∀ (f : Causalean.ValuesOn I Ω) (g : Causalean.ValuesOn J Ω), (∀ (v : M) (hI : v ∈ I) (hJ : v ∈ J), f ⟨v, hI⟩ = g ⟨v, hJ⟩) → f ≍ g","l":93,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesProjection_overrideOn","k":"theorem","m":"Causalean.SCM.ID.Density.PiUnion","a":"SCM/ID/Density","t":2,"d":"Projecting an override back to the overridden coordinates returns the replacement assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {I W : Finset (Causalean.SWIGNode N)} (hWI : W ⊆ I) (x : Causalean.ValuesOn I (Causalean.swigΩ Ω)) (y : Causalean.ValuesOn W (Causalean.swigΩ Ω)), Causalean.valuesProjection hWI (Causalean.SCM.overrideOn x y) = y","l":81,"f":"Causalean/SCM/ID/Density/PiUnion.lean"},{"n":"Causalean.SCM.valuesProjection_randomToObserved_evalMap_meas","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.WitnessBridge","a":"SCM/Do/Rule2Kernel","t":2,"d":"The pulled-back observed-coordinate projection through randomToObserved ∘ evalMap s is the latent projection of the evaluation, valuesProjection hY ∘ randomToObserved ∘ E. This is definitional unfolding, exposed so downstream condDistrib_map_comp rewrites can name the composite.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {Y : Finset (Causalean.SWIGNode N)} (hY : Y ⊆ M.observed) (s : M.FixedValues), Measurable fun ℓ => Causalean.valuesProjection hY (M.randomToObserved (M.evalMap s ℓ))","l":193,"f":"Causalean/SCM/Do/Rule2Kernel/WitnessBridge.lean"},{"n":"Causalean.SCM.valuesUnionEquiv","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For two sets of graph nodes that are disjoint, the value-assignment equivalence for their union is a measurable bijection between assignments on their union and pairs consisting of an assignment on each set.","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → [inst_1 : (n : N) → MeasurableSpace (Ω n)] → {A B : Finset (Causalean.SWIGNode N)} → Disjoint A B → Causalean.ValuesOn (A ∪ B) (Causalean.swigΩ Ω) ≃ᵐ Causalean.ValuesOn A (Causalean.swigΩ Ω) × Causalean.ValuesOn B (Causalean.swigΩ Ω)","l":100,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.valuesUnionEquiv_extendObsPrefix","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"The successor-prefix extension is inverse to the union-equivalence view of the successor prefix.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) {n : ℕ} (hn : n < M.observed.card) (p : Causalean.ValuesOn (M.prefixNodes n) (Causalean.swigΩ Ω) × Causalean.swigΩ Ω ↑(M.observedAt ⟨n, hn⟩)), (Causalean.SCM.valuesUnionEquiv (Causalean.SCM.prefixNodes_disjoint_singleton_next M hn)) ((Causalean.valuesEquivOfEq (Causalean.SCM.prefixNodes_succ M hn)) (M.extendObsPrefix hn p)) = (p.1, Causalean.SCM.singletonValues p.2)","l":472,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.valuesUnionEquiv_valuesEquivOfEq_symm_valuesUnionMk","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Transporting a combined assignment to an equal index set and back, then splitting the disjoint union, recovers the original pair of assignments.","s":"∀ {N : Type u_1} [inst : DecidableEq N] {Ω : N → Type u_2} [inst_1 : (n : N) → MeasurableSpace (Ω n)] {A B C : Finset (Causalean.SWIGNode N)} (hDisj : Disjoint A B) (hUnion : C = A ∪ B) (p : Causalean.ValuesOn A (Causalean.swigΩ Ω) × Causalean.ValuesOn B (Causalean.swigΩ Ω)), (Causalean.SCM.valuesUnionEquiv hDisj) ((Causalean.valuesEquivOfEq hUnion) ((Causalean.valuesEquivOfEq hUnion).symm (Causalean.valuesUnionMk p.1 p.2))) = p","l":460,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SCM.valuesUnionMk_comm","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":1,"d":"Union commutativity. When A and B are disjoint, reindexing valuesUnionMk a b (a block on A ∪ B) along A ∪ B = B ∪ A yields valuesUnionMk b a.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A B : Finset M}, Disjoint A B → ∀ (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω), (Causalean.valuesEquivOfEq (Finset.union_comm A B)) (Causalean.valuesUnionMk a b) = Causalean.valuesUnionMk b a","l":33,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_comm_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Union commutativity, HEq form. Supersedes the ad-hoc per-proof HEq construction: a one-line corollary of valuesUnionMk_comm.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A B : Finset M}, Disjoint A B → ∀ (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω), Causalean.valuesUnionMk a b ≍ Causalean.valuesUnionMk b a","l":70,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_empty_right","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Right ∅ collapse. Reindexing valuesUnionMk a e (with e the trivial block on ∅) along A ∪ ∅ = A recovers a.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A : Finset M} (a : Causalean.ValuesOn A Ω) (e : Causalean.ValuesOn ∅ Ω), (Causalean.valuesEquivOfEq (Finset.union_empty A)) (Causalean.valuesUnionMk a e) = a","l":50,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.valuesUnionMk_empty_right_heq","k":"theorem","m":"Causalean.SCM.Do.ValuesReindex","a":"SCM/Do","t":2,"d":"Right ∅ collapse, HEq form. valuesUnionMk a e (trivial ∅ block) is HEq to a; a one-line corollary of valuesUnionMk_empty_right.","s":"∀ {M : Type u_3} [inst : DecidableEq M] {Ω : M → Type u_4} [inst_1 : (m : M) → MeasurableSpace (Ω m)] {A : Finset M} (a : Causalean.ValuesOn A Ω) (e : Causalean.ValuesOn ∅ Ω), Causalean.valuesUnionMk a e ≍ a","l":79,"f":"Causalean/SCM/Do/ValuesReindex.lean"},{"n":"Causalean.SCM.withDensity_obsDensity_eq","k":"theorem","m":"Causalean.SCM.ID.Density.ReferenceMeasure","a":"SCM/ID/Density","t":1,"d":"In a structural causal model whose observational law is absolutely continuous with respect to the joint reference measure on the observed nodes, weighting that joint reference measure by the observational density recovers the observational law exactly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (ref : Causalean.SCM.ReferenceMeasures Ω), M.DominatedObs ref → ∀ (s : M.FixedValues), (Causalean.SCM.jointRef ref M.observed).withDensity (M.obsDensity ref s) = M.obsKernel s","l":86,"f":"Causalean/SCM/ID/Density/ReferenceMeasure.lean"},{"n":"Causalean.SCM.xRandomAsFixed","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For a set of treatment-variable names and an assignment to their observed random copies, the random-to-intervention value map is the assignment to the corresponding intervention copies that for an intervention copy returns the value of its matching observed random copy and has no possible random-copy case](step:2).","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → {X : Finset N} → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed X) (Causalean.swigΩ Ω)","l":196,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.zFixedAsRandom","k":"def","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":1,"d":"For a set of treatment-variable names and an assignment to their intervention copies, the intervention-to-random value map is the assignment to the corresponding observed random copies that for a random copy returns the value of its matching intervention copy and has no possible fixed-copy case](step:2).","s":"{N : Type u_1} → [inst : DecidableEq N] → {Ω : N → Type u_2} → {Z : Finset N} → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.fixed Z) (Causalean.swigΩ Ω) → Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random Z) (Causalean.swigΩ Ω)","l":43,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SCM.zFixedAsRandom_proj_fixSetExtend","k":"theorem","m":"Causalean.SCM.Do.Rule2Kernel.Helpers","a":"SCM/Do/Rule2Kernel","t":2,"d":"Relabeling the treatment part of the extended fixed assignment back to observed coordinates recovers the treatment assignment.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (s0 : M.FixedValues) (t : Causalean.ValuesOn (Finset.image Causalean.SWIGNode.random X) (Causalean.swigΩ Ω)), Causalean.SCM.zFixedAsRandom (Causalean.valuesProjection (Causalean.SCM.fixSet_image_fixed_subset M X hObs hFix) (M.fixSetExtend X hObs hFix s0 t)) = t","l":282,"f":"Causalean/SCM/Do/Rule2Kernel/Helpers.lean"},{"n":"Causalean.SWIGGraph","k":"structure","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"A Single-World Intervention Graph (SWIG), G = (S, V, U, E, ι) (Definition 4 from Basic Concepts.tex): a directed acyclic graph on the SWIG nodes whose vertices are partitioned into fixed intervention nodes, observed random nodes, and unobserved random nodes, where every fixed node is genuinely of fixed form, every observed node is of random form, every unobserved node is of random form, and the observed and unobserved sets are disjoint. Every edge of the graph has both endpoints classified as fixed, observed, or unobserved; the map sending each fixed intervention node to its random counterpart lands inside the observed nodes; fixed nodes and unobserved nodes have no parents; a fixed-form node absent from the fixed set is isolated, with neither parents nor children; and every child of a classified node is observed.","s":"(N : Type u_2) → [DecidableEq N] → [Fintype N] → Type u_2","l":367,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For two single-world intervention graphs, graph equivalence ignoring topological order holds exactly when they have the same directed edges, the same fixed nodes, the same observed nodes, and the same unobserved nodes.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGGraph N → Prop","l":464,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.parents_eq","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Equivalent SWIGGraphs have the same parents Finset at every node.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H : Causalean.SWIGGraph N}, (∀ (u v : Causalean.SWIGNode N), G.dag.edge u v ↔ H.dag.edge u v) → ∀ (v : Causalean.SWIGNode N), G.dag.parents v = H.dag.parents v","l":521,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.refl","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is reflexive.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), G.Equivalent G","l":481,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.splitMono_congr","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Congruence of splitMono under SWIGGraph.Equivalent.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {G₁ G₂ : Causalean.SWIGGraph N}, G₁.Equivalent G₂ → ∀ (X : Finset N) (hObs₁ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G₁.observed) (hFix₁ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G₁.fixed) (hObs₂ : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G₂.observed) (hFix₂ : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G₂.fixed), (G₁.splitMono X hObs₁ hFix₁).Equivalent (G₂.splitMono X hObs₂ hFix₂)","l":636,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.Equivalent.symm","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is symmetric.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H : Causalean.SWIGGraph N}, G.Equivalent H → H.Equivalent G","l":488,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.Equivalent.trans","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"SWIG graph equivalence is transitive.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] {G H K : Causalean.SWIGGraph N}, G.Equivalent H → H.Equivalent K → G.Equivalent K","l":503,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.activePath_node_not_fixed","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Every node on an active path conditioned on the union of a conditioning set and the fixed intervention nodes, with observed endpoints, is not fixed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {R X Y Z : Finset (Causalean.SWIGNode N)} {x y v : Causalean.SWIGNode N} {p : List (Causalean.SWIGNode N)}, X ⊆ R ∩ G.observed → Y ⊆ R ∩ G.observed → x ∈ X → y ∈ Y → G.dag.IsActivePath (Z ∪ G.fixed) p → p.head? = some x → p.getLast? = some y → v ∈ p → v ∉ G.fixed","l":246,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.backdoorCriterion","k":"def","m":"Causalean.SCM.ID.BackdoorCriterion","a":"SCM/ID","t":1,"d":"Given a SWIG graph, a finite treatment-variable set $X$, the condition that the random copy of every treatment variable is observed, the condition that the fixed copy of no treatment variable is already fixed, a finite outcome-vertex set $Y$, and a finite candidate adjustment-vertex set $Z$, the backdoor criterion holds precisely when every vertex in $Z$ is observed, $Z$ is disjoint from $Y$, $Z$ is disjoint from the random copies of the treatment variables, no vertex in $Z$ is a descendant of a random treatment copy, and in the graph obtained by splitting the treatment variables, $Y$ is d-separated from their random copies after conditioning on $Z$ and their fixed copies.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N) → Prop","l":69,"f":"Causalean/SCM/ID/BackdoorCriterion.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph and a starting split node, the breadth-first bidirected-reachability set is the set reached by repeatedly adding observed nodes directly confounded with the current frontier, starting from the given node when it is observed and otherwise returning the empty set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":92,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_closed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Closure at saturation. Under the BFS invariants — the visited set is observed, and every *already-expanded* visited node (one outside the frontier) has all its directly-confounded neighbors in visited — and given enough remaining fuel (card observed - card visited ≤ fuel), the result of go is closed under the directly-confounded relation: every neighbor of a node in the result is again in the result.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (fuel : ℕ) (frontier visited : Finset (Causalean.SWIGNode N)), visited ⊆ G.observed → (∀ a ∈ visited, a ∉ frontier → ∀ (b : Causalean.SWIGNode N), G.directlyConfounded a b → b ∈ visited) → G.observed.card - visited.card ≤ fuel → ∀ a ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel, ∀ (b : Causalean.SWIGNode N), G.directlyConfounded a b → b ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel","l":299,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Soundness of BFS. Every node produced by bidirectedBFS.go from a frontier and visited set all of whose elements are bidirected-reachable from start is itself bidirected-reachable from start. Fully proved by induction on the fuel.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start : Causalean.SWIGNode N} (fuel : ℕ) (frontier visited : Finset (Causalean.SWIGNode N)), (∀ y ∈ visited, G.bidirectedReachable start y) → (∀ y ∈ frontier, G.bidirectedReachable start y) → ∀ z ∈ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel, G.bidirectedReachable start z","l":265,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_go_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"If the visited set and frontier are within observed, so is the result of go.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (frontier visited : Finset (Causalean.SWIGNode N)) (fuel : ℕ), visited ⊆ G.observed → Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel ⊆ G.observed","l":228,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedBFS_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The bidirected BFS from start is contained in observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (start : Causalean.SWIGNode N), G.bidirectedBFS start ⊆ G.observed","l":247,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedNeighbors","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph and a split node, the bidirected-neighbor set consists exactly of the graph's observed nodes that are directly confounded with that node.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":78,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedNeighbors_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The bidirected neighbors of a node are observed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.bidirectedNeighbors v ⊆ G.observed","l":210,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable","k":"inductive","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a finite collection of base variables with decidable equality and a single-world intervention graph, the bidirected-reachability relation on split nodes is generated by relating each observed split node to itself and by extending any such relation across a direct-confounding link from its current endpoint. Two observed nodes are therefore related exactly through a finite chain of direct-confounding links.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":82,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_head","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Prepend a directly-confounded step at the head of a reachability chain: if u and v are directly confounded and v reaches w, then u reaches w. Proved by induction on the v-to-w derivation.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v w : Causalean.SWIGNode N}, G.directlyConfounded u v → G.bidirectedReachable v w → G.bidirectedReachable u w","l":174,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_observed_left","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Both endpoints of a bidirected-reachability derivation are observed (left endpoint).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → u ∈ G.observed","l":154,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_observed_right","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Both endpoints of a bidirected-reachability derivation are observed (right endpoint).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → v ∈ G.observed","l":162,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_symm","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Bidirected reachability is symmetric.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, G.bidirectedReachable u v → G.bidirectedReachable v u","l":192,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.bidirectedReachable_trans","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Bidirected reachability is transitive.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v w : Causalean.SWIGNode N}, G.bidirectedReachable u v → G.bidirectedReachable v w → G.bidirectedReachable u w","l":200,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph and a split node, the c-component associated with that node is its breadth-first bidirected-reachability set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":112,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_eq_of_mem_cComponentSet","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"A node in a listed c-component has that component as its computed c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v : Causalean.SWIGNode N}, v ∈ C → G.cComponentOf v = C","l":513,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_eq_of_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Two c-components seeded by reachable observed nodes are equal.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v w : Causalean.SWIGNode N}, G.bidirectedReachable v w → G.cComponentOf v = G.cComponentOf w","l":497,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentOf_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The c-component of v is contained in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.cComponentOf v ⊆ G.observed","l":455,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph, the canonical set of c-components is the set of c-components associated with its observed nodes, with duplicate components identified.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Finset (Causalean.SWIGNode N))","l":134,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_biUnion","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"The c-components cover exactly the observed nodes: their union recovers the set of observed nodes exactly.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), G.cComponentSet.biUnion id = G.observed","l":474,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_pairwise_disjoint","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Distinct c-components are pairwise disjoint: no observed node belongs to two different c-components.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), (↑G.cComponentSet).PairwiseDisjoint id","l":601,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponentSet_subset_observed","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"Every c-component (in the canonical set) is contained in the observed nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N), ∀ C ∈ G.cComponentSet, C ⊆ G.observed","l":465,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.cComponents","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph, the array of c-components is formed by scanning its observed nodes and appending a node's c-component exactly when that node is not already contained in a previously appended component.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Array (Finset (Causalean.SWIGNode N))","l":122,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.child_classified","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If w is a child of u in G, then w is classified (fixed, observed, or unobserved).","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u w : Causalean.SWIGNode N}, w ∈ G.dag.children u → w ∈ G.fixed ∪ G.observed ∪ G.unobserved","l":539,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.dSep_union_fixed_of_induce_dSep","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":1,"d":"D-separation in an ancestral induced SWIG lifts to the ambient SWIG once fixed intervention nodes are included in the conditioning set. Fix a SWIG G and a node set R that is closed under G's ancestral relation, with X, Y, and Z each contained in the observed nodes of R. If X and Y are d-separated by Z in the graph induced on R, then X and Y are d-separated by Z together with G's fixed intervention nodes, in the ambient graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R X Y Z : Finset (Causalean.SWIGNode N)), X ⊆ R ∩ G.observed → Y ⊆ R ∩ G.observed → Z ⊆ R ∩ G.observed → G.dag.ancestralSet R ⊆ R → (G.induce R).dag.dSep X Y Z → G.dag.dSep X Y (Z ∪ G.fixed)","l":300,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.decDirectlyConfounded","k":"instance","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"For a finite collection of base variables with decidable equality, a single-world intervention graph, and two split nodes, the decision procedure for direct confounding determines whether the nodes are distinct and have a common unobserved parent in the graph.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (v₁ v₂ : Causalean.SWIGNode N) → Decidable (G.directlyConfounded v₁ v₂)","l":73,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.decIsAncestrallyClosed","k":"instance","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"For a finite single-world intervention graph, the decision procedure for ancestral closure determines whether every parent of an observed node that is itself a random node is also observed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → Decidable G.isAncestrallyClosed","l":55,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.directlyConfounded","k":"def","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"For a single-world intervention graph and two split nodes, the direct-confounding relation holds exactly when the nodes are distinct and some unobserved node has a directed edge to each of them.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":62,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.directlyConfounded_symm","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The directly-confounded relation is symmetric: if v₁ and v₂ share an unobserved parent, then so do v₂ and v₁ (the shared parent and the observed-ness conditions are symmetric in the two arguments).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v₁ v₂ : Causalean.SWIGNode N}, G.directlyConfounded v₁ v₂ → G.directlyConfounded v₂ v₁","l":146,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.frontdoorCriterion","k":"def","m":"Causalean.SCM.ID.Frontdoor","a":"SCM/ID","t":1,"d":"For a finite node population, a SWIG graph, a treatment-variable set whose random copies are observed and whose fixed copies are not already fixed, a mediator-variable set whose random copies are observed and whose fixed copies are not already fixed, and an outcome-node set, the frontdoor criterion holds exactly when the mediator random copies d-separate the outcomes from the treatment fixed copies after intervening on the treatments, the empty set satisfies the back-door criterion for treatment and mediator, the treatment random copies satisfy the back-door criterion for mediator and outcome, and the mediator random copies are disjoint from both treatment random copies and outcomes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → (W : Finset N) → (∀ D ∈ W, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ W, Causalean.SWIGNode.fixed D ∉ G.fixed) → Finset (Causalean.SWIGNode N) → Prop","l":65,"f":"Causalean/SCM/ID/Frontdoor.lean"},{"n":"Causalean.SWIGGraph.induce","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"For a single-world intervention graph and a retained set of split nodes, the induced single-world intervention graph has as its observed nodes the retained observed nodes, fixed nodes precisely those original fixed nodes whose random counterparts remain observed, unobserved nodes precisely those original unobserved nodes with an edge into the retained observed nodes, and directed edges precisely the original edges with both endpoints among these retained nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGGraph N","l":109,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.induce_cComponentOf_eq_of_shared_unobserved_parent","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"If a latent node has edges into two observed nodes retained by an induced graph, those observed nodes seed the same induced c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) {u v w : Causalean.SWIGNode N}, u ∈ G.unobserved → v ∈ R → w ∈ R → G.dag.edge u v → G.dag.edge u w → (G.induce R).cComponentOf v = (G.induce R).cComponentOf w","l":560,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.induce_isAncestor_mem_R","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"In the induced subgraph, every vertex with a proper ancestor lies in the retained observed support.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) {u v : Causalean.SWIGNode N}, (G.induce R).dag.isAncestor u v → v ∈ R ∩ G.observed","l":249,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"For a single-world intervention graph and a set of active split nodes, the induced directed acyclic graph retains exactly the original directed edges whose two endpoints are active.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.DAG (Causalean.SWIGNode N)","l":65,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_children_subset","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"Every child in the restricted DAG is also a child in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (u : Causalean.SWIGNode N), (G.inducedDag active).children u ⊆ G.dag.children u","l":88,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_edge_iff","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"For a set of active nodes and vertices u, v, u and v are joined by an edge of the DAG restricted to the active nodes exactly when they are joined by an edge of the original DAG and both are active.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (u v : Causalean.SWIGNode N), (G.inducedDag active).edge u v ↔ G.dag.edge u v ∧ u ∈ active ∧ v ∈ active","l":74,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_isAncestor_mem_active","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"If (G.inducedDag active).isAncestor u v, then both endpoints belong to active.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) {u v : Causalean.SWIGNode N}, (G.inducedDag active).isAncestor u v → u ∈ active ∧ v ∈ active","l":95,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedDag_parents_subset","k":"theorem","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"Every parent in the restricted DAG is also a parent in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (active : Finset (Causalean.SWIGNode N)) (v : Causalean.SWIGNode N), (G.inducedDag active).parents v ⊆ G.dag.parents v","l":81,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedEdge","k":"def","m":"Causalean.Graph.Induce","a":"Graph","t":1,"d":"For a single-world intervention graph, a set of active split nodes, and two split nodes, the induced edge relation holds exactly when the original graph has the directed edge, the first endpoint belongs to the active set, and the second endpoint belongs to the active set.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":54,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.inducedEdge_decidable","k":"instance","m":"Causalean.Graph.Induce","a":"Graph","t":2,"d":"For a finite collection of base variables with decidable equality, a single-world intervention graph, and a set of active split nodes, the decision procedure for the induced edge relation determines, for every ordered pair of split nodes, whether the original graph joins them by an edge and both endpoints are active.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (active : Finset (Causalean.SWIGNode N)) → DecidableRel (G.inducedEdge active)","l":58,"f":"Causalean/Graph/Induce.lean"},{"n":"Causalean.SWIGGraph.iota","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a single-world intervention graph and a fixed intervention node in that graph, the canonical link map returns its random counterpart, together with the fact that this counterpart is observed.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → ↥G.fixed → ↥G.observed","l":435,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaN","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a single-world intervention graph and a base variable whose fixed copy belongs to its fixed nodes, the base-variable link map returns the same base variable together with the fact that its random copy is observed.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → { n // Causalean.SWIGNode.fixed n ∈ G.fixed } → { n // Causalean.SWIGNode.random n ∈ G.observed }","l":448,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaNode","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a single-world intervention graph and a fixed intervention node in that graph, the node-level canonical link is that node's random counterpart, with the membership certification omitted.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → ↥G.fixed → Causalean.SWIGNode N","l":440,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.iotaNode_eq_iotaMap","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Forgetting the membership proof in the graph-level link map gives the node-level link map.","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (s : ↥G.fixed), G.iotaNode s = Causalean.iotaMap ↑s","l":444,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.isAncestrallyClosed","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"For a single-world intervention graph, ancestral closure holds exactly when every observed node's parent that is a random node is itself observed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Prop","l":46,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.isStandard","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a single-world intervention graph, the standard-graph property holds exactly when it contains no fixed intervention nodes.","s":"{N : Type u_2} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Prop","l":457,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.mem_bidirectedBFS_iff_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"BFS computes bidirected reachability. Fix a single-world intervention graph G and a node start that is observed in G. Then a node w is found by the breadth-first search from start exactly when w is bidirected-reachable from start — connected to it by a chain of directly-confounded (shared-unobserved-parent) pairs.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start w : Causalean.SWIGNode N}, start ∈ G.observed → (w ∈ G.bidirectedBFS start ↔ G.bidirectedReachable start w)","l":409,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_bidirectedBFS_self","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The start node belongs to its own BFS result (when observed).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {start : Causalean.SWIGNode N}, start ∈ G.observed → start ∈ G.bidirectedBFS start","l":256,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_cComponentOf_iff_reachable","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":1,"d":"Membership in a c-component is exactly bidirected reachability from its seed. For a node v that is observed in G, a node w belongs to the c-component seeded at v exactly when w is bidirected-reachable from v.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v w : Causalean.SWIGNode N}, v ∈ G.observed → (w ∈ G.cComponentOf v ↔ G.bidirectedReachable v w)","l":490,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.mem_cComponentOf_self","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"An observed node belongs to its own c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N}, v ∈ G.observed → v ∈ G.cComponentOf v","l":460,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.no_shared_unobserved_parent_of_mem_cComponentSet_of_not_mem","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"No latent root can be a shared parent of a c-component node and a node outside that c-component.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v w u : Causalean.SWIGNode N}, v ∈ C → w ∉ C → u ∈ G.unobserved → G.dag.edge u v → G.dag.edge u w → False","l":544,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.nodeIndex","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable base-variable labels, a SWIG graph, a finite set of graph nodes, and a node belonging to that set, the node-position selector returns that node's position when the set is sorted in graph topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → ↥D → Fin D.card","l":45,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.nodesAt","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable base-variable labels, a SWIG graph, a finite set of graph nodes, and a position from zero through one less than the set's size, the node-at-position selector returns the node occupying that position when the set is sorted in graph topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → (D : Finset (Causalean.SWIGNode N)) → Fin D.card → ↥D","l":38,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.nonDescIn","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"For a single-world intervention graph, a selected finite node set, and a target node, the induced non-descendant set is the selected set with the target and all of its proper descendants in the induced graph removed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":86,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.nonDescIn_subset","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Every induced non-descendant lies in the selected set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.nonDescIn R v₀ ⊆ R","l":122,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.nonDescIn_subset_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Every induced non-descendant lies in the selected set with the target node removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.nonDescIn R v₀ ⊆ R.erase v₀","l":116,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.not_directlyConfounded_of_mem_cComponentSet_of_not_mem","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"A node outside a c-component is not directly confounded with a node inside it.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {C : Finset (Causalean.SWIGNode N)}, C ∈ G.cComponentSet → ∀ {v w : Causalean.SWIGNode N}, v ∈ C → w ∉ C → ¬G.directlyConfounded v w","l":525,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.not_mem_fixed_of_mem_observed","k":"theorem","m":"Causalean.Graph.DSep.InduceTransport","a":"Graph/DSep","t":2,"d":"Observed SWIG vertices are not members of the fixed intervention set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {v : Causalean.SWIGNode N}, v ∈ G.observed → v ∉ G.fixed","l":59,"f":"Causalean/Graph/DSep/InduceTransport.lean"},{"n":"Causalean.SWIGGraph.observedPredecessors","k":"def","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":1,"d":"For a finite collection of distinguishable base-variable labels, a SWIG graph, and a graph node, the observed-predecessor set consists exactly of the observed nodes that occur strictly before that node in the graph's topological order.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":31,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SWIGGraph.observedPredecessors_subset_observed","k":"theorem","m":"Causalean.SCM.Factored.ObsChainKernel","a":"SCM/Factored","t":2,"d":"Observed predecessors are observed by construction.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), G.observedPredecessors v ⊆ G.observed","l":37,"f":"Causalean/SCM/Factored/ObsChainKernel.lean"},{"n":"Causalean.SWIGGraph.parent_classified","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If u is a parent of v in G, then u is classified (fixed, observed, or unobserved).","s":"∀ {N : Type u_2} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) {u v : Causalean.SWIGNode N}, u ∈ G.dag.parents v → u ∈ G.fixed ∪ G.observed ∪ G.unobserved","l":532,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGGraph.prefixIn","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable base-variable labels, a SWIG graph, a finite set of graph nodes, and a nonnegative integer, the topological prefix consists of the nodes in that set whose topological positions are strictly less than $n$.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → ℕ → Finset (Causalean.SWIGNode N)","l":52,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.prefixIn_obsParentClosed","k":"theorem","m":"Causalean.SCM.ID.Density.QFactor","a":"SCM/ID/Density","t":2,"d":"Topological prefixes of an observed-parent-closed set remain observed-parent-closed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] {Ω : N → Type u_2} [inst_2 : (n : N) → MeasurableSpace (Ω n)] (M : Causalean.SCM N Ω) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ M.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ M.fixed) (Y : Finset (Causalean.SWIGNode N)) (k : ℕ), (M.fixSet X hObs hFix).ObsParentClosed (((M.fixSet X hObs hFix).induce (Causalean.SCM.ID.fixAncestralSet M X hObs hFix Y)).prefixIn (Causalean.SCM.ID.fixObservedAncestralSet M X hObs hFix Y) k)","l":41,"f":"Causalean/SCM/ID/Density/QFactor.lean"},{"n":"Causalean.SWIGGraph.prefixIn_subset","k":"theorem","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":2,"d":"Every node in a topological prefix of a selected node set belongs to the selected node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (D : Finset (Causalean.SWIGNode N)) (n : ℕ), G.prefixIn D n ⊆ D","l":57,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.properDescIn","k":"def","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"For a single-world intervention graph, a selected finite node set, and a target node, the proper-descendant set within the induced graph is the finite set of descendants of the target in the graph induced by the selected nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Causalean.SWIGNode N → Finset (Causalean.SWIGNode N)","l":76,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_disjoint_nonDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"The proper-descendant set and non-descendant set are disjoint inside the selected nodes.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), Disjoint (G.properDescIn R v₀) (G.nonDescIn R v₀)","l":128,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_subset_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"Proper descendants lie in the selected set with the target node removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.properDescIn R v₀ ⊆ R.erase v₀","l":106,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.properDescIn_union_nonDescIn_eq_erase","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":1,"d":"Within a selected node set R of a SWIG graph G, the proper descendants of a node v₀ together with its non-descendants exhaust the selected nodes other than v₀ itself.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), G.properDescIn R v₀ ∪ G.nonDescIn R v₀ = R.erase v₀","l":138,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.qFactorParents","k":"def","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":1,"d":"For a SWIG graph and a finite set of its vertices, the conditioning-parent set is the union of the graph's observed predecessors of the vertices in that set, with the set itself removed.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N) → Finset (Causalean.SWIGNode N)","l":78,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SWIGGraph.qFactorParents_subset_observed","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.CComponentFactor","a":"SCM/ID/GraphicalThms","t":2,"d":"qFactorParents C consists of observed nodes outside C.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (C : Finset (Causalean.SWIGNode N)), G.qFactorParents C ⊆ G.observed","l":88,"f":"Causalean/SCM/ID/GraphicalThms/CComponentFactor.lean"},{"n":"Causalean.SWIGGraph.randomVars","k":"def","m":"Causalean.SCM.Model.SCM","a":"SCM/Model","t":1,"d":"For a single-world intervention graph, its random-node set is the union of its observed nodes and its unobserved latent nodes.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset (Causalean.SWIGNode N)","l":93,"f":"Causalean/SCM/Model/SCM.lean"},{"n":"Causalean.SWIGGraph.splitMono","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"For a finite vertex set with decidable equality, a SWIG graph, and a finite set of vertices selected for splitting, provided that the random copy of every selected vertex is an observed vertex of the graph and the fixed copy of every selected vertex is not already among the graph's fixed vertices, the monolithic multi-target split SWIG graph is obtained by rerouting every outgoing edge of each selected random copy to leave the corresponding fixed copy, while adding those fixed copies to the fixed vertices and retaining the observed and unobserved vertices.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.SWIGGraph N) → (X : Finset N) → (∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) → (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) → Causalean.SWIGGraph N","l":231,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoDAG","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"For a finite vertex set with decidable equality, a SWIG graph, and a finite set of vertices selected for splitting, the monolithically split directed acyclic graph is the directed acyclic graph obtained by rerouting, in one operation, every edge from the random copy of a selected vertex to instead leave that vertex's fixed copy.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset N → Causalean.DAG (Causalean.SWIGNode N)","l":145,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoEdgeRel","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"For a finite vertex set with decidable equality, a directed-edge relation on its random and fixed copies, and a finite set of vertices selected for splitting, the monolithic split edge relation removes every edge leaving the random copy of a selected vertex and otherwise retains the corresponding original edge, while each fixed copy of a selected vertex inherits the outgoing edges of its random copy and every other fixed copy retains its original outgoing edges.","s":"{N : Type u_1} → [DecidableEq N] → (Causalean.SWIGNode N → Causalean.SWIGNode N → Prop) → Finset N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":75,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoEdgeRel_decidable","k":"instance","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"For a collection of base variables with decidable equality, a directed-edge relation on split nodes for which every proposed edge can be decided, and a finite set of vertices selected for splitting, the decision procedure for the monolithic split edge relation determines, for every ordered pair of split nodes, whether that pair is joined after the split.","s":"{N : Type u_1} → [inst : DecidableEq N] → (dagEdge : Causalean.SWIGNode N → Causalean.SWIGNode N → Prop) → [DecidableRel dagEdge] → (X : Finset N) → DecidableRel (Causalean.SWIGGraph.splitMonoEdgeRel dagEdge X)","l":97,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMonoTopo","k":"def","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"For a finite vertex set with decidable equality, a SWIG graph, and a finite set of vertices selected for splitting, the topological-order assignment for the monolithically split graph assigns each random copy twice its original topological rank plus one, and assigns each selected fixed copy twice the original rank of its random copy, while assigning each unselected fixed copy twice its own original rank plus one.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → Finset N → Causalean.SWIGNode N → ℕ","l":122,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_empty_parents","k":"theorem","m":"Causalean.SCM.Model.InterventionSet","a":"SCM/Model","t":2,"d":"Splitting no variables leaves every parent set unchanged.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (v : Causalean.SWIGNode N), (G.splitMono ∅ (fun x hD => absurd hD (Finset.notMem_empty x)) fun x hD => absurd hD (Finset.notMem_empty x)).dag.parents v = G.dag.parents v","l":125,"f":"Causalean/SCM/Model/InterventionSet.lean"},{"n":"Causalean.SWIGGraph.splitMono_fixed","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting adds the fixed copies of the target variables to the fixed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).fixed = G.fixed ∪ Finset.image Causalean.SWIGNode.fixed X","l":517,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_observed","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting preserves the observed node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).observed = G.observed","l":505,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_parents_char","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":1,"d":"Characterization of parents in splitMono. Fix a SWIG G and a set X of variables to split, where the random copy of every variable in X is observed in G and the fixed copy of every variable in X is not already among G's fixed nodes. Then, for any node v, a node x is a parent of v in the graph obtained by monolithically splitting X exactly when either x is a parent of v in the original graph and is not the random copy of any variable in X, or x is the fixed copy of some variable D ∈ X whose random copy is a parent of v in the original graph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v x : Causalean.SWIGNode N), x ∈ (G.splitMono X hObs hFix).dag.parents v ↔ (x ∈ G.dag.parents v ∧ ∀ D ∈ X, x ≠ Causalean.SWIGNode.random D) ∨ ∃ D ∈ X, x = Causalean.SWIGNode.fixed D ∧ Causalean.SWIGNode.random D ∈ G.dag.parents v","l":527,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_parents_eq_of_no_fixed_parent","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Parent-set coincidence at non-.fixed-targeted vertices.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed) (v : Causalean.SWIGNode N), (∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ (G.splitMono X hObs hFix).dag.parents v) → (G.splitMono X hObs hFix).dag.parents v = G.dag.parents v","l":601,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.splitMono_unobserved","k":"theorem","m":"Causalean.Graph.SWIGSplitMono","a":"Graph","t":2,"d":"Monolithic splitting preserves the unobserved node set.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (X : Finset N) (hObs : ∀ D ∈ X, Causalean.SWIGNode.random D ∈ G.observed) (hFix : ∀ D ∈ X, Causalean.SWIGNode.fixed D ∉ G.fixed), (G.splitMono X hObs hFix).unobserved = G.unobserved","l":511,"f":"Causalean/Graph/SWIGSplitMono.lean"},{"n":"Causalean.SWIGGraph.subset_bidirectedBFS_go","k":"theorem","m":"Causalean.Graph.CComponents","a":"Graph","t":2,"d":"The visited set only grows: it is contained in the result of go.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (frontier visited : Finset (Causalean.SWIGNode N)) (fuel : ℕ), visited ⊆ Causalean.SWIGGraph.bidirectedBFS.go G frontier visited fuel","l":216,"f":"Causalean/Graph/CComponents.lean"},{"n":"Causalean.SWIGGraph.topoLinearOrder","k":"def","m":"Causalean.SCM.ID.Density.IdentifyMass","a":"SCM/ID/Density","t":1,"d":"For a finite collection of distinguishable base-variable labels and a SWIG graph, the topological linear order orders its nodes by their positions in the graph's topological ordering.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.SWIGGraph N → LinearOrder (Causalean.SWIGNode N)","l":34,"f":"Causalean/SCM/ID/Density/IdentifyMass.lean"},{"n":"Causalean.SWIGGraph.v₀_not_mem_nonDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"The target node is not in its induced non-descendant set because it is explicitly removed.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), v₀ ∉ G.nonDescIn R v₀","l":101,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGGraph.v₀_not_mem_properDescIn","k":"theorem","m":"Causalean.SCM.ID.GraphicalThms.InducedSubgraph","a":"SCM/ID/GraphicalThms","t":2,"d":"A node is not a proper descendant of itself inside the induced subgraph.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.SWIGGraph N) (R : Finset (Causalean.SWIGNode N)) (v₀ : Causalean.SWIGNode N), v₀ ∉ G.properDescIn R v₀","l":95,"f":"Causalean/SCM/ID/GraphicalThms/InducedSubgraph.lean"},{"n":"Causalean.SWIGNode","k":"inductive","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a collection of base variables, the population of nodes in a single-world intervention graph consists of a random-node constructor that assigns each base variable its natural random copy and a fixed-node constructor that assigns each base variable its intervention copy. Thus it is the disjoint union of two copies of the base-variable population.","s":"Type u_1 → Type u_1","l":73,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.equiv","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The split-node equivalence bijects each random copy of a base variable with the first copy of that variable and each fixed copy with the second copy.","s":"{N : Type u_1} → Causalean.SWIGNode N ≃ N ⊕ N","l":102,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.fixed_injective","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The fixed-node constructor is injective: equal fixed SWIG nodes come from the same base variable.","s":"∀ {N : Type u_1}, Function.Injective Causalean.SWIGNode.fixed","l":97,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.instFintype","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"For a finite collection of base variables, the finite enumeration of its split SWIG nodes contains exactly the random and fixed copy of every base variable.","s":"{N : Type u_1} → [Fintype N] → Fintype (Causalean.SWIGNode N)","l":113,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SWIGNode.random_injective","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The random-node constructor is injective: equal random SWIG nodes come from the same base variable.","s":"∀ {N : Type u_1}, Function.Injective Causalean.SWIGNode.random","l":92,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.SameImmoralities","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex population and two directed acyclic graphs on it, the same-immoralities condition holds exactly when, for every ordered triple of vertices, it is an immorality in the first graph if and only if it is an immorality in the second graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":54,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.SameImmoralities.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameImmoralities is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameImmoralities G₁ G₂ → Causalean.SameImmoralities G₂ G₁","l":31,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameImmoralities.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameImmoralities is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.SameImmoralities G₁ G₂ → Causalean.SameImmoralities G₂ G₃ → Causalean.SameImmoralities G₁ G₃","l":35,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameSkeleton","k":"def","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex population and two directed acyclic graphs on it, the same-skeleton condition holds exactly when, for every pair of vertices, a directed edge joins the pair in either direction in the first graph if and only if one does in the second graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Prop","l":51,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.SameSkeleton.symm","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameSkeleton is symmetric.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameSkeleton G₂ G₁","l":23,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.SameSkeleton.trans","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"SameSkeleton is transitive.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ G₃ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameSkeleton G₂ G₃ → Causalean.SameSkeleton G₁ G₃","l":27,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.Stat.AsymptoticEquicont","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"For a score family, a reference parameter, a population probability measure, a sample-space probability measure, and an independent and identically distributed sample from the population, class-level asymptotic equicontinuity means that for every $\\varepsilon>0$ and every $\\eta>0$, there is a $\\delta>0$ such that, for all sufficiently large sample sizes, the sample-space probability that some parameter within distance $\\delta$ of the reference parameter has centered empirical-process norm exceeding $\\varepsilon$ is at most $\\eta$.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → (E → X → E) → E → (P : MeasureTheory.Measure X) → (μ : MeasureTheory.Measure Ω) → Causalean.Stat.IIDSample Ω X μ P → Prop","l":68,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.Concentration.AntitoneOn.leftRiemann_sum_le_integral","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"For an antitone grid and antitone integrand, the full left Riemann sum is bounded by the interval integral.","s":"∀ (n : ℕ) (f : ℕ → ℝ) (g : ℝ → ℝ), Antitone f → AntitoneOn g (Set.Icc (f n) (f 0)) → ∑ j, (f ↑j - f (↑j + 1)) * g (f ↑j) ≤ ∫ (x : ℝ) in f n..f 0, g x","l":1596,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.BinaryFactoredVCClass","k":"structure","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"A binary factorization of a real-valued function class through a Boolean labeling π, such that on every finite sample each function's values factor through the Boolean labels, and whose realized Boolean patterns have VC dimension at most d on every finite sample.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → ℝ) → ℕ → Type (max u v)","l":40,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.BinaryTraceEntropyControl","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a Boolean classifier family and a natural-number bound, binary trace-entropy control holds exactly when either every finite sample has trace-family VC dimension at most $d$, or every sample of size $m$ has at most $(m+1)^d$ realized label patterns.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → Bool) → ℕ → Prop","l":392,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.CoeffBox","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given an index set of coefficients and a real bound, the coefficient box is the set of real coefficient vectors indexed by that set for which every coordinate has absolute value at most $B$.","s":"Type w → ℝ → Type (max 0 w)","l":22,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.EmpiricalFunctionSpace","k":"structure","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"An element of the empirical function space is an index picking out one function of the class; the structure packages the indexed class of functions equipped with the empirical pseudometric induced by a fixed sample.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → (ι → 𝒳 → ℝ) → (Fin n → 𝒳) → Type u","l":143,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.EmpiricalFunctionSpace.coe_apply","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"Coercing an empirical function-space element gives the indexed function it stores.","s":"∀ {𝒳 : Type v} {n : ℕ} {ι : Type u} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} (q : Causalean.Stat.Concentration.EmpiricalFunctionSpace F S), F q.index = F q.index","l":157,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.ClosedBallParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, the closed-ball parameter space is the set of pairs consisting of a point in that real Euclidean space and a nonnegative real radius.","s":"ℕ → Type","l":205,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, the Euclidean point space is the real Euclidean space of that dimension.","s":"ℕ → Type","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2Cover.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Enlarging the positive envelope of a polynomial L² covering certificate preserves the certificate and its polynomial constants.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2Cover F V","l":236,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2Cover.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Pulling a polynomial-cover class back along an arbitrary parameter map preserves a polynomial L² covering certificate when the new parameter type is nonempty. Cover centers are replaced by occupied-class representatives, which costs only a factor two in radius.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialL2Cover (fun k => F (e k)) U","l":99,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2CoverWith.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Enlarging an envelope preserves named entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V A p","l":254,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPolynomialL2CoverWith.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Pullback preserves named entropy witnesses, with the explicit factor-two radius cost used by the ordinary pullback construction.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun k => F (e k)) U (2 * A) p","l":162,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPseudoDimAtMost.compDomain","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Precomposing every function in a real-valued class with a fixed map cannot increase its pseudo-dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {𝒴 : Type w} {F : ι → 𝒳 → ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (g : 𝒴 → 𝒳), Causalean.Stat.Concentration.HasPseudoDimAtMost (fun i y => F i (g y)) d","l":69,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasPseudoDimAtMost.reindex","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Reparameterizing a real-valued class by an arbitrary map cannot increase its pseudo-dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPseudoDimAtMost (fun k => F (e k)) d","l":55,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasUniformPolynomialL2CoverOver.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A parameterwise envelope enlargement preserves uniform named witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {F : (s : S) → ι s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → (∀ (s : S), U s ≤ V s) → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F V","l":269,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasUniformPolynomialL2CoverOver.pullback","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A parameterwise pullback preserves entropy witnesses uniformly over the outer parameter family.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {U : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → (∀ (s : S), Nonempty (κ s)) → ∀ (e : (s : S) → κ s → ι s), Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s k => F s (e s k)) U","l":221,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasVCAtMost.compDomain","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Precomposing the observation argument of a Boolean class cannot increase its finite-trace VC dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {𝒴 : Type w} {π : ι → 𝒳 → Bool} {d : ℕ}, Causalean.Stat.Concentration.HasVCAtMost π d → ∀ (g : 𝒴 → 𝒳), Causalean.Stat.Concentration.HasVCAtMost (fun i y => π i (g y)) d","l":46,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.HasVCAtMost.reindex","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Reparameterizing a Boolean class by an arbitrary map cannot increase its finite-trace VC dimension.","s":"∀ {𝒳 : Type u} {ι : Type v} {κ : Type w} {π : ι → 𝒳 → Bool} {d : ℕ}, Causalean.Stat.Concentration.HasVCAtMost π d → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasVCAtMost (fun k => π (e k)) d","l":33,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension and a maximal natural-number degree, the radial-monomial parameter space consists of a center in that Euclidean space paired with a degree from zero through $p$.","s":"ℕ → ℕ → Type","l":284,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialPolynomialParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a maximal degree, and a coefficient bound, the radial-polynomial parameter space consists of a Euclidean center paired with a coefficient vector whose every coordinate has absolute value at most $B$.","s":"ℕ → ℕ → ℝ → Type","l":33,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialResidualScoreParam","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a maximal degree, a coefficient bound, and an arm index set, the radial residual-score parameter space consists of a radial-polynomial parameter paired with one arm index.","s":"ℕ → ℕ → ℝ → Type v → Type (max 0 v)","l":151,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, the point-radius space is the set of pairs consisting of a point in the corresponding real Euclidean space and an arbitrary real radius threshold.","s":"ℕ → Type","l":28,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.abs_boundedRadialPolynomial_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every boxed shared-center radial polynomial is bounded by the number of basis functions times the coefficient bound times the monomial envelope.","s":"∀ (d p : ℕ) {q a b B : ℝ}, 0 < q → 0 ≤ a → a ≤ b → 0 ≤ B → ∀ (x : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (β : Causalean.Stat.Concentration.CoeffBox (Fin (p + 1)) B) (z : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), |Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomial d p q a b B x β z| ≤ Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope b p B","l":51,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.abs_radialAnnulusMonomial_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every degree k ≤ p radial monomial on an ordered nonnegative annulus is bounded in absolute value by radialMonomialEnvelope b p.","s":"∀ (d p k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → k ≤ p → ∀ (x z : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), |Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x z| ≤ Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p","l":46,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombinationVCBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a number of Boolean component classes and a common VC-dimension bound, the Boolean-combination VC bound is the explicit natural number $2^{m(d+1)+1}$.","s":"ℕ → ℕ → ℕ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombination_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"VC bound for a point-dependent Boolean combination of classes. Given m independently parameterized Boolean classifier families, each of VC dimension at most d, applying to them any combining rule that may itself depend on the sampled point still yields a Boolean class of VC dimension at most booleanCombinationVCBound m d.","s":"∀ {𝒳 : Type u} {m d : ℕ} {ι : Fin m → Type v} (π : (j : Fin m) → ι j → 𝒳 → Bool), (∀ (j : Fin m), Causalean.Stat.Concentration.HasVCAtMost (π j) d) → ∀ (combine : 𝒳 → (Fin m → Bool) → Bool), Causalean.Stat.Concentration.HasVCAtMost (fun θ x => combine x fun j => π j (θ j) x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.booleanCombinationVCBound m d)","l":423,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomial","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a maximal degree, a bandwidth and two relative annulus endpoints, a coefficient bound, a center, a boxed coefficient vector, and an evaluation point, the bounded radial polynomial is the sum, over degrees zero through $p$, of each coefficient times the corresponding radial annulus monomial about the shared center.","s":"(d p : ℕ) → ℝ → ℝ → ℝ → (B : ℝ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.CoeffBox (Fin (p + 1)) B → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given an observation space equipped with a σ-algebra, a natural dimension, a maximal degree, a Euclidean location map, a bandwidth and two relative annulus endpoints, a coefficient bound, a radial-polynomial parameter, and an observation, the observation-indexed bounded radial polynomial is the bounded radial polynomial evaluated at the Euclidean location of that observation.","s":"{Ω : Type u} → [MeasurableSpace Ω] → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → ℝ → ℝ → ℝ → (B : ℝ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialPolynomialParam d p B → Ω → ℝ","l":40,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Boxed finite radial polynomials with one moving center shared by all degrees have a uniform polynomial L² covering certificate.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b B : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → 0 < B → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.boundedRadialPolynomialOn d p loc q a b B) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope b p B)","l":92,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBallClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a closed-ball parameter, and a point in the corresponding Euclidean space, the closed-ball classifier returns true exactly when the point lies in the closed ball specified by that parameter.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.ClosedBallParam d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Bool","l":211,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBall_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Closed balls in d-dimensional Euclidean space have VC dimension at most d + 2. This neutral Causalean-only statement replaces the paper-local planar ball lemma with a finite-dimensional bound.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.euclideanClosedBallClassifier d) (d + 2)","l":259,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Every finite family of measurable functions bounded by a positive common envelope has a uniform polynomial L² covering certificate.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Fintype ι] (F : ι → 𝒳 → ℝ) {U : ℝ}, 0 < U → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":282,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteClass_hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A finite bounded measurable class has named entropy witnesses depending only on its cardinality.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [inst_1 : Fintype ι] (F : ι → 𝒳 → ℝ) {U : ℝ}, 0 < U → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U (max 1 ↑(Fintype.card ι)) 1","l":312,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteSignedArmRadial_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Multiplying a moving-center radial monomial by any bounded measurable finite family of signed arms preserves a uniform polynomial cover. Boolean arms and the two signs are obtained by taking a finite arm type and values in {0,1} or {-1,1}.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) {q a b : ℝ}, Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ ω => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b θ.1 ω * arm θ.2 ω) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":427,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnionVCBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a number of component classes and a common VC-dimension bound, the finite-union VC bound is the explicit natural number $2^{m(d+1)+1}$.","s":"ℕ → ℕ → ℕ","l":29,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnion_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A finite union of m Boolean classes of VC dimension at most d has finite VC dimension bounded by finiteUnionVCBound m d.","s":"∀ {𝒳 : Type u} {K : Type v} [inst : Fintype K] {ι : K → Type w} {d : ℕ} (π : (k : K) → ι k → 𝒳 → Bool), (∀ (k : K), Causalean.Stat.Concentration.HasVCAtMost (π k) d) → Causalean.Stat.Concentration.HasVCAtMost (fun θ => π θ.fst θ.snd) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.finiteUnionVCBound (Fintype.card K) d)","l":443,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.fixedRadialPseudoDimBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, the fixed-degree radial pseudo-dimension bound is the explicit Boolean-combination bound $2^{3(d+3)+1}$.","s":"ℕ → ℕ","l":296,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a finite feature index set, a real-valued feature family, a real coefficient vector, and an evaluation point, the homogeneous linear-sign classifier returns true exactly when the coefficient-feature inner product at that point is strictly positive.","s":"{𝒳 : Type u} → {K : Type v} → [Fintype K] → (K → 𝒳 → ℝ) → (K → ℝ) → 𝒳 → Bool","l":78,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Trace","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Homogeneous linear threshold classifiers in K real coordinates have VC dimension at most the number of coordinates.","s":"∀ {𝒳 : Type u} {K : Type v} [inst : Fintype K] (φ : K → 𝒳 → ℝ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.linearSignClass φ) (Fintype.card K)","l":83,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Trace.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExteriorClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a center point, and a point together with a radius threshold, the moving-center exterior classifier returns true exactly when the point lies at distance strictly greater than the threshold from the center.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint d → Bool","l":39,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExterior_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Variable-radius strict-exterior tests indexed by a moving Euclidean center have VC dimension at most d + 2.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterBallExteriorClassifier d) (d + 2)","l":199,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBallClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a center point, and a point together with a radius threshold, the moving-center closed-ball classifier returns true exactly when the point lies at distance no greater than the threshold from the center.","s":"(d : ℕ) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadiusPoint d → Bool","l":34,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBall_hasVCAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Variable-radius closed-ball tests indexed by a moving Euclidean center have VC dimension at most d + 2.","s":"∀ (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.movingCenterClosedBallClassifier d) (d + 2)","l":186,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a bandwidth, two relative annulus endpoints, a natural-number degree, a center point, and an evaluation point, the radial annulus monomial equals $(\\operatorname{dist}(z,x)/q)^k$ when the distance from the evaluation point to the center lies between $aq$ and $bq$, inclusive, and equals zero otherwise.","s":"(d : ℕ) → ℝ → ℝ → ℝ → ℕ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":277,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A fixed degree k ≤ p of the moving-center radial class has the same uniform polynomial cover and envelope as the full degree vector.","s":"∀ (d p k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → k ≤ p → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun x => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":359,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"For every positive bandwidth and ordered nonnegative annulus, the class of fixed-degree radial monomials with moving center has pseudo-dimension bounded solely by the Euclidean dimension.","s":"∀ (d k : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPseudoDimAtMost (fun x => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.fixedRadialPseudoDimBound d)","l":337,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial_measurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"A compactly supported Euclidean radial monomial is Borel measurable in the observation for every fixed center and degree.","s":"∀ (d k : ℕ) (q a b : ℝ) (x : Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d), Measurable (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialAnnulusMonomial d q a b k x)","l":32,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension, a maximal degree, a bandwidth and two relative annulus endpoints, a center-and-degree parameter, and a Euclidean evaluation point, the radial-monomial class maps that parameter and point to the corresponding radial annulus monomial.","s":"(d p : ℕ) → ℝ → ℝ → ℝ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam d p → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d → ℝ","l":291,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"Moving-center radial monomials of every degree from zero through p have a polynomial L²(Q) cover with a positive constant envelope, uniformly over every probability measure Q.","s":"∀ (d p : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass d p q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":342,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Pseudo-dimension bound for the moving-center radial-monomial class. For a positive bandwidth q, a nonnegative annulus inner radius a, and inner radius at most outer radius b, allowing both the Euclidean center and the monomial degree — ranging from zero through p — to vary gives the radial-monomial class a pseudo-dimension of at most radialPseudoDimBound d p.","s":"∀ (d p : ℕ) {q a b : ℝ}, 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPseudoDimAtMost (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialClass d p q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound d p)","l":409,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"For an upper relative radius $b$ and a nonnegative integer $p$, the radial-monomial envelope is $(\\max\\{1,b\\})^p$.","s":"ℝ → ℕ → ℝ","l":25,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given an observation space $\\Omega$, a Euclidean dimension $d$, a maximum degree $p$, a location map from observations to $d$-dimensional Euclidean space, a bandwidth $q$, annulus radii $a$ and $b$, a radial-monomial parameter θ, and an observation $\\omega$, the radial monomial evaluated at that observation is the corresponding radial-monomial class function evaluated at the mapped location.","s":"{Ω : Type u} → [MeasurableSpace Ω] → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → ℝ → ℝ → ℝ → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialMonomialParam d p → Ω → ℝ","l":372,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Covering certificate transported through a location map. For a measurable map from the underlying observation space into d-dimensional Euclidean space, given a positive bandwidth q, a nonnegative annulus inner radius a, and inner radius at most outer radius b, composing the moving-center radial-monomial class of degree at most p with the location map still carries a uniform polynomial L² covering certificate at envelope radialMonomialEnvelope b p.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p)","l":384,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn_hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Cover","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"The moving-center radial class after a measurable location map has the canonical named VC-subgraph entropy witnesses.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) {q a b : ℝ}, Measurable loc → 0 < q → 0 ≤ a → a ≤ b → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialOn d p loc q a b) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialMonomialEnvelope b p) 16 (8 * (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound d p + 1))","l":407,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Cover.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPolynomialEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given an outer relative radius, a maximal degree, and a coefficient bound, the radial-polynomial envelope is $(p+1)B$ times the radial-monomial envelope at that outer radius and degree.","s":"ℝ → ℕ → ℝ → ℝ","l":47,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialPseudoDimBound","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Geometry","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given a natural dimension and a maximal degree, the radial pseudo-dimension bound is the explicit finite-union bound $2^{(p+1)(2^{3(d+3)+1}+1)+1}$ obtained by combining $p+1$ fixed-degree bounds.","s":"ℕ → ℕ → ℕ","l":300,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Geometry.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given an observation space equipped with a σ-algebra, an arm index set, a natural dimension, a maximal degree, a Euclidean location map, an arm-valued function, a response function, a bandwidth and two relative annulus endpoints, a coefficient bound, a degree index, a residual-score parameter, and an observation, the radial residual score is the product of the arm value, the selected radial annulus monomial, and the residual obtained by subtracting the shared-center boxed radial-polynomial fit from the response.","s":"{Ω : Type u} → [MeasurableSpace Ω] → {A : Type v} → (d p : ℕ) → (Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) → (A → Ω → ℝ) → (Ω → ℝ) → ℝ → ℝ → ℝ → (B : ℝ) → Fin (p + 1) → Causalean.Stat.Concentration.EuclideanRadialPolynomial.RadialResidualScoreParam d p B A → Ω → ℝ","l":159,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope","k":"def","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Given an outer relative radius, a maximal degree, a coefficient bound, and a response bound, the radial residual-score envelope is the radial-monomial envelope times the sum of $R$ and the radial-polynomial envelope.","s":"ℝ → ℕ → ℝ → ℝ → ℝ","l":171,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":1,"d":"Covering certificate for a bounded finite-arm residual score. Given a measurable Euclidean location map, a finite family of arms, each measurable and bounded in absolute value by 1, a measurable response variable that is bounded in absolute value by R, together with a positive bandwidth q, a nonnegative annulus inner radius a, inner radius at most outer radius b, a positive polynomial-coefficient bound B, and a positive response bound R, then the score formed by multiplying a bounded arm function, a radial monomial, and the residual of the response against a boxed radial-polynomial fit carries a positive-envelope uniform polynomial L²(Q) covering certificate — with envelope radialResidualScoreEnvelope b p B R — over every probability measure Q, including atomic ones.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] (d p : ℕ) (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) (response : Ω → ℝ) {q a b B R : ℝ} (j : Fin (p + 1)), Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → Measurable response → (∀ (ω : Ω), |response ω| ≤ R) → 0 < q → 0 ≤ a → a ≤ b → 0 < B → 0 < R → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore d p loc arm response q a b B j) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope b p B R)","l":188,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore_hasUniformPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EuclideanRadialPolynomial.Score","a":"Stat/Concentration/Covering/EuclideanRadialPolynomial","t":2,"d":"The radial residual-score construction admits entropy witnesses depending only on the Euclidean dimension, polynomial degree, and finite arm type. In particular, the witnesses precede all location maps, arm functions, responses, bandwidths, annuli, and envelope radii.","s":"∀ {Ω : Type u} [inst : MeasurableSpace Ω] {A : Type v} [Fintype A] [Nonempty A] (d p : ℕ), ∃ C n, ∀ (loc : Ω → Causalean.Stat.Concentration.EuclideanRadialPolynomial.EuclideanPoint d) (arm : A → Ω → ℝ) (response : Ω → ℝ) {q a b B R : ℝ}, Measurable loc → (∀ (s : A), Measurable (arm s)) → (∀ (s : A) (ω : Ω), |arm s ω| ≤ 1) → Measurable response → (∀ (ω : Ω), |response ω| ≤ R) → 0 < q → 0 ≤ a → a ≤ b → 0 < B → 0 < R → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun θ => Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScore d p loc arm response q a b B θ.2 θ.1) (Causalean.Stat.Concentration.EuclideanRadialPolynomial.radialResidualScoreEnvelope b p B R) C n","l":295,"f":"Causalean/Stat/Concentration/Covering/EuclideanRadialPolynomial/Score.lean"},{"n":"Causalean.Stat.Concentration.F_on","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an observation space and class-index set, a function class $F$ and a finite set of its indices, the restricted function class is indexed by precisely those indices in the finite set and assigns each retained index its original function.","s":"{Z : Type v} → {ι : Type u} → (ι → Z → ℝ) → (f : Finset ι) → ↥f → Z → ℝ","l":190,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.HasCountableEmpiricalSupReduction","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given a measure $\\mu$ on an observation space and a family of real-valued functions indexed by a set $\\iota$, the countable empirical-supremum reduction property holds exactly when every member of the family is measurable and there is a sequence of indices whose associated countable subfamily has, for every sample size, the same supremum of absolute centered empirical averages as the full family almost surely under the corresponding product measure.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → Prop","l":33,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given a real-valued function class on an observation space, an envelope $U$, an entropy base $A$, and an exponent $v$, the polynomial empirical $L^2$ covering property holds exactly when, for every positive sample size and every sample of that size, and for every radius $\\varepsilon$ with $0<\\varepsilon\\le 1$, there is a finite index set forming an $L^2$ cover under the empirical measure at radius $\\varepsilon U$ and its cardinality is at most $(A/\\varepsilon)^v$.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → ℝ → ℝ → Prop","l":23,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EmpiricalCover","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Covering-number bound from an empirical cover. Suppose F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, every member of F is measurable, the envelope U is strictly positive, the sample S has positive size n, and the relative scale ε lies in the interval $(0,1]$ ; then Dudley's empirical covering number of the class at radius ε * U, taken in the totally bounded empirical pseudometric space this cover furnishes, is at most the real power (A / ε) ^ v.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A v : ℝ} (hcover : Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v) (hmeas : ∀ (i : ι), Measurable (F i)) (hU : 0 < U) {n : ℕ} (S : Fin n → 𝒳) (hn : 0 < n) (ε : ℝ), 0 < ε → ε ≤ 1 → have htot := Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.totallyBounded hcover hmeas hU S hn; ↑(Causalean.Stat.Concentration.coveringNumber htot (ε * U)) ≤ (A / ε).rpow v","l":62,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EmpiricalCover.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.enlargeEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Enlarging the envelope of an empirical polynomial cover preserves its entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → U ≤ V → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F V A v","l":37,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.monoEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Enlarging the envelope in a polynomial empirical-cover certificate keeps the same centers, entropy base, and exponent.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → U ≤ V → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F V A v","l":119,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.pullback","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Restricting a measurable polynomial empirical-cover class along a nonempty parameter map preserves its exponent and costs only a factor two in the entropy base. The replacement centers are chosen inside the restricted class.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} [Nonempty κ] {F : ι → 𝒳 → ℝ} {U A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → (∀ (i : ι), Measurable (F i)) → ∀ (e : κ → ι), Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover (fun k => F (e k)) U (2 * A) v","l":51,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover.totallyBounded","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EmpiricalCover","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"A positive-envelope polynomial empirical covering certificate makes the class image totally bounded in the empirical root-mean-square pseudometric on every nonempty finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A v : ℝ}, Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → (∀ (i : ι), Measurable (F i)) → 0 < U → ∀ {n : ℕ} (S : Fin n → 𝒳), 0 < n → TotallyBounded Set.univ","l":21,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EmpiricalCover.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover","k":"structure","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A class of functions has uniform polynomial L² entropy with envelope U when U is positive, every function in the class is measurable and pointwise bounded in absolute value by U, and the class admits one polynomial covering-number bound, in the relative radius, holding simultaneously for every probability measure and every relative radius in (0,1] .","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → Prop","l":192,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Pointwise addition of two independently indexed polynomial-entropy classes has uniform polynomial L² entropy with summed envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover G V → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x + G p.2 x) (U + V)","l":270,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.finProd","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A fixed finite product of independently indexed polynomial-entropy classes again has uniform polynomial L² entropy, with the product envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : K → Type v} {F : (k : K) → ι k → 𝒳 → ℝ} {U : K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasPolynomialL2Cover (F k) (U k)) → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ x => ∏ k, F k (θ k) x) (∏ k, U k)","l":413,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.finSum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A fixed finite sum of independently indexed polynomial-entropy classes again has uniform polynomial L² entropy, with the sum of the envelopes.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : K → Type v} {F : (k : K) → ι k → 𝒳 → ℝ} {U : K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasPolynomialL2Cover (F k) (U k)) → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun θ x => ∑ k, F k (θ k) x) (∑ k, U k)","l":355,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.hasPolynomialEmpiricalL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Empirical covering from population covering. If a family F of real-valued functions with envelope U admits a uniform polynomial $L^2$ covering-number bound over every probability measure, then there exist a base A at least Euler's number and an exponent v at least one such that F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → ∃ A v, Real.exp 1 ≤ A ∧ 1 ≤ v ∧ Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v","l":161,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Pointwise multiplication of two independently indexed bounded polynomial-entropy classes has uniform polynomial L² entropy with product envelope.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover G V → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x * G p.2 x) (U * V)","l":311,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.mulIndicator","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Multiplying a polynomial-entropy real class by a measurable finite-VC family of indicators preserves uniform polynomial L² entropy.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {π : κ → 𝒳 → Bool} {U : ℝ} {d : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → (∀ (j : κ), MeasurableSet {x | π j x = true}) → Causalean.Stat.Concentration.HasVCAtMost π d → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun p x => F p.1 x * if π p.2 x = true then 1 else 0) U","l":247,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Negating every member of a polynomial-entropy class preserves its envelope and uniform polynomial L² entropy.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → Causalean.Stat.Concentration.HasPolynomialL2Cover (fun i x => -F i x) U","l":242,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2Cover.varianceAdaptiveExpectedMaximal_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.ExpectedMaximal","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"An existing HasPolynomialL2Cover certificate yields some explicit polynomial constants for which the countable class satisfies the variance-adaptive expected maximal inequality at every positive sample size.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ : ℝ}, Causalean.Stat.Concentration.HasPolynomialL2Cover F U → 0 < σ → σ < U → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → ∃ A v, Real.exp 1 ≤ A ∧ 1 ≤ v ∧ ∀ (n : ℕ), 0 < n → (∫ (S : Fin n → 𝒳), Causalean.Stat.Concentration.countableEmpiricalSup P F S ∂MeasureTheory.Measure.pi fun x => P) ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":73,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/ExpectedMaximal.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith","k":"structure","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"A polynomial L² covering certificate with named entropy witnesses. Unlike HasPolynomialL2Cover, this form retains the particular base and integer exponent, so a family of constructions can share witnesses before its observation-specific parameters are introduced.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → {ι : Type v} → (ι → 𝒳 → ℝ) → ℝ → ℝ → ℕ → Prop","l":471,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Addition combines named polynomial-cover witnesses by the same explicit formula as the ordinary cover algebra.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V A B : ℝ} {p q : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith G V B q → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun z x => F z.1 x + G z.2 x) (U + V) (8 * A * B) (p + q + 2)","l":566,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.enlargeEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Enlarging a named envelope preserves its entropy witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U V A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → U ≤ V → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V A p","l":491,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.forget","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Forgetting the named witnesses gives the ordinary polynomial-cover certificate.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":483,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.hasPolynomialEmpiricalL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"Named arbitrary-law entropy witnesses give correspondingly named empirical witnesses after the canonical Euler-base and positive-exponent normalization.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A₀ : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A₀ p → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U (max (Real.exp 1) (2 * A₀)) ↑(p + 1)","l":220,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Multiplication combines named polynomial-cover witnesses by the same explicit formula as the ordinary cover algebra.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {κ : Type w} {F : ι → 𝒳 → ℝ} {G : κ → 𝒳 → ℝ} {U V A B : ℝ} {p q : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith G V B q → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun z x => F z.1 x * G z.2 x) (U * V) (8 * A * B) (p + q + 2)","l":609,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Negation preserves named polynomial-cover witnesses.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {U A : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U A p → Causalean.Stat.Concentration.HasPolynomialL2CoverWith (fun i x => -F i x) U A p","l":541,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPolynomialL2CoverWith.tightenEnvelopeBy","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"If an assembled class has envelope c * V but a direct pointwise bound by V, shrinking the envelope by the fixed factor c costs the same factor in the named entropy base.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} {F : ι → 𝒳 → ℝ} {V A c : ℝ} {p : ℕ}, Causalean.Stat.Concentration.HasPolynomialL2CoverWith F (c * V) A p → 1 ≤ c → 0 < V → (∀ (i : ι) (x : 𝒳), |F i x| ≤ V) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F V (c * A) p","l":655,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a real-valued function class on an observation space and a nonnegative integer $d$, the pseudo-dimension-at-most-$d$ property holds exactly when, for every finite collection of observation-threshold pairs, the VC dimension of the strict-subgraph labelings induced by the class on that collection is at most $d$.","s":"{𝒳 : Type u} → {ι : Type v} → (ι → 𝒳 → ℝ) → ℕ → Prop","l":28,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost.hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Bounded finite-pseudo-dimension classes admit a polynomial L² cover. If a real-valued function class has pseudo-dimension at most d, every member is measurable, and the class is uniformly bounded by a positive envelope U, then the class carries a uniform polynomial L² covering certificate at envelope U.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d : ℕ} {U : ℝ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → (∀ (i : ι), Measurable (F i)) → 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2Cover F U","l":226,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasPseudoDimAtMost.hasPolynomialL2CoverWith","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A bounded measurable pseudo-dimension class has the canonical named entropy witnesses used by the VC-subgraph covering theorem.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d : ℕ} {U : ℝ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → (∀ (i : ι), Measurable (F i)) → 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → Causalean.Stat.Concentration.HasPolynomialL2CoverWith F U 16 (8 * (d + 1))","l":511,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF","k":"structure","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":1,"d":"A random variable X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ if, for every t with b · |t| < 1, exp (t * X) is μ-integrable and the moment-generating function obeys mgf X μ t ≤ exp (v t² / 2).","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → (Ω → ℝ) → NNReal → NNReal → autoParam (MeasureTheory.Measure Ω) Causalean.Stat.Concentration.HasSubexponentialMGF._auto_1 → Prop","l":46,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.add_of_indepFun","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Independent sub-exponential variables add: X + Y has parameters (vX + vY, max bX bY).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X Y : Ω → ℝ} {vX bX vY bY : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X vX bX μ → Causalean.Stat.Concentration.HasSubexponentialMGF Y vY bY μ → ProbabilityTheory.IndepFun X Y μ → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => X ω + Y ω) (vX + vY) (max bX bY) μ","l":114,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.congr","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality transfers along an a.e.-equality.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {Y : Ω → ℝ}, X =ᵐ[μ] Y → Causalean.Stat.Concentration.HasSubexponentialMGF Y v b μ","l":84,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.const_mul","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality is preserved under scaling: r • X has parameters (r² v, |r| b).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ (r : ℝ), Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => r * X ω) (⟨r ^ 2, sq_nonneg r⟩ * v) (⟨|r|, abs_nonneg r⟩ * b) μ","l":94,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.isFiniteMeasure","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"A sub-exponential random variable lives on a finite measure (integrability of exp (0 · X) = 1).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → MeasureTheory.IsFiniteMeasure μ","l":62,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.measure_abs_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Chernoff bound. If the random variable X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ and ε is nonnegative, then the probability that |X| is at least ε is at most $2\\exp(-ε^2/(2(v+bε)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |X ω|} ≤ 2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε)))","l":44,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.measure_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":1,"d":"Chernoff bound for the right tail of a sub-exponential random variable. If X has a sub-exponential moment-generating function with parameters (v, b) with respect to μ and ε is nonnegative, then the probability that X is at least ε is at most $\\exp(-ε^2/(2(v+bε)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ X ω} ≤ Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε)))","l":154,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.neg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"Sub-exponentiality is preserved under negation (same parameters).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {v b : NNReal}, Causalean.Stat.Concentration.HasSubexponentialMGF X v b μ → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => -X ω) v b μ","l":69,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.sum_range_of_iIndepFun","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"A sum of n independent sub-exponential variables sharing common parameters (v, b) is sub-exponential with parameters (n • v, b) (the b-parameters coincide, so they do not grow).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Z : ℕ → Ω → ℝ}, ProbabilityTheory.iIndepFun Z μ → (∀ (i : ℕ), AEMeasurable (Z i) μ) → ∀ {v b : NNReal} {n : ℕ}, (∀ i < n, Causalean.Stat.Concentration.HasSubexponentialMGF (Z i) v b μ) → Causalean.Stat.Concentration.HasSubexponentialMGF (fun ω => ∑ i ∈ Finset.range n, Z i ω) (n • v) b μ","l":202,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasSubexponentialMGF.zero","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.SubExponential","a":"Stat/Concentration/TailBounds","t":2,"d":"The constant 0 is sub-exponential with parameters (0, b) (any b).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {b : NNReal} [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.Concentration.HasSubexponentialMGF (fun x => 0) 0 b μ","l":148,"f":"Causalean/Stat/Concentration/TailBounds/SubExponential.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given an observation space, an auxiliary parameter set $S$, an index set depending on the parameter, a parameter-indexed family of real-valued function classes, and a parameter-indexed envelope, the uniform polynomial $L^2$ covering property over $S$ holds exactly when there are one real entropy base and one nonnegative-integer exponent such that every parameter's class has the corresponding polynomial $L^2$ covering certificate with its designated envelope.","s":"{𝒳 : Type u} → [MeasurableSpace 𝒳] → (S : Type u_1) → {ι : S → Type v} → ((s : S) → ι s → 𝒳 → ℝ) → (S → ℝ) → Prop","l":686,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.add","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise addition.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {G : (s : S) → κ s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S G V → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s z x => F s z.1 x + G s z.2 x) fun s => U s + V s","l":707,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.finSum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A nonempty finite sum preserves entropy witnesses uniformly over all auxiliary parameters.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {K : Type w} [inst_1 : Fintype K] [Nonempty K] {ι : S → K → Type v} {F : (s : S) → (k : K) → ι s k → 𝒳 → ℝ} {U : S → K → ℝ}, (∀ (k : K), Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s => F s k) fun s => U s k) → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s θ x => ∑ k, F s k (θ k) x) fun s => ∑ k, U s k","l":735,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise multiplication.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {κ : S → Type w} {F : (s : S) → ι s → 𝒳 → ℝ} {G : (s : S) → κ s → 𝒳 → ℝ} {U V : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S G V → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s z x => F s z.1 x * G s z.2 x) fun s => U s * V s","l":721,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver.neg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Algebra","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Uniform named covers are stable under pointwise negation.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {S : Type u_1} {ι : S → Type v} {F : (s : S) → ι s → 𝒳 → ℝ} {U : S → ℝ}, Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S F U → Causalean.Stat.Concentration.HasUniformPolynomialL2CoverOver S (fun s i x => -F s i x) U","l":697,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Algebra.lean"},{"n":"Causalean.Stat.Concentration.HasVCAtMost","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a Boolean classifier family and a natural number, the property of having VC dimension at most that number means that, for every finite sample, the VC dimension of the label patterns realized on that sample is at most $d$.","s":"{𝒳 : Type u} → {κ : Type w} → (κ → 𝒳 → Bool) → ℕ → Prop","l":167,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasVCAtMost.indicatorClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A Boolean VC class, viewed as a real-valued zero-one indicator class, has the same pseudo-dimension bound.","s":"∀ {𝒳 : Type u} [MeasurableSpace 𝒳] {κ : Type w} (π : κ → 𝒳 → Bool) (d : ℕ), Causalean.Stat.Concentration.HasVCAtMost π d → Causalean.Stat.Concentration.HasPseudoDimAtMost (fun j x => if π j x = true then 1 else 0) d","l":171,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.HasVCUniformEntropy","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"For a measure on a measurable sample space, a class of real-valued functions indexed by a set, an envelope bound, a radius, a covering constant, and an entropy exponent, the class has uniform VC-type entropy exactly when (1) the radius is positive, (2) the radius is strictly smaller than the envelope bound, (3) the covering constant is at least $e$, (4) the entropy exponent is at least one, (5) every function in the class is measurable, (6) every function is bounded in absolute value by the envelope bound at every sample point, (7) every function has population $L^2$ distance at most the radius from the zero function, and (8) for every sequence of indices, the resulting sequence of functions has the stipulated polynomial empirical $L^2$ covering property.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → ℝ → ℝ → ℝ → ℝ → Prop","l":23,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean","k":"def","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"The centered empirical mean for a measure (a probability law in intended use), a Hilbert-valued feature map, and a sample with the stated size and observations is given by its empirical mean minus its population mean.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → MeasureTheory.Measure X → (X → H) → (m : ℕ) → (Fin m → X) → H","l":43,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMeanNorm","k":"def","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"The norm statistic for a centered empirical mean under a measure (a probability law in intended use), a Hilbert-valued feature map, and a sample with the stated size and observations is given by the Hilbert norm of that centered mean.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → MeasureTheory.Measure X → (X → H) → (m : ℕ) → (Fin m → X) → ℝ","l":51,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMeanNorm_boundedDifference","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a Hilbert-valued feature map with unit-norm bound at every observation, a positive sample size, a coordinate, a sample, and a replacement observation, changing that coordinate changes the centered-mean norm by at most two divided by the sample size.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) (f : X → H), (∀ (x : X), ‖f x‖ ≤ 1) → ∀ {m : ℕ}, 0 < m → ∀ (i : Fin m) (z : Fin m → X) (x' : X), |Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMeanNorm P f m z - Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMeanNorm P f m (Function.update z i x')| ≤ 2 / ↑m","l":108,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_eq_inv_smul_sum_centered","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a Hilbert-valued feature map, a positive sample size, and a sample, the centered empirical mean equals the inverse sample size times the sum of centered observations.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H) {m : ℕ}, 0 < m → ∀ (z : Fin m → X), Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean P f m z = (↑m)⁻¹ • ∑ r, (f (z r) - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f)","l":75,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_norm_highProbability","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Tail","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"For a probability law, a strongly measurable Hilbert-valued feature map with unit-norm bound at every observation, a nonempty sample size, and a confidence level strictly between zero and one, the centered empirical mean lies within the standard dimension-free radius with probability at least one minus the confidence level.","s":"∀ {X : Type u_2} {H : Type u_3} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → (∀ (x : X), ‖f x‖ ≤ 1) → ∀ {m : ℕ}, 1 ≤ m → ∀ {δ : ℝ}, 0 < δ → δ < 1 → 1 - ENNReal.ofReal δ ≤ (MeasureTheory.Measure.pi fun x => P) {z | ‖(↑m)⁻¹ • ∑ r, f (z r) - ∫ (x : X), f x ∂P‖ ≤ 1 / √↑m + √(2 * Real.log (1 / δ) / ↑m)}","l":193,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Tail.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_norm_integral_le_inv_sqrt","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Tail","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable Hilbert-valued feature map with unit-norm bound at every observation, and a nonempty sample size, the expected norm of the centered empirical mean is at most one divided by the square root of the sample size.","s":"∀ {X : Type u_2} {H : Type u_3} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → (∀ (x : X), ‖f x‖ ≤ 1) → ∀ {m : ℕ}, 1 ≤ m → (∫ (z : Fin m → X), ‖(↑m)⁻¹ • ∑ r, f (z r) - ∫ (x : X), f x ∂P‖ ∂MeasureTheory.Measure.pi fun x => P) ≤ 1 / √↑m","l":44,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Tail.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_norm_tail_le","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Tail","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable Hilbert-valued feature map with unit-norm bound at every observation, a nonempty sample size, and a confidence level strictly between zero and one, the probability that the centered empirical mean exceeds the standard dimension-free radius is at most the confidence level.","s":"∀ {X : Type u_2} {H : Type u_3} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → (∀ (x : X), ‖f x‖ ≤ 1) → ∀ {m : ℕ}, 1 ≤ m → ∀ {δ : ℝ}, 0 < δ → δ < 1 → ((MeasureTheory.Measure.pi fun x => P) {z | 1 / √↑m + √(2 * Real.log (1 / δ) / ↑m) < ‖(↑m)⁻¹ • ∑ r, f (z r) - ∫ (x : X), f x ∂P‖}).toReal ≤ δ","l":112,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Tail.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_secondMoment_eq","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable square-integrable Hilbert-valued feature map, and a positive sample size, the centered empirical mean has second moment equal to the inverse sample size times the population centered second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∀ {m : ℕ}, 0 < m → (∫ (z : Fin m → X), ‖Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean P f m z‖ ^ 2 ∂MeasureTheory.Measure.pi fun x => P) = (↑m)⁻¹ * ∫ (x : X), ‖f x - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f‖ ^ 2 ∂P","l":314,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean_secondMoment_le","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"For a probability law, a strongly measurable square-integrable Hilbert-valued feature map, and a positive sample size, the centered empirical mean has second moment at most the inverse sample size times the raw population second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∀ {m : ℕ}, 0 < m → (∫ (z : Fin m → X), ‖Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMean P f m z‖ ^ 2 ∂MeasureTheory.Measure.pi fun x => P) ≤ (↑m)⁻¹ * ∫ (x : X), ‖f x‖ ^ 2 ∂P","l":337,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredSum_secondMoment_eq","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable square-integrable Hilbert-valued feature map, and a sample size, the second moment of the unscaled centered sum equals the sample size times the population centered second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∀ (m : ℕ), (∫ (z : Fin m → X), ‖∑ r, (f (z r) - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f)‖ ^ 2 ∂MeasureTheory.Measure.pi fun x => P) = ↑m * ∫ (x : X), ‖f x - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f‖ ^ 2 ∂P","l":258,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.empiricalMean","k":"def","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"The empirical mean of a Hilbert-valued feature map on a sample with the stated size and observations is given by the inverse sample size times its finite coordinate sum. For sample size zero the inverse is zero by convention, so the value is zero.","s":"{X : Type u_1} → {H : Type u_2} → [inst : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → (X → H) → (m : ℕ) → (Fin m → X) → H","l":36,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.integrable_of_norm_le_one","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"A strongly measurable Hilbert-valued feature map whose norm is at most one at every observation under a probability law is Bochner integrable.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → (∀ (x : X), ‖f x‖ ≤ 1) → MeasureTheory.Integrable f P","l":59,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.integral_inner_centered_coordinates_eq_zero","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable square-integrable Hilbert-valued feature map, and two distinct product coordinates, the expected inner product of their centered values is zero.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∀ {m : ℕ} {i j : Fin m}, i ≠ j → (∫ (z : Fin m → X), inner ℝ (f (z i) - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f) (f (z j) - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f) ∂MeasureTheory.Measure.pi fun x => P) = 0","l":185,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.integral_norm_centered_coordinate_sq_eq","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable square-integrable Hilbert-valued feature map, and a product coordinate, the coordinate's centered second moment equals the population centered second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∀ {m : ℕ} (i : Fin m), (∫ (z : Fin m → X), ‖f (z i) - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f‖ ^ 2 ∂MeasureTheory.Measure.pi fun x => P) = ∫ (x : X), ‖f x - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f‖ ^ 2 ∂P","l":166,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.integral_norm_le_sqrt_integral_norm_sq","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Tail","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"Under a probability law, a strongly measurable Hilbert-valued random variable with integrable squared norm has expected norm at most the square root of its expected squared norm.","s":"∀ {Ω : Type u_1} {H : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → H), MeasureTheory.StronglyMeasurable g → MeasureTheory.Integrable (fun ω => ‖g ω‖ ^ 2) μ → ∫ (ω : Ω), ‖g ω‖ ∂μ ≤ √(∫ (ω : Ω), ‖g ω‖ ^ 2 ∂μ)","l":30,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Tail.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.integral_sub_populationMean_eq_zero","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law and an integrable Hilbert-valued feature map, the integral of the map centered at its population mean is zero.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.Integrable f P → ∫ (x : X), f x - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f ∂P = 0","l":157,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.measurable_centeredEmpiricalMeanNorm","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law, a strongly measurable Hilbert-valued feature map, and a sample size, the norm of its centered empirical mean is measurable on the finite product sample space.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) (f : X → H), MeasureTheory.StronglyMeasurable f → ∀ (m : ℕ), Measurable (Causalean.Stat.Concentration.HilbertEmpiricalMean.centeredEmpiricalMeanNorm P f m)","l":92,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.memLp_two_of_norm_le_one","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"A strongly measurable Hilbert-valued feature map whose norm is at most one at every observation under a probability law belongs to the square-integrable class.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → (∀ (x : X), ‖f x‖ ≤ 1) → MeasureTheory.MemLp f 2 P","l":67,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.norm_sum_sq_eq_sum_inner","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a finite family of Hilbert vectors, the squared norm of its sum is the double sum of its pairwise inner products.","s":"∀ {H : Type u_2} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_3 : MeasurableSpace H] [BorelSpace H] {ι : Type u_3} [inst_5 : Fintype ι] (v : ι → H), ‖∑ i, v i‖ ^ 2 = ∑ i, ∑ j, inner ℝ (v i) (v j)","l":149,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean","k":"def","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":1,"d":"The population mean of a Hilbert-valued feature map under a measure (a probability law in intended use; none is required) is given by its Bochner integral, which is zero when the feature map is not integrable.","s":"{X : Type u_1} → {H : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup H] → [InnerProductSpace ℝ H] → MeasureTheory.Measure X → (X → H) → H","l":30,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.HilbertEmpiricalMean.population_centered_secondMoment_le","k":"theorem","m":"Causalean.Stat.Concentration.HilbertEmpiricalMean.Basic","a":"Stat/Concentration/HilbertEmpiricalMean","t":2,"d":"For a probability law and a strongly measurable square-integrable Hilbert-valued feature map, centering does not increase its population second moment.","s":"∀ {X : Type u_1} {H : Type u_2} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace ℝ H] [CompleteSpace H] [inst_4 : MeasurableSpace H] [BorelSpace H] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (f : X → H), MeasureTheory.StronglyMeasurable f → MeasureTheory.MemLp f 2 P → ∫ (x : X), ‖f x - Causalean.Stat.Concentration.HilbertEmpiricalMean.populationMean P f‖ ^ 2 ∂P ≤ ∫ (x : X), ‖f x‖ ^ 2 ∂P","l":226,"f":"Causalean/Stat/Concentration/HilbertEmpiricalMean/Basic.lean"},{"n":"Causalean.Stat.Concentration.IsL2Cover","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a measure $Q$ on an observation space, a real-valued function class, a radius $r$, and a finite set of class indices, the open $L^2(Q)$ cover property holds exactly when every class member lies at strictly less than distance $r$ from a member indexed by that finite set.","s":"{𝒳 : Type u} → {ι : Type v} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → ℝ → Finset ι → Prop","l":49,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.L2CoveringNumberLe","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a measure $Q$ on an observation space, a real-valued function class, a radius $r$, and a nonnegative integer $N$, the $L^2(Q)$ covering-number-at-most-$N$ property holds exactly when there is a finite set of at most $N$ class indices that forms an open $L^2(Q)$ cover at radius $r$.","s":"{𝒳 : Type u} → {ι : Type v} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → ℝ → ℕ → Prop","l":56,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.LipschitzAt0","k":"def","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"For a real-valued transformation and a real constant, the transformation is Lipschitz at zero with constant $L$ exactly when (1) it maps zero to zero and (2) for every two real numbers $x$ and $y$, its increment has absolute value at most $L|x-y|$.","s":"(ℝ → ℝ) → ℝ → Prop","l":56,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.LocalizedGramGood","k":"def","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":1,"d":"For a real lower-eigenvalue threshold, a real-valued observation-weight function, a finite indexed family of real-valued features, and a finite sample, the localized empirical Gram matrix is good exactly when (1) the local count is positive and (2) for every real coefficient vector indexed by the features, its quadratic form is at least one half of the threshold times the local count times the squared Euclidean norm of that vector.","s":"{N : ℕ} → {κ : Type u_1} → {X : Type u_2} → [Fintype κ] → ℝ → (X → ℝ) → (κ → X → ℝ) → (Fin N → X) → Prop","l":226,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.LocalizedRegime","k":"structure","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"The localized regime packages the assumptions needed for a critical-radius uniform-deviation bound over a bounded loss class: a non-negative uniform bound b such that every loss in the class is bounded in absolute value by b on the sample, together with a sub-root function ψ that upper-bounds the localized Rademacher complexity of the class at every sample size.","s":"(Ω : Type u_1) → (ι : Type u_2) → (𝒳 : Type u_3) → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → Type","l":54,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses","k":"structure","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"The deterministic prerequisites used to run the Dudley entropy-integral step after localization: a samplewise $L^2$ radius bound on the localized star-hull-zeroed class, needed to run Dudley with $c = r$, and a total-boundedness (covering-number) precondition on that same localized class.","s":"{ι : Type u} → {𝒳 : Type v} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → Prop","l":54,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.X","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an observation space and class-index set, a function class $F$, a sample $S$ of $m$ observations, and a class index, the aggregated Rademacher variable maps each sign vector to the sum of its $m$ coordinate increments.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → ι → Signs m → ℝ","l":148,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.Y","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an observation space and class-index set, a function class $F$, a sample $S$ of $m$ observations, a sample coordinate, and a class index, the Rademacher increment is the function that maps each Rademacher sign vector to $m^{-1}$ times the selected sign times the selected function's value at the selected sample coordinate.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → Fin m → ι → Signs m → ℝ","l":139,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.r","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an observation space and class-index set, a function class $F$, a sample $S$ of $m$ observations, a nonempty finite set of class indices, and a sample coordinate, the finite-class coordinate envelope is $m^{-1}$ times the largest absolute function value at that coordinate among the selected indices.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → (f : Finset ι) → f.Nonempty → Fin m → ℝ","l":155,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.r'","k":"def","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an observation space and class-index set, a function class $F$, a sample $S$ of $m$ observations, a sample coordinate, and a class index, the pointwise coordinate radius is $m^{-1}$ times the absolute value of the selected function at that coordinate.","s":"{Z : Type v} → {m : ℕ} → {ι : Type u} → (ι → Z → ℝ) → (Fin m → Z) → Fin m → ι → ℝ","l":163,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MassartNotation.xy_identity","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The aggregate Rademacher variable is exactly the sum of its coordinate increments.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} (F : ι → Z → ℝ) (S : Fin m → Z) (j : ι), Causalean.Stat.Concentration.MassartNotation.X F S j = ∑ i, Causalean.Stat.Concentration.MassartNotation.Y F S i j","l":172,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.MonotoneOn.leftRiemann_sum_le_integral_antitoneOn","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"For a monotone grid and an antitone integrand, a left Riemann term is bounded by the corresponding interval integral.","s":"∀ (n : ℕ) (f : ℕ → ℝ) (g : ℝ → ℝ), Monotone f → AntitoneOn g (Set.Icc (f 0) (f n)) → ∀ (j : Fin n), (f (↑j + 1) - f ↑j) * g (f (↑j + 1)) ≤ ∫ (x : ℝ) in f ↑j..f (↑j + 1), g x","l":1536,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.aggregateScore","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a finite coordinate index and sample space, natural-valued coordinate counts, a multiplier and logarithmic level, nonnegative coordinate means, and a sample point, the aggregate score is the sum of all coordinate deviation-plus-radius scores.","s":"{ι : Type u_1} → {Ω : Type u_2} → [Fintype ι] → (ι → Ω → ℕ) → ℝ → ℝ → (ι → NNReal) → Ω → ℝ","l":26,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.badAny","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a finite coordinate index and sample space, natural-valued coordinate counts, a multiplier and logarithmic level, and nonnegative coordinate means, the aggregate bad event occurs when at least one coordinate lies in its self-normalized bad event.","s":"{ι : Type u_1} → {Ω : Type u_2} → [Fintype ι] → (ι → Ω → ℕ) → ℝ → ℝ → (ι → NNReal) → Set Ω","l":18,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a multiplier and logarithmic level and a nonnegative Poisson mean, the self-normalized bad event contains counts whose absolute deviation exceeds one quarter of the multiplier times their random radius.","s":"ℝ → ℝ → NNReal → Set ℕ","l":36,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent_subset_remote_bernstein","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a Poisson law with a nonnegative mean and a logarithmic level of at least one, the universal self-normalized bad event is contained in the Bernstein event at forty times that level.","s":"∀ (lambda : NNReal) {L : ℝ}, 1 ≤ L → Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda ⊆ {w | Causalean.Stat.Concentration.PoissonSelfNormalized.deviation lambda w > √(2 * ↑lambda * (Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay * L)) + 2 * (Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay * L)}","l":423,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.deviation","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a nonnegative mean and a natural-valued count, the absolute deviation of the count from its mean is their absolute difference.","s":"NNReal → ℕ → ℝ","l":21,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.forty_le_scalarDecay","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"The scalar exponential decay rate is at least forty, and hence is positive.","s":"40 ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay","l":52,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.independent_poisson_badAny_moment","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"On a measurable sample space with a finite coordinate index, let a probability law carry natural-valued coordinate counts with nonnegative coordinate means. If the counts are measurable, each count has its stated Poisson law, and the counts are mutually independent, then under a fixed coordinate-cardinality cap, for moment order one, two, or four and a logarithmic level of at least one, the aggregate score moment on the union of bad coordinates decays exponentially while retaining the local scale determined by the sum of the means.","s":"∀ {ι : Type u_1} {Ω : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (W : ι → Ω → ℕ) (lambda : ι → NNReal), (∀ (i : ι), Measurable (W i)) → (∀ (i : ι), ProbabilityTheory.HasLaw (W i) (ProbabilityTheory.poissonMeasure (lambda i)) μ) → ProbabilityTheory.iIndepFun W μ → ∀ {r t : ℕ}, Fintype.card ι ≤ r → t = 1 ∨ t = 2 ∨ t = 4 → ∀ {L : ℝ}, 1 ≤ L → ∫ (ω : Ω), (Causalean.Stat.Concentration.PoissonSelfNormalized.badAny W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda).indicator (fun ω => Causalean.Stat.Concentration.PoissonSelfNormalized.aggregateScore W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda ω ^ t) ω ∂μ ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant r t * Real.exp (-20 * L) * (√((∑ i, ↑(lambda i)) * L) + L) ^ t","l":306,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.independent_poisson_badAny_moment_four","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"On a measurable sample space, let a probability law carry four natural-valued coordinate counts with nonnegative coordinate means. If the counts are measurable, each count has its stated Poisson law, and the counts are mutually independent, then for moment order one, two, or four and a logarithmic level of at least one, the four-coordinate aggregate score moment on the union of bad coordinates has the corresponding exponentially decaying local scale bound.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ) (lambda : Fin 4 → NNReal), (∀ (i : Fin 4), Measurable (W i)) → (∀ (i : Fin 4), ProbabilityTheory.HasLaw (W i) (ProbabilityTheory.poissonMeasure (lambda i)) μ) → ProbabilityTheory.iIndepFun W μ → ∀ {t : ℕ}, t = 1 ∨ t = 2 ∨ t = 4 → ∀ {L : ℝ}, 1 ≤ L → ∫ (ω : Ω), (Causalean.Stat.Concentration.PoissonSelfNormalized.badAny W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda).indicator (fun ω => Causalean.Stat.Concentration.PoissonSelfNormalized.aggregateScore W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda ω ^ t) ω ∂μ ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant 4 t * Real.exp (-20 * L) * (√((∑ i, ↑(lambda i)) * L) + L) ^ t","l":512,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.independent_poisson_normalized_badAny_moment","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"On a measurable sample space with a finite coordinate index, let a probability law carry natural-valued coordinate counts with nonnegative coordinate intensities and a strictly positive exposure. If the counts are measurable, each count is Poisson with mean exposure times intensity, and the counts are mutually independent, then under a fixed coordinate-cardinality cap, for moment order one, two, or four and a logarithmic level of at least one, the normalized aggregate score moment on the union of bad coordinates decays exponentially at the local scale determined by total intensity and exposure.","s":"∀ {ι : Type u_1} {Ω : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (W : ι → Ω → ℕ) (q : ι → NNReal) (m : NNReal), 0 < m → (∀ (i : ι), Measurable (W i)) → (∀ (i : ι), ProbabilityTheory.HasLaw (W i) (ProbabilityTheory.poissonMeasure (m * q i)) μ) → ProbabilityTheory.iIndepFun W μ → ∀ {r t : ℕ}, Fintype.card ι ≤ r → t = 1 ∨ t = 2 ∨ t = 4 → ∀ {L : ℝ}, 1 ≤ L → ∫ (ω : Ω), (Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedBadAny W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L m q).indicator (fun ω => Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedAggregateScore W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L m q ω ^ t) ω ∂μ ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant r t * Real.exp (-20 * L) * (√((∑ i, ↑(q i)) * L / ↑m) + L / ↑m) ^ t","l":81,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.independent_poisson_normalized_badAny_moment_four","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"On a measurable sample space, let a probability law carry four natural-valued coordinate counts with nonnegative coordinate intensities and a strictly positive exposure. If the counts are measurable, each count is Poisson with mean exposure times intensity, and the counts are mutually independent, then for moment order one, two, or four and a logarithmic level of at least one, the normalized four-coordinate bad-event moment has the corresponding exponentially decaying local-scale bound.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (W : Fin 4 → Ω → ℕ) (q : Fin 4 → NNReal) (m : NNReal), 0 < m → (∀ (i : Fin 4), Measurable (W i)) → (∀ (i : Fin 4), ProbabilityTheory.HasLaw (W i) (ProbabilityTheory.poissonMeasure (m * q i)) μ) → ProbabilityTheory.iIndepFun W μ → ∀ {t : ℕ}, t = 1 ∨ t = 2 ∨ t = 4 → ∀ {L : ℝ}, 1 ≤ L → ∫ (ω : Ω), (Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedBadAny W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L m q).indicator (fun ω => Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedAggregateScore W Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L m q ω ^ t) ω ∂μ ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant 4 t * Real.exp (-20 * L) * (√((∑ i, ↑(q i)) * L / ↑m) + L / ↑m) ^ t","l":205,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.integrable_deviation","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"The absolute deviation under the Poisson law with a nonnegative mean is integrable.","s":"∀ (lambda : NNReal), MeasureTheory.Integrable (Causalean.Stat.Concentration.PoissonSelfNormalized.deviation lambda) (ProbabilityTheory.poissonMeasure lambda)","l":86,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.integrable_exp_centered_poisson","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"Under a Poisson law with a nonnegative mean, the exponential of its centered count at a real tilt is integrable.","s":"∀ (lambda : NNReal) (theta : ℝ), MeasureTheory.Integrable (fun w => Real.exp (theta * (↑w - ↑lambda))) (ProbabilityTheory.poissonMeasure lambda)","l":18,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.integrable_radius","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Under a Poisson law with a nonnegative mean, the self-normalizing radius at a nonnegative logarithmic level is integrable.","s":"∀ (lambda : NNReal) (L : ℝ), 0 ≤ L → MeasureTheory.Integrable (Causalean.Stat.Concentration.PoissonSelfNormalized.radius L) (ProbabilityTheory.poissonMeasure lambda)","l":94,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.integrable_score_pow","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"Under a Poisson law with a nonnegative mean, at a logarithmic level of at least one, every moment order through four gives an integrable power of the universal score.","s":"∀ (lambda : NNReal) {L : ℝ}, 1 ≤ L → ∀ {t : ℕ}, t ≤ 4 → MeasureTheory.Integrable (fun w => Causalean.Stat.Concentration.PoissonSelfNormalized.score Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda w ^ t) (ProbabilityTheory.poissonMeasure lambda)","l":341,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.integral_exp_centered_poisson","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a Poisson law with a nonnegative mean and a real tilt, the centered moment-generating function equals the exponential of the mean times the exponential remainder.","s":"∀ (lambda : NNReal) (theta : ℝ), ∫ (w : ℕ), Real.exp (theta * (↑w - ↑lambda)) ∂ProbabilityTheory.poissonMeasure lambda = Real.exp (↑lambda * (Real.exp theta - 1 - theta))","l":42,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.localScale","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a nonnegative Poisson mean and a logarithmic level, the deterministic local scale is the square root of mean times level, plus the level.","s":"NNReal → ℝ → ℝ","l":31,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.measurableSet_badEvent","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"Given a multiplier and logarithmic level and a nonnegative Poisson mean, the self-normalized bad event is measurable.","s":"∀ (H L : ℝ) (lambda : NNReal), MeasurableSet (Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent H L lambda)","l":66,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.measurable_deviation","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"The absolute-deviation function for a nonnegative mean is measurable.","s":"∀ (lambda : NNReal), Measurable (Causalean.Stat.Concentration.PoissonSelfNormalized.deviation lambda)","l":56,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.measurable_radius","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"The self-normalizing radius at a fixed logarithmic level is measurable.","s":"∀ (L : ℝ), Measurable (Causalean.Stat.Concentration.PoissonSelfNormalized.radius L)","l":61,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedAggregateScore","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a finite coordinate index and sample space, natural-valued coordinate counts, a multiplier and logarithmic level, a nonnegative exposure, nonnegative coordinate intensities, and a sample point, the normalized aggregate score sums the normalized deviation-plus-radius scores across coordinates.","s":"{ι : Type u_1} → {Ω : Type u_2} → [Fintype ι] → (ι → Ω → ℕ) → ℝ → ℝ → NNReal → (ι → NNReal) → Ω → ℝ","l":39,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedBadAny","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a finite coordinate index and sample space, natural-valued coordinate counts, a multiplier and logarithmic level, a nonnegative exposure, and nonnegative coordinate intensities, the normalized aggregate bad event occurs when at least one coordinate exceeds its normalized self-normalized threshold.","s":"{ι : Type u_1} → {Ω : Type u_2} → [Fintype ι] → (ι → Ω → ℕ) → ℝ → ℝ → NNReal → (ι → NNReal) → Set Ω","l":30,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedDeviation","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a nonnegative exposure, a nonnegative intensity, and a natural-valued count, the normalized deviation is the absolute difference between count divided by exposure and intensity.","s":"NNReal → NNReal → ℕ → ℝ","l":18,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.normalizedRadius","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Scaling","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a nonnegative exposure, a logarithmic level, and a natural-valued count, the normalized self-normalizing radius is the square root of normalized count times normalized level, plus normalized level.","s":"NNReal → ℝ → ℕ → ℝ","l":24,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Scaling.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poissonMeasure_zero_ae","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"Under a zero-mean Poisson law, the count is almost surely zero.","s":"(fun w => w) =ᵐ[ProbabilityTheory.poissonMeasure 0] 0","l":115,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_abs_bernstein","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"For a Poisson law with a nonnegative mean and a nonnegative deviation level, the probability that the absolute deviation exceeds the Bernstein threshold is at most twice the exponential of minus that level, including when the mean is zero.","s":"∀ (lambda : NNReal) {z : ℝ}, 0 ≤ z → (ProbabilityTheory.poissonMeasure lambda) {w | Causalean.Stat.Concentration.PoissonSelfNormalized.deviation lambda w > √(2 * ↑lambda * z) + 2 * z} ≤ 2 * ENNReal.ofReal (Real.exp (-z))","l":361,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_badEvent_probability","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a Poisson law with a nonnegative mean and a logarithmic level of at least one, the universal self-normalized bad event has probability at most twice the exponential of minus forty times that level.","s":"∀ (lambda : NNReal) {L : ℝ}, 1 ≤ L → (ProbabilityTheory.poissonMeasure lambda) (Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda) ≤ 2 * ENNReal.ofReal (Real.exp (-(Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay * L)))","l":482,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_lower_bernstein","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a Poisson law with a nonnegative mean and a nonnegative deviation level, the probability of exceeding the lower Gaussian threshold is at most the exponential of minus that level.","s":"∀ (lambda : NNReal) {z : ℝ}, 0 ≤ z → (ProbabilityTheory.poissonMeasure lambda) {w | ↑lambda - ↑w > √(2 * ↑lambda * z)} ≤ ENNReal.ofReal (Real.exp (-z))","l":200,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_quarter_overflow_le_exp","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"For a positive sample size, a Poisson count with mean one quarter of that size overshoots the sample size with probability at most exp (-n/8).","s":"∀ (n : ℕ), 0 < n → (ProbabilityTheory.poissonMeasure (↑n / 4)).real (Set.Ioi n) ≤ Real.exp (-↑n / 8)","l":514,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_quarter_overflow_le_inv","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a positive sample size, a Poisson count with mean one quarter of that size overshoots the sample size with probability at most 8 / n.","s":"∀ (n : ℕ), 0 < n → (ProbabilityTheory.poissonMeasure (↑n / 4)).real (Set.Ioi n) ≤ 8 / ↑n","l":551,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_score_moment","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"Under a Poisson law with a nonnegative mean, at a logarithmic level of at least one, every moment order through four has its universal score moment bounded by a constant times the matching local-scale power.","s":"∀ (lambda : NNReal) {L : ℝ}, 1 ≤ L → ∀ {t : ℕ}, t ≤ 4 → ∫ (w : ℕ), Causalean.Stat.Concentration.PoissonSelfNormalized.score Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda w ^ t ∂ProbabilityTheory.poissonMeasure lambda ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.scalarMomentConstant t * Causalean.Stat.Concentration.PoissonSelfNormalized.localScale lambda L ^ t","l":566,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_upper_bernstein","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Chernoff","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a Poisson law with a nonnegative mean and a nonnegative deviation level, the probability of exceeding the upper Bernstein threshold is at most the exponential of minus that level.","s":"∀ (lambda : NNReal) {z : ℝ}, 0 ≤ z → (ProbabilityTheory.poissonMeasure lambda) {w | ↑w - ↑lambda > √(2 * ↑lambda * z) + 2 * z} ≤ ENNReal.ofReal (Real.exp (-z))","l":80,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Chernoff.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.poisson_weighted_bad_moment","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Under a Poisson law with a nonnegative mean, at a logarithmic level of at least one, each moment order one, two, or four has its universal score truncated to the bad event bounded by an exponentially decaying multiple of the matching local-scale power.","s":"∀ (lambda : NNReal) {L : ℝ}, 1 ≤ L → ∀ {t : ℕ}, t = 1 ∨ t = 2 ∨ t = 4 → ∫ (w : ℕ), (Causalean.Stat.Concentration.PoissonSelfNormalized.badEvent Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda).indicator (fun w => Causalean.Stat.Concentration.PoissonSelfNormalized.score Causalean.Stat.Concentration.PoissonSelfNormalized.universalH L lambda w ^ t) w ∂ProbabilityTheory.poissonMeasure lambda ≤ Causalean.Stat.Concentration.PoissonSelfNormalized.scalarBadMomentConstant t * Real.exp (-(Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay * L)) * Causalean.Stat.Concentration.PoissonSelfNormalized.localScale lambda L ^ t","l":441,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a coordinate-cardinality cap and a moment order, the finite-product moment constant is a fixed power of ten depending only on those two quantities.","s":"ℕ → ℕ → ℝ","l":34,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant_pos","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Product","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a coordinate-cardinality cap and moment order, the finite-product moment constant is strictly positive.","s":"∀ (r t : ℕ), 0 < Causalean.Stat.Concentration.PoissonSelfNormalized.productMomentConstant r t","l":38,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Product.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.radius","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a logarithmic level and a natural-valued count, the self-normalizing random radius is the square root of count times level, plus the level.","s":"ℝ → ℕ → ℝ","l":26,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.scalarBadMomentConstant","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a moment order, the uniform scalar bad-event moment constant is a fixed power of ten large enough for all orders through four.","s":"ℕ → ℝ","l":18,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.scalarBadMomentConstant_pos","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a moment order, the scalar bad-event moment constant is strictly positive.","s":"∀ (t : ℕ), 0 < Causalean.Stat.Concentration.PoissonSelfNormalized.scalarBadMomentConstant t","l":26,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.scalarDecay","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"The pre-aggregation exponential decay rate is 40.","s":"ℝ","l":45,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.scalarMomentConstant","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a moment order, the uniform unconditioned scalar moment constant is a fixed power of ten large enough for all orders through four.","s":"ℕ → ℝ","l":22,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.scalarMomentConstant_pos","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"For a moment order, the unconditioned scalar moment constant is strictly positive.","s":"∀ (t : ℕ), 0 < Causalean.Stat.Concentration.PoissonSelfNormalized.scalarMomentConstant t","l":30,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.score","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Moments","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"Given a radius multiplier and logarithmic level, a nonnegative Poisson mean, and a natural-valued count, the scalar deviation-plus-radius score adds the absolute deviation to the multiplied self-normalizing radius.","s":"ℝ → ℝ → NNReal → ℕ → ℝ","l":35,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Moments.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.universalH","k":"def","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":1,"d":"The universal self-normalization multiplier is 1024.","s":"ℝ","l":42,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.PoissonSelfNormalized.universalH_pos","k":"theorem","m":"Causalean.Stat.Concentration.PoissonSelfNormalized.Basic","a":"Stat/Concentration/PoissonSelfNormalized","t":2,"d":"The universal self-normalization multiplier is strictly positive.","s":"0 < Causalean.Stat.Concentration.PoissonSelfNormalized.universalH","l":48,"f":"Causalean/Stat/Concentration/PoissonSelfNormalized/Basic.lean"},{"n":"Causalean.Stat.Concentration.ProbabilityTheory.Kernel.iIndepFun.ae_iIndepFun_real","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"Kernel independence of a finite family of measurable real random variables specializes to ordinary independence on almost every probability fiber. The countable rational half-line generator is what permits one common outer null set for all measurable events.","s":"∀ {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace Ω] {κ : ProbabilityTheory.Kernel α Ω} {ν : MeasureTheory.Measure α} [Finite ι] {X : ι → Ω → ℝ}, (∀ (i : ι), Measurable (X i)) → ProbabilityTheory.Kernel.iIndepFun X κ ν → ∀ᵐ (a : α) ∂ν, ProbabilityTheory.iIndepFun X (κ a)","l":28,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.RademacherUpperBound","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For a measurable sample space, an index set, an observation domain, a class of real-valued functions, a real-valued functional on such functions, a measure on the sample space, a random observation map from that sample space, a sample size, and a real-valued radius envelope, the radius envelope is a deterministic upper bound for localized Rademacher complexity exactly when, for every nonnegative real radius, the population Rademacher complexity of the zero-out localized star-hull family at that radius is at most the envelope evaluated at that radius.","s":"{Ω : Type u_1} → {ι : Type u_2} → {𝒳 : Type u_3} → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → ℕ → (ℝ → ℝ) → Prop","l":138,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.centered_condDistrib_mgf_le","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A probability law P, measurable design, and measurable outcome that is almost surely nonnegative and at most one, together with a measurable regression equal almost surely to its conditional expectation given the design, yield the conditional Hoeffding exponential-moment bound on almost every design fibre.","s":"∀ {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [inst_2 : MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → (∀ᵐ (omega : Omega) ∂P, 0 ≤ Y omega) → (∀ᵐ (omega : Omega) ∂P, Y omega ≤ 1) → ∀ (mD : D → ℝ), Measurable mD → P[Y | MeasurableSpace.comap design inferInstance] =ᵐ[P] mD ∘ design → ∀ᵐ (d : D) ∂MeasureTheory.Measure.map design P, ∀ (s : ℝ), ∫ (y : ℝ), Real.exp (s * (y - mD d)) ∂(ProbabilityTheory.condDistrib Y design P) d ≤ Real.exp (s ^ 2 / 8)","l":113,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.measurableSet_selfNormalizedDeviation","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A measurable design, measurable outcome, measurable regression, and measurable complete-design weights make the event that the absolute self-normalized sum exceeds the fixed threshold measurable.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] {design : Omega → D}, Measurable design → ∀ {Y : Omega → ℝ}, Measurable Y → ∀ {mD : D → ℝ}, Measurable mD → ∀ {w : (Fin N → D) → Fin N → ℝ}, Measurable w → ∀ (t : ℝ), MeasurableSet {z | t ≤ |Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.selfNormalizedWeightedCenteredSum design Y mD w z|}","l":99,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.measurable_realizedWeightEnergy","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"The sum of squared coefficients selected by a measurable complete-design weight array is itself measurable.","s":"∀ {N : ℕ} {D : Type u_1} [inst : MeasurableSpace D] {w : (Fin N → D) → Fin N → ℝ}, Measurable w → Measurable (Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w)","l":44,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.measurable_selfNormalizedWeightedCenteredSum","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A measurable design, measurable outcome, measurable regression, and measurable complete-design weights make the self-normalized weighted centered sum measurable, including at zero energy.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] {design : Omega → D}, Measurable design → ∀ {Y : Omega → ℝ}, Measurable Y → ∀ {mD : D → ℝ}, Measurable mD → ∀ {w : (Fin N → D) → Fin N → ℝ}, Measurable w → Measurable (Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.selfNormalizedWeightedCenteredSum design Y mD w)","l":85,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.measurable_weightedCenteredMarkSum","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A measurable regression and measurable complete-design weights make the weighted centered mark sum jointly measurable.","s":"∀ {N : ℕ} {D : Type u_1} [inst : MeasurableSpace D] {mD : D → ℝ}, Measurable mD → ∀ {w : (Fin N → D) → Fin N → ℝ}, Measurable w → Measurable fun p => Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum mD w p.1 p.2","l":54,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.measurable_weightedCenteredSum","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A measurable design, measurable outcome, measurable regression, and measurable complete-design weights make the finite-sample weighted centered sum measurable.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] {design : Omega → D}, Measurable design → ∀ {Y : Omega → ℝ}, Measurable Y → ∀ {mD : D → ℝ}, Measurable mD → ∀ {w : (Fin N → D) → Fin N → ℝ}, Measurable w → Measurable (Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w)","l":68,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_observation_law_map_design_outcome","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Tail","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"A probability law P, measurable design, and measurable outcome map an i.i.d. observation product to the retained-design law with the finite product of its one-observation conditional mark kernels.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [inst_2 : MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → MeasureTheory.Measure.map (fun z => (Causalean.Mathlib.Probability.designVector design z, fun i => Y (z i))) (MeasureTheory.Measure.pi fun x => P) = Causalean.Stat.attachKernel (MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map design P) (Causalean.Stat.finProductKernel N (ProbabilityTheory.condDistrib Y design P))","l":125,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Tail.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_weighted_centered_attachKernel_conditional_mgf_le","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"Under a probability design law and Markov marking kernel whose conditional mean is the measurable regression, whose marks lie in the unit interval almost surely on almost every fibre, and whose complete-design weight array is measurable, the weighted centered sum has the conditional Hoeffding exponential-moment bound on almost every complete-design fibre.","s":"∀ {N : ℕ} {D : Type u_1} [inst : MeasurableSpace D] (Q : MeasureTheory.Measure D) [MeasureTheory.IsProbabilityMeasure Q] (K : ProbabilityTheory.Kernel D ℝ) [ProbabilityTheory.IsMarkovKernel K] (mD : D → ℝ), Measurable mD → (∀ᵐ (d : D) ∂Q, ∫ (y : ℝ), y ∂K d = mD d) → (∀ᵐ (d : D) ∂Q, ∀ᵐ (y : ℝ) ∂K d, y ∈ Set.Icc 0 1) → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → ∀ᵐ (d : Fin N → D) ∂MeasureTheory.Measure.pi fun x => Q, ∀ (s : ℝ), ∫ (y : Fin N → ℝ), Real.exp (s * Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum mD w d y) ∂(Causalean.Stat.finProductKernel N K) d ≤ Real.exp (s ^ 2 * Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w d / 8)","l":257,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_weighted_centered_attachKernel_hasSubgaussianMGF","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"Under a probability design law and Markov marking kernel whose conditional mean is the measurable regression, whose marks lie in the unit interval almost surely on almost every fibre, and whose complete-design weight array is measurable, the weighted centered mark sum is sub-Gaussian on almost every complete-design fibre with one quarter of its realized weight energy as variance proxy.","s":"∀ {N : ℕ} {D : Type u_1} [inst : MeasurableSpace D] (Q : MeasureTheory.Measure D) [MeasureTheory.IsProbabilityMeasure Q] (K : ProbabilityTheory.Kernel D ℝ) [ProbabilityTheory.IsMarkovKernel K] (mD : D → ℝ), Measurable mD → (∀ᵐ (d : D) ∂Q, ∫ (y : ℝ), y ∂K d = mD d) → (∀ᵐ (d : D) ∂Q, ∀ᵐ (y : ℝ) ∂K d, y ∈ Set.Icc 0 1) → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → ∀ᵐ (d : Fin N → D) ∂MeasureTheory.Measure.pi fun x => Q, ProbabilityTheory.HasSubgaussianMGF (Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum mD w d) (Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w d / 4).toNNReal ((Causalean.Stat.finProductKernel N K) d)","l":172,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_weighted_centered_attachKernel_tail_le","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Tail","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":2,"d":"Under a probability design law and Markov marking kernel whose conditional mean is the measurable regression, whose marks lie in the unit interval almost surely on almost every fibre, with measurable complete-design weights of positive realized energy almost surely, every nonnegative threshold obeys the two-sided self-normalized Hoeffding tail bound for the retained-design law.","s":"∀ {N : ℕ} {D : Type u_1} [inst : MeasurableSpace D] (Q : MeasureTheory.Measure D) [MeasureTheory.IsProbabilityMeasure Q] (K : ProbabilityTheory.Kernel D ℝ) [ProbabilityTheory.IsMarkovKernel K] (mD : D → ℝ), Measurable mD → (∀ᵐ (d : D) ∂Q, ∫ (y : ℝ), y ∂K d = mD d) → (∀ᵐ (d : D) ∂Q, ∀ᵐ (y : ℝ) ∂K d, y ∈ Set.Icc 0 1) → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → (∀ᵐ (d : Fin N → D) ∂MeasureTheory.Measure.pi fun x => Q, 0 < Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w d) → ∀ {t : ℝ}, 0 ≤ t → (Causalean.Stat.attachKernel (MeasureTheory.Measure.pi fun x => Q) (Causalean.Stat.finProductKernel N K)).real {p | t * √(Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w p.1) ≤ |Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum mD w p.1 p.2|} ≤ 2 * Real.exp (-2 * t ^ 2)","l":169,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Tail.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_weighted_centered_conditional_mgf_le","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":1,"d":"A probability law P, measurable design, and measurable outcome that is almost surely in the unit interval, together with a measurable regression equal almost surely to the conditional expectation given the design and measurable complete-design weights, give the conditional Hoeffding exponential-moment bound on almost every complete design vector.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [inst_2 : MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → (∀ᵐ (omega : Omega) ∂P, 0 ≤ Y omega) → (∀ᵐ (omega : Omega) ∂P, Y omega ≤ 1) → ∀ (mD : D → ℝ), Measurable mD → P[Y | MeasurableSpace.comap design inferInstance] =ᵐ[P] mD ∘ design → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → ∀ᵐ (d : Fin N → D) ∂MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map design P, ∀ (s : ℝ), ∫ (y : Fin N → ℝ), Real.exp (s * Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum mD w d y) ∂(Causalean.Stat.finProductKernel N (ProbabilityTheory.condDistrib Y design P)) d ≤ Real.exp (s ^ 2 * Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w d / 8)","l":287,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.product_weighted_centered_tail_le","k":"theorem","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Tail","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":1,"d":"A probability law P, measurable design, and measurable outcome that is almost surely in the unit interval, together with a measurable regression equal almost surely to the conditional expectation given the design, measurable complete-design weights of positive realized energy almost surely, and a nonnegative threshold, yield the two-sided self-normalized Hoeffding tail bound for the finite i.i.d. sample.","s":"∀ {N : ℕ} {Omega : Type u_1} {D : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace D] (P : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure P] (design : Omega → D), Measurable design → ∀ (Y : Omega → ℝ), Measurable Y → (∀ᵐ (omega : Omega) ∂P, 0 ≤ Y omega) → (∀ᵐ (omega : Omega) ∂P, Y omega ≤ 1) → ∀ (mD : D → ℝ), Measurable mD → P[Y | MeasurableSpace.comap design inferInstance] =ᵐ[P] mD ∘ design → ∀ (w : (Fin N → D) → Fin N → ℝ), Measurable w → (∀ᵐ (z : Fin N → Omega) ∂MeasureTheory.Measure.pi fun x => P, 0 < Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w (Causalean.Mathlib.Probability.designVector design z)) → ∀ {t : ℝ}, 0 ≤ t → (MeasureTheory.Measure.pi fun x => P).real {z | t * √(Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy w (Causalean.Mathlib.Probability.designVector design z)) ≤ |Causalean.Mathlib.Probability.weightedCenteredSum design Y mD w z|} ≤ 2 * Real.exp (-2 * t ^ 2)","l":286,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Tail.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.realizedWeightEnergy","k":"def","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":1,"d":"For a coefficient array determined by a complete design vector and a complete design vector, the realized weight energy is the sum of the squares of the coefficients selected by that design vector.","s":"{N : ℕ} → {D : Type u_1} → ((Fin N → D) → Fin N → ℝ) → (Fin N → D) → ℝ","l":26,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.selfNormalizedWeightedCenteredSum","k":"def","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":1,"d":"For a design map, a real-valued outcome map, a real-valued conditional mean function of the design, a coefficient array determined by a complete design vector, and a finite vector of sampled units, the self-normalized weighted centered sum is the weighted centered sum divided by the square root of the realized weight energy of the sampled design vector; when that energy is zero, this value is zero.","s":"{N : ℕ} → {Omega : Type u_1} → {D : Type u_2} → (Omega → D) → (Omega → ℝ) → (D → ℝ) → ((Fin N → D) → Fin N → ℝ) → (Fin N → Omega) → ℝ","l":37,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.weightedCenteredMarkSum","k":"def","m":"Causalean.Stat.Concentration.RandomDesignWeightedHoeffding.Basic","a":"Stat/Concentration/RandomDesignWeightedHoeffding","t":1,"d":"For a real-valued conditional mean function of a design value, a coefficient array determined by a complete design vector, a complete design vector, and a vector of real marks, the weighted centered mark sum is the sum over sample positions of the selected coefficient times the mark minus its conditional mean at the corresponding design value.","s":"{N : ℕ} → {D : Type u_1} → (D → ℝ) → ((Fin N → D) → Fin N → ℝ) → (Fin N → D) → (Fin N → ℝ) → ℝ","l":31,"f":"Causalean/Stat/Concentration/RandomDesignWeightedHoeffding/Basic.lean"},{"n":"Causalean.Stat.Concentration.SubRoot","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For a real-valued radius envelope, the sub-root condition holds exactly when (1) the envelope is nonnegative at every nonnegative radius, (2) for every two nonnegative radii with the first no larger than the second, the envelope at the first is no larger than the envelope at the second, and (3) for every two positive radii with the first no larger than the second, the envelope divided by the radius is no smaller at the first than at the second.","s":"(ℝ → ℝ) → Prop","l":176,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.absInner_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Signed empirical averages of a uniformly bounded function class are bounded above by the common absolute bound.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (H : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |H i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun i => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * H i (S k)|)","l":802,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.absInner_le_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"A signed empirical average of a uniformly bounded function has absolute value no larger than the same uniform bound.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (H : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |H i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳) (σ : Signs n) (i : ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * H i (S k)| ≤ M","l":778,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.abs_inner_starHullZeroOut_mono","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Per-(p, σ, S) zero-out inner-expression bound. For r₁ ≤ r₂, the inner expression of empiricalRademacherComplexity for the zero-out family at r₁ is dominated in absolute value by that at r₂. This is the precise ingredient the helper above consumes for localRademacherComplexity_mono_r.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → ∀ {n : ℕ} (S : Fin n → 𝒳) (σ : Signs n) (p : Causalean.Stat.Concentration.starHullParam ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₁ p (S k)| ≤ |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p (S k)|","l":162,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_bound","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"A zeroed-out star-hull value inherits a bound on its corresponding base-family value.","s":"∀ {𝒳 : Type u_4} {ι : Type u_5} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {b r : ℝ}, 0 ≤ b → ∀ {p : Causalean.Stat.Concentration.starHullParam ι} {x : 𝒳}, |F p.2 x| ≤ b → |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ b","l":346,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If every function in a class is pointwise bounded in absolute value by a common constant, then every zeroed localized star-hull function has the same bound at every point.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B r : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → ∀ (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ B","l":71,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_le_starHullEval","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise dominance by the inclusion. The zero-out family is dominated coordinatewise (in absolute value) by the un-localised inclusion starHullEval F.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r p x| ≤ |Causalean.Stat.Concentration.starHullEval F p x|","l":217,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.abs_starHullZeroOut_mono","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise monotonicity of the zero-out family in the radius. For r₁ ≤ r₂, |starHullZeroOut F norm r₁ p x| ≤ |starHullZeroOut F norm r₂ p x| coordinatewise.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → ∀ (p : Causalean.Stat.Concentration.starHullParam ι) (x : 𝒳), |Causalean.Stat.Concentration.starHullZeroOut F norm r₁ p x| ≤ |Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p x|","l":202,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.absolute_dudley_vc_starHullZeroOut_linear_residual_shared","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Shared absolute-form Dudley/VC bridge for the localized star-hull class.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1203,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.ae_ae_condExpKernel_of_ae","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"An ambient almost-sure proposition holds on almost every fiber of the regular conditional distribution.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {p : Ω → Prop}, (∀ᵐ (ω : Ω) ∂μ, p ω) → ∀ᵐ (ω : Ω) ∂μ.trim hm, ∀ᵐ (ω' : Ω) ∂(ProbabilityTheory.condExpKernel μ m) ω, p ω'","l":149,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.ae_condExpKernel_projection_energy_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalProjectionTail","a":"Stat/Concentration","t":1,"d":"Fix a sub-σ-algebra m of the ambient σ-algebra and a finite family eps of real random variables on Ω that is measurable and almost surely bounded by 1 in absolute value under μ; suppose further that, on almost every m-conditioning fiber, each eps i has conditional mean zero under the regular conditional kernel given m, and that the family eps is conditionally independent given m. Let Pi be a matrix-valued map that is m-measurable entrywise and, on almost every m-conditioning fiber, symmetric, idempotent (Pi·Pi = Pi), and of rank at most r, and fix a positive tolerance zeta. Then, on almost every m-conditioning fiber, the conditional probability — under the regular conditional kernel given m — that the projected noise's squared energy $\\sum_i(\\sum_j \\mathrm{Pi}_{ij}\\,\\mathrm{eps}_j)^2$ exceeds the threshold $8(r\\log 5+\\log(2/\\zeta))$ is at most zeta.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {n r : ℕ} (eps : Fin n → Ω → ℝ) (Pi : Ω → Matrix (Fin n) (Fin n) ℝ), (∀ (i : Fin n), Measurable (eps i)) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |eps i ω| ≤ 1) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ.trim hm, ∫ (ω' : Ω), eps i ω' ∂(ProbabilityTheory.condExpKernel μ m) ω = 0) → ProbabilityTheory.iCondIndepFun m hm eps μ → (∀ (i j : Fin n), Measurable fun ω => Pi ω i j) → (∀ᵐ (ω : Ω) ∂μ.trim hm, (Pi ω).transpose = Pi ω) → (∀ᵐ (ω : Ω) ∂μ.trim hm, Pi ω * Pi ω = Pi ω) → (∀ᵐ (ω : Ω) ∂μ.trim hm, (Pi ω).rank ≤ r) → ∀ {zeta : ℝ}, 0 < zeta → ∀ᵐ (ω : Ω) ∂μ.trim hm, ((ProbabilityTheory.condExpKernel μ m) ω).real {ω' | 8 * (↑r * Real.log 5 + Real.log (2 / zeta)) < ∑ i, (∑ j, Pi ω' i j * eps j ω') ^ 2} ≤ zeta","l":27,"f":"Causalean/Stat/Concentration/ConditionalProjectionTail.lean"},{"n":"Causalean.Stat.Concentration.ae_eq_const_condExpKernel_of_measurable","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":2,"d":"A random variable measurable with respect to the conditioning σ-algebra equals its observed value on almost every conditional fiber.","s":"∀ {Ω : Type u_1} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {β : Type u_2} [inst_2 : MeasurableSpace β] [MeasurableEq β] {f : Ω → β}, Measurable f → ∀ᵐ (ω : Ω) ∂μ.trim hm, ∀ᵐ (ω' : Ω) ∂(ProbabilityTheory.condExpKernel μ m) ω, f ω' = f ω","l":161,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.averaging_exists_zero_count","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative finite probability weighting with average count below one must assign count zero to at least one index.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (μ : ι → ℝ) (cnt : ι → ℕ), ∑ i, μ i = 1 → (∀ (i : ι), 0 ≤ μ i) → ∑ i, μ i * ↑(cnt i) < 1 → ∃ i, cnt i = 0","l":153,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount","k":"def","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"For an independent and identically distributed sample with a specified sample-space law and population law, a real-valued statistic on the observation space, and a nonnegative integer $m$, the Bernoulli count is the function that maps every sample-space outcome to the sum of the statistic over its first $m$ sampled observations.","s":"{Ω : Type u_1} → {𝒳 : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace 𝒳] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure 𝒳} → Causalean.Stat.IIDSample Ω 𝒳 μ P → (𝒳 → ℝ) → ℕ → Ω → ℝ","l":44,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"A count of m observations of a statistic bounded by one never exceeds m.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, (∀ (x : 𝒳), f x ≤ 1) → ∀ (m : ℕ) (ω : Ω), Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ ↑m","l":63,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_lower_tail","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"Lower multiplicative tail for the count of an i.i.d. [0,1]-valued statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in the unit interval. If p is nonnegative, the population mean of f is at least p, and twice the threshold a is less than m times p, then the probability that the sum of f over the first m draws is at most a is at most $\\exp(-mp/8)$.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, 0 ≤ p → p ≤ ∫ (x : 𝒳), f x ∂P → ∀ {m : ℕ}, 2 * a < ↑m * p → μ.real {ω | Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ a} ≤ Real.exp (-(↑m * p) / 8)","l":297,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_measurable","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → ∀ (m : ℕ), Measurable (Causalean.Stat.Concentration.bernoulliCount S f m)","l":49,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"A count built from a nonnegative statistic is nonnegative.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, (∀ (x : 𝒳), 0 ≤ f x) → ∀ (m : ℕ) (ω : Ω), 0 ≤ Causalean.Stat.Concentration.bernoulliCount S f m ω","l":56,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernoulliCount_upper_tail","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":1,"d":"Upper multiplicative tail for the count of an i.i.d. [0,1]-valued statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in the unit interval. If the population mean of f is at most p and m times p is less than half the threshold a, then the probability that the sum of f over the first m draws exceeds a is at most $\\exp(-a(\\log 2 - 1/2))$.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ {m : ℕ}, ↑m * p < a / 2 → μ.real {ω | a < Causalean.Stat.Concentration.bernoulliCount S f m ω} ≤ Real.exp (-a * (Real.log 2 - 1 / 2))","l":272,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bernsteinCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Given a real range-bound constant $c$, a real standard-deviation proxy $\\sigma$, a natural-number sample size $n$, and a real confidence level $\\delta$, the two-sided Bernstein confidence-interval half-width is $2\\sigma\\sqrt{\\log(2/\\delta)/n}+2c\\log(2/\\delta)/n$.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":65,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernsteinCIHalfWidth_mono_sigma","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The Bernstein confidence-interval half-width does not decrease when its standard-deviation input is increased, so an upper variance bound gives a conservative interval.","s":"∀ {c : ℝ} {n : ℕ} {δ σ σ' : ℝ}, σ ≤ σ' → Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ ≤ Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ' n δ","l":373,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.bernstein_abs_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Bernstein inequality for the sample mean of a bounded statistic. Given an i.i.d. sample S and a statistic f that is measurable and integrable under the population law P, suppose c is nonnegative, f stays within c of its population mean m = ∫ f ∂P, P-almost everywhere, and the population variance of f is at most σ². Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws deviates from m by at least ε in absolute value is at most $2\\exp(-n\\varepsilon^2/(2(2\\sigma^2+c\\varepsilon)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X' : Type u_2} [inst : MeasurableSpace X'] {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ 2 * Real.exp (-↑n * ε ^ 2 / (2 * (2 * σ ^ 2 + c * ε)))","l":223,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ci_cover","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Bernstein confidence interval (coverage form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable and P-integrable, if f is almost-everywhere within c of m, for some nonnegative c, if the variance of f is bounded by σ², for some positive σ, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at least 1 − δ the sample mean X̄ₙ of f over S's first n draws satisfies |X̄ₙ − m| < w, where w = bernsteinCIHalfWidth c σ n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → 0 < σ → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 1 - δ ≤ μ.real {ω | |S.sampleMean f n ω - ∫ (x : X'), f x ∂P| < Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ}","l":208,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Bernstein confidence interval (miss-probability form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable and P-integrable, if f is almost-everywhere within c of m, for some nonnegative c, if the variance of f is bounded by σ², for some positive σ, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at most δ the sample mean X̄ₙ of f over S's first n draws satisfies w ≤ |X̄ₙ − m|, where w = bernsteinCIHalfWidth c σ n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → 0 < σ → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.bernsteinCIHalfWidth c σ n δ ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ δ","l":119,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.bernstein_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"One-sided Bernstein inequality for the sample mean of a bounded statistic. Given an i.i.d. sample S and a statistic f that is measurable and integrable under the population law P, suppose c is nonnegative, f stays within c of its population mean m = ∫ f ∂P, P-almost everywhere, and the population variance of f is at most σ². Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws exceeds m by at least ε is at most $\\exp(-n\\varepsilon^2/(2(2\\sigma^2+c\\varepsilon)))$.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X' : Type u_2} [inst : MeasurableSpace X'] {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → MeasureTheory.Integrable f P → ∀ {c σ : ℝ}, 0 ≤ c → (∀ᵐ (x : X') ∂P, |f x - ∫ (y : X'), f y ∂P| ≤ c) → ∫ (x : X'), (f x - ∫ (y : X'), f y ∂P) ^ 2 ∂P ≤ σ ^ 2 → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ S.sampleMean f n ω - ∫ (x : X'), f x ∂P} ≤ Real.exp (-↑n * ε ^ 2 / (2 * (2 * σ ^ 2 + c * ε)))","l":154,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_lower_tail_of_tilt","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"Chernoff lower tail at an arbitrary nonpositive tilt. For a measurable [0,1]-valued statistic with population mean at least p, the probability that the count of the first m observations falls at or below a level a is at most exp (-s * a + m * p * (exp s - 1)), for every nonpositive s. Optimising over s recovers the usual multiplicative Chernoff bounds.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, p ≤ ∫ (x : 𝒳), f x ∂P → ∀ (m : ℕ), ∀ s ≤ 0, μ.real {ω | Causalean.Stat.Concentration.bernoulliCount S f m ω ≤ a} ≤ Real.exp (-s * a + ↑m * (p * (Real.exp s - 1)))","l":231,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_mgf_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"If the population mean of a measurable [0,1]-valued statistic is at most p, then at every nonnegative tilt s the moment generating function of the count of the first m observations is at most exp (m * p * (exp s - 1)).","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ (m : ℕ) (s : ℝ), 0 ≤ s → ProbabilityTheory.mgf (Causalean.Stat.Concentration.bernoulliCount S f m) μ s ≤ Real.exp (↑m * (p * (Real.exp s - 1)))","l":176,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_mgf_le_integral","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"The moment generating function of the count of the first m observations of a measurable [0,1]-valued statistic is at most exp (m * mean * (exp s - 1)), where mean is the statistic's population mean. This is the i.i.d. tensorisation of the one-observation bound and holds at every tilt s.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ (m : ℕ) (s : ℝ), ProbabilityTheory.mgf (Causalean.Stat.Concentration.bernoulliCount S f m) μ s ≤ Real.exp (↑m * ((∫ (x : 𝒳), f x ∂P) * (Real.exp s - 1)))","l":145,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.boundedCount_upper_tail_of_tilt","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"Chernoff upper tail at an arbitrary nonnegative tilt. For a measurable [0,1]-valued statistic with population mean at most p, the probability that the count of the first m observations exceeds a level a is at most exp (-s * a + m * p * (exp s - 1)), for every nonnegative s. Optimising over s recovers the usual multiplicative Chernoff bounds.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure 𝒳} (S : Causalean.Stat.IIDSample Ω 𝒳 μ P) {f : 𝒳 → ℝ}, Measurable f → (∀ (x : 𝒳), f x ∈ Set.Icc 0 1) → ∀ {p a : ℝ}, ∫ (x : 𝒳), f x ∂P ≤ p → ∀ (m : ℕ) (s : ℝ), 0 ≤ s → μ.real {ω | a < Causalean.Stat.Concentration.bernoulliCount S f m ω} ≤ Real.exp (-s * a + ↑m * (p * (Real.exp s - 1)))","l":195,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.bounded_hasSubexponentialMGF","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"A bounded mean-zero random variable is sub-exponential with parameters (2σ², c). The MGF branch is bounded_mgf_le_exp_sq (note 2σ² · t²/2 = σ²t²), valid on c|t| < 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure μ] {c σ : ℝ} (hc : 0 ≤ c), AEMeasurable X μ → ∫ (x : Ω), X x ∂μ = 0 → (∀ᵐ (ω : Ω) ∂μ, |X ω| ≤ c) → ∫ (x : Ω), (fun ω => X ω ^ 2) x ∂μ ≤ σ ^ 2 → Causalean.Stat.Concentration.HasSubexponentialMGF X ⟨2 * σ ^ 2, mul_nonneg (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (Even.pow_nonneg (even_two_mul 1) σ)⟩ ⟨c, hc⟩ μ","l":131,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.bounded_mgf_le_exp_sq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"MGF bound for a bounded mean-zero random variable on the validity interval. For a mean-zero X with |X| ≤ c μ-a.e. and second moment E[X²] ≤ σ², the moment-generating function obeys mgf X μ t ≤ exp(σ² t²) whenever c |t| ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure μ] {c σ : ℝ}, 0 ≤ c → AEMeasurable X μ → ∫ (x : Ω), X x ∂μ = 0 → (∀ᵐ (ω : Ω) ∂μ, |X ω| ≤ c) → ∫ (x : Ω), (fun ω => X ω ^ 2) x ∂μ ≤ σ ^ 2 → ∀ {t : ℝ}, c * |t| ≤ 1 → ProbabilityTheory.mgf X μ t ≤ Real.exp (σ ^ 2 * t ^ 2)","l":77,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.card_growthFamily_le_sum_choose","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"A finite set family with bounded VC dimension has cardinality controlled by the Sauer-Shelah binomial sum on the sample.","s":"∀ {n d : ℕ} (𝒜 : Finset (Finset (Fin n))), 𝒜.vcDim ≤ d → 𝒜.card ≤ ∑ k ≤ d, n.choose k","l":63,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.card_subsample_family_le_succ_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A sampled Boolean trace family with bounded VC dimension has polynomially many realized restricted patterns.","s":"∀ {n m d : ℕ} (J : Fin m → Fin n) (P : Finset (Fin n → Bool)), (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d → (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).card ≤ (m + 1) ^ d","l":224,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.centeredEmpiricalAverage","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given a measure $\\mu$ on an observation space, a sample of $n$ observations, and a real-valued function, the centered empirical average is its sample average minus its integral with respect to $\\mu$. For an empty sample the average term is zero by the inverse-of-zero convention, so the value is minus the integral.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → {n : ℕ} → (Fin n → Ω) → (Ω → ℝ) → ℝ","l":25,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.ciSup_mul_const_of_le_one","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"A supremum commutes with multiplication by a nonnegative constant.","s":"∀ {A : Type u_4} (c : A → ℝ) (b : ℝ), 0 ≤ b → ⨆ a, c a * b = (⨆ a, c a) * b","l":66,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.ciSup_prod_eq_of_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"The bounded supremum of a real-valued quantity indexed by two choices is unchanged when the two choices are optimized one after the other.","s":"∀ {A : Type u_1} {B : Type u_2} [Nonempty A] [Nonempty B] (f : A → B → ℝ), BddAbove (Set.range fun p => f p.1 p.2) → ⨆ p, f p.1 p.2 = ⨆ b, ⨆ a, f a b","l":67,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.collision_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"When two Boolean vectors are separated under nonnegative coordinate weights, their probability of agreeing on every coordinate of a repeated weighted sample decays exponentially with the sample length.","s":"∀ {n m : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε W : ℝ), 0 < r → W = ∑ j, w j → 0 < W → W ≤ r ^ 2 → ∀ (a b : Fin n → Bool), ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b → (∑ j with a j = b j, w j / W) ^ m ≤ Real.exp (-↑m * ε ^ 2 / r ^ 2)","l":176,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.completeSpace_matrix_linftyOp","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Completeness of finite matrices under the ℓ∞-operator norm follows from the coordinatewise function-space uniformity. Needed to invoke the Neumann/geometric-series unit API.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] [CompleteSpace R], CompleteSpace (Matrix α β R)","l":49,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.countableEmpiricalProcessSup","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"For a measure on a measurable sample space, a class of real-valued functions, a countable enumeration of that class, and a finite sample, the countable empirical-process supremum is the extended nonnegative real supremum, over the enumerated functions, of the absolute centered empirical average.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (ι → Ω → ℝ) → (ℕ → ι) → {n : ℕ} → (Fin n → Ω) → ENNReal","l":34,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.countableEmpiricalSup","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given a measure $P$ on an observation space, a real-valued function class indexed by a set $\\iota$, and a sample of $n$ observations, the countable empirical supremum is the supremum, over the class, of the absolute difference between the sample average and the population mean.","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → {ι : Type v} → MeasureTheory.Measure 𝒳 → (ι → 𝒳 → ℝ) → {n : ℕ} → (Fin n → 𝒳) → ℝ","l":153,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset","k":"def","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"For a totally bounded subset $A$ of a pseudometric space and a strictly positive real radius $\\varepsilon$, the chosen finite $\\varepsilon$-net for $A$ is a finite set whose cardinality attains the covering number.","s":"{X : Type u_1} → [inst : PseudoMetricSpace X] → {A : Set X} → TotallyBounded A → {ε : ℝ} → ε > 0 → Finset X","l":99,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The chosen covering finset has cardinality equal to the covering number.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), (Causalean.Stat.Concentration.coveringFinset ha hε).card = Causalean.Stat.Concentration.coveringNumber ha ε","l":117,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringFinset_cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The chosen covering finset covers the target set by balls of the requested positive radius.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), A ⊆ ⋃ y ∈ Causalean.Stat.Concentration.coveringFinset ha hε, Metric.ball y ε","l":107,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber","k":"def","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"For a totally bounded subset $A$ of a pseudometric space and a real radius $\\varepsilon$, the covering number of $A$ at radius $\\varepsilon$ is the least cardinality of a finite family of open $\\varepsilon$-balls covering $A$ when $\\varepsilon>0$, and is zero when $\\varepsilon\\leq0$.","s":"{X : Type u_1} → [inst : PseudoMetricSpace X] → {A : Set X} → TotallyBounded A → ℝ → ℕ","l":39,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_aemeasurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"The covering-number function of the radius is almost-everywhere measurable on the real line.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) (μ : MeasureTheory.Measure ℝ), AEMeasurable (Causalean.Stat.Concentration.coveringNumber ha) μ","l":83,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_antitone","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"Covering numbers weakly decrease as the positive covering radius grows.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A), AntitoneOn (Causalean.Stat.Concentration.coveringNumber ha) (Set.Ioi 0)","l":58,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":1,"d":"Value of the covering number at a positive radius. For a totally bounded subset of a pseudometric space, at any positive covering radius ε, the covering number equals the least cardinality of a finite ε-net witnessing total boundedness, as selected by coveringNumber_exists.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X} (ha : TotallyBounded A) {ε : ℝ} (hε : ε > 0), Causalean.Stat.Concentration.coveringNumber ha ε = Nat.find (Causalean.Stat.Concentration.coveringNumber_exists ha hε)","l":49,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_exists","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"A totally bounded set has a finite positive-radius net, with the net size recorded as a natural number.","s":"∀ {X : Type u_1} {A : Set X} [inst : PseudoMetricSpace X], TotallyBounded A → ∀ {ε : ℝ}, ε > 0 → ∃ n t, t.card = n ∧ A ⊆ ⋃ y ∈ t, Metric.ball y ε","l":23,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.coveringNumber_nonzero","k":"theorem","m":"Causalean.Stat.Concentration.Covering.CoveringNumber","a":"Stat/Concentration/Covering","t":2,"d":"A nonempty totally bounded set has a positive covering number at every positive radius.","s":"∀ {X : Type u_1} [inst : PseudoMetricSpace X] {A : Set X}, A.Nonempty → ∀ (ha : TotallyBounded A) {ε : ℝ}, ε > 0 → 0 < Causalean.Stat.Concentration.coveringNumber ha ε","l":74,"f":"Causalean/Stat/Concentration/Covering/CoveringNumber.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For a real-valued radius envelope, the critical radius is the infimum of the positive real radii $δ$ for which $ψ(δ) ≤ δ^2$.","s":"(ℝ → ℝ) → ℝ","l":147,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_fp_of_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Fixed-point property at the critical radius. If ψ is sub-root and its critical radius criticalRadius ψ is positive, then the critical radius is itself a solution of its own defining inequality: ψ (criticalRadius ψ) ≤ (criticalRadius ψ) ^ 2.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → 0 < Causalean.Stat.Concentration.criticalRadius ψ → ψ (Causalean.Stat.Concentration.criticalRadius ψ) ≤ Causalean.Stat.Concentration.criticalRadius ψ ^ 2","l":254,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_le","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Any δ > 0 with ψ δ ≤ δ² upper-bounds the critical radius.","s":"∀ {ψ : ℝ → ℝ} {δ : ℝ}, 0 < δ → ψ δ ≤ δ ^ 2 → Causalean.Stat.Concentration.criticalRadius ψ ≤ δ","l":165,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"The critical radius of a positive-slope linear envelope equals its slope, giving the exact fixed-point scale for a localized empirical-process bound.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) = C","l":38,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The critical radius of a positive-slope linear envelope is at most its slope.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) ≤ C","l":337,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_linear_sq_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The squared critical radius of a positive-slope linear envelope is at most the squared slope.","s":"∀ {C : ℝ}, 0 < C → (Causalean.Stat.Concentration.criticalRadius fun r => C * r) ^ 2 ≤ C ^ 2","l":344,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The critical radius is non-negative.","s":"∀ (ψ : ℝ → ℝ), 0 ≤ Causalean.Stat.Concentration.criticalRadius ψ","l":153,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_pos_of_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Positivity of the critical radius. If ψ is sub-root, r₀ is a positive radius with ψ r₀ ≤ r₀ ^ 2, and ψ grows strictly faster than the square near the origin, i.e. there is some ε with 0 < ε < r₀ and ε ^ 2 < ψ ε, then the critical radius of ψ is strictly positive.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ∀ {r₀ : ℝ}, 0 < r₀ → ψ r₀ ≤ r₀ ^ 2 → (∃ ε > 0, ε < r₀ ∧ ε ^ 2 < ψ ε) → 0 < Causalean.Stat.Concentration.criticalRadius ψ","l":300,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC critical radius is bounded by the finite-VC slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":354,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_sq_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC squared critical radius is bounded by the squared slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2","l":361,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.criticalRadius_vcLocalizedPsi_sq_le_rate","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Rate bound for the finite-VC critical radius. For a nonnegative localization constant K and a positive sample size n, the squared critical radius of the finite-VC localized envelope vcLocalizedPsi K d n is at most 36·(K·d·log(n+1)+1)/n — the advertised (d·log n)/n-order bound.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":369,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.designInv00_perturb","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":1,"d":"The (0,0) specialization of the entrywise inverse perturbation bound. If S is invertible, every row of S⁻¹ has its absolute entries summing to at most c, every entry of M differs from the corresponding entry of S by at most η, and the product c · ((p+1) · η) is at most 1/2, then M is also invertible and its (0,0) inverse entry differs from S's by at most 2 · c² · ((p+1) · η).","s":"∀ {p : ℕ} (S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), IsUnit S.det → ∀ {η c : ℝ}, (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → IsUnit M.det ∧ |M⁻¹ 0 0 - S⁻¹ 0 0| ≤ 2 * c ^ 2 * (↑(p + 1) * η)","l":233,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.designInv_perturb","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Entrywise perturbation of an inverse entry. Let S be an invertible (p+1)×(p+1) real matrix whose inverse has rows with absolute sums bounded by c, and let M be a matrix all of whose entries are within η of S (|Mⱼₖ − Sⱼₖ| ≤ η). If c·(p+1)·η ≤ 1/2, then M is invertible and every selected pair of inverse entries satisfies","s":"∀ {p : ℕ} (S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), IsUnit S.det → ∀ {η c : ℝ}, (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → ∀ {i j : Fin (p + 1)}, IsUnit M.det ∧ |M⁻¹ i j - S⁻¹ i j| ≤ 2 * c ^ 2 * (↑(p + 1) * η)","l":102,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.designMatrix_inv_concentration","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.DesignInverse","a":"Stat/Concentration/Matrix","t":1,"d":"Matrix-inverse concentration of the random design moment matrix. Let g j k : Ω → ℝ be the per-entry design statistics, each square-integrable under μ, so the empirical moment matrix is M(ω) = fun j k => ∑ᵢ g j k (ωᵢ) and, given that the population matrix S equals N times its entrywise expectation, S j k = N·𝔼[g j k], suppose S is invertible with inverse row sums bounded by c ≥ 0, η is positive, and the scale satisfies c·(p+1)·η ≤ 1/2. Then the event on which M fails to be invertible *or* its leverage (M⁻¹)₀₀ is farther than 2 c² (p+1) η from (S⁻¹)₀₀ has probability at most the union-bound tail ∑_{j,k} N·Var[g j k]/η². This is the high-probability statement that the random design is non-degenerate with O(1/(Nh)) leverage on the good event.","s":"∀ {N p : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Fin (p + 1) → Fin (p + 1) → Ω → ℝ), (∀ (j k : Fin (p + 1)), MeasureTheory.MemLp (g j k) 2 μ) → ∀ (S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), (∀ (j k : Fin (p + 1)), S j k = ↑N * ∫ (x : Ω), g j k x ∂μ) → IsUnit S.det → ∀ {η c : ℝ}, 0 < η → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → c * (↑(p + 1) * η) ≤ 1 / 2 → (MeasureTheory.Measure.pi fun x => μ) {ω | ¬(IsUnit (Matrix.of fun j k => ∑ i, g j k (ω i)).det ∧ |(Matrix.of fun j k => ∑ i, g j k (ω i))⁻¹ 0 0 - S⁻¹ 0 0| ≤ 2 * c ^ 2 * (↑(p + 1) * η))} ≤ ∑ a, ENNReal.ofReal (↑N * ProbabilityTheory.variance (g a.1 a.2) μ / η ^ 2)","l":38,"f":"Causalean/Stat/Concentration/Matrix/DesignInverse.lean"},{"n":"Causalean.Stat.Concentration.dist_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"","s":"∀ {𝒳 : Type v} {n : ℕ} {ι : Type u} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} (f g : Causalean.Stat.Concentration.EmpiricalFunctionSpace F S), dist f g = Causalean.Stat.Concentration.empiricalNorm S (F f.index - F g.index)","l":167,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.dudley_entropy_integral_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":1,"d":"Dudley entropy-integral bound, without the outer absolute value. Fix a positive scale ε that is strictly less than half the common empirical-norm envelope c, where the sample size m is positive, every member of the class has empirical norm on the sample at most c, and the sample-restricted function class is totally bounded in the empirical pseudometric. Then the empirical Rademacher complexity computed without the outer absolute value is at most 4ε + (12/√m) ∫_ε^(c/2) √(log(coveringNumber x)) dx, the usual Dudley chaining bound in terms of the covering-number entropy integral.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c ε : ℝ}, 0 < ε → ∀ (h' : TotallyBounded Set.univ), 0 < m → (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ε < c / 2 → empiricalRademacherComplexity_without_abs m F S ≤ 4 * ε + 12 / √↑m * ∫ (x : ℝ) in ε..c / 2, √(Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' x))","l":1840,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.empiricalBernsteinCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Given an independent and identically distributed sample, a real-valued statistic, real interval endpoints $a$ and $b$, a real range-bound constant $c$, a natural-number sample size $n$, and a real confidence level $\\delta$, the data-driven Bernstein confidence-interval half-width as a function of the sample outcome is the Bernstein half-width with standard-deviation input equal to the square root of the observed sample variance plus the empirical-variance slack.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℝ → ℝ → ℝ → ℕ → ℝ → Ω → ℝ","l":362,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"For a sample of $n$ observations and two real-valued functions on its observation space, the empirical distance between the functions is the empirical norm of their pointwise difference on that sample.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → (𝒳 → ℝ) → (𝒳 → ℝ) → ℝ","l":41,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_comm","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical distance is symmetric.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f g : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalDist S f g = Causalean.Stat.Concentration.empiricalDist S g f","l":106,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"For a fixed sample S and functions f and g, the empirical distance between f and g equals the empirical norm of their pointwise difference.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f g : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalDist S f g = Causalean.Stat.Concentration.empiricalNorm S (f - g)","l":48,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_proj","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"A single sample coordinate is bounded by the empirical norm up to the sample-size scaling.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f : 𝒳 → ℝ) (i : Fin n), |f (S i)| / √↑n ≤ Causalean.Stat.Concentration.empiricalNorm S f","l":112,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalDist_sq_eq_weightedHammingSq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedCovering","a":"Stat/Concentration/Covering","t":1,"d":"Empirical L² distance as weighted Hamming distance. For a real-valued class that factors through a Boolean classifier at each sample coordinate, F i (S j) = φ j (π i (S j)), the squared empirical L²(Pₙ) distance between two members F i and F i' equals the weighted Hamming distance² of their induced Boolean sample-patterns, with per-coordinate weight (φ j true − φ j false)² / n.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} {n : ℕ} (F : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) (π : ι → 𝒳 → Bool) (φ : Fin n → Bool → ℝ), (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (i i' : ι), Causalean.Stat.Concentration.empiricalDist S (F i) (F i') ^ 2 = Causalean.Stat.Concentration.weightedHammingSq (fun j => (φ j true - φ j false) ^ 2 / ↑n) (fun j => π i (S j)) fun j => π i' (S j)","l":16,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedCovering.lean"},{"n":"Causalean.Stat.Concentration.empiricalFunctionSpace_dist_le_of_forall_abs_sub_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If two functions differ by at most a nonnegative amount at every observation in a finite sample, then their empirical distance is at most that amount.","s":"∀ {ι : Type u} {𝒳 : Type v} {G : Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ} {n : ℕ} {S : Fin n → 𝒳} (q q' : Causalean.Stat.Concentration.EmpiricalFunctionSpace G S) {η : ℝ}, 0 ≤ η → (∀ (j : Fin n), |G q.index (S j) - G q'.index (S j)| ≤ η) → dist q q' ≤ η","l":126,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalL2Radius_measurable","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Rademacher","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"For a countably indexed family of measurable functions, the largest empirical L² norm attained over the family is a measurable function of the sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Countable ι] (F : ι → 𝒳 → ℝ), (∀ (i : ι), Measurable (F i)) → ∀ (n : ℕ), Measurable fun S => Causalean.Stat.Concentration.empiricalL2Radius✝ F S","l":589,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"For a sample of $n$ observations and a real-valued function on its observation space, the empirical norm of the function on that sample is the square root of the average of its squared values at the sample observations.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → (𝒳 → ℝ) → ℝ","l":27,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_const_mul","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Scaling a function by a constant scales its empirical root-mean-square norm by the absolute value of that constant.","s":"∀ {𝒳 : Type u_2} {n : ℕ} (S : Fin n → 𝒳) (c : ℝ) (f : 𝒳 → ℝ), (Causalean.Stat.Concentration.empiricalNorm S fun x => c * f x) = |c| * Causalean.Stat.Concentration.empiricalNorm S f","l":620,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_def","k":"theorem","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"The empirical norm unfolds to the square root of the average squared sample values.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) (f : 𝒳 → ℝ), Causalean.Stat.Concentration.empiricalNorm S f = √(1 / ↑n * ∑ i, f (S i) ^ 2)","l":34,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalNorm_le_of_forall_abs_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A function whose absolute value is bounded at every point of a finite sample has empirical root-mean-square norm no larger than that bound.","s":"∀ {𝒳 : Type v} {n : ℕ} (S : Fin n → 𝒳) {f : 𝒳 → ℝ} {η : ℝ}, 0 ≤ η → (∀ (j : Fin n), |f (S j)| ≤ η) → Causalean.Stat.Concentration.empiricalNorm S f ≤ η","l":91,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalPMet","k":"def","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":1,"d":"For a sample of $n$ observations, the empirical pseudometric on real-valued functions on the observation space is the pseudometric whose distance between two functions is their empirical distance on that sample.","s":"{𝒳 : Type v} → {n : ℕ} → (Fin n → 𝒳) → PseudoMetricSpace (𝒳 → ℝ)","l":56,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_F_on_univ_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Restricting a finite function family to its full index set leaves its empirical Rademacher complexity unchanged.","s":"∀ {ι' : Type u_2} {Z : Type u_3} [inst : Fintype ι'] {m : ℕ} (H : ι' → Z → ℝ) (S' : Fin m → Z), empiricalRademacherComplexity m (Causalean.Stat.Concentration.F_on H Finset.univ) S' = empiricalRademacherComplexity m H S'","l":587,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_congr_sample","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The empirical Rademacher complexity sees the function family only through its values on the observed sample: two families that agree at every sample point have equal empirical Rademacher complexity.","s":"∀ {𝒳 : Type u_1} {ι : Type u_2} (n : ℕ) (f f' : ι → 𝒳 → ℝ) (sample : Fin n → 𝒳), (∀ (i : ι) (k : Fin n), f i (sample k) = f' i (sample k)) → empiricalRademacherComplexity n f sample = empiricalRademacherComplexity n f' sample","l":73,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_contraction_abs_of_bddAbove","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Ledoux–Talagrand contraction over an arbitrary (possibly infinite) index. For an L-Lipschitz φ with φ 0 = 0 and a class F with a uniform bound M, R̂_n(φ ∘ F) ≤ 2L · R̂_n(F) for *any* nonempty index ι. The infinite-index case is reduced to the finite-index rademacher_contraction_abs by choosing, for each of the finitely many sign vectors, an ε-approximate maximizer; their finite collection is a finite subindex on which the Fintype contraction applies.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → ∀ (F : ι → 𝒳 → ℝ) {M : ℝ}, 0 ≤ M → (∀ (i : ι) (x : 𝒳), |F i x| ≤ M) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => φ (F i x)) S ≤ 2 * L * empiricalRademacherComplexity n F S","l":841,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_eq","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The empirical Rademacher complexity of a function family on a fixed sample is the average, over all sign vectors of the sample length, of the largest absolute sign-weighted sample average attained across the family.","s":"∀ {𝒳 : Type u_1} {ι : Type u_2} (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳), empiricalRademacherComplexity n f S = (↑(Fintype.card (Signs n)))⁻¹ * ∑ σ, ⨆ i, |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * f i (S k)|","l":52,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_measurable_countable","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Rademacher","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":2,"d":"For a countably indexed family of measurable functions, the empirical Rademacher complexity of the family is a measurable function of the sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Countable ι] (F : ι → 𝒳 → ℝ), (∀ (i : ι), Measurable (F i)) → ∀ (n : ℕ), Measurable fun S => empiricalRademacherComplexity n F S","l":552,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_mono_of_inner","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Per-σ, per-i inner-expression bound implies empirical Rademacher comparison.","s":"∀ {𝒳 : Type u_3} {ι' : Type u_4} {n : ℕ} (f g : ι' → 𝒳 → ℝ) (S : Fin n → 𝒳), (∀ (σ : Signs n), BddAbove (Set.range fun i => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * g i (S k)|)) → (∀ (σ : Signs n) (i : ι'), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * f i (S k)| ≤ |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * g i (S k)|) → empiricalRademacherComplexity n f S ≤ empiricalRademacherComplexity n g S","l":134,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_smul_class","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Scalar-multiplication law for empirical Rademacher complexity. Scaling each element of a function class by c scales the absolute-value empirical Rademacher complexity by |c|.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (c : ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => c * F i x) S = |c| * empiricalRademacherComplexity n F S","l":728,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_sub_le","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Empirical Rademacher complexity is sub-additive over differences of classes. For classes F, G sharing the index ι, each with a uniform bound, the complexity of fun i x => F i x - G i x is at most the sum of the individual complexities.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (F G : ι → 𝒳 → ℝ) {MF MG : ℝ}, 0 ≤ MF → 0 ≤ MG → (∀ (i : ι) (x : 𝒳), |F i x| ≤ MF) → (∀ (i : ι) (x : 𝒳), |G i x| ≤ MG) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => F i x - G i x) S ≤ empiricalRademacherComplexity n F S + empiricalRademacherComplexity n G S","l":813,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacherComplexity_without_abs_smul_class","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Without-abs analogue of empiricalRademacherComplexity_smul_class: the signed scaling carries c, not |c|.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (c : ℝ), 0 ≤ c → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity_without_abs n (fun i x => c * F i x) S = c * empiricalRademacherComplexity_without_abs n F S","l":437,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empiricalRademacher_withAbs_finiteClass_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The absolute empirical Rademacher complexity of a finite class is bounded by a Massart logarithmic factor times a common radius.","s":"∀ {ι' : Type u_2} {Z : Type u_3} {m : ℕ}, 0 < m → ∀ (H : ι' → Z → ℝ) (S' : Fin m → Z) (f : Finset ι'), f.Nonempty → ∀ (ρ : ℝ), (∀ i ∈ f, √(∑ k, ((↑m)⁻¹ * |H i (S' k)|) ^ 2) ≤ ρ) → empiricalRademacherComplexity m (Causalean.Stat.Concentration.F_on H f) S' ≤ ρ * √(2 * Real.log (2 * ↑f.card))","l":450,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.empiricalVarianceSlack","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Given real interval endpoints $a$ and $b$, a natural-number sample size $n$, and a real confidence level $\\delta$, the empirical-variance slack is $\\max\\{|a|,|b|\\}^2\\sqrt{\\log(4/\\delta)/(2n)}+2\\max\\{|a|,|b|\\}(b-a)\\sqrt{\\log(4/\\delta)/(2n)}$.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":155,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empirical_abs_withZero_le_no_abs_plus_neg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"If a function class contains the zero function, its empirical Rademacher complexity with absolute values is bounded by the sum of the corresponding unsigned complexities for the class and its negation.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Finite ι] (F : Option ι → 𝒳 → ℝ), (∀ (x : 𝒳), F none x = 0) → ∀ (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n F S ≤ empiricalRademacherComplexity_without_abs n F S + empiricalRademacherComplexity_without_abs n (fun i x => -F i x) S","l":616,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.empirical_bernstein_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Data-driven (empirical) Bernstein confidence interval, miss-probability form. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a,b] with a < b, with population mean m = ∫ f ∂P. Suppose c is a nonnegative bound with f deviating from m by at most c, P-almost everywhere, the population variance σ² = ∫ (f − m)² ∂P is strictly positive, the sample size n is positive, and the confidence level δ lies in (0, 1]. Then the population mean m falls outside the random, data-driven interval [X̄ₙ − ŵ(ω), X̄ₙ + ŵ(ω)] — whose half-width ŵ(ω) = empiricalBernsteinCIHalfWidth S f a b c n δ ω is computed from the observed sample variance via √(V̂ₙ(ω) + τ) — with probability at most 2δ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ {c : ℝ}, 0 ≤ c → (∀ᵐ (x : X) ∂P, |f x - ∫ (y : X), f y ∂P| ≤ c) → 0 < ∫ (x : X), (f x - ∫ (y : X), f y ∂P) ^ 2 ∂P → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.empiricalBernsteinCIHalfWidth S f a b c n δ ω ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ 2 * δ","l":381,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.empirical_dist_eq_zero_of_factor_pattern","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"If two functions induce the same Boolean pattern at every observation in a finite sample, then their empirical distance is zero whenever their values factor through those patterns.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ {i i' : ι}, Causalean.Stat.Concentration.restrictionPattern (π i') S = Causalean.Stat.Concentration.restrictionPattern (π i) S → dist { index := i } { index := i' } = 0","l":80,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.empirical_variance_concentration","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"Data-driven population-variance bound. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. For any confidence level δ in (0, 1] and any positive sample size n, the event that the observed sample variance plus the deterministic slack empiricalVarianceSlack a b n δ falls short of the population variance σ² = ∫ (f − ∫ f)² ∂P has probability at most δ; equivalently, with probability at least 1 − δ the population variance is bounded by the observed sample variance plus that slack.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | S.sampleVariance f n ω + Causalean.Stat.Concentration.empiricalVarianceSlack a b n δ < ∫ (x : X), (f x - ∫ (y : X), f y ∂P) ^ 2 ∂P} ≤ δ","l":178,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.entrywise_gram_lower_bound","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"If an empirical matrix is entrywise close to a scaled population matrix with nonnegative tolerance, each empirical quadratic form is bounded below by the population form minus the dimension-scaled error.","s":"∀ {κ : Type u_1} [inst : Fintype κ] (G M : κ → κ → ℝ) (v : κ → ℝ) {scale epsilon : ℝ}, 0 ≤ epsilon → (∀ (j k : κ), |G j k - scale * M j k| ≤ epsilon) → scale * ∑ j, ∑ k, v j * v k * M j k - ↑(Fintype.card κ) * epsilon * ∑ j, v j ^ 2 ≤ ∑ j, ∑ k, v j * v k * G j k","l":275,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.entrywise_to_quadratic","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"If finite-matrix entries are bounded by a nonnegative tolerance, the absolute quadratic-form error is at most dimension times tolerance times squared vector norm.","s":"∀ {κ : Type u_1} [inst : Fintype κ] (E : κ → κ → ℝ) (v : κ → ℝ) {epsilon : ℝ}, 0 ≤ epsilon → (∀ (j k : κ), |E j k| ≤ epsilon) → |∑ j, ∑ k, v j * v k * E j k| ≤ ↑(Fintype.card κ) * epsilon * ∑ j, v j ^ 2","l":234,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.erm_excess_le_two_uniformDeviation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":2,"d":"ERM basic inequality (deterministic). If ihat beats the comparator istar in empirical risk on the sample X ∘ ω (Rₙ(ihat) ≤ Rₙ(istar)), then its excess population risk is at most twice the uniform deviation of the loss class on that sample.","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (X : Ω → 𝒳), (∀ (i : ι), AEMeasurable (f i ∘ X) μ) → ∀ (ω : Fin n → Ω) {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → ∀ (ihat istar : ι), (↑n)⁻¹ * ∑ k, f ihat (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k)) → ∫ (x : Ω), (fun ω' => f ihat (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ ≤ 2 * uniformDeviation n f μ X (X ∘ ω)","l":38,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.erm_oracle_inequality","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Generic ERM oracle inequality (Rademacher). Consider a countable hypothesis class indexed by ι, evaluated through measurable loss functions f i composed with a measurable data map X, where every loss value is bounded in absolute value by a nonnegative constant b. Let ihat assign to each sample of size n an index that attains empirical risk no larger than that of a fixed comparator istar. Then, provided the McDiarmid tail parameter t satisfies t·b² ≤ 1/2 and ε is nonnegative, the probability, over the n-fold product sample, that the excess population risk of ihat over istar exceeds 4·𝔯ₙ + 2ε — where 𝔯ₙ is the Rademacher complexity of the loss class — is at most exp(−ε²·t·n).","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Countable ι] [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (f i)) → ∀ (X : Ω → 𝒳), Measurable X → ∀ {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → ∀ {t : ℝ}, t * b ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ihat : (Fin n → Ω) → ι) (istar : ι), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, f (ihat ω) (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 • rademacherComplexity n f μ X + 2 * ε < ∫ (x : Ω), (fun ω' => f (ihat ω) (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":118,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.erm_oracle_inequality_separable","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ERMOracle","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Generic ERM oracle inequality (separable class). As in erm_oracle_inequality, but the hypothesis index ι need only be a separable, first-countable topological space rather than countable — the form that covers the (uncountable but separable) L²/L¹-ball linear classes. Given measurable loss functions f i composed with a measurable data map X, with every loss value bounded in absolute value by a nonnegative constant b and each loss value f i x depending continuously on the index i, let ihat assign to each sample of size n an index that attains empirical risk no larger than that of a fixed comparator istar. Then, provided the McDiarmid tail parameter t satisfies t·b² ≤ 1/2 and ε is nonnegative, the probability, over the n-fold product sample, that the excess population risk of ihat over istar exceeds 4·𝔯ₙ + 2ε — where 𝔯ₙ is the Rademacher complexity of the loss class — is at most exp(−ε²·t·n).","s":"∀ {n : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] {ι : Type u_2} {𝒳 : Type u_3} {μ : MeasureTheory.Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [inst_3 : TopologicalSpace ι] [TopologicalSpace.SeparableSpace ι] [FirstCountableTopology ι] [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (f i)) → ∀ (X : Ω → 𝒳), Measurable X → ∀ {b : ℝ}, 0 ≤ b → (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) → (∀ (x : 𝒳), Continuous fun i => f i x) → ∀ {t : ℝ}, t * b ^ 2 ≤ 1 / 2 → ∀ {ε : ℝ}, 0 ≤ ε → ∀ (ihat : (Fin n → Ω) → ι) (istar : ι), (∀ (ω : Fin n → Ω), (↑n)⁻¹ * ∑ k, f (ihat ω) (X (ω k)) ≤ (↑n)⁻¹ * ∑ k, f istar (X (ω k))) → ((MeasureTheory.Measure.pi fun x => μ) fun ω => 4 • rademacherComplexity n f μ X + 2 * ε < ∫ (x : Ω), (fun ω' => f (ihat ω) (X ω')) x ∂μ - ∫ (x : Ω), (fun ω' => f istar (X ω')) x ∂μ).toReal ≤ Real.exp (-ε ^ 2 * t * ↑n)","l":167,"f":"Causalean/Stat/Concentration/UniformDeviation/ERMOracle.lean"},{"n":"Causalean.Stat.Concentration.exists_separating_subsample","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Honest isolated core: the weighted random-coordinate extraction.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < r → 0 < ε → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), 2 ≤ P.card → (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → ∃ m J, ↑m ≤ 1 + 2 * r ^ 2 / ε ^ 2 * Real.log ↑P.card ∧ (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d ∧ Set.InjOn (Causalean.Stat.Concentration.subsamplePattern J) ↑P","l":527,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.exp_le_one_add_add_sq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"Pointwise elementary bound exp u ≤ 1 + u + u² valid for |u| ≤ 1.","s":"∀ {u : ℝ}, |u| ≤ 1 → Real.exp u ≤ 1 + u + u ^ 2","l":66,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.exp_mul_le_secant","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"On the unit interval the exponential function stays below the chord joining its values at the two endpoints: for a number x between zero and one and any tilt s, exp (s * x) is at most 1 + x * (exp s - 1). This is the convexity step behind every Bernoulli-type Chernoff bound.","s":"∀ {x s : ℝ}, x ∈ Set.Icc 0 1 → Real.exp (s * x) ≤ 1 + x * (Real.exp s - 1)","l":74,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.exp_neg_log_two_div","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":2,"d":"For a positive confidence level, exponentiating the negative logarithm of twice its inverse returns half that confidence level.","s":"∀ {δ : ℝ}, 0 < δ → Real.exp (-Real.log (2 / δ)) = δ / 2","l":80,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.finiteSampleMeasure","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"For a sample of $n$ observations in a measurable space, the finite-sample measure assigns equal mass $1/n$ to each observation, equivalently forming the normalized sum of point masses at the observations (with the displayed formula also determining the value when $n=0$).","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → {n : ℕ} → (Fin n → 𝒳) → MeasureTheory.Measure 𝒳","l":23,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.finiteSampleMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The empirical law of a positive-size finite sample is a probability measure.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Concentration.finiteSampleMeasure S)","l":30,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.finite_averaging_exists_separating_subsample","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Finite averaging core for the weighted random-coordinate extraction.","s":"∀ {n : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < r → 0 < ε → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), 2 ≤ P.card → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → ∃ m J, ↑m ≤ 1 + 2 * r ^ 2 / ε ^ 2 * Real.log ↑P.card ∧ Set.InjOn (Causalean.Stat.Concentration.subsamplePattern J) ↑P","l":318,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.finite_iSup_eq_value","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"A real-valued function on a nonempty finite population attains a largest value, and its supremum is that value.","s":"∀ {α : Type u_3} [Nonempty α] [Finite α] (f : α → ℝ), ∃ a, ⨆ x, f x = f a ∧ ∀ (x : α), f x ≤ f a","l":102,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.growthFamily","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"For a family of Boolean-valued classifiers and a sample of $n$ observations, the growth family of the classifier family on the sample is the finite collection of all restriction patterns realized by at least one classifier in the family.","s":"{𝒳 : Type v} → {ι : Type u} → {n : ℕ} → (ι → 𝒳 → Bool) → (Fin n → 𝒳) → Finset (Finset (Fin n))","l":38,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.growthFamilyRep","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a Boolean classifier family, a finite sample, and a realized label pattern on that sample, a representative classifier index is chosen whose labels realize that pattern.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ↥(Causalean.Stat.Concentration.growthFamily π S) → ι","l":649,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.growthFamilyRep_spec","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The chosen growth-family representative realizes the pattern it represents.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳) (A : ↥(Causalean.Stat.Concentration.growthFamily π S)), Causalean.Stat.Concentration.restrictionPattern (π (Causalean.Stat.Concentration.growthFamilyRep π S A)) S = ↑A","l":655,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.growthFamily_card_le_succ_pow_of_trace","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A Boolean class whose trace family has bounded VC dimension, or a direct trace-size bound, realizes no more patterns on a finite sample than a polynomial of degree d in one plus the sample size.","s":"∀ {ι : Type u} {𝒳 : Type v} (π : ι → 𝒳 → Bool) (d n : ℕ), Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → ∀ (S : Fin n → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (n + 1) ^ d","l":405,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.hasCountableEmpiricalSupReduction_of_pointwise_dense","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Separability","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Countable supremum reduction from pointwise density. Let μ be a σ-finite measure on Ω, g : ι → Ω → ℝ a family of functions, and g0 : ℕ → ι a countable subfamily. Suppose S is a μ-conull subset of Ω, on S, every g i is the pointwise limit, along some subsequence, of the countable subfamily g ∘ g0, each g i is measurable, and there is a single μ-integrable envelope G dominating |g i| uniformly in i. Then the countable subfamily indexed by g0 realizes the full continuum empirical-process supremum of g almost surely under every finite product law of μ.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (g : ι → Ω → ℝ) (g0 : ℕ → ι) (S : Set Ω), (∀ᵐ (z : Ω) ∂μ, z ∈ S) → (∀ (i : ι), ∃ kseq, ∀ z ∈ S, Filter.Tendsto (fun m => g (g0 (kseq m)) z) Filter.atTop (nhds (g i z))) → (∀ (i : ι), Measurable (g i)) → (∃ G, MeasureTheory.Integrable G μ ∧ ∀ (i : ι) (z : Ω), |g i z| ≤ G z) → Causalean.Stat.Concentration.HasCountableEmpiricalSupReduction μ g","l":47,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Separability.lean"},{"n":"Causalean.Stat.Concentration.hasPseudoDimAtMost_iff_growthFamily","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The pseudo-dimension certificate unfolds to the existing finite Boolean growth-family VC certificate on every thresholded sample.","s":"∀ {𝒳 : Type u} {ι : Type v} (F : ι → 𝒳 → ℝ) (d : ℕ), Causalean.Stat.Concentration.HasPseudoDimAtMost F d ↔ ∀ (n : ℕ) (T : Fin n → 𝒳 × ℝ), (Causalean.Stat.Concentration.growthFamily (Causalean.Stat.Concentration.subgraphClassifier F) T).vcDim ≤ d","l":35,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.hasSubgaussianMGF_linearCombination_of_iIndep","k":"theorem","m":"Causalean.Stat.Concentration.ConditionalKernel","a":"Stat/Concentration","t":1,"d":"For a finite family of real random variables eps and real coefficients v, if each eps i is a.e. measurable, each eps i is a.s. bounded by 1 in absolute value, each eps i has mean zero, and the family eps is independent, then the linear combination $\\sum_i v_i \\cdot \\mathrm{eps}_i$ is sub-Gaussian with variance proxy $\\sum_i v_i^2$.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] (eps : ι → Ω → ℝ) (v : ι → ℝ), (∀ (i : ι), AEMeasurable (eps i) P) → (∀ (i : ι), ∀ᵐ (ω : Ω) ∂P, |eps i ω| ≤ 1) → (∀ (i : ι), ∫ (ω : Ω), eps i ω ∂P = 0) → ProbabilityTheory.iIndepFun eps P → ProbabilityTheory.HasSubgaussianMGF (fun ω => ∑ i, v i * eps i ω) ⟨∑ i, v i ^ 2, Finset.sum_nonneg fun i x => Even.pow_nonneg (even_two_mul 1) (v i)⟩ P","l":97,"f":"Causalean/Stat/Concentration/ConditionalKernel.lean"},{"n":"Causalean.Stat.Concentration.hoeffdingCIHalfWidth","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Given real endpoints $a$ and $b$, a natural-number sample size $n$, and a real confidence level $\\delta$, the two-sided Hoeffding confidence-interval half-width is $(b-a)\\sqrt{\\log(2/\\delta)/(2n)}$.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":58,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffdingCIHalfWidth_half","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"Halving the confidence level in a Hoeffding half-width replaces its logarithmic factor with log(4/δ), so the resulting half-width can be used in split-confidence and union-bound calculations.","s":"∀ (a b : ℝ) (n : ℕ) {δ : ℝ}, Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n (δ / 2) = (b - a) * √(Real.log (4 / δ) / (2 * ↑n))","l":167,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_abs_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":1,"d":"Two-sided Hoeffding inequality for the sample mean of a bounded statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws deviates from its population mean $E[f]$ by at least ε in absolute value is at most $2\\exp(-2n\\varepsilon^2/(b-a)^2)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ 2 * Real.exp (-2 * ↑n * ε ^ 2 / (b - a) ^ 2)","l":161,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ci_cover","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Hoeffding confidence interval (coverage form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable, if f is almost-everywhere valued in the interval [a, b] with a < b, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at least 1 − δ the sample mean X̄ₙ of f over S's first n draws satisfies |X̄ₙ − m| < w, where w = hoeffdingCIHalfWidth a b n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X') ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 1 - δ ≤ μ.real {ω | |S.sampleMean f n ω - ∫ (x : X'), f x ∂P| < Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n δ}","l":180,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ci_miss","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Finite-sample Hoeffding confidence interval (miss-probability form). Let S be an i.i.d. sample drawn from P, and write m = ∫ f dP for the population mean. If f is measurable, if f is almost-everywhere valued in the interval [a, b] with a < b, if the sample size n is positive, and if the confidence level δ lies in (0, 1], then with probability at most δ the sample mean X̄ₙ of f over S's first n draws satisfies w ≤ |X̄ₙ − m|, where w = hoeffdingCIHalfWidth a b n δ.","s":"∀ {Ω : Type u_1} {X' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X'] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X'} (S : Causalean.Stat.IIDSample Ω X' μ P) {f : X' → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X') ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → μ.real {ω | Causalean.Stat.Concentration.hoeffdingCIHalfWidth a b n δ ≤ |S.sampleMean f n ω - ∫ (x : X'), f x ∂P|} ≤ δ","l":87,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.hoeffding_ge","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":1,"d":"One-sided Hoeffding inequality for the sample mean of a bounded statistic. Let S be an i.i.d. sample and let f be a measurable statistic taking values in an interval [a, b] with a < b P-almost everywhere. Then for any sample size n ≥ 1 and any threshold ε ≥ 0, the probability that the sample mean of f over n draws exceeds its population mean $E[f]$ by at least ε is at most $\\exp(-2n\\varepsilon^2/(b-a)^2)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ∀ {a b : ℝ}, a < b → (∀ᵐ (x : X) ∂P, f x ∈ Set.Icc a b) → ∀ (n : ℕ), 0 < n → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ S.sampleMean f n ω - ∫ (x : X), f x ∂P} ≤ Real.exp (-2 * ↑n * ε ^ 2 / (b - a) ^ 2)","l":89,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.iSup_abs_le_iSup_add_iSup_neg_of_exists_zero","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"If a finite collection of real numbers includes zero, its largest absolute value is at most the sum of its largest value and the largest value after negation.","s":"∀ {α : Type u_3} [Finite α] (x : α → ℝ), (∃ a, x a = 0) → ⨆ a, |x a| ≤ (⨆ a, x a) + ⨆ a, -x a","l":530,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.iid_sum_bernstein_union_bound","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Bernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"For a positive finite sample size, a finite family of measurable, integrable statistics under a probability law, and nonnegative coordinatewise envelopes and variance proxies, suppose each positive coordinate threshold is compared with a statistic whose centered values are bounded by its envelope and whose centered second moment is bounded by its variance proxy. Then the probability that any coordinate sum differs from its population total by at least its own threshold is at most the sum of the corresponding two-sided Bernstein tails.","s":"∀ {N : ℕ} {ι : Type u_3} {X : Type u_4} [inst : Fintype ι] [inst_1 : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (g : ι → X → ℝ), (∀ (a : ι), Measurable (g a)) → (∀ (a : ι), MeasureTheory.Integrable (g a) P) → ∀ (b sigma2 eta : ι → ℝ), (∀ (a : ι), 0 ≤ b a) → (∀ (a : ι), 0 ≤ sigma2 a) → (∀ (a : ι), 0 < eta a) → 0 < N → (∀ (a : ι), ∀ᵐ (x : X) ∂P, |g a x - ∫ (y : X), g a y ∂P| ≤ b a) → (∀ (a : ι), ∫ (x : X), (g a x - ∫ (y : X), g a y ∂P) ^ 2 ∂P ≤ sigma2 a) → (MeasureTheory.Measure.pi fun x => P).real {omega | ∃ a, eta a ≤ |∑ i, g a (omega i) - ↑N * ∫ (x : X), g a x ∂P|} ≤ ∑ a, 2 * Real.exp (-eta a ^ 2 / (2 * (2 * ↑N * sigma2 a + b a * eta a)))","l":261,"f":"Causalean/Stat/Concentration/TailBounds/Bernstein.lean"},{"n":"Causalean.Stat.Concentration.iid_sum_chebyshev","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":1,"d":"Chebyshev concentration of an iid sum. Combining the iid expectation and variance laws with Chebyshev's inequality: for a fixed square-integrable statistic g and any positive threshold ε, the sum ∑ᵢ g(Xᵢ) over an iid sample of size N deviates from its mean N·𝔼[g] by at least ε with probability at most N·Var[g]/ε². For the design weight g = K((·−t)/h) this is the concentration of the total kernel weight M₀₀ around N·𝔼[K((A−t)/h)] = Θ(Nh).","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.MemLp g 2 μ → ∀ {ε : ℝ}, 0 < ε → (MeasureTheory.Measure.pi fun x => μ) {ω | ε ≤ |∑ i, g (ω i) - ↑N * ∫ (x : Ω), g x ∂μ|} ≤ ENNReal.ofReal (↑N * ProbabilityTheory.variance g μ / ε ^ 2)","l":78,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.iid_sum_union_bound","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InverseUnionBound","a":"Stat/Concentration/Matrix","t":1,"d":"Union bound for iid coordinate sums. For a finite family g : ι → Ω → ℝ such that each g a is square-integrable under μ, evaluated on an iid sample of size N drawn from the product law Measure.pi, and for any positive deviation threshold η, the probability that *some* index a has its sample sum ∑ᵢ g a (ωᵢ) deviate from its mean N·𝔼[g a] by at least η is bounded by the sum of the per-index Chebyshev bounds N·Var[g a]/η².","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Ω : Type u_2} [inst_1 : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : ι → Ω → ℝ), (∀ (a : ι), MeasureTheory.MemLp (g a) 2 μ) → ∀ {η : ℝ}, 0 < η → (MeasureTheory.Measure.pi fun x => μ) {ω | ∃ a, η ≤ |∑ i, g a (ω i) - ↑N * ∫ (x : Ω), g a x ∂μ|} ≤ ∑ a, ENNReal.ofReal (↑N * ProbabilityTheory.variance (g a) μ / η ^ 2)","l":28,"f":"Causalean/Stat/Concentration/Matrix/InverseUnionBound.lean"},{"n":"Causalean.Stat.Concentration.instCoeFunEmpiricalFunctionSpaceForallReal","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"For every observation space, sample size, index set, real-valued function class on that space, and sample of that size, the function-evaluation coercion identifies each empirical-function-space element with the function selected by its index; its evaluation rule returns that selected function.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → CoeFun (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S) fun x => 𝒳 → ℝ","l":150,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.instDistEmpiricalFunctionSpace","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"For every observation space, sample size, index set, real-valued function class on that space, and sample of that size, the distance structure on the empirical function space assigns to two selected functions their empirical distance on the sample; its distance rule is that empirical distance.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → Dist (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":163,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.instMeasurableSingletonClassSigns_causalean","k":"instance","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"For every sign-vector length, the measurable-singleton structure for the space of sign vectors of that length certifies that every singleton set of sign vectors is measurable under the product σ-algebra; its measurability rule supplies this certification for each sign vector.","s":"∀ {m : ℕ}, MeasurableSingletonClass (Signs m)","l":35,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.instNonemptySubtypeIntMemFinsetInsertNegOfNatSingleton_causalean","k":"instance","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"For every nonnegative integer, the nonemptiness structure for the two-point sign set $\\{-1,1\\}$ certifies that this set contains at least one integer.","s":"Nonempty ↥{-1, 1}","l":29,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.instPseudoMetricSpaceEmpiricalFunctionSpace","k":"instance","m":"Causalean.Stat.Concentration.Covering.EmpiricalPseudoMetric","a":"Stat/Concentration/Covering","t":2,"d":"For every observation space, sample size, index set, real-valued function class on that space, and sample of that size, the pseudometric-space structure on the empirical function space is the one induced by the empirical pseudometric of the selected functions on that sample.","s":"{𝒳 : Type v} → {n : ℕ} → {ι : Type u} → {F : ι → 𝒳 → ℝ} → {S : Fin n → 𝒳} → PseudoMetricSpace (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":171,"f":"Causalean/Stat/Concentration/Covering/EmpiricalPseudoMetric.lean"},{"n":"Causalean.Stat.Concentration.integral_finiteSampleMeasure","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"Integration against the empirical law is the arithmetic average of a measurable real-valued function over the sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → ∀ {f : 𝒳 → ℝ}, Measurable f → ∫ (x : 𝒳), f x ∂Causalean.Stat.Concentration.finiteSampleMeasure S = 1 / ↑n * ∑ i, f (S i)","l":44,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.integral_log_div_eq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"Exact evaluation of the logarithmic ratio integral on [ε, δ].","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, Real.log (δ / x) = δ - ε - ε * Real.log (δ / ε)","l":90,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.integral_log_div_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"The logarithmic ratio integral over [ε, δ] is at most the interval length.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, Real.log (δ / x) ≤ δ - ε","l":116,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.integral_sum_pi_eq","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":2,"d":"Expectation of an iid sum. For an iid sample of size N drawn from a probability measure μ (modelled by the product measure Measure.pi), the expectation of the sum ∑ᵢ g(Xᵢ) of a fixed integrable statistic g equals N · 𝔼[g].","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.Integrable g μ → (∫ (ω : Fin N → Ω), ∑ i, g (ω i) ∂MeasureTheory.Measure.pi fun x => μ) = ↑N * ∫ (ω : Ω), g ω ∂μ","l":31,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.intervalIntegrable_log_div","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"When ε is positive and no larger than δ, the logarithm of the ratio δ divided by x is interval-integrable from ε to δ with respect to Lebesgue measure.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → IntervalIntegrable (fun x => Real.log (δ / x)) MeasureTheory.volume ε δ","l":74,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.intervalIntegrable_sqrt_log_div","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"When ε is positive and no larger than δ, the square-root log-ratio kernel is integrable on the interval from ε to δ with respect to Lebesgue measure.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → IntervalIntegrable (fun x => √(Real.log (δ / x))) MeasureTheory.volume ε δ","l":48,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a finite index set, a real-valued feature family, a coefficient bound, a coefficient vector in the corresponding box, and an evaluation point, the linear parameter class evaluation is the sum of each coefficient times its corresponding feature value at that point.","s":"{𝒳 : Type u} → {K : Type w} → [Fintype K] → (K → 𝒳 → ℝ) → (B : ℝ) → Causalean.Stat.Concentration.CoeffBox K B → 𝒳 → ℝ","l":26,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass_hasPolynomialL2Cover","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Covering certificate for a bounded-coefficient linear class. Given a finite family of measurable real-valued features, each bounded in absolute value by M, where B is a positive coefficient bound and M is positive, the class of linear combinations of the features with each coefficient constrained to [-B,B] carries a uniform polynomial L² covering certificate at envelope |K|·B·M.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {K : Type w} [inst_1 : Fintype K] [Nonempty K] (φ : K → 𝒳 → ℝ) {B M : ℝ}, 0 < B → 0 < M → (∀ (k : K), Measurable (φ k)) → (∀ (k : K) (x : 𝒳), |φ k x| ≤ M) → Causalean.Stat.Concentration.HasPolynomialL2Cover (Causalean.Stat.Concentration.linearParameterClass φ B) (↑(Fintype.card K) * B * M)","l":133,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linearParameterClass_hasPseudoDimAtMost","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Parametric","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A finite-dimensional linear class has pseudo-dimension bounded by the number of coordinates (the stated +1 leaves room for the affine threshold).","s":"∀ {𝒳 : Type u} [MeasurableSpace 𝒳] {K : Type w} [inst : Fintype K] (φ : K → 𝒳 → ℝ) (B : ℝ), Causalean.Stat.Concentration.HasPseudoDimAtMost (Causalean.Stat.Concentration.linearParameterClass φ B) (Fintype.card K + 1)","l":31,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Parametric.lean"},{"n":"Causalean.Stat.Concentration.linear_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative linear function is sub-root.","s":"∀ {C : ℝ}, 0 ≤ C → Causalean.Stat.Concentration.SubRoot fun r => C * r","l":318,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_abs_entry_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"Each coefficient norm is dominated by the ℓ∞-operator (max-row-sum) norm.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] (A : Matrix α β R) (i : α) (j : β), ‖A i j‖ ≤ ‖A‖","l":56,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_norm_le_of_entry","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"The ℓ∞-operator norm of an entrywise-η-bounded rectangular matrix is at most its number of columns times η.","s":"∀ {p q : ℕ} {A : Matrix (Fin (p + 1)) (Fin (q + 1)) ℝ} {η : ℝ}, (∀ (i : Fin (p + 1)) (j : Fin (q + 1)), |A i j| ≤ η) → ‖A‖ ≤ ↑(q + 1) * η","l":88,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.linftyOp_norm_le_of_rowsum","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.InversePerturbation","a":"Stat/Concentration/Matrix","t":2,"d":"A row-sum upper bound for the ℓ∞-operator norm: if every row's coefficient-norm sum is ≤ c, then ‖A‖ ≤ c.","s":"∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : NormedAddCommGroup R] {A : Matrix α β R} {c : ℝ}, 0 ≤ c → (∀ (i : α), ∑ j, ‖A i j‖ ≤ c) → ‖A‖ ≤ c","l":70,"f":"Causalean/Stat/Concentration/Matrix/InversePerturbation.lean"},{"n":"Causalean.Stat.Concentration.lipschitzAt0_neg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"Negating a function that fixes zero preserves the same Lipschitz constant.","s":"∀ (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → Causalean.Stat.Concentration.LipschitzAt0 (fun x => -φ x) L","l":521,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.localCount","k":"def","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":1,"d":"For a real-valued observation-weight function and a finite sample, the unnormalised local count is the sum of the weights assigned to all observations in the sample.","s":"{N : ℕ} → {X : Type u_1} → (X → ℝ) → (Fin N → X) → ℝ","l":211,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localGramEntry","k":"def","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":1,"d":"For a real-valued observation-weight function, a finite indexed family of real-valued features, a finite sample, and two feature indices, the corresponding entry of the unnormalised weighted empirical Gram matrix is the sum, over observations, of the weight times the two selected feature values.","s":"{N : ℕ} → {κ : Type u_1} → {X : Type u_2} → [Fintype κ] → (X → ℝ) → (κ → X → ℝ) → (Fin N → X) → κ → κ → ℝ","l":215,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localGramQuadratic","k":"def","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":1,"d":"For a real-valued observation-weight function, a finite indexed family of real-valued features, a finite sample, and a real coefficient vector indexed by the features, the quadratic form of the unnormalised weighted empirical Gram matrix is the double sum of the product of two coefficients and their weighted empirical Gram-matrix entry.","s":"{N : ℕ} → {κ : Type u_1} → {X : Type u_2} → [Fintype κ] → (X → ℝ) → (κ → X → ℝ) → (Fin N → X) → (κ → ℝ) → ℝ","l":220,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localGram_centered_envelope","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A localized Gram coordinate with a local weight between zero and one and features bounded by a nonnegative envelope differs from its population mean by at most twice the squared envelope almost everywhere.","s":"∀ {X : Type u_1} {κ : Type u_2} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ) (phi : κ → X → ℝ), (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → ∀ {B : ℝ}, 0 ≤ B → (∀ (j : κ), ∀ᵐ (x : X) ∂P, |phi j x| ≤ B) → ∀ (j k : κ), ∀ᵐ (x : X) ∂P, |q x * phi j x * phi k x - ∫ (y : X), q y * phi j y * phi k y ∂P| ≤ 2 * B ^ 2","l":82,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localGram_centered_secondMoment_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A localized Gram coordinate from measurable functions, a unit-interval local weight, bounded features, and the stated local mass has centered second moment bounded by envelope-to-the-fourth times local mass.","s":"∀ {X : Type u_1} {κ : Type u_2} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ) (phi : κ → X → ℝ), Measurable q → (∀ (j : κ), Measurable (phi j)) → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → ∀ {B p : ℝ}, 0 ≤ B → (∀ (j : κ), ∀ᵐ (x : X) ∂P, |phi j x| ≤ B) → ∫ (x : X), q x ∂P = p → ∀ (j k : κ), ∫ (x : X), (q x * phi j x * phi k x - ∫ (y : X), q y * phi j y * phi k y ∂P) ^ 2 ∂P ≤ B ^ 4 * p","l":119,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localGram_integrable","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A localized Gram coordinate from measurable functions, a unit-interval local weight, and bounded features is integrable under the population law.","s":"∀ {X : Type u_1} {κ : Type u_2} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ) (phi : κ → X → ℝ), Measurable q → (∀ (j : κ), Measurable (phi j)) → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → ∀ {B : ℝ}, 0 ≤ B → (∀ (j : κ), ∀ᵐ (x : X) ∂P, |phi j x| ≤ B) → ∀ (j k : κ), MeasureTheory.Integrable (fun x => q x * phi j x * phi k x) P","l":187,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity","k":"def","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"Given a real-valued family of functions indexed by a set of labels, a real-valued functional measuring the size of such functions, a measure on a sample space equipped with a σ-algebra, a sample-space-valued covariate map, a nonnegative integer sample size, and a real radius, the localized Rademacher complexity is the Rademacher complexity of the star-hull family after functions whose size exceeds that radius are replaced by zero.","s":"{Ω : Type u_1} → {ι : Type u_2} → {𝒳 : Type u_3} → [inst : MeasurableSpace Ω] → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → MeasureTheory.Measure Ω → (Ω → 𝒳) → ℕ → ℝ → ℝ","l":88,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_le_upperBound","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Upper-bound consumption. A RademacherUpperBound ψ directly bounds the localized Rademacher complexity by ψ r for every r ≥ 0 — by definition, since RademacherUpperBound is stated in terms of starHullZeroOut, the same integrand defining localRademacherComplexity.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {μ : MeasureTheory.Measure Ω} {X : Ω → 𝒳} {n : ℕ} {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n ψ → ∀ {r : ℝ}, 0 ≤ r → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r ≤ ψ r","l":268,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_mono_r","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The localized Rademacher complexity is monotone in the radius.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) {r₁ r₂ : ℝ}, r₁ ≤ r₂ → (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r₂ p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r₂) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r₁ ≤ Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r₂","l":229,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacherComplexity_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The localized Rademacher complexity is non-negative.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) (r : ℝ), 0 ≤ Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r","l":98,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localRademacher_le_critical_radius","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"Localized inequality (headline). Suppose the envelope ψ is sub-root: nonnegative, non-decreasing, and with the ratio ψ r / r non-increasing in r, and ψ upper-bounds the localized population Rademacher complexity of the class F (measured by norm, under the sampling law μ, map X, and sample size n) at every nonnegative radius. Writing δ* := criticalRadius ψ, suppose δ* is positive and it satisfies the fixed-point bound ψ δ* ≤ δ*². Then for every radius r ≥ δ*, the localized Rademacher complexity at radius r is at most r · δ*.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {μ : MeasureTheory.Measure Ω} {X : Ω → 𝒳} {n : ℕ} {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n ψ → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius ψ ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius ψ → ψ (Causalean.Stat.Concentration.criticalRadius ψ) ≤ Causalean.Stat.Concentration.criticalRadius ψ ^ 2 → Causalean.Stat.Concentration.localRademacherComplexity F norm μ X n r ≤ r * Causalean.Stat.Concentration.criticalRadius ψ","l":282,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.localWeight_centered_envelope","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A local weight between zero and one almost everywhere, with the stated population mean, differs from that mean by at most one almost everywhere.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ), (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → ∀ {p : ℝ}, ∫ (x : X), q x ∂P = p → ∀ᵐ (x : X) ∂P, |q x - p| ≤ 1","l":35,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localWeight_centered_secondMoment_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A measurable local weight between zero and one almost everywhere, with the stated population mean, has centered second moment at most that local mass.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ), Measurable q → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → ∀ {p : ℝ}, ∫ (x : X), q x ∂P = p → ∫ (x : X), (q x - ∫ (y : X), q y ∂P) ^ 2 ∂P ≤ p","l":57,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localWeight_integrable","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"A measurable local weight between zero and one almost everywhere is integrable under the probability law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ), Measurable q → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → MeasureTheory.Integrable q P","l":26,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localizedGramRate","k":"def","m":"Causalean.Stat.Concentration.Matrix.LocalizedGram","a":"Stat/Concentration/Matrix","t":1,"d":"For a finite feature index set, a real coercivity level, and a real envelope bound, the localized Gram-matrix exponential rate is the smaller of $1/20$ and $\\lambda^2/[16d(4dB^4+B^2\\lambda)]$, where $d$ is the number of feature indices.","s":"(κ : Type u_1) → [Fintype κ] → ℝ → ℝ → ℝ","l":23,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGram.lean"},{"n":"Causalean.Stat.Concentration.localizedGram_failure_subset_deviations","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":2,"d":"Under positive sample size, mass, and coercivity and population coercivity, failure of empirical local-Gram coercivity implies either a large count error or a large entrywise Gram error.","s":"∀ {N : ℕ} {κ : Type u_1} {X : Type u_2} [inst : Fintype κ] [Nonempty κ] (q : X → ℝ) (phi : κ → X → ℝ) (M : κ → κ → ℝ) {p lambda : ℝ}, 0 < N → 0 < p → 0 < lambda → (∀ (v : κ → ℝ), lambda * p * ∑ j, v j ^ 2 ≤ ∑ j, ∑ k, v j * v k * M j k) → {omega | ¬Causalean.Stat.Concentration.LocalizedGramGood lambda q phi omega} ⊆ {omega | ↑N * p / 2 ≤ |Causalean.Stat.Concentration.localCount q omega - ↑N * p| ∨ ∃ j k, lambda * ↑N * p / (4 * ↑(Fintype.card κ)) ≤ |Causalean.Stat.Concentration.localGramEntry q phi omega j k - ↑N * M j k|}","l":391,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localizedGram_good_of_entrywise","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGramBasic","a":"Stat/Concentration/Matrix","t":1,"d":"Given positive sample size, mass, and coercivity, population coercivity, a small count error, and small entrywise Gram errors, the realised local count is positive and the empirical Gram is coercive relative to it.","s":"∀ {N : ℕ} {κ : Type u_1} {X : Type u_2} [inst : Fintype κ] [Nonempty κ] (q : X → ℝ) (phi : κ → X → ℝ) (M : κ → κ → ℝ) {p lambda : ℝ}, 0 < N → 0 < p → 0 < lambda → (∀ (v : κ → ℝ), lambda * p * ∑ j, v j ^ 2 ≤ ∑ j, ∑ k, v j * v k * M j k) → ∀ (omega : Fin N → X), |Causalean.Stat.Concentration.localCount q omega - ↑N * p| < ↑N * p / 2 → (∀ (j k : κ), |Causalean.Stat.Concentration.localGramEntry q phi omega j k - ↑N * M j k| < lambda * ↑N * p / (4 * ↑(Fintype.card κ))) → Causalean.Stat.Concentration.LocalizedGramGood lambda q phi omega","l":316,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGramBasic.lean"},{"n":"Causalean.Stat.Concentration.localizedVCDudleyHypotheses_of_empiricalRadius","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Assemble the deterministic localized Dudley hypotheses for a bounded binary-factored VC class once the samplewise empirical-radius bridge is known.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → (∀ {n : ℕ} (S : Fin n → 𝒳) {r : ℝ}, 0 ≤ r → ∀ (p : Causalean.Stat.Concentration.starHullParam ι), Causalean.Stat.Concentration.empiricalNorm S (Causalean.Stat.Concentration.starHullZeroOut F norm r p) ≤ r) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm","l":229,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.localized_empiricalGram_coercive","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGram","a":"Stat/Concentration/Matrix","t":1,"d":"Given measurable local weights and features, positive sample size, nonnegative local mass, and positive population coercivity, a nonnegative feature envelope, a unit-interval local weight, features bounded by that envelope, the stated local mass, and population Gram coercivity, failure of positive local count or realised-count-relative empirical Gram coercivity has an explicit Bernstein bound with exponent proportional to sample size times local mass.","s":"∀ {N : ℕ} {κ : Type u_1} {X : Type u_2} [inst : Fintype κ] [Nonempty κ] [inst_2 : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ) (phi : κ → X → ℝ), Measurable q → (∀ (j : κ), Measurable (phi j)) → ∀ {p lambda B : ℝ}, 0 < N → 0 ≤ p → 0 < lambda → 0 ≤ B → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → (∀ (j : κ), ∀ᵐ (x : X) ∂P, |phi j x| ≤ B) → ∫ (x : X), q x ∂P = p → (∀ (v : κ → ℝ), lambda * p * ∑ j, v j ^ 2 ≤ ∑ j, ∑ k, v j * v k * ∫ (x : X), q x * phi j x * phi k x ∂P) → (MeasureTheory.Measure.pi fun x => P).real {omega | ¬Causalean.Stat.Concentration.LocalizedGramGood lambda q phi omega} ≤ 2 * (1 + ↑(Fintype.card κ) * ↑(Fintype.card κ)) * Real.exp (-↑N * p * Causalean.Stat.Concentration.localizedGramRate κ lambda B)","l":235,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGram.lean"},{"n":"Causalean.Stat.Concentration.localized_empiricalGram_coercive_of_pos","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.LocalizedGram","a":"Stat/Concentration/Matrix","t":2,"d":"Given measurable local weights and features, positive sample size, local mass, and population coercivity, a nonnegative feature envelope, a unit-interval local weight, features bounded by that envelope, the stated local mass, and population Gram coercivity, failure of positive local count or realised-count-relative empirical Gram coercivity has an explicit Bernstein bound with exponent proportional to sample size times local mass.","s":"∀ {N : ℕ} {κ : Type u_1} {X : Type u_2} [inst : Fintype κ] [Nonempty κ] [inst_2 : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (q : X → ℝ) (phi : κ → X → ℝ), Measurable q → (∀ (j : κ), Measurable (phi j)) → ∀ {p lambda B : ℝ}, 0 < N → 0 < p → 0 < lambda → 0 ≤ B → (∀ᵐ (x : X) ∂P, 0 ≤ q x ∧ q x ≤ 1) → (∀ (j : κ), ∀ᵐ (x : X) ∂P, |phi j x| ≤ B) → ∫ (x : X), q x ∂P = p → (∀ (v : κ → ℝ), lambda * p * ∑ j, v j ^ 2 ≤ ∑ j, ∑ k, v j * v k * ∫ (x : X), q x * phi j x * phi k x ∂P) → (MeasureTheory.Measure.pi fun x => P).real {omega | ¬Causalean.Stat.Concentration.LocalizedGramGood lambda q phi omega} ≤ 2 * (1 + ↑(Fintype.card κ) * ↑(Fintype.card κ)) * Real.exp (-↑N * p * Causalean.Stat.Concentration.localizedGramRate κ lambda B)","l":32,"f":"Causalean/Stat/Concentration/Matrix/LocalizedGram.lean"},{"n":"Causalean.Stat.Concentration.localized_offset_expectation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.LocalizedEnvelopeExpectation","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized offset expectation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], a sample size n at least 1, and a positive upper bound ρ on the positive critical radius criticalRadius (R.ψ n), satisfying the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, with the star-hull Rademacher process almost-surely bounded and integrable at every radius r ≥ ρ and a slack-domination condition bounding the McDiarmid tail by ρ² across the diameter shells up to Rmax. Suppose further that every norm (F i) lies in [0, Rmax], so the sharp deviation bound applies uniformly over the class, that the exponent κ lies strictly between 0 and 1, that the coupling constant A is nonnegative, that the regret radius Δ i is nonnegative for every i, and that the localization radius is dominated by the regret via the margin coupling norm (F i) ≤ A · (Δ i) ^ κ. Then the expectation over the n-fold sample of the supremum over i of the positive part of 2 · |(Pₙ−P)F i| − Δ i / 4 is at most offsetPeelingConstantC (1/8) κ · (16·ρ·A)^{1/(1−κ)} + 10·ρ² + 4·R.b·δ.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {ρ Rmax : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ ρ → 0 < ρ → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (r : ℝ), ρ ≤ r → ∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → (∀ (r : ℝ), ρ ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)) → (∀ (K : ℕ), Rmax ≤ ρ * 2 ^ K → R.b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑n) ≤ ρ ^ 2) → (∀ (i : ι), 0 ≤ norm (F i)) → (∀ (i : ι), norm (F i) ≤ Rmax) → ∀ (Δ : ι → ℝ) (κ A : ℝ), 0 < κ → κ < 1 → 0 ≤ A → (∀ (i : ι), 0 ≤ Δ i) → (∀ (i : ι), norm (F i) ≤ A * Δ i ^ κ) → (∫ (ω : Fin n → Ω), ⨆ i, max 0 (2 * |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| - Δ i / 4) ∂MeasureTheory.Measure.pi fun x => μ) ≤ Causalean.Mathlib.OffsetPeeling.offsetPeelingConstantC (1 / 8) κ * (16 * ρ * A) ^ (1 / (1 - κ)) + 10 * ρ ^ 2 + 4 * R.b * δ","l":255,"f":"Causalean/Stat/Concentration/UniformDeviation/LocalizedEnvelopeExpectation.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized uniform deviation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], and a sample size n at least 1. If the radius r restricting the class to {i : norm (F i) ≤ r} is at least the population critical radius criticalRadius (R.ψ n), itself positive, the envelope satisfies the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, and the star-hull Rademacher process at radius r is almost-surely bounded and its empirical complexity integrable, then there is a measurable event of probability at least 1 − δ on which, simultaneously for every i with norm (F i) ≤ r, the empirical mean of F i deviates from its population mean by at most 4 · r · criticalRadius (R.ψ n) + R.b · √(2 · log(1/δ) / n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), norm (F i) ≤ r → |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| ≤ 4 * r * Causalean.Stat.Concentration.criticalRadius (R.ψ n) + R.b * √(2 * Real.log (1 / δ) / ↑n)","l":94,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation_expectation","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.LocalizedEnvelopeExpectation","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Localized uniform deviation, in expectation. Fix a localized regime R built from measurable losses F i composed with a measurable data map X, and a sample size n at least 1. If the radius r restricting the class to {i : norm (F i) ≤ r} is at least the population critical radius criticalRadius (R.ψ n), itself positive, the envelope R.ψ n satisfies the sub-root self-bounding condition R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2 at that radius, and the empirical Rademacher complexity of the radius-r star-hull is integrable, then the expectation, over the n-fold sample, of the uniform deviation of the radius-r-restricted class is at most 2·r·criticalRadius (R.ψ n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) (n : ℕ), 0 < n → ∀ {r : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ r → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → (∫ (ω : Fin n → Ω), uniformDeviation n (fun i ω' => if norm (F i) ≤ r then F i (X ω') else 0) μ id (id ∘ ω) ∂MeasureTheory.Measure.pi fun x => μ) ≤ 2 * r * Causalean.Stat.Concentration.criticalRadius (R.ψ n)","l":67,"f":"Causalean/Stat/Concentration/UniformDeviation/LocalizedEnvelopeExpectation.lean"},{"n":"Causalean.Stat.Concentration.localized_uniform_deviation_sharp","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.UniformDeviationLocalized","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Sharp localized uniform deviation. Fix a localized regime R built from measurable losses F i composed with a measurable map X, a confidence level δ in (0,1], and a sample size n at least 1. Let ρ be a positive upper bound on the positive population critical radius criticalRadius (R.ψ n), satisfying the sub-root fixed-point bound R.ψ n (criticalRadius (R.ψ n)) ≤ criticalRadius (R.ψ n) ^ 2, and suppose the star-hull Rademacher process is almost-surely bounded and its empirical complexity integrable at every radius r ≥ ρ. If, further, for every peeling level K covering the diameter cap Rmax ≤ ρ · 2^K, the McDiarmid slack R.b · √(2 · log(2(K+1)/δ) / n) at confidence 1 − δ is itself dominated by ρ², then there is a measurable event of probability at least 1 − δ on which, simultaneously for every i with 0 ≤ norm (F i) ≤ Rmax, the empirical mean of F i deviates from its population mean by at most 8 · ρ · norm (F i) + 5 · ρ².","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], Measurable X → (∀ (i : ι), Measurable (F i)) → ∀ (R : Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X) {δ : ℝ}, 0 < δ → δ ≤ 1 → ∀ (n : ℕ), 0 < n → ∀ {ρ Rmax : ℝ}, Causalean.Stat.Concentration.criticalRadius (R.ψ n) ≤ ρ → 0 < ρ → 0 < Causalean.Stat.Concentration.criticalRadius (R.ψ n) → R.ψ n (Causalean.Stat.Concentration.criticalRadius (R.ψ n)) ≤ Causalean.Stat.Concentration.criticalRadius (R.ψ n) ^ 2 → (∀ (r : ℝ), ρ ≤ r → ∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → (∀ (r : ℝ), ρ ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)) → (∀ (K : ℕ), Rmax ≤ ρ * 2 ^ K → R.b * √(2 * Real.log (2 * (↑K + 1) / δ) / ↑n) ≤ ρ ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ Rmax → |(↑n)⁻¹ * ∑ k, F i (X (ω k)) - ∫ (x : Ω), (fun ω' => F i (X ω')) x ∂μ| ≤ 8 * ρ * norm (F i) + 5 * ρ ^ 2","l":341,"f":"Causalean/Stat/Concentration/UniformDeviation/UniformDeviationLocalized.lean"},{"n":"Causalean.Stat.Concentration.log_coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"Logarithmic Sauer-Shelah covering-number bound. Under the binary-factoring hypothesis F i (S j) = φ j (π i (S j)), with the sample-restricted class totally bounded in the empirical pseudometric, a positive covering radius ε, a positive sample size n, and VC dimension at most d for the induced Boolean growth family on the sample, the logarithm of the empirical covering number at radius ε is at most log(d+1) + d·log n.","s":"∀ {𝒳 : Type v} {ι : Type u} {n d : ℕ} [Nonempty ι] {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → 0 < n → (Causalean.Stat.Concentration.growthFamily π S).vcDim ≤ d → Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' ε) ≤ Real.log ↑(d + 1) + ↑d * Real.log ↑n","l":234,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.log_coveringNumber_le_of_card_bound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"A direct cardinality bound on the Boolean growth family gives the same logarithmic bound on the empirical covering number.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} [Nonempty ι] {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → ∀ {N : ℕ}, (Causalean.Stat.Concentration.growthFamily π S).card ≤ N → Real.log ↑(Causalean.Stat.Concentration.coveringNumber h' ε) ≤ Real.log ↑N","l":195,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.log_div_nonneg_of_mem_Icc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"On the interval [ε, δ], the log-ratio kernel is nonnegative.","s":"∀ {δ ε x : ℝ}, 0 < ε → x ∈ Set.Icc ε δ → 0 ≤ Real.log (δ / x)","l":27,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.log_two_div_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.ConfidenceInterval","a":"Stat/Concentration/UniformDeviation","t":2,"d":"For a confidence level between zero and one, the logarithm of twice its inverse is nonnegative.","s":"∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → 0 ≤ Real.log (2 / δ)","l":73,"f":"Causalean/Stat/Concentration/UniformDeviation/ConfidenceInterval.lean"},{"n":"Causalean.Stat.Concentration.log_two_growth_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"When a Boolean growth family on a sample has positive cardinality and at most the degree-d polynomial number of patterns in one plus the sample size, twice the logarithm of twice its cardinality is no greater than twice that degree times the logarithm of one plus the sample size, plus two.","s":"∀ {ι : Type u} {𝒳 : Type v} (π : ι → 𝒳 → Bool) (d n : ℕ) (S : Fin n → 𝒳), 0 < (Causalean.Stat.Concentration.growthFamily π S).card → (Causalean.Stat.Concentration.growthFamily π S).card ≤ (n + 1) ^ d → 2 * Real.log (2 * ↑(Causalean.Stat.Concentration.growthFamily π S).card) ≤ 2 * ↑d * Real.log (↑n + 1) + 2","l":419,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.massart_lemma_pmf","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":1,"d":"Massart's finite-class lemma. Given a nonempty finite subset f of the index set selecting finitely many functions from the class, the empirical Rademacher complexity (without absolute value) of that finite subclass, evaluated at the sample S of size m, is at most the largest per-function coordinate $\\ell^2$-radius $\\sqrt{\\sum_i (F_j(S_i)/m)^2}$ over j ∈ f, times $\\sqrt{2\\log|f|}$.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} (F : ι → Z → ℝ) (S : Fin m → Z) (f : Finset ι) (hs : f.Nonempty), empiricalRademacherComplexity_pmf_without_abs m (Causalean.Stat.Concentration.F_on F f) S ≤ (f.sup' hs fun j => √(∑ i, ((↑m)⁻¹ * |F j (S i)|) ^ 2)) * √(2 * Real.log ↑f.card)","l":215,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.massart_lemma_pmf.sign_mean_zero","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"A single Rademacher-signed sample value has mean zero.","s":"∀ {Z : Type v} {m : ℕ} (f : Z → ℝ) (S : Fin m → Z) (a : Fin m), ∫ (ω : Signs m), ↑↑(ω a) * f (S a) ∂(signVecPMF m).toMeasure = 0","l":197,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.matrix_rank_le_of_gram_factor","k":"theorem","m":"Causalean.Stat.Concentration.ProjectionMatrixTail","a":"Stat/Concentration","t":2,"d":"A matrix admitting a Gram factorization through r vectors has rank at most r.","s":"∀ {R : Type u_1} {ι : Type u_2} [Nontrivial R] [inst : CommRing R] [inst_1 : Fintype ι] {r : ℕ} (Pi : Matrix ι ι R) (vectors : Fin r → ι → R), (∀ (i j : ι), Pi i j = ∑ k, vectors k i * vectors k j) → Pi.rank ≤ r","l":29,"f":"Causalean/Stat/Concentration/ProjectionMatrixTail.lean"},{"n":"Causalean.Stat.Concentration.measurablespace_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The sign-vector measurable space agrees with the product measurable space.","s":"∀ {m : ℕ}, instMeasurableSpaceSigns m = MeasurableSpace.pi","l":64,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.measureL2Dist","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a measure $Q$ on an observation space and two real-valued functions on that space, their measure-based $L^2$ semidistance is $\\sqrt{\\int (f-g)^2\\,dQ}$.","s":"{𝒳 : Type u} → [inst : MeasurableSpace 𝒳] → MeasureTheory.Measure 𝒳 → (𝒳 → ℝ) → (𝒳 → ℝ) → ℝ","l":43,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.measureL2Dist_finiteSampleMeasure_eq_empiricalDist","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"L² distance under the empirical law agrees exactly with the empirical root-mean-square distance used by the existing Dudley API.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {n : ℕ} (S : Fin n → 𝒳), 0 < n → ∀ {f g : 𝒳 → ℝ}, Measurable f → Measurable g → Causalean.Stat.Concentration.measureL2Dist (Causalean.Stat.Concentration.finiteSampleMeasure S) f g = Causalean.Stat.Concentration.empiricalDist S f g","l":59,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.measureReal_abs_dev_le_two_sided","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"Generic two-sided assembly: the two-sided deviation event {ε ≤ |T − m|} is covered by the two one-sided events, so its measure is at most the sum of their one-sided bounds. Reused by the Hoeffding and Bernstein two-sided tail bounds.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (T : Ω → ℝ) (m Bup Blow ε : ℝ), μ.real {ω | ε ≤ T ω - m} ≤ Bup → μ.real {ω | ε ≤ -T ω + m} ≤ Blow → μ.real {ω | ε ≤ |T ω - m|} ≤ Bup + Blow","l":141,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.measure_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Massart","a":"Stat/Concentration/TailBounds","t":2,"d":"The Rademacher sign-vector law agrees with the product of uniform two-point coordinate laws.","s":"∀ {m : ℕ}, (signVecPMF m).toMeasure ≍ MeasureTheory.Measure.pi fun x => (PMF.uniformOfFintype ↥{-1, 1}).toMeasure","l":73,"f":"Causalean/Stat/Concentration/TailBounds/Massart.lean"},{"n":"Causalean.Stat.Concentration.measure_exists_abs_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Maximal inequality (existential form). If every member Y i, i ∈ t, of a finite family indexed by t has a sub-exponential moment-generating function with common parameters (v, b) with respect to μ and ε is nonnegative, then the probability that |Y i| ≥ ε for at least one i ∈ t is at most the union bound card t · 2 exp(−ε² / (2 (v + b ε))).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (t : Finset ι) (Y : ι → Ω → ℝ) {v b : NNReal}, (∀ i ∈ t, Causalean.Stat.Concentration.HasSubexponentialMGF (Y i) v b μ) → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ∃ i ∈ t, ε ≤ |Y i ω|} ≤ ↑t.card * (2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε))))","l":69,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.measure_projection_energy_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.ProjectionMatrixTail","a":"Stat/Concentration","t":1,"d":"Projected bounded-noise tail bound. Let eps : Fin n → Ω → ℝ be coordinate noise terms such that each eps i is measurable, each is bounded by 1 in absolute value almost surely, each has mean zero, and the coordinates are mutually independent. If the n × n matrix Pi is symmetric and idempotent — so it is an orthogonal projection — and has rank at most r, then for any nonnegative t, the squared Euclidean norm of the projected vector Pi · eps exceeds 4 * t ^ 2 with probability at most 5 ^ r * exp (-t ^ 2 / 2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n r : ℕ} (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] (eps : Fin n → Ω → ℝ) (Pi : Matrix (Fin n) (Fin n) ℝ), (∀ (i : Fin n), AEMeasurable (eps i) P) → (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂P, |eps i ω| ≤ 1) → (∀ (i : Fin n), ∫ (ω : Ω), eps i ω ∂P = 0) → ProbabilityTheory.iIndepFun eps P → Pi.transpose = Pi → Pi * Pi = Pi → Pi.rank ≤ r → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 4 * t ^ 2 < ∑ i, (∑ j, Pi i j * eps j ω) ^ 2} ≤ 5 ^ r * Real.exp (-t ^ 2 / 2)","l":45,"f":"Causalean/Stat/Concentration/ProjectionMatrixTail.lean"},{"n":"Causalean.Stat.Concentration.measure_sup'_ge_le","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.MaximalInequality","a":"Stat/Concentration/TailBounds","t":1,"d":"Maximal inequality (sup' form). If t is a nonempty finite index set, every member Y i, i ∈ t, has a sub-exponential moment-generating function with common parameters (v, b) with respect to μ, and ε is nonnegative, then the probability that the pointwise maximum of |Y i| over i ∈ t is at least ε is at most the union bound card t · 2 exp(−ε² / (2 (v + b ε))).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (t : Finset ι) (ht : t.Nonempty) (Y : ι → Ω → ℝ) {v b : NNReal}, (∀ i ∈ t, Causalean.Stat.Concentration.HasSubexponentialMGF (Y i) v b μ) → ∀ {ε : ℝ}, 0 ≤ ε → μ.real {ω | ε ≤ t.sup' ht fun i => |Y i ω|} ≤ ↑t.card * (2 * Real.exp (-ε ^ 2 / (2 * (↑v + ↑b * ε))))","l":91,"f":"Causalean/Stat/Concentration/TailBounds/MaximalInequality.lean"},{"n":"Causalean.Stat.Concentration.mem_growthFamily_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"Membership in the growth family means that some classifier realizes that restriction pattern on the sample.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {π : ι → 𝒳 → Bool} {S : Fin n → 𝒳} {A : Finset (Fin n)}, A ∈ Causalean.Stat.Concentration.growthFamily π S ↔ ∃ i, Causalean.Stat.Concentration.restrictionPattern (π i) S = A","l":47,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.mem_starHull_self","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"Each F i belongs to its own star hull (take α = 1).","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), F i ∈ Causalean.Stat.Concentration.starHull F","l":48,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.mgf_eq_of_mem_zero_one","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"For a statistic that only takes the values zero and one, the moment generating function is exactly 1 + p * (exp s - 1), where p is the probability that the statistic equals one.","s":"∀ {𝒳 : Type u_2} [inst : MeasurableSpace 𝒳] {P : MeasureTheory.Measure 𝒳} [MeasureTheory.IsProbabilityMeasure P] {f : 𝒳 → ℝ}, AEMeasurable f P → (∀ᵐ (x : 𝒳) ∂P, f x = 0 ∨ f x = 1) → ∀ (p s : ℝ), ∫ (x : 𝒳), f x ∂P = p → ProbabilityTheory.mgf f P s = 1 + p * (Real.exp s - 1)","l":122,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.mgf_le_of_mem_Icc_zero_one","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.BinomialCount","a":"Stat/Concentration/TailBounds","t":2,"d":"The moment generating function of a measurable statistic taking values in the unit interval is at most exp (mean * (exp s - 1)), where mean is the statistic's population mean. This holds at every tilt s, positive or negative.","s":"∀ {𝒳 : Type u_2} [inst : MeasurableSpace 𝒳] {P : MeasureTheory.Measure 𝒳} [MeasureTheory.IsProbabilityMeasure P] {f : 𝒳 → ℝ}, AEMeasurable f P → (∀ᵐ (x : 𝒳) ∂P, f x ∈ Set.Icc 0 1) → ∀ (s : ℝ), ProbabilityTheory.mgf f P s ≤ Real.exp ((∫ (x : 𝒳), f x ∂P) * (Real.exp s - 1))","l":87,"f":"Causalean/Stat/Concentration/TailBounds/BinomialCount.lean"},{"n":"Causalean.Stat.Concentration.norm_unitInv_sub_unitInv_le","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.Resolvent","a":"Stat/Concentration/Matrix","t":1,"d":"Resolvent inequality. For a seminormed ring R and units a and b of R, the norm of the difference of their inverses is bounded above by the product of the norms of the two inverses and the norm of their difference.","s":"∀ {R : Type u_1} [inst : SeminormedRing R] (a b : Rˣ), ‖↑a⁻¹ - ↑b⁻¹‖ ≤ ‖↑a⁻¹‖ * ‖↑b⁻¹‖ * ‖↑a - ↑b‖","l":27,"f":"Causalean/Stat/Concentration/Matrix/Resolvent.lean"},{"n":"Causalean.Stat.Concentration.partB.mem_incrementPairFinset_repr","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"Every increment-pair representative comes from two adjacent chain approximations for some function in the class.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [inst : Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} (c_pos : 0 < c) (h' : TotallyBounded Set.univ) (n : ℕ) (j : Fin n), ∀ hk ∈ Causalean.Stat.Concentration.incrementPairFinset✝ c_pos h' n j, ∃ fh, (Causalean.Stat.Concentration.chainApprox✝ c_pos h' fh (↑j + 1), Causalean.Stat.Concentration.chainApprox✝ c_pos h' fh ↑j) = hk","l":1065,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.patternCover","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"For a family of real-valued functions, a family of Boolean-valued classifiers, and a sample of $n$ observations, the pattern cover contains one empirical-function representative for each Boolean restriction pattern realized by the classifier family on that sample.","s":"{𝒳 : Type v} → {ι : Type u} → {n : ℕ} → {F : ι → 𝒳 → ℝ} → (ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → Finset (Causalean.Stat.Concentration.EmpiricalFunctionSpace F S)","l":100,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.patternCover_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The pattern representative cover has no more elements than the realized Boolean growth family.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳), (Causalean.Stat.Concentration.patternCover π S).card ≤ (Causalean.Stat.Concentration.growthFamily π S).card","l":112,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.patternCover_covers","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The pattern representative cover covers every empirical function at every positive radius when the real class factors through the Boolean pattern.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ {ε : ℝ}, 0 < ε → Set.univ ⊆ ⋃ y ∈ Causalean.Stat.Concentration.patternCover π S, Metric.ball y ε","l":128,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.per_pair_collision_sum","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"The weighted sum over all fixed-length coordinate selections that agree under two Boolean patterns equals the corresponding power of the total weight of their agreeing coordinates.","s":"∀ {n m : ℕ} (w : Fin n → ℝ) (a b : Fin n → Bool), (∑ J, (∏ t, w (J t)) * if ∀ (t : Fin m), a (J t) = b (J t) then 1 else 0) = (∑ j with a j = b j, w j) ^ m","l":113,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.pointwise_bound_from_empirical_norm","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"If every function has empirical norm at most a given bound on a positive-size sample, then its absolute value at each sampled observation is at most the sample-size square root times that bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (i : Fin m), 0 < m → ∀ (f : ι), |F f (S i)| ≤ √↑m * c","l":159,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.probability_abs_sub_mean_gt_le","k":"theorem","m":"Causalean.Stat.Concentration.Chebyshev","a":"Stat/Concentration","t":1,"d":"For a finite measure Q and a statistic F on the underlying space, if F is square-integrable under Q, the deviation threshold a is positive, F has mean equal to mean under Q, and the variance of F under Q is at most the envelope v, then the probability that F deviates from mean by more than a in absolute value is at most v/a².","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure Q] (F : Ω → ℝ) (mean v a : ℝ), MeasureTheory.MemLp F 2 Q → 0 < a → ∫ (x : Ω), F x ∂Q = mean → ProbabilityTheory.variance F Q ≤ v → (Q {x | a < |F x - mean|}).toReal ≤ v / a ^ 2","l":23,"f":"Causalean/Stat/Concentration/Chebyshev.lean"},{"n":"Causalean.Stat.Concentration.pseudoDim_gives_finite_subgraph_vc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The real-valued theorem genuinely reuses the existing finite VC combinatorics: its hypothesis gives the exact threshold-trace certificate needed by VCCovering and HausslerPacking.","s":"∀ {𝒳 : Type u} {ι : Type v} {F : ι → 𝒳 → ℝ} {d n : ℕ}, Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ (T : Fin n → 𝒳 × ℝ), (Causalean.Stat.Concentration.growthFamily (Causalean.Stat.Concentration.subgraphClassifier F) T).vcDim ≤ d","l":607,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_congr_ae","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":1,"d":"The population Rademacher complexity is unchanged when each member of the function family is replaced by an almost-everywhere-equal version: if for every index i, f i ∘ sample equals f' i ∘ sample almost everywhere with respect to μ, then the two families f and f' have equal Rademacher complexity.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [Countable ι] (n : ℕ) (f f' : ι → 𝒳 → ℝ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (sample : Ω → 𝒳), (∀ (i : ι), (fun ω => f i (sample ω)) =ᵐ[μ] fun ω => f' i (sample ω)) → rademacherComplexity n f μ sample = rademacherComplexity n f' μ sample","l":92,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_congr_ae_all","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The population Rademacher complexity is unchanged under an almost-everywhere agreement of the whole family at once: if almost every ω satisfies f i (sample ω) = f' i (sample ω) for all indices i simultaneously, the two families have equal Rademacher complexity.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] (n : ℕ) (f f' : ι → 𝒳 → ℝ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] (sample : Ω → 𝒳), (∀ᵐ (ω : Ω) ∂μ, ∀ (i : ι), f i (sample ω) = f' i (sample ω)) → rademacherComplexity n f μ sample = rademacherComplexity n f' μ sample","l":120,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_eq","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Rademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"The population Rademacher complexity of a function family is the mean, over samples of independent draws, of the empirical Rademacher complexity of that family on the drawn sample.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] (n : ℕ) (f : ι → 𝒳 → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳), rademacherComplexity n f μ X = ∫ (ω : Fin n → Ω), empiricalRademacherComplexity n f (X ∘ ω) ∂MeasureTheory.Measure.pi fun x => μ","l":63,"f":"Causalean/Stat/Concentration/Rademacher/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacherComplexity_zeroOut_le_starHullZeroOut","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.LocalRademacher","a":"Stat/Concentration/Rademacher","t":2,"d":"Bridge lemma: ι-indexed zero-out class ≤ starHullParam ι-indexed zero-out class.","s":"∀ {Ω : Type u_1} {ι : Type u_2} {𝒳 : Type u_3} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) (n : ℕ) {r : ℝ}, (∀ (S : Fin n → 𝒳) (σ : Signs n), BddAbove (Set.range fun p => |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S k)|)) → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (X ∘ ω)) (MeasureTheory.Measure.pi fun x => μ) → rademacherComplexity n (fun i ω => if norm (F i) ≤ r then F i (X ω) else 0) μ id ≤ rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X","l":314,"f":"Causalean/Stat/Concentration/Rademacher/LocalRademacher.lean"},{"n":"Causalean.Stat.Concentration.rademacher_contraction","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"Ledoux–Talagrand contraction principle (signed form). If L is nonnegative and φ : ℝ → ℝ is L-Lipschitz, i.e. |φ x - φ y| ≤ L * |x - y| for all x, y, then composing each function of the family F with φ does not increase the signed (without-abs) empirical Rademacher complexity on the sample S by more than the factor L.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L : ℝ}, 0 ≤ L → (∀ (x y : ℝ), |φ x - φ y| ≤ L * |x - y|) → ∀ (F : ι → 𝒳 → ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity_without_abs n (fun i x => φ (F i x)) S ≤ L * empiricalRademacherComplexity_without_abs n F S","l":500,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.rademacher_contraction_abs","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":1,"d":"Contraction principle, absolute-value form. If φ fixes 0 and is L-Lipschitz, i.e. φ 0 = 0 and |φ x - φ y| ≤ L * |x - y| for all x, y, then composing each function of the family F with φ multiplies the (absolute-value) empirical Rademacher complexity on the sample S by at most 2 * L.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L : ℝ}, Causalean.Stat.Concentration.LipschitzAt0 φ L → ∀ (F : ι → 𝒳 → ℝ) (n : ℕ) (S : Fin n → 𝒳), empiricalRademacherComplexity n (fun i x => φ (F i x)) S ≤ 2 * L * empiricalRademacherComplexity n F S","l":663,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_dudley_example","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Direct application to the fixed-sample Dudley bound. For a measurable class of pseudo-dimension at most d, uniformly bounded by a positive envelope U, and a positive scale δ strictly less than U/2, evaluated on a sample S of positive size n, the class's empirical restriction is totally bounded, its covering number at every relative radius ε in (0,1] obeys the same polynomial bound vcSubgraphCoverBound d ε, and consequently its empirical Rademacher complexity without the outer absolute value is at most the Dudley entropy-integral bound 4δ + (12/√n) ∫_δ^(U/2) √(log(coveringNumber x)) dx.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U δ : ℝ}, 0 < U → 0 < δ → δ < U / 2 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → ∃ (htot : TotallyBounded Set.univ), (∀ (ε : ℝ), 0 < ε → ε ≤ 1 → Causalean.Stat.Concentration.coveringNumber htot (ε * U) ≤ Causalean.Stat.Concentration.vcSubgraphCoverBound d ε) ∧ empiricalRademacherComplexity_without_abs n F S ≤ 4 * δ + 12 / √↑n * ∫ (x : ℝ) in δ..U / 2, √(Real.log ↑(Causalean.Stat.Concentration.coveringNumber htot x))","l":165,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_coveringNumber_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The empirical covering number used by Dudley is polynomially bounded at every envelope-relative radius, with the same arbitrary-measure constants.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [inst_1 : Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ} (hmeas : ∀ (i : ι), Measurable (F i)) (hpdim : Causalean.Stat.Concentration.HasPseudoDimAtMost F d) {U ε : ℝ} (hU : 0 < U), 0 < ε → ε ≤ 1 → ∀ (henvelope : ∀ (i : ι) (x : 𝒳), |F i x| ≤ U) (S : Fin n → 𝒳) (hn : 0 < n), have htot := Causalean.Stat.Concentration.real_vcSubgraph_empirical_totallyBounded hmeas hpdim hU henvelope S hn; Causalean.Stat.Concentration.coveringNumber htot (ε * U) ≤ Causalean.Stat.Concentration.vcSubgraphCoverBound d ε","l":123,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_l2_covering","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"The arbitrary-measure VC-subgraph theorem specializes to the empirical law of every positive-size finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U ε : ℝ}, 0 < U → 0 < ε → ε ≤ 1 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → Causalean.Stat.Concentration.L2CoveringNumberLe (Causalean.Stat.Concentration.finiteSampleMeasure S) F (ε * U) (Causalean.Stat.Concentration.vcSubgraphCoverBound d ε)","l":69,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_empirical_totallyBounded","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Empirical","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A pseudo-dimension certificate makes the existing empirical function space totally bounded on every positive-size finite sample.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] {F : ι → 𝒳 → ℝ} {d n : ℕ}, (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U : ℝ}, 0 < U → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (S : Fin n → 𝒳), 0 < n → TotallyBounded Set.univ","l":85,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Empirical.lean"},{"n":"Causalean.Stat.Concentration.real_vcSubgraph_l2_covering","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Polynomial L²(Q) covering number from a pseudo-dimension bound. For a family of measurable real-valued functions of pseudo-dimension at most d, uniformly bounded by a positive envelope U, and a relative radius ε with 0 < ε ≤ 1, the L²(Q) covering number at radius ε·U is at most vcSubgraphCoverBound d ε, uniformly over every probability measure Q on the domain.","s":"∀ {𝒳 : Type u} {ι : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] (F : ι → 𝒳 → ℝ) (d : ℕ), (∀ (i : ι), Measurable (F i)) → Causalean.Stat.Concentration.HasPseudoDimAtMost F d → ∀ {U ε : ℝ}, 0 < U → 0 < ε → ε ≤ 1 → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → ∀ (Q : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure Q], Causalean.Stat.Concentration.L2CoveringNumberLe Q F (ε * U) (Causalean.Stat.Concentration.vcSubgraphCoverBound d ε)","l":544,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.restrictionPattern","k":"def","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"For a Boolean-valued classifier and a sample of $n$ observations, the restriction pattern of the classifier on the sample is the finite set of sample coordinates at which the classifier is true.","s":"{𝒳 : Type v} → {n : ℕ} → (𝒳 → Bool) → (Fin n → 𝒳) → Finset (Fin n)","l":30,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.restrictionPattern_mem_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"Membership in a restriction pattern means the classifier is true at that sample coordinate.","s":"∀ {𝒳 : Type v} {n : ℕ} {p : 𝒳 → Bool} {S : Fin n → 𝒳} {j : Fin n}, j ∈ Causalean.Stat.Concentration.restrictionPattern p S ↔ p (S j) = true","l":55,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.sampleMean_sub_ge_setEq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"For n > 0, the centered sample-mean tail event coincides with the centered-sum tail event: ε ≤ X̄ₙ − m ⟺ n ε ≤ ∑_{i<n} (f (Z i) − m). Reused by both the Hoeffding and Bernstein tail bounds.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (m : ℝ) {n : ℕ}, 0 < n → ∀ (ε : ℝ), {ω | ε ≤ S.sampleMean f n ω - m} = {ω | ↑n * ε ≤ ∑ i ∈ Finset.range n, (f (S.Z i ω) - m)}","l":67,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.Concentration.sample_eq_growthFamilyRep_of_pattern","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"When a function class factorizes samplewise through Boolean labels, any function with a given observed Boolean pattern agrees on the sample with that pattern's chosen representative.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} {F : ι → 𝒳 → ℝ} {π : ι → 𝒳 → Bool} {S : Fin n → 𝒳} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (A : ↥(Causalean.Stat.Concentration.growthFamily π S)) {i : ι}, Causalean.Stat.Concentration.restrictionPattern (π i) S = ↑A → ∀ (k : Fin n), F i (S k) = F (Causalean.Stat.Concentration.growthFamilyRep π S A) (S k)","l":758,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.self_log_solve","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A self-referential logarithmic inequality implies an explicit linear-log upper bound.","s":"∀ {a L : ℝ} {d : ℕ}, 1 ≤ d → 2 ≤ a → 0 ≤ L → L ≤ ↑d * Real.log (2 + a * L) → L ≤ 1 + 2 * ↑d * Real.log (2 * a * ↑d)","l":546,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sharp_vc_weightedHamming_packing_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":2,"d":"A finite Boolean VC class with nonnegative coordinate weights has a dimension-free-base polynomial packing bound in weighted Hamming distance.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < ε → ε ≤ r → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → P.card ≤ ⌈(16 * r ^ 2 / ε ^ 2) ^ (2 * (d + 1))⌉₊","l":172,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.signed_sum_le_empiricalDist","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"A signed sum of pointwise differences is at most the sample size times the empirical distance between the two functions.","s":"∀ {Z : Type v} {m : ℕ} {S : Fin m → Z} (f g : Z → ℝ) (σ : Signs m), ∑ i, ↑↑(σ i) * (f (S i) - g (S i)) ≤ ↑m * Causalean.Stat.Concentration.empiricalDist S f g","l":572,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.splitBound.bddAbove_increment_term","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"The sum of chaining increment terms has a finite upper bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} {c_pos : 0 < c}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (h : TotallyBounded Set.univ) (n : ℕ), ¬m = 0 → ∀ (i : Signs m), BddAbove (Set.range fun fh => ∑ x_1, ∑ i_1, ↑↑(i i_1) * (Causalean.Stat.Concentration.chainApprox✝ c_pos h fh (↑x_1 + 1) (S i_1) - Causalean.Stat.Concentration.chainApprox✝ c_pos h fh (↑x_1) (S i_1)))","l":266,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.splitBound.bddAbove_main_term","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"The main remainder term in the Dudley chain has a finite upper bound.","s":"∀ {Z : Type v} {m : ℕ} {ι : Type u} [Nonempty ι] {F : ι → Z → ℝ} {S : Fin m → Z} {c : ℝ} {c_pos : 0 < c}, (∀ (f : ι), Causalean.Stat.Concentration.empiricalNorm S (F f) ≤ c) → ∀ (h : TotallyBounded Set.univ) (n : ℕ), ¬m = 0 → ∀ (i : Signs m), BddAbove (Set.range fun fh => ∑ i_1, ↑↑(i i_1) * (F fh (S i_1) - Causalean.Stat.Concentration.chainApprox✝ c_pos h fh n (S i_1)))","l":217,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.sqrtLog_integral_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":1,"d":"Dudley entropy-integral evaluation. For a positive lower limit ε that is at most the upper limit δ, the integral of √(log(δ/x)) over [ε,δ] is at most δ - ε.","s":"∀ {δ ε : ℝ}, 0 < ε → ε ≤ δ → ∫ (x : ℝ) in ε..δ, √(Real.log (δ / x)) ≤ δ - ε","l":127,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.sqrt_le_add_one_div_two","k":"theorem","m":"Causalean.Stat.Concentration.Covering.SqrtLogIntegral","a":"Stat/Concentration/Covering","t":2,"d":"For nonnegative t, the square root is bounded by the arithmetic mean of t and 1.","s":"∀ {t : ℝ}, 0 ≤ t → √t ≤ (t + 1) / 2","l":18,"f":"Causalean/Stat/Concentration/Covering/SqrtLogIntegral.lean"},{"n":"Causalean.Stat.Concentration.sqrt_sum_inv_abs_sq_eq_empiricalNorm_div_sqrt","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"For a positive sample size, the square root of the sum of squared sample-normalized absolute function values equals the empirical norm divided by the square root of the sample size.","s":"∀ {𝒳 : Type u_2} {n : ℕ}, 0 < n → ∀ (S : Fin n → 𝒳) (g : 𝒳 → ℝ), √(∑ k, ((↑n)⁻¹ * |g (S k)|) ^ 2) = Causalean.Stat.Concentration.empiricalNorm S g / √↑n","l":899,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHull","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Given a real-valued family of functions on a covariate space, the star hull of that family about zero is the set of all functions obtained by multiplying one member of the family by a real coefficient between zero and one, inclusive.","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Set (𝒳 → ℝ)","l":42,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullBall","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For an index set, an observation domain, a class of real-valued functions, a real-valued functional on such functions, and a real radius, the star-hull ball is the set of all star-hull functions whose functional value is at most the radius.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → Set (𝒳 → ℝ)","l":53,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullEval","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Given a real-valued family of functions on a covariate space, the star-hull evaluation map assigns to each admissible scaling coefficient and family index the corresponding rescaled function, whose value at each covariate point is the coefficient times that function's value.","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ","l":102,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_mem_starHull","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"The evaluation map lands inside the (set-level) star hull.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (p : Causalean.Stat.Concentration.starHullParam ι), Causalean.Stat.Concentration.starHullEval F p ∈ Causalean.Stat.Concentration.starHull F","l":111,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_one","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"At parameter (1, i), evaluation recovers F i.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), Causalean.Stat.Concentration.starHullEval F (⟨1, of_eq_true (Eq.trans Set.mem_Icc._simp_1 (Eq.trans (congr (congrArg And zero_le_one._simp_1) (Std.le_refl._simp_1 1)) (and_self True)))⟩, i) = F i","l":118,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullEval_zero","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"At parameter (0, i), evaluation is the zero function.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι), Causalean.Stat.Concentration.starHullEval F (⟨0, of_eq_true (Eq.trans Set.mem_Icc._simp_1 (Eq.trans (congr (congrArg And (Std.le_refl._simp_1 0)) zero_le_one._simp_1) (and_self True)))⟩, i) = fun x => 0","l":124,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullIndex","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Given a real-valued family of functions on a covariate space, the star-hull index set is the collection of precisely those functions that belong to its star hull.","s":"{ι : Type u_1} → {𝒳 : Type u_2} → (ι → 𝒳 → ℝ) → Type u_2","l":77,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullParam","k":"def","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Given an index set, the star-hull parameter set consists of every pair of an index from that set and a real scaling coefficient between zero and one, inclusive.","s":"Type u_1 → Type u_1","l":92,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullParam.instNonempty","k":"instance","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"starHullParam inherits Nonempty from its ι factor (the [0,1] factor is always nonempty).","s":"∀ {ι : Type u_1} [Nonempty ι], Nonempty (Causalean.Stat.Concentration.starHullParam ι)","l":130,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternClass","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a real-valued function family, a localization norm, a Boolean factorization family, a finite sample, and a localization radius, the star-hull pattern class assigns to each realized label pattern the representative function for that pattern, multiplied by its star-hull pattern coefficient.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ℝ → ↥(Causalean.Stat.Concentration.growthFamily π S) → 𝒳 → ℝ","l":670,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternCoeff","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a real-valued function family, a localization norm, a Boolean factorization family, a finite sample, a localization radius, and a realized label pattern, the star-hull pattern coefficient is the supremum of the active zeroed-star-hull scale coefficients among functions realizing that pattern on the sample.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → {n : ℕ} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (S : Fin n → 𝒳) → ℝ → ↥(Causalean.Stat.Concentration.growthFamily π S) → ℝ","l":662,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullPatternCoeff_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The coefficient assigned to any observed Boolean pattern by the localized star-hull pattern class is nonnegative.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (S : Fin n → 𝒳) (r : ℝ) (A : ↥(Causalean.Stat.Concentration.growthFamily π S)), 0 ≤ Causalean.Stat.Concentration.starHullPatternCoeff F norm π S r A","l":718,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For an index set, an observation domain, a class of real-valued functions, a real-valued functional on such functions, and a real radius, the zero-out localized star-hull family assigns to each base function and each scalar between zero and one their scalar product when its functional value is at most the radius, and assigns the zero function otherwise.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → Causalean.Stat.Concentration.starHullParam ι → 𝒳 → ℝ","l":58,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff","k":"def","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"For an index set, an observation domain, a class of real-valued functions, a real-valued functional on such functions, a real radius, and a base-function index, the zero-out scale coefficient is the supremum over scalars in $[0,1]$ of that scalar when the corresponding scaled function has functional value at most the radius, and zero otherwise.","s":"{ι : Type u_2} → {𝒳 : Type u_3} → (ι → 𝒳 → ℝ) → ((𝒳 → ℝ) → ℝ) → ℝ → ι → ℝ","l":72,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_le_one","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The star-hull zero-out scale coefficient is at most one.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (i : ι), Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i ≤ 1","l":78,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_mul_empiricalNorm_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Under the localized VC hypotheses, a function's localized scale coefficient times its empirical norm over the sample is at most the localization radius.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) {r : ℝ}, 0 ≤ r → ∀ (i : ι), Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i * Causalean.Stat.Concentration.empiricalNorm S (F i) ≤ r","l":937,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOutScaleCoeff_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The largest active scalar in a zeroed star hull is nonnegative because the zero scalar is always available.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) (i : ι), 0 ≤ Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i","l":678,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_bddAbove_of_bound","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The signed empirical average over the zero-out star hull is bounded above by the uniform bound on the base family at the sampled points: if |F i (S_fin k)| ≤ b for every index and sample coordinate, then for any sign vector the family of signed averages indexed by star-hull parameters has b as an upper bound.","s":"∀ {𝒳 : Type u_4} {ι : Type u_5} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) {b : ℝ}, 0 ≤ b → ∀ (m : ℕ) (r : ℝ) (S_fin : Fin m → 𝒳), (∀ (i : ι) (k : Fin m), |F i (S_fin k)| ≤ b) → ∀ (σ : Signs m), BddAbove (Set.range fun p => |(↑m)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r p (S_fin k)|)","l":365,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_le_growthFamily","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The empirical Rademacher complexity of a zero-augmented localized star hull is bounded by its radius times a logarithmic factor determined by the number of distinct Boolean patterns in the sample.","s":"∀ {ι : Type u} {𝒳 : Type v} [Nonempty ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ {n : ℕ}, 0 < n → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ r * √(2 * Real.log (2 * ↑(Causalean.Stat.Concentration.growthFamily π S).card) / ↑n)","l":1062,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_le_patternClass","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If function values on every finite sample depend only on Boolean labels, then the empirical Rademacher complexity of the zeroed localized star hull is no greater than that of its finite sample-pattern class.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} [Nonempty ι] {n : ℕ} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (S : Fin n → 𝒳) (r : ℝ), empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullPatternClass F norm π S r) S","l":774,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_empirical_rademacher_massart_vc","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Massart finite-realization bound for the localized star-hull zero-out class under binary trace entropy control.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (d n : ℕ), Causalean.Stat.Concentration.BinaryTraceEntropyControl π d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ r * √((2 * ↑d * Real.log (↑n + 1) + 2) / ↑n)","l":1147,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_inner_sup_eq","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Supremizing over the star-hull scalar collapses to the largest active coefficient times the base-class inner Rademacher term.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) {n : ℕ} (ω : Fin n → 𝒳) (σ : Signs n) (i : ι), ⨆ a, |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r (a, i) (ω k)| = Causalean.Stat.Concentration.starHullZeroOutScaleCoeff F norm r i * |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * F i (ω k)|","l":121,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_inner_term_eq","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"The inner Rademacher term for a fixed star-hull scalar factors into that scalar, or zero when the radius test fails.","s":"∀ {ι : Type u_2} {𝒳 : Type u_3} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (r : ℝ) {n : ℕ} (ω : Fin n → 𝒳) (σ : Signs n) (a : ↑(Set.Icc 0 1)) (i : ι), |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * Causalean.Stat.Concentration.starHullZeroOut F norm r (a, i) (ω k)| = (if norm (Causalean.Stat.Concentration.starHullEval F (a, i)) ≤ r then ↑a else 0) * |(↑n)⁻¹ * ∑ k, ↑↑(σ k) * F i (ω k)|","l":93,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.starHullZeroOut_totallyBounded_of_bounded","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"A uniformly bounded function class has a totally bounded empirical image on every finite sample after taking its zeroed star hull.","s":"∀ {ι : Type u} {𝒳 : Type v} {F : ι → 𝒳 → ℝ} {norm : (𝒳 → ℝ) → ℝ} {B : ℝ}, (∀ (i : ι) (x : 𝒳), |F i x| ≤ B) → ∀ {n : ℕ} (S : Fin n → 𝒳) {r : ℝ}, TotallyBounded Set.univ","l":140,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.starHull_congr","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"Pointwise-equal families have the same star hull.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} {F G : ι → 𝒳 → ℝ}, (∀ (i : ι) (x : 𝒳), F i x = G i x) → Causalean.Stat.Concentration.starHull F = Causalean.Stat.Concentration.starHull G","l":62,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.starHull_smul_mem","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":1,"d":"Star-hull rescaling. For any coefficient α satisfying 0 ≤ α and α ≤ 1, the rescaled function α • F i lies in the star hull of the family F.","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} (F : ι → 𝒳 → ℝ) (i : ι) {α : ℝ}, 0 ≤ α → α ≤ 1 → α • F i ∈ Causalean.Stat.Concentration.starHull F","l":70,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.Concentration.subRoot_continuousOn_Ioi","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":2,"d":"Sub-root continuity. If ψ is sub-root, then ψ is continuous on the open ray (0, ∞). The non-increasing ratio condition ψ(r)/r ↘ supplies the linear squeeze bounds, while monotonicity of ψ supplies the opposite side of the squeeze.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ContinuousOn ψ (Set.Ioi 0)","l":205,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.subRoot_homogeneity","k":"theorem","m":"Causalean.Stat.Concentration.UniformDeviation.CriticalRadius","a":"Stat/Concentration/UniformDeviation","t":1,"d":"Sub-root inequality. If ψ is sub-root and δ* is a positive radius with ψ δ* ≤ δ* ^ 2, then for every radius r at least δ*, ψ r is at most r · δ*.","s":"∀ {ψ : ℝ → ℝ}, Causalean.Stat.Concentration.SubRoot ψ → ∀ {δ_star r : ℝ}, 0 < δ_star → δ_star ≤ r → ψ δ_star ≤ δ_star ^ 2 → ψ r ≤ r * δ_star","l":184,"f":"Causalean/Stat/Concentration/UniformDeviation/CriticalRadius.lean"},{"n":"Causalean.Stat.Concentration.subgraphClassifier","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a real-valued function class on an observation space, a class index, and an observation-threshold pair, the strict subgraph classifier returns true exactly when the threshold is strictly below the selected function's value at the observation.","s":"{𝒳 : Type u} → {ι : Type v} → (ι → 𝒳 → ℝ) → ι → 𝒳 × ℝ → Bool","l":22,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern","k":"def","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"Given a map from a finite sample-coordinate set into a finite original-coordinate set and a Boolean-valued vector on the original-coordinate set, the subsample pattern is the finite set of sample coordinates whose mapped original coordinates have Boolean value true.","s":"{n m : ℕ} → (Fin m → Fin n) → (Fin n → Bool) → Finset (Fin m)","l":26,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern_eq_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Equality of sampled Boolean patterns means that the two patterns agree at every coordinate selected by the sample.","s":"∀ {n m : ℕ} {J : Fin m → Fin n} {a b : Fin n → Bool}, Causalean.Stat.Concentration.subsamplePattern J a = Causalean.Stat.Concentration.subsamplePattern J b ↔ ∀ (t : Fin m), a (J t) = b (J t)","l":84,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsamplePattern_mem_iff","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A sampled coordinate belongs to a Boolean pattern exactly when the original Boolean vector is true at the coordinate from which it was sampled.","s":"∀ {n m : ℕ} {J : Fin m → Fin n} {a : Fin n → Bool} {i : Fin m}, i ∈ Causalean.Stat.Concentration.subsamplePattern J a ↔ a (J i) = true","l":47,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.subsample_image_vcDim_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Restricting a Boolean set family along a coordinate map does not increase VC dimension.","s":"∀ {n m : ℕ} (d : ℕ) (J : Fin m → Fin n) (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (Finset.image (Causalean.Stat.Concentration.subsamplePattern J) P).vcDim ≤ d","l":234,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sum_choose_le_succ_mul_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The Sauer-Shelah binomial sum is bounded by the usual polynomial upper bound in sample size and VC dimension.","s":"∀ {n d : ℕ}, 0 < n → ↑(∑ k ≤ d, n.choose k) ≤ ↑(d + 1) * ↑n ^ d","l":218,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.sum_choose_le_succ_pow","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"A partial sum of binomial coefficients is no larger than a polynomial power, providing the elementary growth bound used in VC estimates.","s":"∀ (n d : ℕ), ∑ k ≤ d, n.choose k ≤ (n + 1) ^ d","l":55,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.sup_pair_lipschitz_scaled","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.Contraction","a":"Stat/Concentration/Rademacher","t":2,"d":"For a finite nonempty collection, the sum of the largest values obtained by adding and subtracting a nonnegative multiple of a transformation with a given Lipschitz constant is no greater than the corresponding sum using that linear bound.","s":"∀ {ι : Type u_3} [Nonempty ι] [Finite ι] (φ : ℝ → ℝ) {L c : ℝ}, 0 ≤ c → (∀ (x y : ℝ), |φ x - φ y| ≤ L * |x - y|) → ∀ (a b : ι → ℝ), (⨆ i, a i + c * φ (b i)) + ⨆ i, a i - c * φ (b i) ≤ (⨆ i, a i + c * (L * b i)) + ⨆ i, a i - c * (L * b i)","l":111,"f":"Causalean/Stat/Concentration/Rademacher/Contraction.lean"},{"n":"Causalean.Stat.Concentration.term_le_total_sum_of_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.DudleyEntropy","a":"Stat/Concentration/Covering","t":2,"d":"A nonnegative term in a finite sum is bounded by the full sum of nonnegative terms.","s":"∀ {α : Type u} [inst : Fintype α] {M : Type u_1} [inst_1 : AddCommMonoid M] [inst_2 : Preorder M] [IsOrderedAddMonoid M] (j : α) (f : α → M), (∀ (j : α), 0 ≤ f j) → f j ≤ ∑ i, f i","l":39,"f":"Causalean/Stat/Concentration/Covering/DudleyEntropy.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveExpectedMaximal_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.ExpectedMaximal","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Variance-adaptive expected maximal inequality. Let P be a probability measure on the sample space and F a countable family of real-valued functions on it. Suppose the population $L^2$ radius σ is strictly positive and strictly less than the envelope U, the covering-entropy base A is at least Euler's number and the exponent v is at least one, every function in F is measurable and bounded in absolute value by U, each function's population $L^2$ distance from the zero function is at most σ, F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, and the sample size n is positive. Then the expected empirical supremum of F over an n-point i.i.d. sample drawn from P is at most the universal constant varianceAdaptiveVCConstant times the variance-adaptive rate σ √(v log(AU/σ)/n) + v U log(AU/σ)/n.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ A v : ℝ}, 0 < σ → σ < U → Real.exp 1 ≤ A → 1 ≤ v → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → ∀ (n : ℕ), 0 < n → (∫ (S : Fin n → 𝒳), Causalean.Stat.Concentration.countableEmpiricalSup P F S ∂MeasureTheory.Measure.pi fun x => P) ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":21,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/ExpectedMaximal.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveRademacherComplexity_le","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Rademacher","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Variance-adaptive Rademacher complexity bound. Let P be a probability measure on the sample space and F a countable family of real-valued functions on it. Suppose the population $L^2$ radius σ is strictly positive and strictly less than the envelope U, the covering-entropy base A is at least Euler's number and the exponent v is at least one, every function in F is measurable and bounded in absolute value by U, each function's population $L^2$ distance from the zero function is at most σ, F has polynomial empirical $L^2$ covering numbers with envelope U, base A, and exponent v, and the sample size n is positive. Then the Rademacher complexity of F under P with n i.i.d. observations is at most half the universal constant varianceAdaptiveVCConstant times the variance-adaptive rate σ √(v log(AU/σ)/n) + v U log(AU/σ)/n.","s":"∀ {𝒳 : Type u} [inst : MeasurableSpace 𝒳] {ι : Type v} [Nonempty ι] [Countable ι] (P : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure P] (F : ι → 𝒳 → ℝ) {U σ A v : ℝ}, 0 < σ → σ < U → Real.exp 1 ≤ A → 1 ≤ v → (∀ (i : ι), Measurable (F i)) → (∀ (i : ι) (x : 𝒳), |F i x| ≤ U) → (∀ (i : ι), (Causalean.Stat.Concentration.measureL2Dist P (F i) fun x => 0) ≤ σ) → Causalean.Stat.Concentration.HasPolynomialEmpiricalL2Cover F U A v → ∀ (n : ℕ), 0 < n → rademacherComplexity n F P id ≤ Causalean.Stat.Concentration.varianceAdaptiveVCConstant / 2 * Causalean.Stat.Concentration.vcExpectedMaximalRate U σ A v n","l":704,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Rademacher.lean"},{"n":"Causalean.Stat.Concentration.varianceAdaptiveVCConstant","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"The fixed numerical constant for the variance-adaptive VC-type expected maximal inequality is $16384$.","s":"ℝ","l":147,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.variance_sum_pi_eq","k":"theorem","m":"Causalean.Stat.Concentration.Matrix.IidSums","a":"Stat/Concentration/Matrix","t":2,"d":"Variance of an iid sum. For an iid sample of size N from μ (the product measure Measure.pi), the variance of the sum ∑ᵢ g(Xᵢ) of a fixed L² statistic g equals N · Var[g] — the coordinate copies are independent, so cross-covariances vanish.","s":"∀ {N : ℕ} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ), MeasureTheory.MemLp g 2 μ → ProbabilityTheory.variance (fun ω => ∑ i, g (ω i)) (MeasureTheory.Measure.pi fun x => μ) = ↑N * ProbabilityTheory.variance g μ","l":61,"f":"Causalean/Stat/Concentration/Matrix/IidSums.lean"},{"n":"Causalean.Stat.Concentration.vcEntropy_chaining_bound","k":"theorem","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.EntropyChaining","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Dudley chaining bound for VC-type entropy. Let μ be a probability measure on Ω, g : ι → Ω → ℝ a family of functions, and g0 : ℕ → ι a sequence of indices (for example an enumeration of a countable dense subfamily). If g has uniform VC-type entropy relative to μ, with envelope U, population $L^2$ radius σ, covering-entropy base A, and exponent v, then there is a constant C > 0, which may depend on U, σ, A and v, such that, for every sample size n ≥ 1, the expectation of the countable empirical-process supremum along the enumeration g0 over the n-fold product of μ is at most C · (σ √(log(U/σ)/n) + U log(U/σ)/n).","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : ι → Ω → ℝ) (g0 : ℕ → ι) (U σ A v : ℝ), Causalean.Stat.Concentration.HasVCUniformEntropy μ g U σ A v → ∃ C, 0 < C ∧ ∀ (n : ℕ), 1 ≤ n → (∫⁻ (w : Fin n → Ω), Causalean.Stat.Concentration.countableEmpiricalProcessSup μ g g0 w ∂MeasureTheory.Measure.pi fun x => μ) ≤ ENNReal.ofReal (C * (σ * √(Real.log (U / σ) / ↑n) + U * Real.log (U / σ) / ↑n))","l":170,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/EntropyChaining.lean"},{"n":"Causalean.Stat.Concentration.vcExpectedMaximalRate","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given an envelope $U$, a variance scale $\\sigma$, an entropy base $A$, an exponent $v$, and a sample size $n$, the variance-adaptive VC-type rate is $\\sigma\\sqrt{vL/n}+vUL/n$, where $L=\\log(\\max\\{e,AU/\\sigma\\})$.","s":"ℝ → ℝ → ℝ → ℝ → ℕ → ℝ","l":138,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedEnvelope","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Fixed-n finite-VC localized envelope package: sub-root envelope, localized Rademacher upper bound, critical-radius bound by the slope, and squared critical-radius rate.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.SubRoot (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ∧ Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ∧ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ∧ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2 ≤ 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":1453,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a tuning constant, a VC-dimension bound, and a sample size, the finite-VC localized envelope maps every radius $r$ to the finite-VC localized slope times $r$.","s":"ℝ → ℕ → ℕ → ℝ → ℝ","l":256,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi_criticalRadius_fp","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"For the finite-VC localized envelope, its value at the critical radius is no larger than the square of that radius.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → Causalean.Stat.Concentration.vcLocalizedPsi K d n (Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n)) ≤ Causalean.Stat.Concentration.criticalRadius (Causalean.Stat.Concentration.vcLocalizedPsi K d n) ^ 2","l":52,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedPsi_subRoot","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC localized envelope is sub-root.","s":"∀ (K : ℝ) (d n : ℕ), Causalean.Stat.Concentration.SubRoot (Causalean.Stat.Concentration.vcLocalizedPsi K d n)","l":331,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRademacherUpperBound","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC localized envelope upper-bounds population localized Rademacher complexity.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K d n)","l":1419,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRademacherUpperBound_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The growth-cardinality localized envelope upper-bounds population localized Rademacher complexity.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.RademacherUpperBound F norm μ X n (Causalean.Stat.Concentration.vcLocalizedPsi K dPi n)","l":1434,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRate_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The rate inside vcLocalizedSlope is nonnegative when K ≥ 0.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 ≤ (K * ↑d * Real.log (↑n + 1) + 1) / ↑n","l":267,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRegime","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a real-valued function class, a function norm, a probability measure, an observation map, a uniform bound that is nonnegative and bounds every function value at every observed point, a tuning constant, a VC-dimension bound, a tuning constant at least one, a binary factorization with VC dimension at most the stated bound, and localized Dudley hypotheses, the localized-regime package consists of this bound and the finite-VC localized envelope with its sub-root and Rademacher upper-bound guarantees.","s":"{Ω : Type u_1} → {ι : Type u} → {𝒳 : Type v} → [inst : MeasurableSpace Ω] → (F : ι → 𝒳 → ℝ) → (norm : (𝒳 → ℝ) → ℝ) → (μ : MeasureTheory.Measure Ω) → (X : Ω → 𝒳) → [MeasureTheory.IsProbabilityMeasure μ] → (b : ℝ) → 0 ≤ b → (∀ (i : ι) (ω : Ω), |F i (X ω)| ≤ b) → (K : ℝ) → (d : ℕ) → 1 ≤ K → Causalean.Stat.Concentration.BinaryFactoredVCClass F d → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X","l":1474,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedRegime_of_card","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a real-valued function class, a function norm, a Boolean factorization family, a probability measure, an observation map, a uniform bound that is nonnegative and bounds every function value at every observed point, a factorization of each finite-sample function value through its Boolean label, a tuning constant, a trace-growth exponent, a tuning constant at least one, a polynomial bound on every finite-sample trace-family cardinality, and localized Dudley hypotheses, the localized-regime package consists of this bound and the localized envelope obtained from the direct trace-cardinality bound.","s":"{Ω : Type u_1} → {ι : Type u} → {𝒳 : Type v} → [inst : MeasurableSpace Ω] → (F : ι → 𝒳 → ℝ) → (norm : (𝒳 → ℝ) → ℝ) → (π : ι → 𝒳 → Bool) → (μ : MeasureTheory.Measure Ω) → (X : Ω → 𝒳) → [MeasureTheory.IsProbabilityMeasure μ] → (b : ℝ) → 0 ≤ b → (∀ (i : ι) (ω : Ω), |F i (X ω)| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → (K : ℝ) → (dPi : ℕ) → 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → Causalean.Stat.Concentration.LocalizedRegime Ω ι 𝒳 F norm μ X","l":1495,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope","k":"def","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":1,"d":"Given a tuning constant, a VC-dimension bound, and a sample size, the finite-VC localized slope is $6\\sqrt{(K d\\log(n+1)+1)/n}$.","s":"ℝ → ℕ → ℕ → ℝ","l":246,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The finite-VC slope is nonnegative.","s":"∀ (K : ℝ) (d n : ℕ), 0 ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":283,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_pos","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"If K ≥ 0 and n > 0, the finite-VC slope is strictly positive.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → 0 < n → 0 < Causalean.Stat.Concentration.vcLocalizedSlope K d n","l":289,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcLocalizedSlope_sq","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The exact square of the finite-VC slope.","s":"∀ {K : ℝ} {d n : ℕ}, 0 ≤ K → Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2 = 36 * ((K * ↑d * Real.log (↑n + 1) + 1) / ↑n)","l":310,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vcMaximalLog","k":"def","m":"Causalean.Stat.Concentration.VarianceAdaptiveVCExpectedMaximal.Basic","a":"Stat/Concentration/VarianceAdaptiveVCExpectedMaximal","t":1,"d":"Given an entropy base $A$, an envelope $U$, and a variance scale $\\sigma$, the normalized logarithmic complexity is $\\log(\\max\\{e,AU/\\sigma\\})$. This normalization makes the logarithm at least one.","s":"ℝ → ℝ → ℝ → ℝ","l":132,"f":"Causalean/Stat/Concentration/VarianceAdaptiveVCExpectedMaximal/Basic.lean"},{"n":"Causalean.Stat.Concentration.vcSubgraphCoverBound","k":"def","m":"Causalean.Stat.Concentration.Covering.RealValuedVCSubgraph.Basic","a":"Stat/Concentration/Covering/RealValuedVCSubgraph","t":1,"d":"Given a nonnegative integer $d$ and a real number $\\varepsilon$, the explicit VC-subgraph cover bound is the least integer no smaller than $(16/\\varepsilon)^{8(d+1)}$.","s":"ℕ → ℝ → ℕ","l":63,"f":"Causalean/Stat/Concentration/Covering/RealValuedVCSubgraph/Basic.lean"},{"n":"Causalean.Stat.Concentration.vc_coveringNumber_le_growth","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":2,"d":"The empirical covering number of a binary-factored class is bounded by the number of realized Boolean restriction patterns.","s":"∀ {𝒳 : Type v} {ι : Type u} {n : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → Causalean.Stat.Concentration.coveringNumber h' ε ≤ (Causalean.Stat.Concentration.growthFamily π S).card","l":163,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.vc_coveringNumber_le_sum_choose","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCCovering","a":"Stat/Concentration/Covering","t":1,"d":"Sauer-Shelah covering-number bound for a binary-factored class. Suppose the real-valued class factors through a Boolean classifier at each sample coordinate: F i (S j) = φ j (π i (S j)), so that the sample-restricted class is totally bounded in the empirical pseudometric, the covering radius ε is positive, and the induced Boolean growth family on the sample has VC dimension at most d. Then the empirical covering number at radius ε is at most the Sauer-Shelah binomial sum ∑_{k≤d} C(n,k).","s":"∀ {𝒳 : Type v} {ι : Type u} {n d : ℕ} {F : ι → 𝒳 → ℝ} {S : Fin n → 𝒳} {π : ι → 𝒳 → Bool} {φ : Fin n → Bool → ℝ}, (∀ (i : ι) (j : Fin n), F i (S j) = φ j (π i (S j))) → ∀ (h' : TotallyBounded Set.univ) {ε : ℝ}, 0 < ε → (Causalean.Stat.Concentration.growthFamily π S).vcDim ≤ d → Causalean.Stat.Concentration.coveringNumber h' ε ≤ ∑ k ≤ d, n.choose k","l":179,"f":"Causalean/Stat/Concentration/Covering/VCCovering.lean"},{"n":"Causalean.Stat.Concentration.vc_localized_deviation_event","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":1,"d":"Finite-VC localized uniform deviation event. For a class of measurable real-valued functions uniformly bounded in absolute value by a nonnegative constant b, with a localization constant K at least 1 and a positive sample size n, suppose the class factors through a binary Boolean family of VC dimension at most d and satisfies the localized Dudley hypotheses relative to the given seminorm. For a failure probability δ in (0,1], writing ρ for the localized slope vcLocalizedSlope K d n, if ρ is at most b and the peeling/log-domination side condition holds at every dyadic scale, then there is a measurable event of probability at least 1 - δ on which every class member i with 0 ≤ norm (F i) ≤ b satisfies the sharp localized deviation bound |n⁻¹ ∑ₖ F i(ωₖ) − 𝔼[F i]| ≤ 8ρ·norm(F i) + 5ρ².","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Stat.Concentration.vcLocalizedSlope K d n ≤ b → (∀ (L : ℕ), b ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n * 2 ^ L → b * √(2 * Real.log (2 * (↑L + 1) / δ) / ↑n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ b → |(↑n)⁻¹ * ∑ k, F i (ω k) - ∫ (x : 𝒳), (fun x => F i x) x ∂μ| ≤ 8 * Causalean.Stat.Concentration.vcLocalizedSlope K d n * norm (F i) + 5 * Causalean.Stat.Concentration.vcLocalizedSlope K d n ^ 2","l":308,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_localized_deviation_event_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":1,"d":"Growth-cardinality localized uniform deviation event. For a class of measurable real-valued functions uniformly bounded in absolute value by a nonnegative constant b that factors through a Boolean classifier π at every finite sample, for a coordinate transform φ with F i (S j) = φ j (π i (S j)), with a localization constant K at least 1 and a positive sample size n, suppose the induced Boolean growth family satisfies the direct cardinality bound #growthFamily π S ≤ (m+1)^dPi on every finite sample of size m and the class satisfies the localized Dudley hypotheses relative to the given seminorm. For a failure probability δ in (0,1], writing ρ for the localized slope vcLocalizedSlope K dPi n, if ρ is at most b and the peeling/log-domination side condition holds at every dyadic scale, then there is a measurable event of probability at least 1 - δ on which every class member i with 0 ≤ norm (F i) ≤ b satisfies the sharp localized deviation bound |n⁻¹ ∑ₖ F i(ωₖ) − 𝔼[F i]| ≤ 8ρ·norm(F i) + 5ρ².","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → 0 < n → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ {δ : ℝ}, 0 < δ → δ ≤ 1 → Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ≤ b → (∀ (L : ℕ), b ≤ Causalean.Stat.Concentration.vcLocalizedSlope K dPi n * 2 ^ L → b * √(2 * Real.log (2 * (↑L + 1) / δ) / ↑n) ≤ Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ^ 2) → ∃ E, MeasurableSet E ∧ (MeasureTheory.Measure.pi fun x => μ) E ≥ 1 - ENNReal.ofReal δ ∧ ∀ ω ∈ E, ∀ (i : ι), 0 ≤ norm (F i) → norm (F i) ≤ b → |(↑n)⁻¹ * ∑ k, F i (ω k) - ∫ (x : 𝒳), (fun x => F i x) x ∂μ| ≤ 8 * Causalean.Stat.Concentration.vcLocalizedSlope K dPi n * norm (F i) + 5 * Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ^ 2","l":393,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_aemeasurable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Measurability bridge for the empirical Rademacher process of the finite-VC localized star hull.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ), (∀ (i : ι), Measurable (F i)) → ∀ (μ : MeasureTheory.Measure 𝒳) (b r : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (n : ℕ), 0 < n → AEMeasurable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":122,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_integrable","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Integrability of the finite-VC localized star-hull empirical Rademacher process follows from the deterministic linear envelope once the residual measurability bridge above is available.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → ∀ (K : ℝ) (d n : ℕ), 1 ≤ K → 0 < n → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), Causalean.Stat.Concentration.vcLocalizedSlope K d n ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":194,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_integrable_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCUniformDeviation","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound variant of the integrability bridge for the localized star-hull empirical Rademacher process.","s":"∀ {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace 𝒳] [Nonempty ι] [Countable ι] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Measurable (F i)) → ∀ (b : ℝ), 0 ≤ b → (∀ (i : ι) (x : 𝒳), |F i x| ≤ b) → (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → 0 < n → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), Causalean.Stat.Concentration.vcLocalizedSlope K dPi n ≤ r → MeasureTheory.Integrable (fun ω => empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) (id ∘ ω)) (MeasureTheory.Measure.pi fun x => μ)","l":246,"f":"Causalean/Stat/Concentration/Covering/VCUniformDeviation.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_le_linear","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The generic finite-VC/Dudley sample-path bridge for the localized star-hull class.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1292,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_empirical_rademacher_le_linear_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound sample-path bridge for the localized star-hull class. This is the direct growth-function analogue of vc_starHullZeroOut_empirical_rademacher_le_linear.","s":"∀ {ι : Type u} {𝒳 : Type v} (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool), (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (S : Fin n → 𝒳) (r : ℝ), 0 ≤ r → empiricalRademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) S ≤ Causalean.Stat.Concentration.vcLocalizedPsi K dPi n r","l":1309,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_population_rademacher_le_linear","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"The generic finite-VC/Dudley population bridge for the localized star-hull class.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (K : ℝ) (d n : ℕ), 1 ≤ K → ∀ (Hvc : Causalean.Stat.Concentration.BinaryFactoredVCClass F d), Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), 0 ≤ r → rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X ≤ Causalean.Stat.Concentration.vcLocalizedPsi K d n r","l":1328,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_starHullZeroOut_population_rademacher_le_linear_of_card","k":"theorem","m":"Causalean.Stat.Concentration.Covering.VCLocalizedRegime","a":"Stat/Concentration/Covering","t":2,"d":"Cardinality-bound population bridge for the localized star-hull class.","s":"∀ {Ω : Type u_1} {ι : Type u} {𝒳 : Type v} [inst : MeasurableSpace Ω] (F : ι → 𝒳 → ℝ) (norm : (𝒳 → ℝ) → ℝ) (π : ι → 𝒳 → Bool) (μ : MeasureTheory.Measure Ω) (X : Ω → 𝒳) [MeasureTheory.IsProbabilityMeasure μ], (∀ {m : ℕ} (S : Fin m → 𝒳), ∃ φ, ∀ (i : ι) (j : Fin m), F i (S j) = φ j (π i (S j))) → ∀ (K : ℝ) (dPi n : ℕ), 1 ≤ K → (∀ (m : ℕ) (S : Fin m → 𝒳), (Causalean.Stat.Concentration.growthFamily π S).card ≤ (m + 1) ^ dPi) → Causalean.Stat.Concentration.LocalizedVCDudleyHypotheses F norm → ∀ (r : ℝ), 0 ≤ r → rademacherComplexity n (Causalean.Stat.Concentration.starHullZeroOut F norm r) μ X ≤ Causalean.Stat.Concentration.vcLocalizedPsi K dPi n r","l":1372,"f":"Causalean/Stat/Concentration/Covering/VCLocalizedRegime.lean"},{"n":"Causalean.Stat.Concentration.vc_weightedHamming_packing_card_le","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"Haussler ε-packing bound, logarithmic form. Let w be a nonnegative weight on the n sample coordinates with total weight at most r², where ε is positive and at most r. If the Boolean family P, viewed as the sets of coordinates where each member is true, has VC dimension at most d and every two distinct members of P are separated by weighted Hamming distance at least ε² in that weighting, then the logarithm of the cardinality of P is at most 1 + 2d·log(4d·r²/ε²) — a bound depending only on d and the ratio r²/ε², with no dependence on the ambient coordinate count n.","s":"∀ {n : ℕ} (d : ℕ) (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (r ε : ℝ), 0 < ε → ε ≤ r → ∑ j, w j ≤ r ^ 2 → ∀ (P : Finset (Fin n → Bool)), (Finset.image (fun a => {j | a j = true}) P).vcDim ≤ d → (∀ a ∈ P, ∀ b ∈ P, a ≠ b → ε ^ 2 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b) → Real.log ↑P.card ≤ 1 + 2 * ↑d * Real.log (4 * ↑d * r ^ 2 / ε ^ 2)","l":620,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.weightedHammingSq","k":"def","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":1,"d":"Given a finite collection of real coordinate weights and two Boolean-valued vectors on the same finite coordinate set, the weighted squared Hamming pseudo-distance is the sum of the weights at exactly those coordinates where the vectors differ.","s":"{n : ℕ} → (Fin n → ℝ) → (Fin n → Bool) → (Fin n → Bool) → ℝ","l":19,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.weightedHammingSq_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.Covering.HausslerPacking","a":"Stat/Concentration/Covering","t":2,"d":"Nonnegative coordinate weights make the weighted Hamming pseudo-distance nonnegative.","s":"∀ {n : ℕ} (w : Fin n → ℝ), (∀ (j : Fin n), 0 ≤ w j) → ∀ (a b : Fin n → Bool), 0 ≤ Causalean.Stat.Concentration.weightedHammingSq w a b","l":35,"f":"Causalean/Stat/Concentration/Covering/HausslerPacking.lean"},{"n":"Causalean.Stat.Concentration.zero_mem_starHull","k":"theorem","m":"Causalean.Stat.Concentration.Rademacher.StarHull","a":"Stat/Concentration/Rademacher","t":2,"d":"The zero function belongs to the star hull of any nonempty family (take α = 0).","s":"∀ {ι : Type u_1} {𝒳 : Type u_2} [Nonempty ι] (F : ι → 𝒳 → ℝ), (fun x => 0) ∈ Causalean.Stat.Concentration.starHull F","l":54,"f":"Causalean/Stat/Concentration/Rademacher/StarHull.lean"},{"n":"Causalean.Stat.DegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Degenerate order-2 kernel. A kernel g : X → X → ℝ is doubly degenerate under the measure P when g is measurable as a function on $X \\times X$, g is symmetric, $g(x,y) = g(y,x)$, integrating g over its second argument against P gives zero for every fixed first argument, and g is square-integrable under the product measure $P \\times P$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → X → ℝ) → Prop","l":95,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.deg'","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"Left degeneracy, from symmetry and right degeneracy.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ}, Causalean.Stat.DegenKernel P g → ∀ (y : X), ∫ (x : X), g x y ∂P = 0","l":129,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"g is L¹ on a finite product measure, from L².","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.DegenKernel P g → MeasureTheory.Integrable (fun p => g p.1 p.2) (P.prod P)","l":122,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.DegenKernel.toOrderDegenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Bridge to the order-m theory (m = 2). If the two-argument kernel g : X → X → ℝ is degenerate: measurable, symmetric, square-integrable under P × P, and with zero conditional mean in each coordinate, then the paired order-2 kernel pairKernel g, defined by z ↦ g(z 0, z 1), is completely degenerate in the order-m sense.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.SigmaFinite P], Causalean.Stat.DegenKernel P g → Causalean.Stat.OrderDegenKernel P (Causalean.Stat.pairKernel g)","l":134,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.E_conditionalMeanAlongMap_comp","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Composing the guarded conditional mean with the coarsening has the same expectation under the original design as the corresponding source value.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (v : B → V), (D.E fun ω => Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h (φ ω) (v (φ ω))) = D.E fun ω => h ω (v (φ ω))","l":187,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.E_map_conditionalMeanAlongMap","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The pushforward expectation of the guarded conditional mean equals the original-design expectation of the source value evaluated at the covariate selected by its coarsening.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [inst_1 : Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (v : B → V), ((D.map φ).E fun b => Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b (v b)) = D.E fun ω => h ω (v (φ ω))","l":177,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.E_map_conditionalMeanAlongMap_sq_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Conditioning over the fibers of any deterministic pushforward weakly contracts covariate-parametric squared loss under an arbitrary finite design.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [inst_1 : Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (v : B → V) (t : ℝ), ((D.map φ).E fun b => (Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b (v b) - t) ^ 2) ≤ D.E fun ω => (h ω (v (φ ω)) - t) ^ 2","l":261,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment","k":"structure","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"A finite uniform-allocation experiment consists of a nonempty admissible allocation set, a normalized nonnegative observation mass for every state and allocation, a finite statistic, and a fallback observation used only to totalize conditioning on null fibers.","s":"(Latent : Type u_5) → (Allocation : Type u_6) → (Observation : Type u_7) → (Statistic : Type u_8) → [Fintype Latent] → [Fintype Allocation] → [Fintype Observation] → [Fintype Statistic] → Type (max (max (max u_5 u_6) u_7) u_8)","l":24,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.CommonConditionalKernel","k":"structure","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"A common finite conditional law assigns normalized nonnegative full-data weights to every statistic value and agrees with each statewise Bayes conditional on every positive-mass fiber.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Type (max (max u_2 u_3) u_4)","l":24,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization","k":"structure","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"A sufficient factorization writes every full-data mass as a nonnegative factor depending on the state and data only through the statistic, times a nonnegative state-independent carrier weight on the full data.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Type (max (max (max u_1 u_2) u_3) u_4)","l":37,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.commonConditionalWeight","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform experiment, a sufficient factorization of its full-data masses, a statistic value, and an allocation--observation pair, the common conditional weight equals the carrier weight divided by the carrier mass when that mass is positive and the pair has the stated statistic, equals zero when the mass is positive but the pair has another statistic, and otherwise is a unit mass at the experiment's designated fallback pair.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} → E.SufficientFactorization → Statistic → Allocation × Observation → ℝ","l":98,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.commonConditionalWeight_eq_conditionalWeight","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"On every positive statewise statistic fiber, factorization identifies the common conditional weight with the guarded Bayes conditional weight.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) (θ : Latent) (s : Statistic), 0 < E.statisticMass θ s → ∀ (z : Allocation × Observation), F.commonConditionalWeight s z = E.conditionalWeight θ s z","l":146,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.commonConditionalWeight_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Factorization-induced common conditional weights are nonnegative on every fiber.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) (s : Statistic) (z : Allocation × Observation), 0 ≤ F.commonConditionalWeight s z","l":108,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.commonConditionalWeight_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Factorization-induced common conditional weights sum to one on every fiber.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) (s : Statistic), ∑ z, F.commonConditionalWeight s z = 1","l":123,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.fiberCarrierMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform experiment, a sufficient factorization of that experiment's full-data masses, and a statistic value, the carrier mass of that statistic fiber is the sum of the factorization's state-independent carrier weights over all allocation--observation pairs having that statistic value.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [DecidableEq Statistic] → {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} → E.SufficientFactorization → Statistic → ℝ","l":58,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.fiberCarrierMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every carrier fiber mass is nonnegative.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) (s : Statistic), 0 ≤ F.fiberCarrierMass s","l":66,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.fiberCarrierMass_pos_of_statisticMass_pos","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Positive statistic mass forces the state-independent carrier fiber to have positive mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) {θ : Latent} {s : Statistic}, 0 < E.statisticMass θ s → 0 < F.fiberCarrierMass s","l":88,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.statisticMass_eq_factor_mul_fiberCarrierMass","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The statistic mass factors as the state-and-statistic factor times the carrier fiber mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} (F : E.SufficientFactorization) (θ : Latent) (s : Statistic), E.statisticMass θ s = F.statisticFactor θ s * F.fiberCarrierMass s","l":75,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.SufficientFactorization.toCommonConditionalKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Sufficiency","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform experiment and a sufficient factorization of its full-data masses, the common conditional kernel induced by that factorization assigns the factorization-induced common conditional weights to every statistic value and allocation--observation pair.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [inst_5 : DecidableEq Observation] → [inst_6 : DecidableEq Statistic] → {E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic} → E.SufficientFactorization → E.CommonConditionalKernel","l":170,"f":"Causalean/Stat/FiniteRaoBlackwell/Sufficiency.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.commonConditionalMarkovKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Given a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with σ-algebras on the allocation, observation, and statistic spaces and measurable singleton statistic values, and a sufficient factorization of its full-data masses, the common conditional Markov kernel assigns to each statistic value the factorization-induced state-independent conditional probability law of the allocation--observation pair.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → [inst_7 : MeasurableSpace Allocation] → [inst_8 : MeasurableSpace Observation] → [inst_9 : MeasurableSpace Statistic] → [MeasurableSingletonClass Statistic] → (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) → E.SufficientFactorization → ProbabilityTheory.Kernel Statistic (Allocation × Observation)","l":98,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.commonConditionalMarkovKernel_isMarkovKernel","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"For a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with distinguishable allocation, observation, and statistic values, σ-algebras on the allocation, observation, and statistic spaces, and measurable singleton statistic values and a sufficient factorization of its full-data probability masses, the factorization-derived common conditional kernel is a Markov kernel: at every statistic value, including a zero-probability fibre, its output law is a probability measure.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Allocation] [inst_8 : MeasurableSpace Observation] [inst_9 : MeasurableSpace Statistic] [inst_10 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization), ProbabilityTheory.IsMarkovKernel (E.commonConditionalMarkovKernel F)","l":111,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.commonConditionalMarkovKernel_singletonReal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The real singleton probability of the factorization-derived common conditional Markov kernel equals its finite common conditional weight.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [inst_14 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization) (s : Statistic) (z : Allocation × Observation), ((E.commonConditionalMarkovKernel F) s).real {z} = F.toCommonConditionalKernel.weight s z","l":182,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.commonRaoBlackwellEstimator","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a common conditional kernel, and a real-valued full-data estimator, the common Rao--Blackwell estimator assigns to each statistic value the finite weighted average of the full-data estimator under that kernel.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [inst_5 : DecidableEq Observation] → [inst_6 : DecidableEq Statistic] → (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) → E.CommonConditionalKernel → (Allocation × Observation → ℝ) → Statistic → ℝ","l":114,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalDesign","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a latent state, and a statistic value, the guarded conditional full-data design is the finite probability design whose probabilities are the guarded conditional weights.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → Statistic → Causalean.Experimentation.DesignBased.FiniteDesign (Allocation × Observation)","l":187,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Given a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with σ-algebras on the allocation, observation, and statistic spaces and measurable singleton statistic values, and a latent state, the guarded conditional Markov kernel assigns to each statistic value the experiment's conditional probability law of the allocation--observation pair at that state, using the fallback law when the statistic fiber has zero probability.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → [inst_7 : MeasurableSpace Allocation] → [inst_8 : MeasurableSpace Observation] → [inst_9 : MeasurableSpace Statistic] → [MeasurableSingletonClass Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → ProbabilityTheory.Kernel Statistic (Allocation × Observation)","l":79,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalKernel_isMarkovKernel","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"For a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with distinguishable allocation, observation, and statistic values, σ-algebras on the allocation, observation, and statistic spaces, and measurable singleton statistic values and a latent state, the guarded conditional kernel is a Markov kernel: at every statistic value, including a zero-probability fibre, its output law is a probability measure.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Allocation] [inst_8 : MeasurableSpace Observation] [inst_9 : MeasurableSpace Statistic] [inst_10 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent), ProbabilityTheory.IsMarkovKernel (E.conditionalKernel θ)","l":87,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalKernel_singletonReal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The real singleton probability of the guarded conditional kernel equals its finite conditional weight.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [inst_14 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic) (z : Allocation × Observation), ((E.conditionalKernel θ) s).real {z} = E.conditionalWeight θ s z","l":170,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalMean","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a real-valued full-data estimator, a latent state, and a statistic value, the conditional mean is the estimator's finite expectation under the guarded conditional design given that state and statistic value.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → (Allocation × Observation → ℝ) → Latent → Statistic → ℝ","l":31,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalMean_eq_commonRaoBlackwellEstimator","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On every positive statewise fiber, the state-indexed conditional mean equals the common Rao--Blackwell estimator supplied by the sufficient conditional kernel.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (K : E.CommonConditionalKernel) (est : Allocation × Observation → ℝ) (θ : Latent) (s : Statistic), 0 < E.statisticMass θ s → E.conditionalMean est θ s = E.commonRaoBlackwellEstimator K est s","l":120,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalMean_sq_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Conditional Jensen for squared loss: the squared error of the conditional mean is at most the conditional mean squared error on every guarded fiber.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (target : ℝ) (est : Allocation × Observation → ℝ) (θ : Latent) (s : Statistic), (E.conditionalMean est θ s - target) ^ 2 ≤ ∑ z, E.conditionalWeight θ s z * (est z - target) ^ 2","l":68,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalWeight","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a latent state, a statistic value, and a full-data point, the guarded conditional weight is the joint mass divided by the statistic mass when that mass is positive and the point has the requested statistic, is zero for other points, and is instead a point mass at the fallback sample when the statistic mass is zero.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → Statistic → Allocation × Observation → ℝ","l":122,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalWeight_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every guarded conditional weight is nonnegative, including on null fibers.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic) (z : Allocation × Observation), 0 ≤ E.conditionalWeight θ s z","l":146,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalWeight_of_eq_zero","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a zero-mass fiber, the guarded conditional weight is the fallback point mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) {θ : Latent} {s : Statistic}, E.statisticMass θ s = 0 → ∀ (z : Allocation × Observation), E.conditionalWeight θ s z = if z = E.fallbackSample then 1 else 0","l":139,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalWeight_of_pos","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a positive fiber, the guarded conditional weight is the usual fiber-restricted Bayes ratio.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) {θ : Latent} {s : Statistic}, 0 < E.statisticMass θ s → ∀ (z : Allocation × Observation), E.conditionalWeight θ s z = if E.sampleStatistic z = s then E.jointMass θ z / E.statisticMass θ s else 0","l":131,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.conditionalWeight_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The guarded conditional weights sum to one on every fiber, with positive fibers normalized by Bayes' formula and null fibers normalized by the fallback point mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic), ∑ z, E.conditionalWeight θ s z = 1","l":161,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.disintegrate_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Every real test function has the same joint expectation as its statistic-marginal expectation of the guarded conditional expectation; null fibers contribute exactly zero.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (f : Allocation × Observation → ℝ), ∑ z, E.jointMass θ z * f z = ∑ s, E.statisticMass θ s * ∑ z, E.conditionalWeight θ s z * f z","l":237,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fallbackAllocation","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, the fallback allocation is a chosen member of its nonempty set of admissible allocations.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Allocation","l":49,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fallbackSample","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, the fallback full-data point pairs its fallback allocation with its fallback observation.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Allocation × Observation","l":53,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fullKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Given a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with σ-algebras on the latent, allocation, and observation spaces and measurable singleton latent states, the full-data Markov kernel assigns to each latent state the experiment's finite joint probability law of the allocation and observation at that state.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → [inst_7 : MeasurableSpace Latent] → [MeasurableSingletonClass Latent] → [inst_9 : MeasurableSpace Allocation] → [inst_10 : MeasurableSpace Observation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → ProbabilityTheory.Kernel Latent (Allocation × Observation)","l":37,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fullKernel_isMarkovKernel","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"For a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with distinguishable allocation, observation, and statistic values, σ-algebras on the latent, allocation, and observation spaces, and measurable singleton latent states, the full-data kernel is a Markov kernel: at every latent state, its output law is a probability measure.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [inst_10 : MeasurableSpace Observation] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic), ProbabilityTheory.IsMarkovKernel E.fullKernel","l":49,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fullKernel_singletonReal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The real singleton probability of the full-data kernel equals the finite joint mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (z : Allocation × Observation), (E.fullKernel θ).real {z} = E.jointMass θ z","l":147,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fullRisk","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a real-valued target indexed by latent state, a real-valued full-data estimator, and a latent state, the full-data squared-error risk is the joint expected value of the estimator's squared error relative to the target at that state.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → (Latent → ℝ) → (Allocation × Observation → ℝ) → Latent → ℝ","l":44,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.fullRisk_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Full-data squared-error risk is nonnegative at every latent state.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (target : Latent → ℝ) (est : Allocation × Observation → ℝ) (θ : Latent), 0 ≤ E.fullRisk target est θ","l":54,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.jointMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a latent state, and a full-data point, the joint mass is the uniform allocation mass of its allocation multiplied by the conditional observation mass of its observation at that state and allocation.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → Allocation × Observation → ℝ","l":67,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.jointMass_eq_statisticMass_mul_conditionalWeight","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every full-data atom factors into its statistic marginal and guarded conditional weight.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (z : Allocation × Observation), E.jointMass θ z = E.statisticMass θ (E.sampleStatistic z) * E.conditionalWeight θ (E.sampleStatistic z) z","l":229,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.jointMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The full-data joint mass is nonnegative at every state and sample point.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (z : Allocation × Observation), 0 ≤ E.jointMass θ z","l":80,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.jointMass_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"At every latent state, the full-data joint mass sums to one.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent), ∑ z, E.jointMass θ z = 1","l":86,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.kernelAverageLoss_posteriorKernel_eq_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Kernel averaging a real latent-state loss through the finite-prior posterior kernel is exactly the finite posterior-weighted sum, matching the real FiniteKernelBayes interface.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [inst_14 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) {Action : Type u_5} (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (loss : Action → Latent → ℝ) (a : Action) (s : Statistic), Causalean.Stat.kernelAverageLoss (E.posteriorKernel prior) loss a s = ∑ θ, E.posteriorWeight prior s θ * loss a θ","l":226,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.kernelAverageLoss_statisticKernel_eq_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Kernel averaging through the statistic kernel is exactly the real finite sum of statistic losses weighted by their statistic masses, matching FiniteKernelBayes.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) {Action : Type u_5} (loss : Action → Statistic → ℝ) (a : Action) (θ : Latent), Causalean.Stat.kernelAverageLoss E.statisticKernel loss a θ = ∑ s, E.statisticMass θ s * loss a s","l":215,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.kernelMean_conditionalKernel_eq_conditionalMean","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"The measure-theoretic conditional-kernel mean of a full-data estimator equals the finite conditional mean, so no ENNReal conversion appears in Rao--Blackwell calculations.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [inst_14 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (est : Allocation × Observation → ℝ) (θ : Latent) (s : Statistic), ∫ (z : Allocation × Observation), est z ∂(E.conditionalKernel θ) s = E.conditionalMean est θ s","l":238,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.measurable_commonRaoBlackwellEstimator","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On finite statistic spaces, the common Rao--Blackwell estimator is measurable for the discrete sigma algebra.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) [inst_7 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (K : E.CommonConditionalKernel) (est : Allocation × Observation → ℝ), Measurable (E.commonRaoBlackwellEstimator K est)","l":131,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.measurable_conditionalMean","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a finite statistic space, the state-indexed conditional mean is measurable for the discrete sigma algebra.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) [inst_7 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (est : Allocation × Observation → ℝ) (θ : Latent), Measurable (E.conditionalMean est θ)","l":36,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.measurable_raoBlackwellEstimator","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The factorization-based Rao--Blackwell estimator is measurable on a finite discrete statistic space.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) [inst_7 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (F : E.SufficientFactorization) (est : Allocation × Observation → ℝ), Measurable (E.raoBlackwellEstimator F est)","l":234,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.minimaxValue_statistic_le_full","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"If the statistic satisfies the finite factorization criterion, its estimator class has minimax squared-risk value no larger than the full-data estimator class.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization) (target : Latent → ℝ), Causalean.Stat.minimaxValue (E.statisticRisk target) ≤ Causalean.Stat.minimaxValue (E.fullRisk target)","l":273,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.minimaxValue_statistic_le_full_of_commonConditionalKernel","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"If every full-data estimator is Rao--Blackwellized through state-independent conditionals, the minimax value over statistic-only estimators is no larger than the full-data minimax value.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (K : E.CommonConditionalKernel) (target : Latent → ℝ), Causalean.Stat.minimaxValue (E.statisticRisk target) ≤ Causalean.Stat.minimaxValue (E.fullRisk target)","l":211,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorDesign","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a finite prior design on latent states, and a statistic value, the guarded posterior design is the finite probability design on latent states whose probabilities are the guarded posterior weights.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent","l":141,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Given a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with σ-algebras on the latent and statistic spaces and measurable singleton statistic values, and a finite prior probability distribution on latent states, the posterior Markov kernel assigns to each statistic value the guarded posterior probability law of the latent state, using the designated fallback law on prior-predictive statistic fibers of zero probability.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → [inst_7 : MeasurableSpace Latent] → [inst_8 : MeasurableSpace Statistic] → [MeasurableSingletonClass Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → ProbabilityTheory.Kernel Statistic Latent","l":128,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorKernel_isMarkovKernel","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"For a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with distinguishable allocation, observation, and statistic values, σ-algebras on the latent and statistic spaces, and measurable singleton statistic values and a finite prior probability distribution on latent states, the guarded posterior kernel is a Markov kernel: at every statistic value, including a prior-predictive zero-probability fibre, its output law is a probability measure.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSpace Statistic] [inst_9 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent), ProbabilityTheory.IsMarkovKernel (E.posteriorKernel prior)","l":135,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorKernel_singletonReal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The real singleton probability of the finite-prior posterior kernel equals the guarded posterior weight of that latent state.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [inst_14 : MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (s : Statistic) (θ : Latent), ((E.posteriorKernel prior) s).real {θ} = E.posteriorWeight prior s θ","l":202,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorWeight","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a finite prior design on latent states, a statistic value, and a latent state, the guarded posterior weight is the prior joint state--statistic mass divided by the prior-predictive statistic mass when the latter is positive, and is the original prior probability of the state when it is zero.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → Statistic → Latent → ℝ","l":68,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorWeight_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every guarded posterior weight is nonnegative.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (s : Statistic) (θ : Latent), 0 ≤ E.posteriorWeight prior s θ","l":90,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorWeight_of_eq_zero","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a null prior-predictive fiber, the guarded posterior is the original prior.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) {s : Statistic}, E.priorStatisticMass prior s = 0 → ∀ (θ : Latent), E.posteriorWeight prior s θ = prior.p θ","l":83,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorWeight_of_pos","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a positive prior-predictive fiber, the guarded posterior is the usual Bayes ratio.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) {s : Statistic}, 0 < E.priorStatisticMass prior s → ∀ (θ : Latent), E.posteriorWeight prior s θ = E.priorJointStatisticMass prior θ s / E.priorStatisticMass prior s","l":75,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posteriorWeight_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Guarded posterior weights sum to one on positive and null prior-predictive fibers.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (s : Statistic), ∑ θ, E.posteriorWeight prior s θ = 1","l":104,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.posterior_disintegrate_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Every real test function of latent state and statistic has the same prior-joint expectation as prior-predictive statistic expectation of posterior conditional expectation; null fibers contribute exactly zero.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (f : Latent → Statistic → ℝ), ∑ θ, ∑ s, E.priorJointStatisticMass prior θ s * f θ s = ∑ s, E.priorStatisticMass prior s * ∑ θ, E.posteriorWeight prior s θ * f θ s","l":155,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorJointStatisticMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a finite prior design on latent states, a latent state, and a statistic value, the prior joint state--statistic mass is the prior probability of the state multiplied by that state's statistic mass at the specified value.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → Latent → Statistic → ℝ","l":27,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorJointStatisticMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every prior state--statistic joint mass is nonnegative.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (θ : Latent) (s : Statistic), 0 ≤ E.priorJointStatisticMass prior θ s","l":32,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorJointStatisticMass_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The prior state--statistic joint masses sum to one.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent), ∑ p, E.priorJointStatisticMass prior p.1 p.2 = 1","l":39,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorRisk_commonRaoBlackwellEstimator_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Under any finite prior on latent states, the common Rao--Blackwell estimator has no larger prior-averaged squared-error risk than the original estimator.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (K : E.CommonConditionalKernel) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (target : Latent → ℝ) (est : Allocation × Observation → ℝ), prior.E (E.statisticRisk target (E.commonRaoBlackwellEstimator K est)) ≤ prior.E (E.fullRisk target est)","l":177,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorRisk_raoBlackwellEstimator_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Under any finite prior, the one statistic-only estimator derived from a sufficient factorization has no larger prior-averaged squared-error risk than the full-data estimator.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (target : Latent → ℝ) (est : Allocation × Observation → ℝ), prior.E (E.statisticRisk target (E.raoBlackwellEstimator F est)) ≤ prior.E (E.fullRisk target est)","l":252,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorStatisticDesign","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment and a finite prior design on latent states, the prior-predictive statistic design is the finite probability design whose probabilities are the prior-predictive statistic masses.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → Causalean.Experimentation.DesignBased.FiniteDesign Statistic","l":148,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorStatisticMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a finite prior design on latent states, and a statistic value, the prior-predictive statistic mass is the sum of the prior joint state--statistic masses over all latent states.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Causalean.Experimentation.DesignBased.FiniteDesign Latent → Statistic → ℝ","l":48,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorStatisticMass_mul_posteriorWeight","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Multiplying the statistic marginal by its guarded posterior weight recovers each state--statistic joint atom, including on null fibers.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (s : Statistic) (θ : Latent), E.priorStatisticMass prior s * E.posteriorWeight prior s θ = E.priorJointStatisticMass prior θ s","l":119,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorStatisticMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every prior-predictive statistic mass is nonnegative.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent) (s : Statistic), 0 ≤ E.priorStatisticMass prior s","l":52,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.priorStatisticMass_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Posterior","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The prior-predictive statistic masses sum to one.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (prior : Causalean.Experimentation.DesignBased.FiniteDesign Latent), ∑ s, E.priorStatisticMass prior s = 1","l":59,"f":"Causalean/Stat/FiniteRaoBlackwell/Posterior.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.raoBlackwellEstimator","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a sufficient factorization, and a real-valued full-data estimator, the Rao--Blackwell estimator is the statistic-only estimator obtained by averaging the full-data estimator with the factorization's derived common conditional law.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [inst_4 : DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) → E.SufficientFactorization → (Allocation × Observation → ℝ) → Statistic → ℝ","l":228,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.sampleStatistic","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment and a full-data point, the sample statistic is the statistic computed from that point's allocation and observation.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Allocation × Observation → Statistic","l":57,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticDesign","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment and a latent state, the statistic design is the finite probability design whose probabilities are the statistic masses.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → Causalean.Experimentation.DesignBased.FiniteDesign Statistic","l":194,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticKernel","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":1,"d":"Given a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with σ-algebras on the latent and statistic spaces and measurable singleton latent states, the statistic Markov kernel assigns to each latent state the experiment's finite marginal probability law of the statistic.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Observation] → [DecidableEq Statistic] → [inst_7 : MeasurableSpace Latent] → [MeasurableSingletonClass Latent] → [inst_9 : MeasurableSpace Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → ProbabilityTheory.Kernel Latent Statistic","l":64,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticKernel_isMarkovKernel","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"For a finite uniform experiment on finite latent, allocation, observation, and statistic spaces, with distinguishable allocation, observation, and statistic values, σ-algebras on the latent and statistic spaces, and measurable singleton latent states, the statistic kernel is a Markov kernel: at every latent state, its output law is a probability measure.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic), ProbabilityTheory.IsMarkovKernel E.statisticKernel","l":70,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticKernel_singletonReal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.KernelBridge","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The real singleton probability of the statistic kernel equals the finite statistic mass.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] [inst_7 : MeasurableSpace Latent] [inst_8 : MeasurableSingletonClass Latent] [inst_9 : MeasurableSpace Allocation] [MeasurableSingletonClass Allocation] [inst_11 : MeasurableSpace Observation] [MeasurableSingletonClass Observation] [inst_13 : MeasurableSpace Statistic] [MeasurableSingletonClass Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic), (E.statisticKernel θ).real {s} = E.statisticMass θ s","l":161,"f":"Causalean/Stat/FiniteRaoBlackwell/KernelBridge.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a latent state, and a statistic value, the statistic mass is the sum of joint masses of all full-data points whose statistic equals that value.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → [DecidableEq Statistic] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Latent → Statistic → ℝ","l":96,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticMass_mul_conditionalWeight","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Multiplying a statistic marginal by its guarded conditional weight recovers the joint mass on that fiber and zero away from it, including when the marginal is zero.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic) (z : Allocation × Observation), E.statisticMass θ s * E.conditionalWeight θ s z = if E.sampleStatistic z = s then E.jointMass θ z else 0","l":200,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every statistic mass is nonnegative.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent) (s : Statistic), 0 ≤ E.statisticMass θ s","l":103,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticMass_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"At every latent state, the statistic masses sum to one.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (θ : Latent), ∑ s, E.statisticMass θ s = 1","l":112,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticRisk","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment, a real-valued target indexed by latent state, a real-valued statistic-only estimator, and a latent state, the statistic-only squared-error risk is the joint expected squared error after applying the estimator to the sample statistic.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → (Latent → ℝ) → (Statistic → ℝ) → Latent → ℝ","l":49,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticRisk_commonRaoBlackwellEstimator_le_fullRisk","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"A common sufficient conditional kernel produces one statistic-only Rao--Blackwell estimator whose squared-error risk is no larger at every latent state.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (K : E.CommonConditionalKernel) (target : Latent → ℝ) (est : Allocation × Observation → ℝ) (θ : Latent), E.statisticRisk target (E.commonRaoBlackwellEstimator K est) θ ≤ E.fullRisk target est θ","l":139,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticRisk_conditionalMean_le_fullRisk","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"At any fixed state, composing that state's conditional mean with the statistic has no larger squared-error risk than the original estimator. This is a fiberwise analytic inequality; the conditional mean in this statement is not asserted to be one estimator shared across states.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (target : Latent → ℝ) (est : Allocation × Observation → ℝ) (θ : Latent), E.statisticRisk target (E.conditionalMean est θ) θ ≤ E.fullRisk target est θ","l":92,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticRisk_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Statistic-only squared-error risk is nonnegative at every latent state.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (target : Latent → ℝ) (est : Statistic → ℝ) (θ : Latent), 0 ≤ E.statisticRisk target est θ","l":61,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.statisticRisk_raoBlackwellEstimator_le_fullRisk","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":1,"d":"A sufficient factorization produces one statistic-only Rao--Blackwell estimator whose squared-error risk is no larger than the full-data estimator at every latent state.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization) (target : Latent → ℝ) (est : Allocation × Observation → ℝ) (θ : Latent), E.statisticRisk target (E.raoBlackwellEstimator F est) θ ≤ E.fullRisk target est θ","l":242,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.uniformAllocationMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite uniform-allocation experiment and an allocation, the uniform allocation mass is the reciprocal of the number of admissible allocations when that allocation is admissible, and zero otherwise.","s":"{Latent : Type u_1} → {Allocation : Type u_2} → {Observation : Type u_3} → {Statistic : Type u_4} → [inst : Fintype Latent] → [inst_1 : Fintype Allocation] → [inst_2 : Fintype Observation] → [inst_3 : Fintype Statistic] → [DecidableEq Allocation] → Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic → Allocation → ℝ","l":61,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.uniformAllocationMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.Core","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The uniform allocation mass is nonnegative at every allocation.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [DecidableEq Observation] [DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (a : Allocation), 0 ≤ E.uniformAllocationMass a","l":71,"f":"Causalean/Stat/FiniteRaoBlackwell/Core.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.worstCaseRisk_commonRaoBlackwellEstimator_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Under state-independent conditional laws, Rao--Blackwellization weakly decreases the finite worst-case squared risk, in the real-valued worstCaseRisk API.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (K : E.CommonConditionalKernel) (target : Latent → ℝ) (est : Allocation × Observation → ℝ), Causalean.Stat.worstCaseRisk (E.statisticRisk target) (E.commonRaoBlackwellEstimator K est) ≤ Causalean.Stat.worstCaseRisk (E.fullRisk target) est","l":191,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment.worstCaseRisk_raoBlackwellEstimator_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.RaoBlackwell","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Under a sufficient factorization, the one statistic-only Rao--Blackwell estimator has no larger finite worst-case squared risk than the full-data estimator.","s":"∀ {Latent : Type u_1} {Allocation : Type u_2} {Observation : Type u_3} {Statistic : Type u_4} [inst : Fintype Latent] [inst_1 : Fintype Allocation] [inst_2 : Fintype Observation] [inst_3 : Fintype Statistic] [inst_4 : DecidableEq Allocation] [inst_5 : DecidableEq Observation] [inst_6 : DecidableEq Statistic] (E : Causalean.Stat.FiniteRaoBlackwell.FiniteUniformExperiment Latent Allocation Observation Statistic) (F : E.SufficientFactorization) (target : Latent → ℝ) (est : Allocation × Observation → ℝ), Causalean.Stat.worstCaseRisk (E.statisticRisk target) (E.raoBlackwellEstimator F est) ≤ Causalean.Stat.worstCaseRisk (E.fullRisk target) est","l":263,"f":"Causalean/Stat/FiniteRaoBlackwell/RaoBlackwell.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a finite count histogram, its histogram fibre is the set of ordered arrays whose length is the histogram total and whose histogram equals the given one.","s":"{X : Type u_1} → [Fintype X] → [DecidableEq X] → (X → ℕ) → Type (max 0 u_1)","l":46,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a finite-alphabet probability law and a Poisson intensity, the count-histogram law is the distribution of the unordered histogram of a finite Poisson sample.","s":"{X : Type u_1} → [Fintype X] → [inst : MeasurableSpace X] → [MeasurableSingletonClass X] → [DecidableEq X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → NNReal → MeasureTheory.Measure (X → ℕ)","l":24,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw_map_histogramTotal","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given a finite-alphabet probability law and a Poisson intensity, the total count under the histogram law has the scalar Poisson distribution with that intensity.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] [inst_3 : DecidableEq X] (P : MeasureTheory.Measure X) [inst_4 : MeasureTheory.IsProbabilityMeasure P] (lam : NNReal), MeasureTheory.Measure.map Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw P lam) = ProbabilityTheory.poissonMeasure lam","l":52,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.finiteSampleHistogram","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a finite ordered sample and an alphabet symbol, its finite-sample histogram count is the number of sample positions equal to that symbol.","s":"{m : ℕ} → {X : Type u_1} → [DecidableEq X] → (Fin m → X) → X → ℕ","l":22,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramFiberFintype","k":"instance","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given a finite count histogram, its histogram fibre has a finite enumeration.","s":"{X : Type u_1} → [inst : Fintype X] → [inst_1 : DecidableEq X] → (c : X → ℕ) → Fintype (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber c)","l":52,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramFiber_nonempty","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Every finite count histogram has at least one compatible ordering.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : DecidableEq X] (c : X → ℕ), Nonempty (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber c)","l":59,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a finite count histogram, its histogram total is the sum of the counts over the alphabet.","s":"{X : Type u_1} → [Fintype X] → (X → ℕ) → ℕ","l":28,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal_finiteSampleHistogram","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"The histogram of a finite ordered sample has total equal to the sample size.","s":"∀ {m : ℕ} {X : Type u_1} [inst : Fintype X] [inst_1 : DecidableEq X] (x : Fin m → X), Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.finiteSampleHistogram x) = m","l":33,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.map_pairRetainedArrays_prod_pi","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.PairingLaw","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given one probability law and a second probability law, coordinatewise pairing sends two independent iid arrays to an iid array from the product law.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (Q : MeasureTheory.Measure Y) [MeasureTheory.IsProbabilityMeasure Q], MeasureTheory.Measure.map (fun z => Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairRetainedArrays z.1 z.2) ((MeasureTheory.Measure.pi fun x => P).prod (MeasureTheory.Measure.pi fun x => Q)) = MeasureTheory.Measure.pi fun x => P.prod Q","l":35,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/PairingLaw.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.measurable_finiteSampleHistogram","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"The map from a finite sample to its histogram is measurable over a finite alphabet with measurable singletons.","s":"∀ {X : Type u_1} [Fintype X] [inst : MeasurableSpace X] [MeasurableSingletonClass X] [inst_2 : DecidableEq X], Measurable fun s => Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.finiteSampleHistogram s.points","l":35,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.measurable_pairRetainedArrays","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.PairingLaw","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Pairing two finite arrays coordinatewise is measurable.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y], Measurable fun z => Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairRetainedArrays z.1 z.2","l":21,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/PairingLaw.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.measurable_pairedPoissonHistogramEstimator","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given a paired-sample estimator and a fallback value, the resulting paired count estimator is measurable on finite measurable alphabets with measurable singletons.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : MeasurableSpace X] [MeasurableSingletonClass X] [inst_3 : DecidableEq X] [inst_4 : Fintype Y] [inst_5 : MeasurableSpace Y] [MeasurableSingletonClass Y] [inst_7 : DecidableEq Y] (est : (Fin n → X × Y) → ℝ) (fallback : ℝ), Measurable (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est fallback)","l":160,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairRetainedArrays","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given one retained array and another retained array, their coordinatewise pairing places the entries with each common index into a pair.","s":"{n : ℕ} → {X : Type u_1} → {Y : Type u_2} → (Fin n → X) → (Fin n → Y) → Fin n → X × Y","l":93,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a paired-sample estimator, two finite count histograms, and certificates that both totals contain the requested sample length, the paired conditional histogram average independently averages the estimator over all compatible orderings, after retaining and coordinatewise pairing the two prefixes.","s":"{n : ℕ} → {X : Type u_1} → {Y : Type u_2} → [inst : Fintype X] → [DecidableEq X] → [inst_2 : Fintype Y] → [DecidableEq Y] → ((Fin n → X × Y) → ℝ) → (cX : X → ℕ) → (cY : Y → ℕ) → n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cX → n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cY → ℝ","l":99,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage_productRisk_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.FixedRisk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given two finite-alphabet probability laws, a paired-sample estimator, a real target, and certificates that both fixed totals contain the requested sample length, independent histogram averaging followed by coordinatewise pairing has no greater squared risk than the paired iid experiment.","s":"∀ {n NX NY : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : MeasurableSpace X] [MeasurableSingletonClass X] [inst_3 : DecidableEq X] [inst_4 : Fintype Y] [inst_5 : MeasurableSpace Y] [MeasurableSingletonClass Y] [inst_7 : DecidableEq Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (Q : MeasureTheory.Measure Y) [MeasureTheory.IsProbabilityMeasure Q] (est : (Fin n → X × Y) → ℝ) (theta : ℝ) (hX : n ≤ NX) (hY : n ≤ NY), ∫ (z : (Fin NX → X) × (Fin NY → Y)), (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage est (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.finiteSampleHistogram z.1) (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.finiteSampleHistogram z.2) (Eq.mpr (id (congrArg (LE.le n) (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal_finiteSampleHistogram z.1))) hX) (Eq.mpr (id (congrArg (LE.le n) (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal_finiteSampleHistogram z.2))) hY) - theta) ^ 2 ∂(MeasureTheory.Measure.pi fun x => P).prod (MeasureTheory.Measure.pi fun x => Q) ≤ ∫ (z : Fin n → X × Y), (est z - theta) ^ 2 ∂MeasureTheory.Measure.pi fun x => P.prod Q","l":348,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/FixedRisk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage_sq_sub_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.FixedRisk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given a paired-sample estimator, a real target, two finite count histograms, and certificates that both totals contain the requested sample length, the squared loss of the independent two-fibre average is at most the uniform average squared loss over both ordering fibres.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : DecidableEq X] [inst_2 : Fintype Y] [inst_3 : DecidableEq Y] (est : (Fin n → X × Y) → ℝ) (theta : ℝ) (cX : X → ℕ) (cY : Y → ℕ) (hX : n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cX) (hY : n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cY), (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage est cX cY hX hY - theta) ^ 2 ≤ (∑ x, ∑ y, (est (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairRetainedArrays (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.retainedHistogramPrefix hX x) (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.retainedHistogramPrefix hY y)) - theta) ^ 2) / (↑(Fintype.card (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber cX)) * ↑(Fintype.card (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber cY)))","l":92,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/FixedRisk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a paired-sample estimator, a fallback value, and two finite count histograms, the paired Poisson-histogram estimator uses the independent compatible-ordering average when both totals are large enough and returns the fallback when either total is too small.","s":"{n : ℕ} → {X : Type u_1} → {Y : Type u_2} → [Fintype X] → [DecidableEq X] → [Fintype Y] → [DecidableEq Y] → ((Fin n → X × Y) → ℝ) → ℝ → (X → ℕ) × (Y → ℕ) → ℝ","l":115,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator_of_total_lt","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given a paired-sample estimator, a fallback value, two finite count histograms, and a certificate that at least one total is too small, the paired count estimator equals its fallback.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : DecidableEq X] [inst_2 : Fintype Y] [inst_3 : DecidableEq Y] (est : (Fin n → X × Y) → ℝ) (fallback : ℝ) (cX : X → ℕ) (cY : Y → ℕ), Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cX < n ∨ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cY < n → Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est fallback (cX, cY) = fallback","l":144,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator_of_totals_ge","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a paired-sample estimator, a fallback value, two finite count histograms, and certificates that both totals contain the requested sample length, the paired count estimator equals the independent compatible-ordering average.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : DecidableEq X] [inst_2 : Fintype Y] [inst_3 : DecidableEq Y] (est : (Fin n → X × Y) → ℝ) (fallback : ℝ) (cX : X → ℕ) (cY : Y → ℕ) (hX : n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cX) (hY : n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal cY), Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est fallback (cX, cY) = Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedHistogramAverage est cX cY hX hY","l":130,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramRisk_le_fixedRisk_add_tails","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given two finite-alphabet probability laws, their Poisson intensities, a paired fixed-sample estimator, a fallback value, and a real target, the paired histogram estimator has integrable squared loss and risk at most the paired fixed-sample risk plus the fallback loss times the sum of the two marginal lower-tail probabilities.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] [inst_3 : DecidableEq X] [inst_4 : Fintype Y] [inst_5 : MeasurableSpace Y] [inst_6 : MeasurableSingletonClass Y] [inst_7 : DecidableEq Y] (P : MeasureTheory.Measure X) [inst_8 : MeasureTheory.IsProbabilityMeasure P] (Q : MeasureTheory.Measure Y) [inst_9 : MeasureTheory.IsProbabilityMeasure Q] (lamP lamQ : NNReal) (est : (Fin n → X × Y) → ℝ) (fallback theta : ℝ), MeasureTheory.Integrable (fun c => (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est fallback c - theta) ^ 2) ((Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw P lamP).prod (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw Q lamQ)) ∧ ∫ (c : (X → ℕ) × (Y → ℕ)), (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est fallback c - theta) ^ 2 ∂(Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw P lamP).prod (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw Q lamQ) ≤ (∫ (z : Fin n → X × Y), (est z - theta) ^ 2 ∂MeasureTheory.Measure.pi fun x => P.prod Q) + (fallback - theta) ^ 2 * ((ProbabilityTheory.poissonMeasure lamP).real {k | k < n} + (ProbabilityTheory.poissonMeasure lamQ).real {k | k < n})","l":150,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramRisk_two_n_exp_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given two finite-alphabet probability laws, a paired fixed-sample estimator, a real target, a target-magnitude bound, and a certificate of that bound, using zero fallback and Poisson means twice the sample size gives the explicit exponential risk penalty supplied by the Poisson lower-tail inequality.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] [inst_3 : DecidableEq X] [inst_4 : Fintype Y] [inst_5 : MeasurableSpace Y] [inst_6 : MeasurableSingletonClass Y] [inst_7 : DecidableEq Y] (P : MeasureTheory.Measure X) [inst_8 : MeasureTheory.IsProbabilityMeasure P] (Q : MeasureTheory.Measure Y) [inst_9 : MeasureTheory.IsProbabilityMeasure Q] (est : (Fin n → X × Y) → ℝ) (theta B : ℝ), |theta| ≤ B → ∫ (c : (X → ℕ) × (Y → ℕ)), (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est 0 c - theta) ^ 2 ∂(Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw P (2 * ↑n)).prod (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw Q (2 * ↑n)) ≤ (∫ (z : Fin n → X × Y), (est z - theta) ^ 2 ∂MeasureTheory.Measure.pi fun x => P.prod Q) + B ^ 2 * Real.exp (-↑n * (1 - Real.log 2)) * 2","l":465,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramRisk_two_n_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Risk","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":2,"d":"Given two finite-alphabet probability laws, a paired fixed-sample estimator, a real target, a target-magnitude bound, and a certificate of that bound, using zero fallback and Poisson means twice the sample size gives a risk penalty no larger than twice the squared bound times the common Poisson lower-tail probability.","s":"∀ {n : ℕ} {X : Type u_1} {Y : Type u_2} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] [inst_3 : DecidableEq X] [inst_4 : Fintype Y] [inst_5 : MeasurableSpace Y] [inst_6 : MeasurableSingletonClass Y] [inst_7 : DecidableEq Y] (P : MeasureTheory.Measure X) [inst_8 : MeasureTheory.IsProbabilityMeasure P] (Q : MeasureTheory.Measure Y) [inst_9 : MeasureTheory.IsProbabilityMeasure Q] (est : (Fin n → X × Y) → ℝ) (theta B : ℝ), |theta| ≤ B → ∫ (c : (X → ℕ) × (Y → ℕ)), (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.pairedPoissonHistogramEstimator est 0 c - theta) ^ 2 ∂(Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw P (2 * ↑n)).prod (Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.countLaw Q (2 * ↑n)) ≤ (∫ (z : Fin n → X × Y), (est z - theta) ^ 2 ∂MeasureTheory.Measure.pi fun x => P.prod Q) + B ^ 2 * (ProbabilityTheory.poissonMeasure (2 * ↑n)).real {k | k < n} * 2","l":415,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Risk.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.retainedHistogramPrefix","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.Basic","a":"Stat/FiniteRaoBlackwell/PairedPoissonHistogram","t":1,"d":"Given a certificate that the requested length does not exceed the histogram total and a compatible ordering, the retained histogram prefix consists of the ordering's first requested entries.","s":"{n : ℕ} → {X : Type u_1} → [inst : Fintype X] → [inst_1 : DecidableEq X] → {c : X → ℕ} → n ≤ Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.histogramTotal c → Causalean.Stat.FiniteRaoBlackwell.PairedPoissonHistogram.HistogramFiber c → Fin n → X","l":84,"f":"Causalean/Stat/FiniteRaoBlackwell/PairedPoissonHistogram/Basic.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite probability design on source points, a deterministic coarsening map, a coarsened value, and a source point, the conditional fiber weight is its design probability divided by the fiber mass when that mass is nonzero and the point maps to the value, and zero otherwise.","s":"{Ω : Type u_1} → {B : Type u_2} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → B) → B → Ω → ℝ","l":55,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Conditional fiber weights are nonnegative on both positive-mass and null fibers.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (b : B) (ω : Ω), 0 ≤ Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight D φ b ω","l":63,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite probability design on source points, a deterministic coarsening map, a real-valued default, a real-valued source function indexed by values, a coarsened value, and an index value, the conditional mean along the map is the design-weighted fiber mean when the fiber has nonzero mass and the supplied default when it has zero mass.","s":"{Ω : Type u_1} → {B : Type u_2} → {V : Type u_3} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → B) → ℝ → (Ω → V → ℝ) → B → V → ℝ","l":122,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap_eq_sum_conditionalFiberWeight","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a nonzero-mass fiber, the guarded conditional mean equals the source-value average under the normalized conditional fiber weights.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (b : B) (v : V), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b ≠ 0 → Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b v = ∑ ω, Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight D φ b ω * h ω v","l":136,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap_mem_Icc","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"If the default lies in a fixed closed interval and every source value lies in that interval, the guarded conditional mean also lies in the interval.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (b : B) (v : V) {l u : ℝ}, d ∈ Set.Icc l u → (∀ (ω : Ω) (v : V), h ω v ∈ Set.Icc l u) → Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b v ∈ Set.Icc l u","l":197,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap_of_fiberMass_eq_zero","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a zero-mass fiber, the guarded conditional mean equals its supplied default.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (b : B) (v : V), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b = 0 → Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b v = d","l":128,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap_sq_le","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"On a nonzero-mass fiber, the squared loss of the conditional mean is at most the conditional fiber average of the source squared losses.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (b : B) (v : V) (t : ℝ), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b ≠ 0 → (Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b v - t) ^ 2 ≤ ∑ ω, Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight D φ b ω * (h ω v - t) ^ 2","l":229,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberMass","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite probability design on source points, a deterministic coarsening map, and a coarsened value, the fiber mass is the total design probability of source points mapped to that value.","s":"{Ω : Type u_1} → {B : Type u_2} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → B) → B → ℝ","l":23,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberMass_eq_map_p","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"A coarsening fiber's total design mass equals the probability of its value under the pushforward design.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [inst_1 : Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (b : B), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b = (D.map φ).p b","l":29,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberMass_nonneg","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"Every coarsening fiber has nonnegative design mass.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (b : B), 0 ≤ Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b","l":36,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberMass_sum","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The masses of all coarsening fibers sum to one.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [inst_1 : Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B), ∑ b, Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b = 1","l":42,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberNumerator","k":"def","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":1,"d":"For a finite probability design on source points, a deterministic coarsening map, a real-valued source function indexed by values, a coarsened value, and an index value, the fiber numerator is the sum of the design probability times the source-function value over source points mapped to that coarsened value.","s":"{Ω : Type u_1} → {B : Type u_2} → {V : Type u_3} → [inst : Fintype Ω] → Causalean.Experimentation.DesignBased.FiniteDesign Ω → (Ω → B) → (Ω → V → ℝ) → B → V → ℝ","l":48,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.fiberNumerator_eq_zero_of_fiberMass_eq_zero","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"If a fiber has zero design mass, the design-weighted numerator of every real-valued source function vanishes on that fiber.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (h : Ω → V → ℝ) (b : B) (v : V), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b = 0 → Causalean.Stat.FiniteRaoBlackwell.fiberNumerator D φ h b v = 0","l":108,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.p_eq_zero_of_mem_fiberMass_eq_zero","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"If a fiber has zero design mass and a source point maps into that fiber, the source point itself has zero design probability.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (b : B), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b = 0 → ∀ (ω : Ω), φ ω = b → D.p ω = 0","l":94,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.sum_conditionalFiberWeight_eq_one","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"When the requested fiber has nonzero mass, its conditional weights sum to one.","s":"∀ {Ω : Type u_1} {B : Type u_2} [inst : Fintype Ω] [Fintype B] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (b : B), Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b ≠ 0 → ∑ ω, Causalean.Stat.FiniteRaoBlackwell.conditionalFiberWeight D φ b ω = 1","l":76,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteRaoBlackwell.sum_fiberMass_mul_conditionalMeanAlongMap","k":"theorem","m":"Causalean.Stat.FiniteRaoBlackwell.DesignPushforward","a":"Stat/FiniteRaoBlackwell","t":2,"d":"The original design-weighted sum equals the sum of each fiber mass times its guarded conditional mean, with null fibers contributing zero regardless of the default.","s":"∀ {Ω : Type u_1} {B : Type u_2} {V : Type u_3} [inst : Fintype Ω] [inst_1 : Fintype B] [Fintype V] (D : Causalean.Experimentation.DesignBased.FiniteDesign Ω) (φ : Ω → B) (d : ℝ) (h : Ω → V → ℝ) (v : B → V), ∑ b, Causalean.Stat.FiniteRaoBlackwell.fiberMass D φ b * Causalean.Stat.FiniteRaoBlackwell.conditionalMeanAlongMap D φ d h b (v b) = ∑ ω, D.p ω * h ω (v (φ ω))","l":152,"f":"Causalean/Stat/FiniteRaoBlackwell/DesignPushforward.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, an arm value, and a category, the arm--category event is the set of observations with both requested labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → Bool → kappa → Set Omega","l":40,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a Boolean arm-label function, an arm value, and a category, the population arm--category mass is the real-valued measure of their joint-label event.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → Bool → kappa → ℝ","l":54,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.armMarkSum","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, a finite sample, an arm value, and a category, the empirical arm--category mark sum is the sum of supported marks over all sample coordinates.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":64,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryArmCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a finite sample, an arm value, and a category, the empirical arm--category count is the number of sample coordinates having both specified labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → Bool → kappa → ℕ","l":30,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a finite sample, and a category, the empirical category count is the number of sample coordinates in that category across both arm values.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → kappa → ℕ","l":35,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryEvent","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function and a category, the category event is the set of observations assigned that category.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → kappa → Set Omega","l":45,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, and a category, the population category mass is the real-valued measure of the category event.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → kappa → ℝ","l":49,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a finite set of categories, an arm-and-category center function, and an arm value, the fixed-stratum arm center target is the sum of category masses multiplied by the corresponding supplied centers.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → Finset kappa → (Bool → kappa → ℝ) → Bool → ℝ","l":113,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget_eq_target","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels, square-integrable supported residuals, cellwise residual centering, and positive arm mass in every selected positive-mass category ensure that the center-weighted target equals the population arm target.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool), Measurable group → Measurable arm → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ k ∈ H, 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget mu group H center a = Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a","l":339,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, an arm-and-category center function, a finite set of categories, an arm value, and a finite sample, the fixed-stratum arm centered noise is the occupancy-weighted sum of totalized cell residual means.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":136,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a Boolean arm-label function, an arm-and-category center function, a finite set of categories, an arm value, and a finite sample, the fixed-stratum arm mass fluctuation is the sum of each center multiplied by empirical category occupancy divided by nominal sample size minus population category mass.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → [DecidableEq kappa] → {m : ℕ} → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":150,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, an arm-and-category center function, a finite set of categories, an arm value, and a finite sample, the fixed-stratum arm missing remainder is the nominal-sample-size-normalized sum of cell centers times category occupancies whose requested arm is absent.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Bool → kappa → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":143,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, a finite set of categories, an arm value, and a finite sample, the fixed-stratum arm score is the sum of totalized arm means weighted by empirical category occupancy divided by the nominal sample size.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → Bool → (Fin m → Omega) → ℝ","l":86,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore_sub_centerTarget_decomposition","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"The single-arm empirical score error around a supplied center target is exactly centered ratio noise plus empirical-mass fluctuation minus the explicit missing-arm remainder.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenterTarget mu group H center a = Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise group arm Y center H a z + Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation mu group arm center H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z","l":305,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a Boolean arm-label function, a real-valued mark, a finite set of categories, and an arm value, the fixed-stratum arm target is the sum of population arm--category means weighted by population category masses.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → Bool → ℝ","l":100,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, a finite set of categories, and a finite sample, the fixed-stratum marked ratio is the fixed-stratum arm score for the true arm minus that for the false arm.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → (Fin m → Omega) → ℝ","l":93,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio_memLp_two","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark together with finite second moments for every supported residual cell imply finite second moments for the treated-minus-control fixed-set score.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H) 2 (MeasureTheory.Measure.pi fun x => mu)","l":175,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedTarget","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a Boolean arm-label function, a real-valued mark, and a finite set of categories, the fixed-stratum marked target is the true-arm target minus the false-arm target.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Finset kappa → ℝ","l":106,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integrable_missingArmCount_mul","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"When the stratum label and the arm assignment are measurable, the product of the counts of missing arm/category cells for two categories is integrable over the product sample, because the two counts are bounded by the sample size.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k l : kappa), MeasureTheory.Integrable (fun z => ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) * ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a l)) (MeasureTheory.Measure.pi fun x => mu)","l":78,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmCenteredNoise_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.CenteredNoiseBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated cellwise second-moment envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the occupancy-weighted centered ratio noise has second moment at most twice the squared envelope times selected mass, divided by safe sample size and overlap.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmCenteredNoise group arm Y center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ (2 * M ^ 2 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon)","l":29,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/CenteredNoiseBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMassFluctuation_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and cell centers bounded in absolute value by the envelope imply that the empirical category-mass fluctuation has second moment at most the squared envelope divided by the safe sample size, including for an empty sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMassFluctuation mu group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m","l":240,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMissingRemainder_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels, cell centers bounded in absolute value by the envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the normalized aggregate empty-arm remainder has second moment bounded by a parametric diagonal term plus the squared exponentially damped missing-arm envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * (1 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":402,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArmMissingRemainder_sq_le_aux","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels, cell centers bounded in absolute value by the envelope, a positive overlap margin, and arm mass at least that margin times category mass imply that the missing-arm remainder has a boundary-safe diagonal-plus-exponential second-moment bound.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmMissingRemainder group arm center H a z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * (1 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":389,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArm_error_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, arm mass at least that margin times category mass, and a deterministic lower bound on every selected category mass imply the boundary-safe one-arm mean-squared-error bound with an inverse-polynomial missing-arm envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon B : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H ^ 2)","l":587,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumArm_error_sq_le_exponential","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, and arm mass at least that margin times category mass imply a uniform one-arm fixed-set mean-squared-error bound with an explicit exponentially damped missing-arm remainder.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (a : Bool) (M epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmScore group arm Y H a z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumArmTarget mu group arm Y H a) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ^ 2)","l":424,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_fixedStratumMarkedRatio_error_sq_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Main","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels and a measurable mark, square-integrable supported residuals in both arms, cellwise residual centering, the stated residual second-moment envelope, bounded cell centers, a positive overlap margin, both arm masses at least that margin times category mass, and a deterministic lower bound on every selected category mass imply that the treated-minus-control fixed-set score obeys the boundary-safe mean-squared-error bound at the same parametric and missing-arm scale, up to the universal two-arm factor.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (H : Finset kappa) (M epsilon B : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, Y omega - center a k ∂mu = 0) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryEvent group arm a k, (Y omega - center a k) ^ 2 ∂mu ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k * M ^ 2) → (∀ (a : Bool) (k : kappa), |center a k| ≤ M) → 0 < epsilon → (∀ (k : kappa), 0 < Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k → ∀ (a : Bool), epsilon * Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → (∫ (z : Fin m → Omega), (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H z - Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedTarget mu group arm Y H) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 4 * M ^ 2 * ((8 * ∑ k ∈ H, Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) / (Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m * epsilon) + 6 / Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize m + 4 * Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H ^ 2)","l":661,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Main.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels imply that the expected category occupancy retained only when one arm is absent equals the sample size times the off-arm category mass times the empty-arm probability for the other coordinates.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) ∂MeasureTheory.Measure.pi fun x => mu) = ↑m * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 1)","l":372,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_mul_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and distinct categories imply that their missing-arm counts have the exact ordered-pair cross moment obtained by excluding the union of the two arm/category cells.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) {k l : kappa}, k ≠ l → (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) * ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a l) ∂MeasureTheory.Measure.pi fun x => mu) = ↑(m.descFactorial 2) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group l - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a l) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a l) ^ (m - 2)","l":537,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_missingArmCount_sq_eq","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable group and arm labels imply that the exact second moment of one missing-arm category count is the sum of its one-coordinate diagonal and ordered two-coordinate contributions.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) = ↑m * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 1) + ↑(m.descFactorial 2) * (Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ 2 * (1 - Causalean.Stat.FiniteStratumMarkedRatioMse.armCategoryMass mu group arm a k) ^ (m - 2)","l":429,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.integral_nested_count_sq_mul_totalized_inverse_le","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.NestedCountBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Measurable outer and inner events, nesting of the inner event inside the outer event, a positive overlap margin, and inner-event mass at least that margin times outer-event mass imply that the expected squared outer count times the zero-safe inverse inner count is at most twice sample size times outer-event mass divided by the margin.","s":"∀ {m : ℕ} {Omega : Type u_1} [inst : MeasurableSpace Omega] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (C R : Set Omega), MeasurableSet C → MeasurableSet R → R ⊆ C → ∀ (epsilon : ℝ), 0 < epsilon → epsilon * (mu C).toReal ≤ (mu R).toReal → (∫ (z : Fin m → Omega), ↑(Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z C).card ^ 2 * if 0 < (Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z R).card then (↑(Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet z R).card)⁻¹ else 0 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 2 * ↑m * (mu C).toReal / epsilon","l":257,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/NestedCountBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a nonnegative integer sample size, a real overlap margin, a real mass lower bound, and a finite set of categories, the lower-mass missing envelope first sets its squared overlap denominator to the squared half-adjusted sample-size overlap margin times the mass lower bound, and then uses the number of selected categories divided by that denominator when it is positive, and their total category mass otherwise. This totalization keeps the envelope defined for small samples, zero overlap, and a zero mass lower bound.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → ℕ → ℝ → ℝ → Finset kappa → ℝ","l":663,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope_eq_of_pos","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"A sample size of at least three, a positive overlap margin, and a positive category-mass lower bound make the boundary-safe lower-mass envelope equal its inverse-polynomial expression.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) {m : ℕ} {epsilon B : ℝ} (H : Finset kappa), 3 ≤ m → 0 < epsilon → 0 < B → Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H = ↑H.card / ((↑(m - 2) / 2 * epsilon) ^ 2 * B)","l":750,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_armMarkSum","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make every empirical arm/category mark sum measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.armMarkSum group arm Y z a k","l":176,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_categoryArmCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make every fixed empirical arm/category count measurable on the product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.categoryArmCount group arm z a k","l":159,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_categoryCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make every fixed empirical category count measurable on the product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.categoryCount group arm z k","l":168,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_fixedStratumMarkedRatio","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make the fixed-set marked ratio score measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (H : Finset kappa), Measurable group → Measurable arm → Measurable Y → Measurable (Causalean.Stat.FiniteStratumMarkedRatioMse.fixedStratumMarkedRatio group arm Y H)","l":211,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_missingArmCount","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels make each missing-arm category count measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount group arm z a k","l":234,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.measurable_totalizedArmMean","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"Measurable group and arm labels and a measurable mark make every totalized empirical arm/category mark mean measurable, including at zero count.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {m : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmMean group arm Y z a k","l":191,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmCount","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a finite sample, an arm value, and a category, the missing-arm count is the category occupancy when the requested arm--category count is zero, and zero otherwise.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin m → Omega) → Bool → kappa → ℕ","l":131,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a nonnegative integer sample size, a real overlap margin, and a finite set of categories, the missing-arm exponential envelope is the sum, over selected categories, of category mass times the exponential decay determined by the sample size, overlap margin, and that mass.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → ℕ → ℝ → Finset kappa → ℝ","l":657,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope_le_lowerMass","k":"theorem","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.MissingMoments","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":2,"d":"A measurable group label, a positive overlap margin, and a deterministic lower bound on every selected category mass ensure that the exponential missing-arm envelope is at most the boundary-safe lower-mass envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa), Measurable group → ∀ (m : ℕ) (epsilon B : ℝ) (H : Finset kappa), 0 < epsilon → (∀ k ∈ H, B ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.categoryMass mu group k) → Causalean.Stat.FiniteStratumMarkedRatioMse.missingArmExponentialEnvelope mu group m epsilon H ≤ Causalean.Stat.FiniteStratumMarkedRatioMse.lowerMassMissingEnvelope mu group m epsilon B H","l":683,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/MissingMoments.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.populationArmMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a measure on the observation space, a category-label function, a Boolean arm-label function, a real-valued mark, an arm value, and a category, the population arm--category mean is the mark integral over that cell divided by its positive mass, and zero when its mass is zero.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Bool → kappa → ℝ","l":78,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.safeSampleSize","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a nonnegative integer sample size, the safe real sample size is one when the sample is empty and otherwise equals the ordinary sample size.","s":"ℕ → ℝ","l":27,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.sampleIndexSet","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.NestedCountBound","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"Given a finite index set, an observation space, an observation assigned to each index, and a set of observations, the sample index set is the finite set of precisely those indices whose assigned observation belongs to that set.","s":"{I : Type u_1} → {Omega : Type u_2} → [Fintype I] → (I → Omega) → Set Omega → Finset I","l":25,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/NestedCountBound.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmMark","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, an arm value, and a category, the supported arm mark equals the mark on the corresponding arm--category event and zero outside it.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → Bool → kappa → Omega → ℝ","l":59,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.supportedArmResidual","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, an arm-and-category center function, an arm value, and a category, the supported arm residual is the mark minus its supplied cell center on that cell and zero outside it.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":119,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, a finite sample, an arm value, and a category, the totalized empirical arm mean is the arm--category mark sum divided by its count when that count is positive, and zero otherwise.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":70,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.FiniteStratumMarkedRatioMse.totalizedArmResidualMean","k":"def","m":"Causalean.Stat.Sample.FiniteStratumMarkedRatioMse.Basic","a":"Stat/Sample/FiniteStratumMarkedRatioMse","t":1,"d":"For a category-label function, a Boolean arm-label function, a real-valued mark, an arm-and-category center function, a finite sample, an arm value, and a category, the totalized empirical residual mean is zero for an empty arm--category cell and otherwise averages its supported centered residuals.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {m : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin m → Omega) → Bool → kappa → ℝ","l":125,"f":"Causalean/Stat/Sample/FiniteStratumMarkedRatioMse/Basic.lean"},{"n":"Causalean.Stat.GMMProblem","k":"structure","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"GMM problem. Bundles, over a probability measure on the data space, a moment function evaluated at the parameter truth θ₀, with a self-adjoint weighting operator and a Jacobian of the population moment at θ₀, verified to be its Fréchet derivative. It asserts the population moment vanishes at the truth and that the moment function at the truth is measurable and square-integrable, and packages a moment covariance operator defined as the second moment of the moment vector together with two-sided inverse witnesses for the covariance operator, for the GMM bread GᵀWG, and for the efficient bread GᵀCov⁻¹G.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst : NormedAddCommGroup F] → [inst_2 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → [MeasurableSpace F] → [inst : MeasurableSpace X] → MeasureTheory.Measure X → Type (max (max u_1 u_2) u_3)","l":83,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.asympVar","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"For finite-dimensional real inner-product parameter and moment spaces, a measurable data space, and a measure on that data space, a bundled GMM problem determines the sandwich asymptotic variance operator, which is $(G^{\\mathsf T}WG)^{-1}G^{\\mathsf T}W\\operatorname{Cov}WG(G^{\\mathsf T}WG)^{-1}$.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E →L[ℝ] E","l":146,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.cov_isPositive","k":"theorem","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":2,"d":"The covariance operator is positive — it is a second moment.","s":"∀ {E : Type u_1} {F : Type u_2} {X : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [inst_7 : MeasurableSpace X] {P : MeasureTheory.Measure X} (prob : Causalean.Stat.GMMProblem P), prob.Cov.IsPositive","l":158,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.effVar","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"For finite-dimensional real inner-product parameter and moment spaces, a measurable data space, and a measure on that data space, a bundled GMM problem determines the efficient asymptotic variance operator, which is $(G^{\\mathsf T}\\operatorname{Cov}^{-1}G)^{-1}$.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E →L[ℝ] E","l":152,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.efficiency","k":"theorem","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"GMM optimal-weighting theorem (Hansen 1982), statistical form. The sandwich asymptotic variance of a GMM problem with an arbitrary symmetric weighting dominates the efficient variance (GᵀCov⁻¹G)⁻¹ in the Löwner order: asympVar − effVar is a positive operator.","s":"∀ {E : Type u_1} {F : Type u_2} {X : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [inst_7 : MeasurableSpace X] {P : MeasureTheory.Measure X} (prob : Causalean.Stat.GMMProblem P), (prob.asympVar - prob.effVar).IsPositive","l":174,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.influence","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"For finite-dimensional real inner-product parameter and moment spaces, a measurable data space, and a measure on that data space, a bundled GMM problem determines the influence function, which maps an observation to $-(G^{\\mathsf T}WG)^{-1}G^{\\mathsf T}Wg(\\theta_0,x)$ using the problem's true parameter and inverse bread matrix.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → X → E","l":139,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GMMProblem.score","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"For finite-dimensional real inner-product parameter and moment spaces, a measurable data space, and a measure on that data space, a bundled GMM problem determines the combined score, which maps a parameter and an observation to $G^{\\mathsf T}Wg(\\theta,x)$ using that problem's Jacobian, weighting operator, and moment function.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [inst_2 : FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [inst_5 : FiniteDimensional ℝ F] → [inst_6 : MeasurableSpace F] → [inst_7 : MeasurableSpace X] → {P : MeasureTheory.Measure X} → Causalean.Stat.GMMProblem P → E → X → E","l":133,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.GlivenkoCantelli","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"For a measurable sample space, measurable observation space, index set, sample-space measure, and observation-space measure, an independent, identically distributed sample and a class of real-valued functions, the weak Glivenko--Cantelli property holds exactly when, for every positive real tolerance, the probability that some class member's empirical mean differs from its population mean by at least that tolerance converges to zero as the sample size tends to infinity.","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → Prop","l":83,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.HasCompactDiscrepancySublevel","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For a covariance model and target matrix, a compact discrepancy sublevel is a nonempty compact cut of the model below some criterion value.","s":"{V : Type u_2} → [inst : Fintype V] → [DecidableEq V] → Set (Causalean.Stat.PositiveCovariance V) → Matrix V V ℝ → Prop","l":265,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Given normed real vector spaces, a map between them and a point, the continuous Hadamard directional differentiability property at that point holds exactly when there exists a continuous directional derivative at that point and that derivative satisfies Hadamard directional differentiability.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → [inst : NormedAddCommGroup F] → [NormedSpace ℝ F] → (E → F) → E → Prop","l":341,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.add","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two real-valued maps are continuously Hadamard directionally differentiable, then their pointwise sum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => f y + g y) x","l":431,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.comp","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If the outer map and the inner map are continuously Hadamard directionally differentiable, then their composition is too.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {G : Type u_3} [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] {f : E → F} {g : F → G} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt g (f x) → Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt (g ∘ f) x","l":369,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.congr_of_eventuallyEq","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two maps agree near a point and one is continuously Hadamard directionally differentiable there, then the other has the same property.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {f g : E → F} {x : E}, f =ᶠ[nhds x] g → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt f x","l":395,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.div","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If the denominator is nonzero and the numerator and denominator maps are continuously Hadamard directionally differentiable, then their pointwise quotient has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, g x ≠ 0 → Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => f y / g y) x","l":479,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.finset_inf'","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If a finite index set is nonempty and every indexed map is continuously Hadamard directionally differentiable, then their pointwise finite infimum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {I : Type u_3} (s : Finset I) (hs : s.Nonempty) {f : I → E → ℝ} {x : E}, (∀ i ∈ s, Causalean.Stat.HasContinuousHadamardDirDerivAt (f i) x) → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => s.inf' hs fun i => f i y) x","l":569,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.finset_sum","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If every map indexed by a finite set is continuously Hadamard directionally differentiable, then their pointwise finite sum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {I : Type u_3} (s : Finset I) {f : I → E → ℝ} {x : E}, (∀ i ∈ s, Causalean.Stat.HasContinuousHadamardDirDerivAt (f i) x) → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => ∑ i ∈ s, f i y) x","l":496,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.finset_sup'","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If a finite index set is nonempty and every indexed map is continuously Hadamard directionally differentiable, then their pointwise finite supremum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {I : Type u_3} (s : Finset I) (hs : s.Nonempty) {f : I → E → ℝ} {x : E}, (∀ i ∈ s, Causalean.Stat.HasContinuousHadamardDirDerivAt (f i) x) → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => s.sup' hs fun i => f i y) x","l":542,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.max","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two real-valued maps are continuously Hadamard directionally differentiable, then their pointwise maximum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => max (f y) (g y)) x","l":512,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.min","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two real-valued maps are continuously Hadamard directionally differentiable, then their pointwise minimum has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => min (f y) (g y)) x","l":527,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.mul","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two real-valued maps are continuously Hadamard directionally differentiable, then their pointwise product has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => f y * g y) x","l":463,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.of_differentiableAt","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If a map is differentiable at a point, then it is continuously Hadamard directionally differentiable there.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {f : E → F} {x : E}, DifferentiableAt ℝ f x → Causalean.Stat.HasContinuousHadamardDirDerivAt f x","l":388,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.of_hasFDerivAt","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If a map has a Fréchet derivative, then it is continuously Hadamard directionally differentiable.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {f : E → F} {L : E →L[ℝ] F} {x : E}, HasFDerivAt f L x → Causalean.Stat.HasContinuousHadamardDirDerivAt f x","l":381,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.prod","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If each component map is continuously Hadamard directionally differentiable, then their product-valued pairing has the same property.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {G : Type u_3} [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] {f : E → F} {g : E → G} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => (f y, g y)) x","l":412,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasContinuousHadamardDirDerivAt.sub","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If two real-valued maps are continuously Hadamard directionally differentiable, then their pointwise difference has the same property.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f g : E → ℝ} {x : E}, Causalean.Stat.HasContinuousHadamardDirDerivAt f x → Causalean.Stat.HasContinuousHadamardDirDerivAt g x → Causalean.Stat.HasContinuousHadamardDirDerivAt (fun y => f y - g y) x","l":447,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasFDerivAt.hasHadamardDirDerivAt","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Fréchet implies Hadamard. If the map φ is Fréchet-differentiable at the point θ, with continuous linear derivative L, then φ is Hadamard directionally differentiable at θ, with the same map L serving as its Hadamard directional derivative.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {φ : E → F} {L : E →L[ℝ] F} {θ : E}, HasFDerivAt φ L θ → Causalean.Stat.HasHadamardDirDerivAt φ (fun x => L x) θ","l":89,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasHadamardDirDerivAt","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Given normed real vector spaces, a map between them, a proposed directional derivative, and a point, the Hadamard directional differentiability property at that point holds exactly when, for every direction, every sequence of directions converging to it, and every strictly positive real step-size sequence converging to zero,","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → [inst : NormedAddCommGroup F] → [NormedSpace ℝ F] → (E → F) → (E → F) → E → Prop","l":72,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasHadamardDirDerivAt.comp","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"If the outer map has a Hadamard directional derivative and the inner map has a Hadamard directional derivative, then their composition has the composed directional derivative.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {G : Type u_3} [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] {f f' : E → F} {g g' : F → G} {x : E}, Causalean.Stat.HasHadamardDirDerivAt g g' (f x) → Causalean.Stat.HasHadamardDirDerivAt f f' x → Causalean.Stat.HasHadamardDirDerivAt (g ∘ f) (g' ∘ f') x","l":348,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.HasIIDSample","k":"def","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"Given a measurable observation space and a measure on it, the existence-of-an-independent, identically distributed-sample assertion states that there is a sample space with a measurable structure and a measure carrying a nonempty collection of independent, identically distributed samples whose common marginal law is the given measure.","s":"{X : Type u} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → Prop","l":235,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.HasL1Bracketing","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"For a measurable observation space and an index set, a class of real-valued functions, and a measure on its domain, the arbitrarily fine finite $L^1$-bracketing property holds exactly when every positive real tolerance admits a finite collection of integrable lower and upper functions whose absolute gaps have integral at most that tolerance and which bracket every class member on one common set of full measure.","s":"{X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace X] → (ι → X → ℝ) → MeasureTheory.Measure X → Prop","l":129,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample","k":"structure","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"An independent and identically distributed sample with marginal law P, realized as a sequence of sample points given by measurable maps on a single ambient probability space: the family is mutually independent, identically distributed, and the law of each point is the population law P.","s":"(Ω : Type u_1) → (X : Type u_2) → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → MeasureTheory.Measure Ω → MeasureTheory.Measure X → Type (max u_1 u_2)","l":32,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.abs_apply_mul_le_norm_sq","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"The pointwise product of two functional-evaluations is dominated by a constant times ‖ψ x‖²: |φ (ψ x) * φ' (ψ x)| ≤ (‖φ‖ * ‖φ'‖) * ‖ψ x‖². Used to derive entrywise integrability of the product from the single hypothesis Integrable (fun x => ‖ψ x‖²) P.","s":"∀ {X : Type u_2} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] (φ φ' : E →L[ℝ] ℝ) (ψ : X → E) (x : X), |φ (ψ x) * φ' (ψ x)| ≤ ‖φ‖ * ‖φ'‖ * ‖ψ x‖ ^ 2","l":107,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapSE","k":"def","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"For a measurable sample space, measurable observation space, sample-space measure, and observation-space measure, an independent, identically distributed sample, a real-valued influence function, and a sample size, the bootstrap standard error is the square root of the bootstrap variance at that sample size.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":57,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapSE_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Consistency of the bootstrap standard error. Along an i.i.d. sample S, if the influence function ψ is measurable, integrable, square-integrable, and has population mean zero, then the bootstrap standard error of √n θ̂ converges in probability to the asymptotic standard deviation $\\sqrt{\\int \\psi^2\\,dP}$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_inProb (S.bootstrapSE ψ) (fun x => √(∫ (x : X), ψ x ^ 2 ∂P)) μ","l":68,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapStudentized","k":"def","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"For a measurable sample space, measurable observation space, sample-space measure, and observation-space measure, a sequence of real-valued estimators, a target real value, an independent, identically distributed sample, a real-valued influence function, and a sample size, the bootstrap studentized statistic is $\\sqrt n(\\widehat\\theta_n-\\theta_0)/\\widehat\\sigma_n$, where the denominator is the bootstrap standard error computed from the full sample of that size.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → (ℕ → Ω → ℝ) → ℝ → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":86,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar","k":"def","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"For an independent and identically distributed sample, a real-valued statistic of one observation, and a nonnegative integer sample size, the nonparametric bootstrap variance is the function that assigns to each sample-space outcome the empirical mean of the statistic squared minus the square of its empirical mean, computed from the first $n$ observations at that outcome.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":60,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_eq_centered","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"Exact bootstrap-variance identity. For an iid sample S, a statistic ψ, a sample size n, and a sample-path outcome ω, the plug-in bootstrap variance of ψ equals the centered empirical second moment of ψ over the first n observations.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → ℝ) (n : ℕ) (ω : Ω), S.bootstrapVar ψ n ω = (↑n)⁻¹ * ∑ i ∈ Finset.range n, (ψ (S.Z i ω) - S.sampleMean ψ n ω) ^ 2","l":81,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_nonneg","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":2,"d":"The bootstrap variance is nonnegative: it is a centered empirical second moment (bootstrapVar_eq_centered).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → ℝ) (n : ℕ) (ω : Ω), 0 ≤ S.bootstrapVar ψ n ω","l":117,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.bootstrapVar_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Bootstrap.Variance","a":"Stat/Bootstrap","t":1,"d":"Consistency of the bootstrap variance. Along the i.i.d. sample S, if the influence function ψ is measurable, integrable, square-integrable, and has population mean zero, then the bootstrap variance converges in probability to the population second moment $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_inProb (S.bootstrapVar ψ) (fun x => ∫ (x : X), ψ x ^ 2 ∂P) μ","l":126,"f":"Causalean/Stat/Bootstrap/Variance.lean"},{"n":"Causalean.Stat.IIDSample.card_injectiveTuples_image_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an ordered injective m-tuple t, the ordered injective tuples with the same image are exactly its m! reorderings.","s":"∀ {m n : ℕ} {t : Fin m → Fin n}, Function.Injective t → {q ∈ Causalean.Stat.injectiveTuples m n | Finset.image q Finset.univ = Finset.image t Finset.univ}.card = m.factorial","l":335,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.card_share_ge_two_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"The number of ordered pairs of injective tuples of length m drawn from n observations that share at least two sample indices is bounded by a polynomial in n determined by the tuple length.","s":"∀ {m n : ℕ}, {tq ∈ Causalean.Stat.injectiveTuples m n ×ˢ Causalean.Stat.injectiveTuples m n | 2 ≤ (Finset.image tq.1 Finset.univ ∩ Finset.image tq.2 Finset.univ).card}.card ≤ m.choose 2 * m ^ 2 * n ^ (2 * m - 2)","l":414,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.charFun_normalizedSum_finset_eq_range_card","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"A normalized finite-set sum has the characteristic function of an equal-size initial block.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ}, Measurable ψ → ∀ (s : Finset ℕ) (t : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map (fun ω => (√↑s.card)⁻¹ * ∑ i ∈ s, ψ (S.Z i ω)) μ) t = MeasureTheory.charFun (MeasureTheory.Measure.map (fun ω => (√↑s.card)⁻¹ * ∑ i ∈ Finset.range s.card, ψ (S.Z i ω)) μ) t","l":83,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedFoldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Fold-B CLT. Along an i.i.d. sample S under a one-shot sample split split, fix a transform ψ : X → ℝ that is measurable, has population mean zero under P, and is square-integrable; provided the fold-B normalized partial sum is almost-everywhere measurable at every sample size, the fold-B normalized partial sum (1/√|B(n)|) Σ_{i∈B(n)} ψ(Z_i) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {ψ : X → ℝ}, Measurable ψ → ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → ∀ (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hSum_meas","l":149,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedSum_vec","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":1,"d":"Multivariate CLT against the concrete Gaussian limit. Along the i.i.d. sample S, with ψ measurable and square-integrable, if ψ has population mean zero, $\\int \\psi\\,dP=0$, then the vector normalised sum converges in distribution to gaussianLimit ψ, the centered Gaussian on E with covariance equal to the second-moment operator of ψ; no abstract target or characteristic-function hypothesis remains.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (S : Causalean.Stat.IIDSample Ω X μ P), ∫ (x : X), ψ x ∂P = 0 → Causalean.Stat.Tendsto_dist_vec (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ fun m => Finset.range m) (Causalean.Stat.gaussianLimit hψ hvar) μ fun n => id (Measurable.aemeasurable (Measurable.const_smul (Finset.measurable_sum ((fun m => Finset.range m) n) fun i x => Measurable.comp hψ (S.meas i)) (√↑((fun m => Finset.range m) n).card)⁻¹))","l":115,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalizedSum_vec_of_charFun","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Multivariate CLT contact. Along the i.i.d. sample S, for an influence function ψ that is measurable, has population mean zero, and is square-integrable, and a target probability measure Q on E whose characteristic function is the Gaussian one $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$ at every point t, then the vector normalised sum converges in distribution to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E} (hψ_meas : Measurable ψ), ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (Q : MeasureTheory.Measure E) [inst_7 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)) → Causalean.Stat.Tendsto_dist_vec (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝) Q μ fun n => id (Measurable.aemeasurable (Measurable.const_smul (Finset.measurable_sum (Causalean.Stat.rng✝ n) fun i x => Measurable.comp hψ_meas (S.meas i)) (√↑(Causalean.Stat.rng✝ n).card)⁻¹))","l":239,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.clt_normalized_sum","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"Central limit theorem for normalised sample sums. Along the i.i.d. sample S, if the transform ψ of the observations is measurable, has population mean zero, and is square-integrable, then the normalised partial sum — the sum of the transformed observations over the first n indices, divided by $\\sqrt n$ — converges in distribution to the centred normal law with variance equal to the population second moment $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ} (hψ_meas : Measurable ψ), ∫ (x : X), ψ x ∂P = 0 → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.normalizedSum S ψ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ fun n => id (Measurable.aemeasurable (Measurable.const_mul (Finset.measurable_sum ((fun m => Finset.range m) n) fun i x => Measurable.comp hψ_meas (S.meas i)) (√↑((fun m => Finset.range m) n).card)⁻¹))","l":137,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_abs_le_zeta","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"The absolute expected product of two kernel evaluations on distinct sample tuples is no larger than the kernel's second moment under the product distribution.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → |∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ| ≤ Causalean.Stat.IIDSample.zetaOrder P g","l":299,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zero_of_image_ne","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Distinct-image cross term. If two ordered injective m-tuples have different images, complete degeneracy kills the cross expectation: some index of t is absent from q, and integrating that coordinate out gives 0.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m) (tail : { k // k ≠ j } → X), ∫ (x : X), g (Causalean.Stat.insertCoord j x tail) ∂P = 0) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Finset.image t Finset.univ ≠ Finset.image q Finset.univ → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":161,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zero_of_shared_le_one","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":1,"d":"Cross-term vanishing. For an i.i.d. sample S, sample size n, and order-m kernel g that is first-order degenerate, let t and q be ordered m-tuples of sample indices that are each injective, and suppose the images of t and q share at most one sample index. Then the expected product of the kernel evaluated along t and along q is zero: zero shared indices give independence with mean zero on each factor, while one shared index reduces, after conditioning on it, to first-order degeneracy of each factor.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → (Finset.image t Finset.univ ∩ Finset.image q Finset.univ).card ≤ 1 → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":279,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_eq_zeta_of_image_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Equal-image cross term. If two ordered injective m-tuples have the same image, then q is a reordering of t, so by symmetry g(Z_q) = g(Z_t) and the cross expectation is ζ_m = zetaOrder P g.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → (∀ (σ : Equiv.Perm (Fin m)) (z : Fin m → X), g (z ∘ ⇑σ) = g z) → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Finset.image t Finset.univ = Finset.image q Finset.univ → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = Causalean.Stat.IIDSample.zetaOrder P g","l":128,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_zero_of_disjoint","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the expected product of terms evaluated on two injective sample-index tuples with disjoint index sets is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → Disjoint (Finset.image t Finset.univ) (Finset.image q Finset.univ) → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":68,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.crossterm_zero_of_shared_one","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the expected product of terms evaluated on two injective sample-index tuples sharing exactly one index is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ} {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → ∀ {a : Fin n}, Finset.image t Finset.univ ∩ Finset.image q Finset.univ = {a} → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω ∂μ = 0","l":114,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.descFactorial_ge","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"When the sample size is at least the order, its falling factorial is at least the sample-size power divided by the order power.","s":"∀ {m n : ℕ}, m ≤ n → ↑n ^ m / ↑m ^ m ≤ ↑(n.descFactorial m)","l":502,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.empProcVec","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Process","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"For an independent and identically distributed sample, a function from one observation to a real normed vector space, a nonnegative integer sample size, and the vector-valued centered empirical process is the function that assigns to each sample-space outcome the sum of the function over the first $n$ observations divided by $\\sqrt n$, minus $\\sqrt n$ times its population integral.","s":"{Ω : Type u_1} → {X : Type u_2} → {E : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → E) → ℕ → Ω → E","l":50,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Process.lean"},{"n":"Causalean.Stat.IIDSample.empProcVec_eq_stochEquicont_gap","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Process","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"For an i.i.d. sample S, a score function ψ, candidate and true parameter values θ and θ₀, sample size n, and outcome ω, the centered empirical-process gap R_n of StochEquicontAt, evaluated at θ, equals empProcVec of the score difference ψ(θ,·) − ψ(θ₀,·).","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : E → X → E) (θ θ₀ : E) (n : ℕ) (ω : Ω), S.empProcVec (fun z => ψ θ z - ψ θ₀ z) n ω = (√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ θ (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • ∫ (z : X), ψ θ z - ψ θ₀ z ∂P","l":68,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Process.lean"},{"n":"Causalean.Stat.IIDSample.empProcess","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"For an independent and identically distributed real-valued sample, a nonnegative integer sample size, a sample-space outcome, and a real threshold, the centered, scaled empirical process is $\\sqrt n$ times the difference between the empirical cumulative distribution function at that threshold and the population cumulative distribution function there.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℕ → Ω → ℝ → ℝ","l":95,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_cell_sandwich","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Pointwise monotone sandwich for the empirical process. Fix n, ω and a bracketing cell a ≤ u ≤ b of local shifts. Writing Gₙ(y) := empProcess n ω y and ya = q₀+a/√n, yb = q₀+b/√n, monotonicity of *both* F̂ₙ and F sandwiches the oscillation at the interior point q₀+u/√n between the two grid-node increments plus the deterministic mesh term:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (ω : Ω) {q₀ a b u : ℝ}, a ≤ u → u ≤ b → |S.empProcess n ω (q₀ + u / √↑n) - S.empProcess n ω q₀| ≤ max |S.empProcess n ω (q₀ + a / √↑n) - S.empProcess n ω q₀| |S.empProcess n ω (q₀ + b / √↑n) - S.empProcess n ω q₀| + √↑n * (↑(ProbabilityTheory.cdf P) (q₀ + b / √↑n) - ↑(ProbabilityTheory.cdf P) (q₀ + a / √↑n))","l":328,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_increment_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed local-shift increment. For a fixed local shift u, the empirical process increment between q₀ and q₀ + u/√n vanishes in probability: Gₙ(q₀ + u/√n) − Gₙ(q₀) →ₚ 0. Elementary second-moment / Chebyshev bound, since the increment is a centered Bernoulli sum of variance pₙ(1−pₙ) → 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (u : ℝ), Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (q₀ + u / √↑n) - S.empProcess n ω q₀) (fun x => 0) μ","l":108,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_node_max_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Finite grid-node maximum tends to zero. The maximum, over a *finite* nonempty index set s of local shifts v i, of the grid-node increments |Gₙ(q₀+v i/√n) − Gₙ(q₀)| vanishes in probability. A finite union of the L2 limits empProcess_increment_tendsto_zero: μ{ε ≤ maxᵢ |Δᵢ|} ≤ Σᵢ μ{ε ≤ |Δᵢ|}, each summand → 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ {ι : Type u_2} (s : Finset ι) (hs : s.Nonempty) (v : ι → ℝ), Causalean.Stat.Tendsto_inProb (fun n ω => s.sup' hs fun i => |S.empProcess n ω (q₀ + v i / √↑n) - S.empProcess n ω q₀|) (fun x => 0) μ","l":415,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_oscillation","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Local oscillation of the sample-quantile empirical process. Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, and a sequence of random endpoints Un that is bounded in probability, $U_n=O_p(1)$, the empirical process $G_n$, evaluated at the shrinking-window point $q_0+U_n/\\sqrt n$ minus its value at $q_0$, converges to zero in probability as $n\\to\\infty$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ {Un : ℕ → Ω → ℝ}, Causalean.Stat.IsBigOp Un (fun x => 1) μ → Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (q₀ + Un n ω / √↑n) - S.empProcess n ω q₀) (fun x => 0) μ","l":541,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_q0_bigO","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed-quantile empirical-process tightness. empProcess(·,q₀) is O_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsBigOp (fun n ω => S.empProcess n ω q₀) (fun x => 1) μ","l":96,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.empProcess_q0_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Fixed-quantile empirical-process CLT. empProcess(·,q₀) = √n(F̂ₙ(q₀) − F(q₀)) ⇒ N(0, τ(1−τ)). Restates empiricalCDF_tendsto_normal at y = q₀, using cdf P q₀ = τ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (hmeas : ∀ (n : ℕ), AEMeasurable (fun ω => S.empProcess n ω q₀) μ), Causalean.Stat.Tendsto_dist (fun n ω => S.empProcess n ω q₀) (Causalean.Stat.gaussianMeasure 0 (τ * (1 - τ))) μ hmeas","l":64,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.empiricalCDF","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"For an independent and identically distributed real-valued sample and a real threshold, the empirical cumulative distribution function maps each nonnegative integer sample size and sample-space outcome to the average of the indicators that the first $n$ observations do not exceed that threshold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℝ → ℕ → Ω → ℝ","l":103,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.IIDSample.empiricalCDF_monotone","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"The empirical cdf is monotone in its real argument y (a sum of monotone lower-ray indicators).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (ω : Ω), Monotone fun y => S.empiricalCDF y n ω","l":131,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure","k":"def","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"For an independent and identically distributed real-valued sample, a nonnegative integer sample size, and a sample-space outcome, the empirical measure is the sum of point masses at the first $n$ observed values, multiplied by the reciprocal of $n$; at sample size zero it is the zero measure.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℕ → Ω → MeasureTheory.Measure ℝ","l":57,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_cdf","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"cdf bridge. For a positive sample size $n$, the cumulative distribution function of the empirical measure S.empiricalMeasure n ω built from an i.i.d. sample at outcome ω coincides pointwise, at every threshold y, with the empirical cdf S.empiricalCDF y n ω.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) (y : ℝ), ↑(ProbabilityTheory.cdf (S.empiricalMeasure n ω)) y = S.empiricalCDF y n ω","l":97,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"For 0 < n the empirical measure is a probability measure.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω), MeasureTheory.IsProbabilityMeasure (S.empiricalMeasure n ω)","l":65,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalMeasure_real_Iic","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":2,"d":"The empirical measure of the lower ray Iic y is F̂ₙ(y) (as ℝ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) (y : ℝ), (S.empiricalMeasure n ω).real (Set.Iic y) = S.empiricalCDF y n ω","l":78,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.empiricalProcess","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"For a measurable sample space, measurable observation space, index set, sample-space measure, and observation-space measure, an independent, identically distributed sample, a class of real-valued functions, one member of that class, and a sample size, the empirical process is the random variable $\\sqrt n\\{n^{-1}\\sum_{k<n}f_i(Z_k)-\\int f_i\\,dP\\}$.","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → ι → ℕ → Ω → ℝ","l":58,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample.iIndepFun_comp","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"Independence of the composed family fun i ↦ f ∘ S.Z i.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, Measurable f → ProbabilityTheory.iIndepFun (fun i ω => f (S.Z i ω)) μ","l":60,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.IIDSample.integrable_injectiveTuples_sum","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The injective-tuple sum of an order-m kernel is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.Integrable (fun ω => ∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) μ","l":428,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderKernelTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An order-m kernel term along an injective tuple is integrable whenever the kernel is integrable under the product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => h fun j => S.Z (↑(t j)) ω) μ","l":194,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An order-m kernel term along an injective tuple is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => g fun j => S.Z (↑(t j)) ω) μ","l":371,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm_mul","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The product of two injective order-m kernel terms is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t q : Fin m → Fin n}, Function.Injective t → Function.Injective q → MeasureTheory.Integrable (fun ω => (g fun j => S.Z (↑(t j)) ω) * g fun j => S.Z (↑(q j)) ω) μ","l":418,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integrable_orderTerm_sq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The square of an order-m kernel term along an injective tuple is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Integrable (fun ω => (g fun j => S.Z (↑(t j)) ω) ^ 2) μ","l":379,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_comp_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.Hoeffding","a":"Stat/Concentration/TailBounds","t":2,"d":"The population mean of a statistic equals its sample-point expectation: ∫ ω, f (S.Z i ω) ∂μ = ∫ x, f x ∂P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {f : X → ℝ}, AEMeasurable f P → ∀ (i : ℕ), ∫ (ω : Ω), f (S.Z i ω) ∂μ = ∫ (x : X), f x ∂P","l":47,"f":"Causalean/Stat/Concentration/TailBounds/Hoeffding.lean"},{"n":"Causalean.Stat.IIDSample.integral_injectiveTuples_sum_sq_degen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Exact second moment of the injective-tuple sum. For an i.i.d. sample S and sample size n, if the order-m kernel g is completely degenerate: symmetric, square-integrable, and with zero conditional mean after integrating out any single coordinate, then the second moment of the sum of g over all ordered injective m-tuples drawn from the first n sample indices equals m! · n^{(m)} · ζ_m, where n^{(m)} is the number of such tuples and ζ_m is the kernel's second moment under the m-fold product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderDegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ = ↑m.factorial * Causalean.Stat.injectiveTupleCount m n * Causalean.Stat.IIDSample.zetaOrder P g","l":375,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.integral_injectiveTuples_sum_sq_le_shared_count","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a first-order-degenerate kernel, the second moment of the unnormalised sum over ordered injective sample tuples is bounded by the number of tuple pairs sharing at least two observations times the kernel's second moment.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ}, ∫ (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ ≤ ↑{tq ∈ Causalean.Stat.injectiveTuples m n ×ˢ Causalean.Stat.injectiveTuples m n | 2 ≤ (Finset.image tq.1 Finset.univ ∩ Finset.image tq.2 Finset.univ).card}.card * Causalean.Stat.IIDSample.zetaOrder P g","l":532,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.integral_offDiag_sum_sq","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"Second moment of the off-diagonal sum. For an i.i.d. sample S and sample size n, if the two-argument kernel g is degenerate, then the second moment of the sum of g(Z_i, Z_j) over all ordered off-diagonal index pairs i ≠ j below n equals 2 · |offDiag| · ζ, where ζ is the kernel's second moment under P × P. The m = 2 case of integral_injectiveTuples_sum_sq_degen, using sum_injectiveTuples_two_eq_offDiag, toOrderDegenKernel, and zeta_eq_zetaOrder (2! = 2, injectiveTupleCount 2 n = |offDiag| = n(n−1)).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), (∑ p ∈ (Finset.range n).offDiag, g (S.Z p.1 ω) (S.Z p.2 ω)) ^ 2 ∂μ = 2 * ↑(Finset.range n).offDiag.card * Causalean.Stat.IIDSample.zeta P g","l":245,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderKernelTerm_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The expectation of an injectively indexed order-m kernel term equals the kernel's product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), h fun j => S.Z (↑(t j)) ω ∂μ = ∫ (z : Fin m → X), h z ∂MeasureTheory.Measure.pi fun x => P","l":209,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderKernelTerm_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed order-m kernel term has mean zero whenever the kernel has zero product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} (S : Causalean.Stat.IIDSample Ω X μ P) {h : (Fin m → X) → ℝ}, Measurable h → ∀ {t : Fin m → Fin n}, Function.Injective t → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), h fun j => S.Z (↑(t j)) ω ∂μ = 0","l":221,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderTerm_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed fully degenerate order-m kernel term has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), g fun j => S.Z (↑(t j)) ω ∂μ = 0","l":329,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_orderTerm_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"An injectively indexed fully degenerate order-m kernel term has mean zero when its product-law mean is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_2 : NeZero m], Causalean.Stat.OrderDegenKernel P g → ∀ {t : Fin m → Fin n}, Function.Injective t → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), g fun j => S.Z (↑(t j)) ω ∂μ = 0","l":320,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The rescaled degenerate U-statistic has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), ∫ (ω : Ω), √↑n * Causalean.Stat.uStatistic S g n ω ∂μ = 0","l":304,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":2,"d":"The rescaled higher-order remainder has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":54,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_sq_degen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Exact variance of the rescaled degenerate fixed-order U-statistic. For an i.i.d. sample S, if the order-m kernel g is completely degenerate and the sample size is at least m, then the second moment of the √n-rescaled order-m U-statistic of g equals n · m! · ζ_m / n^{(m)}, which specializes to 2ζ/(n−1) when m = 2.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_3 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderDegenKernel P g → ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatisticOrder S g n ω) ^ 2 ∂μ = ↑n * ↑m.factorial * Causalean.Stat.IIDSample.zetaOrder P g / Causalean.Stat.injectiveTupleCount m n","l":465,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_order_sq_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":1,"d":"L² bound on the rescaled higher-order remainder. For an i.i.d. sample S, if the order-m kernel g is first-order degenerate, then there is a nonnegative constant C, depending only on the order m and the kernel's second moment ζ_m = E[g²], such that the second moment of the √n-rescaled order-m U-statistic of g is at most C/n for every sample size n ≥ m. This is the keystone estimate; it packages cross-term vanishing, the Cauchy–Schwarz bound |E[g_t g_q]| ≤ ζ_m, the O(n^{2m-2}) count of surviving tuple pairs, and the n · (n^{(m)})⁻² normalization.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.OrderFirstDegenKernel P g → ∃ C, 0 ≤ C ∧ ∀ {n : ℕ}, m ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatisticOrder S g n ω) ^ 2 ∂μ ≤ C / ↑n","l":750,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_sq","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"L² bound on the rescaled degenerate U-statistic. For an i.i.d. sample S, if the two-argument kernel g is degenerate and the sample size is at least two, then the second moment of the √n-rescaled degenerate U-statistic of g equals 2ζ/(n−1), where ζ is the kernel's second moment under P × P. The m = 2 case of integral_rescaled_order_sq_degen (n · 2! · ζ / n^{(2)} = 2ζ/(n−1)), via uStatisticOrder_two_eq_uStatistic and zeta_eq_zetaOrder.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ∫ (ω : Ω), (√↑n * Causalean.Stat.uStatistic S g n ω) ^ 2 ∂μ = 2 * Causalean.Stat.IIDSample.zeta P g / (↑n - 1)","l":273,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_sqrt_mul_uMean","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The rescaled fixed-order U-statistic has mean equal to the same rescaling of the product-law kernel mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S h n ω ∂μ = √↑n * Causalean.Stat.uMeanOrder h P","l":273,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_degenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel gives a mean-zero rescaled fixed-order U-statistic in the nonempty sampling regime m ≤ n. This is a mean statement only; it does not assert the variance bound or negligibility.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":349,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel whose product-law mean is zero gives a mean-zero rescaled fixed-order U-statistic. This is a mean statement only; it does not assert the variance bound or negligibility.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P] [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":337,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_rescaled_uStatisticOrder_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"If an order-m kernel has product-law mean zero, then the rescaled fixed-order U-statistic has mean zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), √↑n * Causalean.Stat.uStatisticOrder S h n ω ∂μ = 0","l":286,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_uMean","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The fixed-order U-statistic is unbiased: its expectation is the product-law kernel mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω ∂μ = Causalean.Stat.uMeanOrder h P","l":230,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_degenKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel gives a mean-zero fixed-order U-statistic in the nonempty sampling regime m ≤ n.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":309,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_degenKernel_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel whose product-law mean is zero gives a mean-zero fixed-order U-statistic. The product-law mean-zero assumption is kept explicit here rather than inferred from coordinatewise degeneracy.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P] [inst_3 : NeZero m], Causalean.Stat.OrderDegenKernel P g → m ≤ n → Causalean.Stat.uMeanOrder g P = 0 → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S g n ω ∂μ = 0","l":299,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.integral_uStatisticOrder_eq_zero_of_uMean_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fixed-order U-statistic with product-law mean zero has expectation zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {h : (Fin m → X) → ℝ}, Measurable h → MeasureTheory.Integrable h (MeasureTheory.Measure.pi fun x => P) → m ≤ n → Causalean.Stat.uMeanOrder h P = 0 → ∫ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω ∂μ = 0","l":262,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.map_eq","k":"theorem","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"For an i.i.d. sample S and any sample index i, the pushforward law of the i-th sample point equals the population law P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (i : ℕ), MeasureTheory.Measure.map (S.Z i) μ = P","l":73,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.map_fintype_tuple_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The joint law of any finite collection of distinct sample coordinates is the corresponding product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ι : Type u_3} [inst_2 : Fintype ι] {n : ℕ} {r : ι → Fin n}, Function.Injective r → MeasureTheory.Measure.map (fun ω i => S.Z (↑(r i)) ω) μ = MeasureTheory.Measure.pi fun x => P","l":36,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.map_tuple_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S, sample size n, and order m, if the index map t : Fin m → Fin n is injective, then the joint law of the sample coordinates selected by t is the m-fold product measure P^m.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m n : ℕ} {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.Measure.map (fun ω j => S.Z (↑(t j)) ω) μ = MeasureTheory.Measure.pi fun x => P","l":62,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.measurable_Z","k":"theorem","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":2,"d":"Each individual sample point of an i.i.d. sample is a measurable map from the ambient probability space to the observation space.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (i : ℕ), Measurable (S.Z i)","l":57,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.measurable_empProcVec","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":2,"d":"The vector empirical process is measurable in ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → ∀ (n : ℕ), Measurable fun ω => S.empProcVec f n ω","l":95,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.measurable_empProcess","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Measurability of empProcess(·,y) (a constant times a measurable sample mean minus a constant).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (n : ℕ) (y : ℝ), Measurable fun ω => S.empProcess n ω y","l":54,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.measurable_sampleMean","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":2,"d":"S.sampleMean g N is measurable for measurable g: it is a finite sum of g ∘ Z i scaled by a constant.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {g : X → ℝ}, Measurable g → ∀ (N : ℕ), Measurable (S.sampleMean g N)","l":43,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.measurable_sampleVariance","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The sample variance is a measurable function of ω.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {f : X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable f → ∀ (n : ℕ), Measurable (S.sampleVariance f n)","l":102,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.memLp_injectiveTuples_sum","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"The injective-tuple sum of an order-m kernel is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => ∑ t ∈ Causalean.Stat.injectiveTuples m n, g fun j => S.Z (↑(t j)) ω) 2 μ","l":440,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_orderTerm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"Each injective order-m kernel term is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ {t : Fin m → Fin n}, Function.Injective t → MeasureTheory.MemLp (fun ω => g fun j => S.Z (↑(t j)) ω) 2 μ","l":407,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_rescaled","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The rescaled degenerate U-statistic is in L².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => √↑n * Causalean.Stat.uStatistic S g n ω) 2 μ","l":293,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.memLp_rescaled_order","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S, order-m kernel g that is measurable and square-integrable under the m-fold product law, and sample size n, the √n-rescaled order-m U-statistic of g is square-integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → MeasureTheory.Integrable (fun z => g z ^ 2) (MeasureTheory.Measure.pi fun x => P) → ∀ (n : ℕ), MeasureTheory.MemLp (fun ω => √↑n * Causalean.Stat.uStatisticOrder S g n ω) 2 μ","l":452,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_quantileIF_eq","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The normalized influence-function sum equals −Gₙ(q₀)/f₀:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, f₀ ≠ 0 → ↑(ProbabilityTheory.cdf P) q₀ = τ → ∀ (n : ℕ) (ω : Ω), (√↑n)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.quantileIF τ q₀ f₀ (S.Z i ω) = -S.empProcess n ω q₀ / f₀","l":328,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_vec_charFun_tendsto","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"Per-direction charFun limit (full-vector form). Convenience wrapper of normalizedSum_vec_charFun_tendsto_of_proj_integrable for the common case where the whole vector ψ is square-integrable: the projection ⟪t,ψ⟫ is then square-integrable by Cauchy–Schwarz, so callers holding the standard full-vector L² condition need not re-establish the per-direction one.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (t : E), ∫ (x : X), inner ℝ t (ψ x) ∂P = 0 → Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝ n) μ) t) Filter.atTop (nhds (Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)))","l":187,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.normalizedSum_vec_charFun_tendsto_of_proj_integrable","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"Per-direction charFun limit (general core). Assuming only that the scalar projection ⟪t,ψ⟫ in the chosen direction t is square-integrable (weaker than requiring the whole vector ψ to be square-integrable) and mean-zero, the characteristic function of the vector normalised sum, evaluated at t, converges to exp(−½ ∫⟪t,ψ⟫² dP). The full-vector convenience form is normalizedSum_vec_charFun_tendsto below.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → E}, Measurable ψ → ∀ (t : E), MeasureTheory.Integrable (fun x => inner ℝ t (ψ x) ^ 2) P → ∫ (x : X), inner ℝ t (ψ x) ∂P = 0 → Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ Causalean.Stat.rng✝ n) μ) t) Filter.atTop (nhds (Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)))","l":102,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IIDSample.orderDegenerateNegligible_of_firstDegen","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":1,"d":"Negligibility of the higher-order remainder. For an i.i.d. sample S, if the order-m kernel g is first-order degenerate: measurable, square-integrable, and mean zero after integrating out any single coordinate, then the √n-rescaled order-m U-statistic of g converges to zero in probability, i.e. it is o_p(1).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → Causalean.Stat.IsLittleOp (fun n ω => √↑n * Causalean.Stat.uStatisticOrder S g n ω) (fun x => 1) μ","l":66,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.IIDSample.orderTerm_diag","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample S and order-m kernel g that is measurable, if the index map t is injective, then the second moment of the kernel term evaluated along the sample coordinates selected by t equals the kernel's second moment ζ_m under the m-fold product law.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {m n : ℕ} {g : (Fin m → X) → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Measurable g → ∀ {t : Fin m → Fin n}, Function.Injective t → ∫ (ω : Ω), (g fun j => S.Z (↑(t j)) ω) ^ 2 ∂μ = Causalean.Stat.IIDSample.zetaOrder P g","l":393,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.permOfImageEq","k":"def","m":"Causalean.Stat.UStatistic.OrderM.ExactVariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Given a nonnegative integer $m$ specifying the size of a source index set, a nonnegative integer $n$ specifying the size of a target index set, two maps from the source index set into the target index set, the assumption that both maps are injective, and the assumption that their images coincide, the selected permutation of the source index set reorders the first map into the second. It first selects, for each target value of the second map, a source index having the same first-map value, and then selects, for each target value of the first map, a source index having the same second-map value.","s":"{m n : ℕ} → {t q : Fin m → Fin n} → Function.Injective t → Function.Injective q → Finset.image t Finset.univ = Finset.image q Finset.univ → Equiv.Perm (Fin m)","l":69,"f":"Causalean/Stat/UStatistic/OrderM/ExactVariance.lean"},{"n":"Causalean.Stat.IIDSample.rescaled_order_normalization_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For a nonnegative second-moment bound and a sample size at least the kernel order, the falling-factorial normalization term is bounded by a constant divided by the sample size.","s":"∀ {m : ℕ} [NeZero m] {n : ℕ}, m ≤ n → ∀ {ζ : ℝ}, 0 ≤ ζ → ↑n * (Causalean.Stat.injectiveTupleCount m n)⁻¹ ^ 2 * (↑(m.choose 2) * ↑m ^ 2 * ↑n ^ (2 * m - 2) * ζ) ≤ ↑(m.choose 2) * ↑m ^ (2 * m + 2) * ζ / ↑n","l":608,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.sampleCov_entry_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":1,"d":"Entrywise covariance-matrix consistency. For an i.i.d. sample S and two continuous linear coordinate functionals φ φ' : E →L[ℝ] ℝ, suppose a vector influence function ψ : X → E is measurable and has square-integrable norm along the sample. Then the empirical mean of the entry product φ(ψ·) · φ'(ψ·) converges in probability to the population integral ∫ x, φ(ψ x) · φ'(ψ x) ∂P.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun ω => ‖ψ (S.Z 0 ω)‖ ^ 2) μ → ∀ (φ φ' : E →L[ℝ] ℝ), Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => φ (ψ x) * φ' (ψ x)) (fun x => ∫ (x : X), φ (ψ x) * φ' (ψ x) ∂P) μ","l":126,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean","k":"def","m":"Causalean.Stat.Sample","a":"Stat/Sample","t":1,"d":"For a measurable sample space carrying a measure, a measurable observation space carrying a population measure, an independent and identically distributed sample from that population, a real-valued statistic of one observation, and a nonnegative integer sample size, the sample mean is the function that assigns each sample-space outcome the average $n^{-1}\\sum_{i<n} f(Z_i)$, with the reciprocal convention also applying when $n=0$.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":66,"f":"Causalean/Stat/Sample.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_mul_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"Empirical mean of a product. For two measurable real-valued statistics g₁, g₂ of an i.i.d. sample whose product is integrable, the empirical mean S.sampleMean (g₁ · g₂) N converges in probability to the population integral ∫ x, g₁ x * g₂ x ∂P. Direct application of the generic WLLN to the product g := fun x => g₁ x * g₂ x.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure P] {g₁ g₂ : X → ℝ}, Measurable g₁ → Measurable g₂ → MeasureTheory.Integrable (fun ω => g₁ (S.Z 0 ω) * g₂ (S.Z 0 ω)) μ → Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => g₁ x * g₂ x) (fun x => ∫ (x : X), g₁ x * g₂ x ∂P) μ","l":62,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_isBigOp","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":1,"d":"Unconditional O_p rate (Lemma A). For an i.i.d. sample S and a statistic f that is measurable and square-integrable under the sampling distribution P, the sample mean over the first n observations, centered at the population mean $\\int f\\,dP$, is stochastically bounded at the rate $\\sqrt{E_P[f^2]/n}$: it is $O_p(n^{-1/2}(E_P[f^2])^{1/2})$:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → Causalean.Stat.IsBigOp (fun n ω => S.sampleMean f n ω - ∫ (x : X), f x ∂P) (fun n => √((∫ (x : X), f x ^ 2 ∂P) / ↑n)) μ","l":702,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_meas_ge_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Chebyshev tail for the centered sample mean. For t > 0,","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → ∀ {n : ℕ}, 0 < n → ∀ {t : ℝ}, 0 < t → μ {ω | t ≤ |S.sampleMean f n ω - ∫ (x : X), f x ∂P|} ≤ ENNReal.ofReal ((∫ (x : X), f x ^ 2 ∂P) / (↑n * t ^ 2))","l":652,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_sub_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Centered sample-mean second moment. For an i.i.d. sample and a square-integrable statistic f, the centered sample mean over the first n points has second moment bounded by E_P[f²]/n:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {f : X → ℝ}, Measurable f → MeasureTheory.MemLp f 2 P → ∀ {n : ℕ}, 0 < n → ∫⁻ (ω : Ω), ENNReal.ofReal ((S.sampleMean f n ω - ∫ (x : X), f x ∂P) ^ 2) ∂μ ≤ ENNReal.ofReal ((∫ (x : X), f x ^ 2 ∂P) / ↑n)","l":524,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IIDSample.sampleMean_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":1,"d":"Generic weak law of large numbers. For an i.i.d. sample S, suppose a real-valued statistic g is measurable and integrable under the population law P. Then the sample mean S.sampleMean g N converges in probability to the population integral ∫ x, g x ∂P as N → ∞.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {g : X → ℝ}, Measurable g → MeasureTheory.Integrable g P → Causalean.Stat.Tendsto_inProb (S.sampleMean g) (fun x => ∫ (x : X), g x ∂P) μ","l":52,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile","k":"def","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"For an independent and identically distributed real-valued sample and a real quantile level, the sample quantile maps each nonnegative integer sample size and sample-space outcome to the lower generalized inverse, at that level, of the empirical cumulative distribution function.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → ℝ → ℕ → Ω → ℝ","l":109,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"For a measurable sample space carrying a measure, a population measure on the real line, a nonnegative integer $k$ specifying a finite collection of quantile coordinates, an independent and identically distributed real-valued sample from that population, and a vector of quantile levels, the sample-quantile vector maps every nonnegative integer sample size and sample-space outcome to the Euclidean vector of the corresponding coordinatewise sample quantiles.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → {k : ℕ} → Causalean.Stat.IIDSample Ω ℝ μ P → (Fin k → ℝ) → ℕ → Ω → EuclideanSpace ℝ (Fin k)","l":70,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec_isAsymLinearVec","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"The sample-quantile vector is asymptotically linear with the joint influence function ψ. Given a SampleQuantileReg regularity bundle at every coordinate j: interior level $\\tau_j$, positive density $f_j$ at the population quantile $q_j$, cdf identification, differentiability of the population cdf, and an atomless population, the vector of sample $\\tau$-quantiles is jointly asymptotically linear at the vector of population quantiles, with influence function the joint quantile influence function $\\psi$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} {k : ℕ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q f : Fin k → ℝ}, (∀ (j : Fin k), Causalean.Stat.SampleQuantileReg P (τ j) (q j) (f j)) → Causalean.Stat.IsAsymLinearVec (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (Causalean.Stat.quantileIFVec τ q f) S fun m => Finset.range m","l":214,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantileVec_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"Joint asymptotic normality of the sample-quantile vector. Given a SampleQuantileReg bundle at every coordinate j, a candidate limit measure Q on the joint quantile space whose characteristic function at every direction t matches $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$, the Gaussian shape determined by the joint influence function $\\psi$, and almost-everywhere measurability of the rescaled estimator sequence at every sample size, then the law of the rescaled sample-quantile vector $\\sqrt n(\\hat q_n(\\tau_\\bullet)-q_\\bullet)$ converges weakly to Q as $n\\to\\infty$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} {k : ℕ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q f : Fin k → ℝ}, (∀ (j : Fin k), Causalean.Stat.SampleQuantileReg P (τ j) (q j) (f j)) → ∀ (Q : MeasureTheory.Measure (EuclideanSpace ℝ (Fin k))) [inst_3 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : EuclideanSpace ℝ (Fin k)), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (z : ℝ), inner ℝ t (Causalean.Stat.quantileIFVec τ q f z) ^ 2 ∂P) / 2)) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (fun m => Finset.range m) n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator (S.sampleQuantileVec τ) (Causalean.Stat.eucl q) (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds ⟨Q, inst_3⟩)","l":313,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_atom_bound","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"Atom bound. If the population cdf $F$ is continuous, i.e. the population is atomless, the sample size $n$ is positive, and the quantile level $\\tau$ is interior, $0<\\tau<1$, then almost surely the empirical cdf evaluated at the sample $\\tau$-quantile $\\hat q_n(\\tau)$ deviates from $\\tau$ by at most $1/n$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω ℝ μ P), (Continuous fun y => ↑(ProbabilityTheory.cdf P) y) → ∀ {n : ℕ}, 0 < n → ∀ {τ : ℝ}, 0 < τ → τ < 1 → ∀ᵐ (ω : Ω) ∂μ, |S.empiricalCDF (S.sampleQuantile τ n ω) n ω - τ| ≤ (↑n)⁻¹","l":149,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_atom_term_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The sample-quantile atom term vanishes. √n (F̂ₙ(q̂ₙ) − τ) →ₚ 0. The switching atom bound |F̂ₙ(q̂ₙ) − τ| ≤ 1/n (a.e.) gives |√n(F̂ₙ(q̂ₙ) − τ)| ≤ 1/√n → 0, so the term converges to 0 even a.e.-deterministically.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => √↑n * (S.empiricalCDF (S.sampleQuantile τ n ω) n ω - τ)) (fun x => 0) μ","l":88,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_inversion","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample-quantile inversion identity. Given a SampleQuantileReg regularity bundle hreg — interior quantile level, positive density $f_0$ at the population quantile $q_0$, cdf identification, differentiability of the population cdf at $q_0$, and an atomless population — the empirical process $G_n$ evaluated at the sample quantile $\\hat q_n(\\tau)$, plus $f_0$ times the rescaled deviation $\\sqrt n(\\hat q_n(\\tau)-q_0)$, converges to zero in probability; equivalently $G_n(\\hat q_n(\\tau)) = -f_0\\sqrt n(\\hat q_n(\\tau)-q_0) + o_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => S.empProcess n ω (S.sampleQuantile τ n ω) + f₀ * (√↑n * (S.sampleQuantile τ n ω - q₀))) (fun x => 0) μ","l":286,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample-quantile asymptotic linearity (derived). Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the sample $\\tau$-quantile $\\hat q_n(\\tau)$ is asymptotically linear at $q_0$ with influence function $\\psi_\\tau$, the Bahadur remainder being proved — not assumed — $o_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsAsymLinear (S.sampleQuantile τ) q₀ (Causalean.Stat.quantileIF τ q₀ f₀) S fun m => Finset.range m","l":382,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_le_iff","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalQuantile","a":"Stat/Quantile","t":1,"d":"Switching relation. For a positive sample size $n$ and an interior quantile level $\\tau\\in(0,1)$, the sample $\\tau$-quantile $\\hat q_n(\\tau)$ is at most a given point x exactly when $\\tau$ is at most the empirical cdf at x.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) {n : ℕ}, 0 < n → ∀ (ω : Ω) {τ : ℝ}, 0 < τ → τ < 1 → ∀ (x : ℝ), S.sampleQuantile τ n ω ≤ x ↔ τ ≤ S.empiricalCDF x n ω","l":120,"f":"Causalean/Stat/Quantile/EmpiricalQuantile.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_quantileRegularity","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Sample quantile satisfies QuantileRegularity (derived Bahadur). Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the sample $\\tau$-quantile sequence itself satisfies the generic QuantileRegularity bundle for these parameters, with the Bahadur remainder now derived rather than assumed.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.QuantileRegularity S (S.sampleQuantile τ) τ q₀ f₀","l":436,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_rate","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Root-n consistency of the sample quantile. Given a SampleQuantileReg regularity bundle hreg for the population $\\tau$-quantile $q_0$ with density $f_0$, the rescaled deviation $\\sqrt n(\\hat q_n(\\tau)-q_0)$ of the sample quantile from the population quantile is bounded in probability, $O_p(1)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.IsBigOp (fun n ω => √↑n * (S.sampleQuantile τ n ω - q₀)) (fun x => 1) μ","l":129,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.IIDSample.sampleQuantile_taylor_remainder_tendsto_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"The Taylor remainder vanishes at the sample quantile. Writing R(y) = F(y) − F(q₀) − f₀(y − q₀) for the first-order Taylor remainder of F = cdf P at q₀, the scaled remainder at the sample quantile vanishes in probability: √n · R(q̂ₙ) →ₚ 0. Mirrors the hRn block of deltaMethod_scalar: the driver Un = √n(q̂ₙ − q₀) is O_p(1), consistency q̂ₙ →ₚ q₀ localizes the derivative little-o |R(y)| ≤ η|y − q₀|, and |√n·R(q̂ₙ)| ≤ η|Un|.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → Causalean.Stat.Tendsto_inProb (fun n ω => √↑n * (↑(ProbabilityTheory.cdf P) (S.sampleQuantile τ n ω) - ↑(ProbabilityTheory.cdf P) q₀ - f₀ * (S.sampleQuantile τ n ω - q₀))) (fun x => 0) μ","l":145,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.IIDSample.sampleSecondMoment_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Limit.WLLN","a":"Stat/Limit","t":1,"d":"Second-moment consistency. For an i.i.d. sample S, suppose a known influence function ψ is measurable and has square-integrable values under the population law P. Then the empirical second moment S.sampleMean (ψ²) N converges in probability to ∫ x, (ψ x)² ∂P. Direct application of the WLLN to g := ψ².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ψ : X → ℝ}, Measurable ψ → MeasureTheory.Integrable (fun x => ψ x ^ 2) P → Causalean.Stat.Tendsto_inProb (S.sampleMean fun x => ψ x ^ 2) (fun x => ∫ (x : X), ψ x ^ 2 ∂P) μ","l":101,"f":"Causalean/Stat/Limit/WLLN.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance","k":"def","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":1,"d":"For an independent and identically distributed sample, a real-valued statistic, and a natural-number sample size $n$, the sample variance as a function of the sample outcome is the average, with divisor $n$, of the squared deviations of the first $n$ observed statistic values from their sample mean.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → ℕ → Ω → ℝ","l":80,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance_eq","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The classical computational form of the sample variance: empirical second moment minus the square of the empirical mean, V̂ₙ(ω) = (1/n) ∑_{i<n} f(Zᵢ ω)² − X̄ₙ(ω)².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (n : ℕ) (ω : Ω), S.sampleVariance f n ω = (↑n)⁻¹ * ∑ i ∈ Finset.range n, f (S.Z i ω) ^ 2 - S.sampleMean f n ω ^ 2","l":118,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sampleVariance_nonneg","k":"theorem","m":"Causalean.Stat.Concentration.TailBounds.EmpiricalBernstein","a":"Stat/Concentration/TailBounds","t":2,"d":"The sample variance is nonnegative: it is 1/n times a sum of squares.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → ℝ) (n : ℕ) (ω : Ω), 0 ≤ S.sampleVariance f n ω","l":92,"f":"Causalean/Stat/Concentration/TailBounds/EmpiricalBernstein.lean"},{"n":"Causalean.Stat.IIDSample.sharedPositions_card_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderSecondMoment","a":"Stat/UStatistic/OrderM","t":2,"d":"For an injective tuple map, the number of positions whose values occur in a second tuple equals the number of values shared by the two tuple images.","s":"∀ {α : Type u_3} {β : Type u_4} [inst : Fintype α] [inst_1 : DecidableEq β] {t q : α → β}, Function.Injective q → {j | q j ∈ Finset.image t Finset.univ}.card = (Finset.image t Finset.univ ∩ Finset.image q Finset.univ).card","l":387,"f":"Causalean/Stat/UStatistic/OrderM/RemainderSecondMoment.lean"},{"n":"Causalean.Stat.IIDSample.supDeviation","k":"def","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"For a measurable sample space, measurable observation space, index set, sample-space measure, and observation-space measure, an independent, identically distributed sample, a class of real-valued functions, and a sample size, the supremum deviation statistic is the supremum, over all class members, of the absolute difference between its empirical and population means.","s":"{Ω : Type u_1} → {X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (ι → X → ℝ) → ℕ → Ω → ℝ","l":68,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.IIDSample.zeta","k":"def","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":1,"d":"For a measurable observation space, a measure on that space, and a real-valued kernel of two observations, the kernel second moment is $\\iint g(x,y)^2\\,dP(x)\\,dP(y)$, evaluated under two independent draws from the given measure.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → X → ℝ) → ℝ","l":217,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zetaOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":1,"d":"For a measurable observation space, a measure on that space, a nonnegative integer kernel order $m$, and a real-valued kernel of $m$ observations, the order-$m$ kernel second moment is $\\int g(z)^2\\,dP^m(z)$, where $P^m$ is the product measure of $m$ independent draws from the given measure.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → ((Fin m → X) → ℝ) → ℝ","l":361,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zetaOrder_nonneg","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"ζ_m is nonnegative.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {g : (Fin m → X) → ℝ}, 0 ≤ Causalean.Stat.IIDSample.zetaOrder P g","l":471,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zeta_eq_zetaOrder","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The order-2 second moment ζ equals the order-m second moment ζ_m of the paired kernel: ∬ g² dP dP = ∫ (pairKernel g)² dP². A change of variables along the Fin 2 → X ≃ X × X measure equivalence.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ} [MeasureTheory.SigmaFinite P], Causalean.Stat.IIDSample.zeta P g = Causalean.Stat.IIDSample.zetaOrder P (Causalean.Stat.pairKernel g)","l":224,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IIDSample.zeta_nonneg","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"ζ ≥ 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {g : X → X → ℝ}, 0 ≤ Causalean.Stat.IIDSample.zeta P g","l":240,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.IndepFun.integrable_finiteSelector_sq","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, and every fixed branch has integrable squared error under the tail law, the pilot-selected squared error is integrable under the ambient probability law.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → MeasureTheory.Integrable (fun omega => err (select (pilot omega)) (tail omega) ^ 2) mu","l":89,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_eq_sum","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, and every fixed branch has integrable squared error under the tail law, the selected risk on that event equals the sum of each selector-cell probability times its fixed-branch tail risk.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [inst_4 : Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → (∀ (i : Iota), MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → ∫ (omega : Omega) in pilot ⁻¹' good, err (select (pilot omega)) (tail omega) ^ 2 ∂mu = ∑ i, (mu (pilot ⁻¹' Causalean.Stat.selectorCell select good i)).toReal * ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu","l":124,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_le","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":1,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, the common risk bound is nonnegative, every branch selected on that event is eligible, eligible branches have integrable squared tail error, and their fixed-branch risks obey the common bound, the pilot-selected risk on the event obeys that same bound.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop} {V : ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → 0 ≤ V → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → (∀ (i : Iota), Eligible i → ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu ≤ V) → ∫ (omega : Omega) in pilot ⁻¹' good, err (select (pilot omega)) (tail omega) ^ 2 ∂mu ≤ V","l":252,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.IndepFun.integral_finiteSelector_sq_le_add_bad","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If pilot and tail coordinates are independent, the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, all branch errors are measurable, the designated pilot event is measurable, the good-event risk bound is nonnegative, the global squared-error envelope is nonnegative, every branch selected on the good event is eligible, eligible branches have integrable squared tail error, their fixed-branch risks obey the good-event bound, the selected squared error obeys the global envelope, and the bad pilot event has probability at most the given tolerance, the full risk is at most the good-event bound plus the envelope times that tolerance.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} [MeasureTheory.IsProbabilityMeasure mu] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop} {V H delta : ℝ}, ProbabilityTheory.IndepFun pilot tail mu → Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), Measurable (err i)) → ∀ {good : Set Alpha}, MeasurableSet good → 0 ≤ V → 0 ≤ H → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → (∀ (i : Iota), Eligible i → ∫ (b : Beta), err i b ^ 2 ∂MeasureTheory.Measure.map tail mu ≤ V) → (∀ (omega : Omega), err (select (pilot omega)) (tail omega) ^ 2 ≤ H) → mu.real (pilot ⁻¹' goodᶜ) ≤ delta → ∫ (omega : Omega), err (select (pilot omega)) (tail omega) ^ 2 ∂mu ≤ V + H * delta","l":354,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.InfluenceFunction","k":"structure","m":"Causalean.Stat.MEstimation.InfluenceFunction","a":"Stat/MEstimation","t":1,"d":"An *influence function* for a parameter θ ∈ E under the law P : Measure X is a measurable, mean-zero, square-integrable function ψ : X → E.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {E : Type u_2} → [inst_1 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [MeasurableSpace E] → MeasureTheory.Measure X → (X → E) → Prop","l":41,"f":"Causalean/Stat/MEstimation/InfluenceFunction.lean"},{"n":"Causalean.Stat.IsAsymLinear","k":"structure","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"An estimator is asymptotically linear when its scaled estimation error equals the normalized empirical average of an influence function that is mean zero and square-integrable under the population law, up to a term that is negligible in probability.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → (ℕ → Ω → ℝ) → ℝ → (X → ℝ) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Finset ℕ) → Prop","l":65,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.normalizedSum","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"For a measurable sample space carrying a measure, a measurable observation space carrying a population measure, an independent and identically distributed sample from that population, a real-valued influence function, a finite index set selected for every nonnegative integer, and a nonnegative integer index, the normalized partial sum maps each sample-space outcome to $|I_n|^{-1/2}\\sum_{i\\in I_n}\\psi(Z_i)$.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → ℝ) → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":103,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.normalizedSum_def","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"The normalised partial sum at sample size index n sends a unit to the sum of the influence-function values over the index block, rescaled by the reciprocal square root of the block size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → ℝ) (I : ℕ → Finset ℕ) (n : ℕ), Causalean.Stat.IsAsymLinear.normalizedSum S ψ I n = fun ω => (√↑(I n).card)⁻¹ * ∑ i ∈ I n, ψ (S.Z i ω)","l":110,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.rescaledEstimator","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"For a sample space, a sequence of real-valued estimators on that space, a real-valued target parameter, a finite index set selected for every nonnegative integer, and a nonnegative integer index, the rescaled estimator maps each sample-space outcome to $\\sqrt{|I_n|}\\,[\\widehat\\theta_n-\\theta_0]$ at that outcome.","s":"{Ω : Type u_1} → (ℕ → Ω → ℝ) → ℝ → (ℕ → Finset ℕ) → ℕ → Ω → ℝ","l":120,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"Given that θn is asymptotically linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable and the rescaled estimator $\\sqrt n(\\theta_n-\\theta_0)$ is a.e. measurable at every sample size, then the rescaled estimator converges in distribution to the centred normal law with variance $\\int \\psi^2\\,dP$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hθn_meas","l":400,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Fold-B asymptotic linearity ⇒ asymptotic normality at rate √|B(n)|. Along an i.i.d. sample S under a one-shot sample split split, if the estimator sequence θn is fold-B asymptotically linear toward θ₀ with influence function ψ that is measurable, and if the rescaled estimator and the fold-B normalized influence-function sum are almost-everywhere measurable at every sample size, then √|B(n)| · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S), Causalean.Stat.IsAsymLinear θn θ₀ ψ S split.foldB → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ split.foldB n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S ψ split.foldB n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ split.foldB) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), ψ x ^ 2 ∂P)) μ hθn_meas","l":210,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IsAsymLinear.tendsto_normal_foldB_sqrt_n","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":1,"d":"Conversion to √n-rate under a fixed split ratio. Along an i.i.d. sample S under a one-shot sample split split with a fold-B asymptotically linear estimator sequence θn (h), suppose the split ratio c is strictly positive, the estimation-fold share |B(n)|/n converges to c, the influence function ψ is measurable, and the √n-rescaled estimator is almost-everywhere measurable at every sample size; then √n · (θn n − θ₀) converges in distribution to the centered Gaussian law with variance (∫ψ²dP)/c.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → Causalean.Stat.IsAsymLinear θn θ₀ ψ S split.foldB → Measurable ψ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (θn n ω - θ₀)) μ), Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (θn n ω - θ₀)) (Causalean.Stat.gaussianMeasure 0 ((∫ (x : X), ψ x ^ 2 ∂P) / c)) μ hθn_meas","l":238,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.IsAsymLinear.toVec","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":2,"d":"Conversely, scalar asymptotic linearity lifts to the vector predicate.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ}, Causalean.Stat.IsAsymLinear θn θ₀ ψ S I → Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S I","l":144,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec","k":"structure","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"A vector-valued estimator sequence is asymptotically linear at a target value when its scaled estimation error equals the normalized empirical average of an influence function that has Bochner mean zero and finite second moment under the population law, up to a remainder that is negligible in probability, along a chosen family of finite index sets selecting which observations enter each empirical average.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → (ℕ → Ω → E) → E → (X → E) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Finset ℕ) → Prop","l":60,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.normalizedSum","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"For a measurable sample space carrying a measure, a measurable observation space carrying a population measure, a normed real vector space, an independent and identically distributed sample from that population, a vector-valued influence function, a finite index set selected for every nonnegative integer, and a nonnegative integer index, the vector normalized partial sum maps each sample-space outcome to $|I_n|^{-1/2}\\sum_{i\\in I_n}\\psi(Z_i)$ in that vector space.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_2 : NormedAddCommGroup E] → [NormedSpace ℝ E] → Causalean.Stat.IIDSample Ω X μ P → (X → E) → (ℕ → Finset ℕ) → ℕ → Ω → E","l":99,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.normalizedSum_def","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":2,"d":"The vector normalised partial sum at sample size index n sends a unit to the sum of the influence-function values over the index block, rescaled by the reciprocal square root of the block size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → E) (I : ℕ → Finset ℕ) (n : ℕ), Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n = fun ω => (√↑(I n).card)⁻¹ • ∑ i ∈ I n, ψ (S.Z i ω)","l":106,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.rescaledEstimator","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"For a sample space, a normed real vector space, a sequence of vector-valued estimators on that space, a target vector, a finite index set selected for every nonnegative integer, and a nonnegative integer index, the rescaled estimator maps each sample-space outcome to $\\sqrt{|I_n|}\\,[\\widehat\\theta_n-\\theta_0]$ in that vector space.","s":"{Ω : Type u_1} → {E : Type u_3} → [inst : NormedAddCommGroup E] → [NormedSpace ℝ E] → (ℕ → Ω → E) → E → (ℕ → Finset ℕ) → ℕ → Ω → E","l":116,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.tendsto_normal_vec","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":1,"d":"Vector asymptotic normality. For a vector-valued estimator sequence θn targeting θ₀ with influence function ψ along the i.i.d. sample S, suppose the remainder between the rescaled estimator and the normalised partial sum is asymptotically negligible (little-o of 1 in norm), the rescaled estimator is a.e. measurable at every sample size, the normalised partial sum is a.e. measurable at every sample size, and the pushforward laws of the normalised partial sum converge to a target probability measure Q on E. Then the pushforward laws of the rescaled estimator likewise converge to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : OpensMeasurableSpace E] {θn : ℕ → Ω → E} {θ₀ : E} {ψ : X → E} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ} (Q : MeasureTheory.ProbabilityMeasure E), Causalean.Stat.IsLittleOp (fun n ω => ‖√↑(I n).card • (θn n ω - θ₀) - (√↑(I n).card)⁻¹ • ∑ i ∈ I n, ψ (S.Z i ω)‖) (fun x => 1) μ → ∀ (_hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ I n) μ) (_hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (_hSum_meas n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ I n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (_hθn_meas n)⟩) Filter.atTop (nhds Q)","l":168,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.tendsto_normal_vec_clt","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Vector asymptotic normality from asymptotic linearity (no CLT hypothesis). Given that θn is vector-asymptotically-linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable, a target probability measure Q on E whose characteristic function is the Gaussian one $\\exp(-\\tfrac12\\int\\langle t,\\psi\\rangle^2\\,dP)$ at every point t, and the rescaled estimator is a.e. measurable at every sample size, then the pushforward laws of the rescaled estimator converge to Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] {θn : ℕ → Ω → E} {θ₀ : E} {ψ : X → E} {S : Causalean.Stat.IIDSample Ω X μ P}, Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S Causalean.Stat.rng✝ → Measurable ψ → ∀ (Q : MeasureTheory.Measure E) [inst_7 : MeasureTheory.IsProbabilityMeasure Q], (∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)) → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ Causalean.Stat.rng✝ n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn θ₀ Causalean.Stat.rng✝ n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds ⟨Q, inst_7⟩)","l":282,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.IsAsymLinearVec.toScalar","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearityVec","a":"Stat/CLT","t":2,"d":"When the parameter space is ℝ, the vector predicate IsAsymLinearVec unfolds to the scalar IsAsymLinear. The forward direction is a direct field-by-field rewrite using ‖x‖ = |x| on ℝ and scalar • = *.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P} {I : ℕ → Finset ℕ}, Causalean.Stat.IsAsymLinearVec θn θ₀ ψ S I → Causalean.Stat.IsAsymLinear θn θ₀ ψ S I","l":127,"f":"Causalean/Stat/CLT/AsymptoticLinearityVec.lean"},{"n":"Causalean.Stat.IsBigOp","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Given a sequence of real-valued random variables, a real-valued rate sequence, and a measure on a measurable sample space, boundedness in probability at that rate means that for every $ arepsilon>0$ there exists a real $M$ such that the limit superior, over indices, of the measure of outcomes satisfying $|X_n|>M r_n$ is at most $ arepsilon$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":153,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.add","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"O_p(rn) + O_p(rn) = O_p(rn).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn rn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω + Yn n ω) rn μ","l":532,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.add'","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Additivity at the sum rate. O_p(rₙ) + O_p(sₙ) = O_p(rₙ + sₙ), for nonnegative rates. (IsBigOp.add is the special case rₙ = sₙ.)","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), 0 ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω + Yn n ω) (fun n => rn n + sn n) μ","l":137,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.const_mul","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Constant multiple. If Xₙ = O_p(rₙ) then c · Xₙ = O_p(rₙ) for any fixed scalar c.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} (c : ℝ), Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp (fun n ω => c * Xn n ω) rn μ","l":110,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.const_mul_tendsto_zero","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Degenerate Slutsky. If a real-valued sequence Xn is bounded in probability, O_p(1), under μ, and a deterministic scalar sequence a converges to 0, then the product sequence a n · Xn is o_p(1). Concretely: the product of a sequence converging to 0 with a tight sequence is o_p(1).","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {a : ℕ → ℝ}, Causalean.Stat.IsBigOp Xn (fun x => 1) μ → Filter.Tendsto a Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => a n * Xn n ω) (fun x => 1) μ","l":256,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.const_rate_collapse","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Collapse a constant rate to 1. For a *fixed* nonnegative N, O_p(fun _ => N) is O_p(fun _ => 1): a constant scale only changes the witness M. Used to normalize the fold-sum O_p bounds to the canonical unit rate consumed by the cross-fitted DML proofs.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {N : ℝ}, 0 ≤ N → Causalean.Stat.IsBigOp Xn (fun x => N) μ → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":98,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.finset_sum","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"A finite sum of O_p(rₙ) sequences is O_p(rₙ) (same rate; constants absorb).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {rn : ℕ → ℝ} {ι : Type u_2} (s : Finset ι) {X : ι → ℕ → Ω → ℝ}, (∀ i ∈ s, Causalean.Stat.IsBigOp (X i) rn μ) → Causalean.Stat.IsBigOp (fun n ω => ∑ i ∈ s, X i n ω) rn μ","l":229,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mono_rate","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Weaken to a larger rate. O_p(rₙ) with 0 ≤ rₙ ≤ sₙ is O_p(sₙ): a larger envelope is a weaker statement.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), rn n ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Xn sn μ","l":64,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mul","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Product rule for stochastic big-O. If Xₙ = O_p(rₙ) and Yₙ = O_p(sₙ) for nonnegative rates, then XₙYₙ = O_p(rₙsₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ rn n) → (∀ (n : ℕ), 0 ≤ sn n) → Causalean.Stat.IsBigOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsBigOp (fun n ω => Xn n ω * Yn n ω) (fun n => rn n * sn n) μ","l":248,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.mul_isLittleOp_one_isLittleOp","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Slutsky-style product: Xn = O_p(1) and Yn = o_p(1) imply Xn · Yn = o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsBigOp Xn (fun x => 1) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun x => 1) μ","l":585,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsBigOp.of_abs_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"If |Xₙ| ≤ |Yₙ| pointwise and Yₙ = O_p(rₙ), then Xₙ = O_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ |Yn n ω|) → Causalean.Stat.IsBigOp Yn rn μ → Causalean.Stat.IsBigOp Xn rn μ","l":211,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.of_sqEnvelope","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Square-envelope O_p lemma. If |X_n|² ≤ c_n μ-a.s. and c_n is O_p(rₙ²) with rₙ > 0, then X_n = O_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn cn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), ∀ᵐ (ω : Ω) ∂μ, Xn n ω ^ 2 ≤ cn n ω) → Causalean.Stat.IsBigOp cn (fun n => rn n ^ 2) μ → Causalean.Stat.IsBigOp Xn rn μ","l":59,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsBigOp.of_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Markov second-moment ⇒ O_p. If each Xₙ is μ-a.e.-measurable and its second moment is bounded by a deterministic envelope, ∫⁻ (Xₙ ω)² dμ ≤ Vₙ with 0 ≤ Vₙ, then Xₙ = O_p(√Vₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {Vn : ℕ → ℝ}, (∀ (n : ℕ), AEMeasurable (Xn n) μ) → (∀ (n : ℕ), 0 ≤ Vn n) → (∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (Xn n ω ^ 2) ∂μ ≤ ENNReal.ofReal (Vn n)) → Causalean.Stat.IsBigOp Xn (fun n => √(Vn n)) μ","l":321,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.scale_rate","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"Absorb a positive constant rate factor. O_p(c · rₙ) with c > 0 is O_p(rₙ); the constant is absorbed into the witness M.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {c : ℝ}, 0 < c → Causalean.Stat.IsBigOp Xn (fun n => c * rn n) μ → Causalean.Stat.IsBigOp Xn rn μ","l":88,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsBigOp.zero","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":2,"d":"The constant-zero sequence is O_p(rₙ) for any rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {rn : ℕ → ℝ}, Causalean.Stat.IsBigOp (fun x x_1 => 0) rn μ","l":220,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.IsCoupling","k":"structure","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"π is a coupling of μ and ν, i.e. a member of the Fréchet class Π(μ, ν), when it is a probability measure on ℝ × ℝ whose first marginal is μ and whose second marginal is ν.","s":"MeasureTheory.Measure (ℝ × ℝ) → MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → Prop","l":32,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.IsDiscrepancyAttainedOn","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For a covariance model and target matrix, discrepancy attainment means that one model member has no larger Gaussian discrepancy than every other member.","s":"{V : Type u_2} → [inst : Fintype V] → [DecidableEq V] → Set (Causalean.Stat.PositiveCovariance V) → Matrix V V ℝ → Prop","l":257,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.IsLittleOp","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Given a sequence of real-valued random variables, a real-valued rate sequence, and a measure on a measurable sample space, negligibility in probability relative to that rate means that for every $ arepsilon>0$, the measure of outcomes satisfying $|X_n|> arepsilon r_n$ tends to zero as the index tends to infinity.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (ℕ → ℝ) → MeasureTheory.Measure Ω → Prop","l":166,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.add_eventually_nonneg_rate","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"The sum of two stochastic little-o terms is stochastic little-o for an eventually nonnegative rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, (∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ rn n) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsLittleOp Yn rn μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω + Yn n ω) rn μ","l":334,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.add_one","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"The sum of two o_p(1) sequences is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω + Yn n ω) (fun x => 1) μ","l":422,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.mul_isBigOp","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"o_p(rn) · O_p(sn) = o_p(rn · sn), assuming positive rates.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn sn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), 0 < sn n) → Causalean.Stat.IsLittleOp Xn rn μ → Causalean.Stat.IsBigOp Yn sn μ → Causalean.Stat.IsLittleOp (fun n ω => Xn n ω * Yn n ω) (fun n => rn n * sn n) μ","l":438,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_abs_le_const_mul","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Domination by a positive constant times a stochastic little-o term preserves the stochastic little-o rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ} {μ : MeasureTheory.Measure Ω} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Yn rn μ → (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ C * |Yn n ω|) → Causalean.Stat.IsLittleOp Xn rn μ","l":398,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_abs_le_const_mul_one","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Domination by a positive constant times an o_p(1) sequence preserves o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ → (∀ (n : ℕ) (ω : Ω), |Xn n ω| ≤ C * |Yn n ω|) → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ","l":430,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.IsLittleOp.of_eq_on_asymptotic","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Truncation reduction. If Xn differs from Yn only on a set whose μ-measure tends to 0, and Yn = o_p(rn), then Xn = o_p(rn).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, Filter.Tendsto (fun n => μ {ω | Xn n ω ≠ Yn n ω}) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp Yn rn μ → Causalean.Stat.IsLittleOp Xn rn μ","l":151,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsLittleOp.of_sqEnvelope","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"Square-envelope o_p lemma. For sequences of random variables X_n, c_n and rates r_n, if each rate r_n is positive, X_n² is bounded above by c_n almost surely, for every n, and c_n is o_p(r_n²), then X_n is o_p(r_n).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn cn : ℕ → Ω → ℝ} {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → (∀ (n : ℕ), ∀ᵐ (ω : Ω) ∂μ, Xn n ω ^ 2 ≤ cn n ω) → Causalean.Stat.IsLittleOp cn (fun n => rn n ^ 2) μ → Causalean.Stat.IsLittleOp Xn rn μ","l":105,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.IsRankedMinimizer","k":"def","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":1,"d":"Given a deterministic tie rank, a criterion vector, and a selected model index, the ranked-minimizer condition says that the selected model has a smaller criterion than each competitor, or an equal criterion and no larger rank.","s":"{ι : Type u_2} → (ι → ℕ) → (ι → ℝ) → ι → Prop","l":83,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.IsRankedMinimizer.le","k":"theorem","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":2,"d":"If a selected model is a ranked minimizer, then, for each competing model, its criterion value is no larger than the competitor's.","s":"∀ {ι : Type u_2} {rank : ι → ℕ} {criterion : ι → ℝ} {selected : ι}, Causalean.Stat.IsRankedMinimizer rank criterion selected → ∀ (j : ι), criterion selected ≤ criterion j","l":90,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.IsUniqueGlobalMax","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"Given a real-valued criterion on a domain and a point in that domain, the assertion that the point is its unique global maximizer means that every domain point has criterion value no larger than this point's value, and that equality can occur only at this point.","s":"{A : Type u_1} → (A → ℝ) → A → Prop","l":26,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.KFoldSplit","k":"structure","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"K-fold sample split. For an i.i.d. sample S, a schedule assigning each sample size n and fold index k a finite index set fold n k, forming a K-fold cross-fitting scheme in which distinct folds are pairwise disjoint at every sample size, the K folds together cover the full index set $\\{0,\\dots,n-1\\}$, every fold grows without bound as $n \\to \\infty$, and each fold's share of the sample converges to $1/K$.","s":"{Ω : Type u_3} → {X : Type u_4} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → ℕ → Type","l":42,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.fold_centered_sum_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Per-fold centered evaluation sum is o_p(1) (K-fold). For an i.i.d. sample and a K-fold split, fixing an evaluation fold k, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training-complement data and the outcome, for every n, measurable with respect to the training-complement σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$, the centered per-fold empirical sum of f over the evaluation fold k — its fold sample average minus its population mean, rescaled by the square root of the fold size — is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (k : Fin K) (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => (√↑(split.fold n k).card)⁻¹ * ∑ i ∈ split.fold n k, (f n ω (S.Z i ω) - ∫ (x : X), f n ω x ∂P)) (fun x => 1) μ","l":742,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.KFoldSplit.fold_disjoint_trainComplement","k":"theorem","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":2,"d":"The evaluation fold is disjoint from its training complement.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (n : ℕ) (k : Fin K), Disjoint (split.fold n k) (split.trainComplement n k)","l":69,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.folds_indep","k":"theorem","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"Independence of evaluation fold and training complement. For a fixed sample size n and fold index k, the sample sub-tuple indexed by the evaluation fold is independent, under μ, of the sub-tuple indexed by the training complement.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (n : ℕ) (k : Fin K), ProbabilityTheory.IndepFun (fun ω i => S.Z (↑i) ω) (fun ω i => S.Z (↑i) ω) μ","l":77,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.KFoldSplit.sqrtFold_integral_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Per-fold bias term is o_p(1) under a nonempty K-fold split. For an i.i.d. sample and a K-fold split with a positive number of folds $K$, given a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$, the per-fold bias term $\\sqrt{|{\\rm fold}(n,k)|}\\cdot\\int f(n,\\omega,\\cdot)\\,dP$ at evaluation fold k is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {K : ℕ} (split : Causalean.Stat.KFoldSplit S K) (k : Fin K), 0 < K → ∀ (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.fold n k).card * ∫ (x : X), f n ω x ∂P) (fun x => 1) μ","l":777,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.KFoldSplit.trainComplement","k":"def","m":"Causalean.Stat.SampleSplit.KFold","a":"Stat/SampleSplit","t":1,"d":"Given a K-fold splitting schedule, a sample size, and a fold index, the training complement is the set of all indices from $0$ through $n-1$ excluding those assigned to that fold.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → {K : ℕ} → Causalean.Stat.KFoldSplit S K → ℕ → Fin K → Finset ℕ","l":63,"f":"Causalean/Stat/SampleSplit/KFold.lean"},{"n":"Causalean.Stat.L1Bracketing","k":"structure","m":"Causalean.Stat.EmpiricalProcess.Basic","a":"Stat/EmpiricalProcess","t":1,"d":"A finite L¹(P) ε-bracketing of a real-valued function class consists of finitely many integrable lower and upper endpoints, a common full-measure support on which every class member is sandwiched by its assigned bracket, and an integrated absolute bracket width at most ε.","s":"{X : Type u_2} → {ι : Type u_3} → [inst : MeasurableSpace X] → (ι → X → ℝ) → MeasureTheory.Measure X → ℝ → Type (max u_2 u_3)","l":98,"f":"Causalean/Stat/EmpiricalProcess/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.FiniteVanTreesModelRegularity","k":"structure","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Main","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Finite-experiment van Trees regularity collects the model-specific normalization, absolute-continuity, derivative, measurability, and integrability conditions for a fixed prior and counting-measure observation model.","s":"(X : Type u_1) → [Fintype X] → [inst : MeasurableSpace X] → [MeasurableSingletonClass X] → ℝ → ℝ → (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (X → ℝ) → Prop","l":244,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Main.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.ac_product_integral_eq_boundary","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.IntegrationByParts","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Two functions on an ordered interval with absolute continuity, almost-everywhere derivative representatives, and an integrable product-rule field satisfy the product-rule integral equals the endpoint change in their product.","s":"∀ {a b : ℝ} {f g df dg : ℝ → ℝ}, a ≤ b → AbsolutelyContinuousOnInterval f a b → AbsolutelyContinuousOnInterval g a b → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt f (df θ) θ) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt g (dg θ) θ) → IntervalIntegrable (fun θ => df θ * g θ + f θ * dg θ) MeasureTheory.volume a b → ∫ (θ : ℝ) in a..b, df θ * g θ + f θ * dg θ = f b * g b - f a * g a","l":22,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/IntegrationByParts.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.ac_weighted_error_integral_eq_boundary","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.IntegrationByParts","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A prior weight, likelihood section, and observation-dependent target with absolute continuity, almost-everywhere derivative representatives, and an integrable weighted-error derivative field satisfy the weighted-error derivative integral equals the weighted endpoint difference.","s":"∀ {a b c : ℝ} {w p g dw dp dg : ℝ → ℝ}, a ≤ b → AbsolutelyContinuousOnInterval w a b → AbsolutelyContinuousOnInterval p a b → AbsolutelyContinuousOnInterval g a b → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt w (dw θ) θ) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt p (dp θ) θ) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt g (dg θ) θ) → IntervalIntegrable (fun θ => (dw θ * p θ + w θ * dp θ) * (c - g θ) - w θ * p θ * dg θ) MeasureTheory.volume a b → ∫ (θ : ℝ) in a..b, (dw θ * p θ + w θ * dp θ) * (c - g θ) - w θ * p θ * dg θ = w b * p b * (c - g b) - w a * p a * (c - g a)","l":52,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/IntegrationByParts.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.average_fisherInformation_le","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"If the prior is nonnegative, normalized, and integrable, prior-weighted Fisher information is integrable, and conditional Fisher information is pointwise bounded wherever the prior is positive, then average Fisher information is bounded by the same constant.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} {ell u I : ℝ} {w : ℝ → ℝ} {p dp : ℝ → X → ℝ}, (∀ (θ : ℝ), 0 ≤ w θ) → ∫ (θ : ℝ), w θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u = 1 → MeasureTheory.Integrable w (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → MeasureTheory.Integrable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → (∀ (θ : ℝ), 0 < w θ → Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ ≤ I) → ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u ≤ I","l":335,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, its derivative, a conditional likelihood density, its derivative, a target function, its parameter derivative, an estimator, and a parameter--observation pair, the derivative-balance field is the parameter derivative of the joint-density-weighted estimation error.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (X → ℝ) → ℝ × X → ℝ","l":87,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.derivative_eq_zero_of_nonnegative_of_eq_zero","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A differentiable nonnegative function has zero derivative at any point where it attains value zero, so the supplied derivative is zero.","s":"∀ {f : ℝ → ℝ} {df θ : ℝ}, (∀ (t : ℝ), 0 ≤ f t) → HasDerivAt f df θ → f θ = 0 → df = 0","l":24,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.errorScoreField","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, its derivative, a conditional likelihood density, its derivative, a target function, an estimator, and a parameter--observation pair, the signed error--joint-score field is estimator error times guarded joint score, weighted by joint density.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (X → ℝ) → ℝ × X → ℝ","l":72,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.errorScoreField_eq_numerator","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"For nonnegative prior and likelihood densities whose derivative numerators vanish wherever the corresponding density is zero, the weighted error--score field equals estimation error times the unguarded product-rule numerator.","s":"∀ {X : Type u_1} {w dw : ℝ → ℝ} {p dp g : ℝ → X → ℝ} {T : X → ℝ} {θ : ℝ} {x : X}, 0 ≤ w θ → 0 ≤ p θ x → (w θ = 0 → dw θ = 0) → (p θ x = 0 → dp θ x = 0) → Causalean.Stat.Limit.ObservationDependentVanTrees.errorScoreField w dw p dp g T (θ, x) = (T x - g θ x) * (dw θ * p θ x + w θ * dp θ x)","l":66,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, a conditional likelihood density, a target function, an estimator, and a parameter--observation pair, the squared-error field is the squared estimator error weighted by the joint density.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → (X → ℝ) → ℝ × X → ℝ","l":96,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.finite_derivative_centering","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If each cell probability has the supplied derivative and the finite likelihood is locally normalized, then the derivative masses sum to zero.","s":"∀ {X : Type u_1} [inst : Fintype X] {p dp : ℝ → X → ℝ} {θ : ℝ}, (∀ (x : X), HasDerivAt (fun t => p t x) (dp θ x) θ) → (∀ᶠ (t : ℝ) in nhds θ, ∑ x, p t x = 1) → ∑ x, dp θ x = 0","l":297,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.finite_likelihoodScore_centered","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If finite likelihood masses are nonnegative, derivatives vanish on zero-mass cells, and the derivative masses are centered, then the likelihood-weighted guarded scores sum to zero.","s":"∀ {X : Type u_1} [inst : Fintype X] {p dp : ℝ → X → ℝ} {θ : ℝ}, (∀ (x : X), 0 ≤ p θ x) → (∀ (x : X), p θ x = 0 → dp θ x = 0) → ∑ x, dp θ x = 0 → ∑ x, Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x * p θ x = 0","l":317,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.finite_likelihoodScore_mul","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If finite likelihood masses are nonnegative and their derivatives vanish on zero-mass cells, then likelihood mass times guarded score equals the cell derivative.","s":"∀ {X : Type u_1} {p dp : ℝ → X → ℝ} {θ : ℝ}, (∀ (x : X), 0 ≤ p θ x) → (∀ (x : X), p θ x = 0 → dp θ x = 0) → ∀ (x : X), Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x * p θ x = dp θ x","l":309,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.finite_likelihood_normalization","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the masses of a finite likelihood sum to one, then its integral against counting measure is one.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : MeasurableSpace X] [MeasurableSingletonClass X] {p : ℝ → X → ℝ} {θ : ℝ}, ∑ x, p θ x = 1 → ∫ (x : X), p θ x ∂MeasureTheory.Measure.count = 1","l":275,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.finite_vanTrees_lower_bound","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Main","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A nondegenerate parameter interval, [finite-model regularity] (hyp:M), continuous differentiability, compact support, a valid prior derivative, nonnegativity, normalization, vanishing boundary errors, measurability and integrability of prior information, a nonnegative sensitivity bound that lies below average sensitivity, upper bounds on average likelihood information and prior information, and positive total information imply a native-real Bayes squared-risk lower bound equal to squared sensitivity divided by the sum of the two information bounds.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] {ell u s I P : ℝ} {w dw : ℝ → ℝ} {p dp g dg : ℝ → X → ℝ} {T : X → ℝ}, ell < u → Causalean.Stat.Limit.ObservationDependentVanTrees.FiniteVanTreesModelRegularity X ell u w dw p dp g dg T → ContDiff ℝ 1 w → Function.support w ⊆ Set.Icc ell u → (∀ (θ : ℝ), HasDerivAt w (dw θ) θ) → (∀ (θ : ℝ), 0 ≤ w θ) → ∫ (θ : ℝ), w θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u = 1 → (∀ᵐ (x : X) ∂MeasureTheory.Measure.count, w u * p u x * (T x - g u x) = 0 ∧ w ell * p ell x * (T x - g ell x) = 0) → MeasureTheory.AEStronglyMeasurable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → MeasureTheory.Integrable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → 0 ≤ s → s ≤ ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField w p dg z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod MeasureTheory.Measure.count → ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation MeasureTheory.Measure.count p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u ≤ I → Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u w dw ≤ P → 0 < Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u w dw + ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation MeasureTheory.Measure.count p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u → s ^ 2 / (I + P) ≤ ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField w p g T z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod MeasureTheory.Measure.count","l":331,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Main.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a measure on a measurable observation space, a conditional likelihood density, its parameter derivative, and a parameter value, the conditional Fisher information is the observation integral of the likelihood times the squared guarded likelihood score at that parameter.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (ℝ → X → ℝ) → (ℝ → X → ℝ) → ℝ → ℝ","l":65,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation_count_eq_sum","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Conditional Fisher information under counting measure equals the finite sum of likelihood mass times squared guarded score.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : MeasurableSpace X] [MeasurableSingletonClass X] (p dp : ℝ → X → ℝ) (θ : ℝ), Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation MeasureTheory.Measure.count p dp θ = ∑ x, p θ x * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x ^ 2","l":327,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.guarded_score_mul_density","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A nonnegative density with a derivative numerator that vanishes whenever the density vanishes has its guarded score times its density equal to that derivative numerator.","s":"∀ {q dq : ℝ}, 0 ≤ q → (q = 0 → dq = 0) → (if 0 < q then dq / q else 0) * q = dq","l":34,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.hasDerivAt_finite_likelihood_sum","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If each cell probability has the supplied parameter derivative, then the derivative of their finite sum is the sum of those derivatives.","s":"∀ {X : Type u_1} [inst : Fintype X] {p dp : ℝ → X → ℝ} {θ : ℝ}, (∀ (x : X), HasDerivAt (fun t => p t x) (dp θ x) θ) → HasDerivAt (fun t => ∑ x, p t x) (∑ x, dp θ x) θ","l":283,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.hasDerivAt_smoothPrior","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Under a positive bandwidth, the explicit derivative representative is the derivative of the smooth quartic prior at every parameter value.","s":"∀ {c a : ℝ}, 0 < a → ∀ (θ : ℝ), HasDerivAt (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a θ) θ","l":325,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.integral_smoothPrior_parameterMeasure","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie in an ambient interval, then the prior has mass one under Lebesgue measure restricted to that interval.","s":"∀ {ell u c a : ℝ}, 0 < a → ell ≤ c - a → c + a ≤ u → ∫ (θ : ℝ), Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u = 1","l":399,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.integral_smoothPrior_volume","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Under a positive bandwidth, the smooth quartic density has total Lebesgue integral one.","s":"∀ {c a : ℝ}, 0 < a → ∫ (θ : ℝ), Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ = 1","l":338,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.jointDensity","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density on the real parameter, a conditional observation density given that parameter, and a parameter--observation pair, the joint density is the product of the prior density and the conditional likelihood density at that pair.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → X → ℝ) → ℝ × X → ℝ","l":31,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.jointScore","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, its derivative, a conditional likelihood density, its derivative, and a parameter--observation pair, the guarded joint score is the derivative of the prior--likelihood product divided by that product when the product is positive, and is zero otherwise.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → ℝ × X → ℝ","l":49,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.jointScore_eq_add","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"At a point where both the prior and likelihood densities are positive, the guarded joint score equals the sum of their guarded scores.","s":"∀ {X : Type u_1} {w dw : ℝ → ℝ} {p dp : ℝ → X → ℝ} {θ : ℝ} {x : X}, 0 < w θ → 0 < p θ x → Causalean.Stat.Limit.ObservationDependentVanTrees.jointScore w dw p dp (θ, x) = Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ + Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x","l":43,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.jointScore_mul_jointDensity","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"For nonnegative prior and likelihood densities whose derivative numerators vanish on their respective zero-density sets, joint density times guarded joint score equals the product-rule derivative numerator.","s":"∀ {X : Type u_1} {w dw : ℝ → ℝ} {p dp : ℝ → X → ℝ} {θ : ℝ} {x : X}, 0 ≤ w θ → 0 ≤ p θ x → (w θ = 0 → dw θ = 0) → (p θ x = 0 → dp θ x = 0) → Causalean.Stat.Limit.ObservationDependentVanTrees.jointScore w dw p dp (θ, x) * Causalean.Stat.Limit.ObservationDependentVanTrees.jointDensity w p (θ, x) = dw θ * p θ x + w θ * dp θ x","l":51,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.joint_score_information_decomposition","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"For nonnegative prior and likelihood densities with zero-set derivative control, likelihood normalization, centered conditional scores, and the stated measurable integrability of all square and cross fields, joint-score information equals prior information plus average conditional Fisher information.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.SigmaFinite μ] {a b : ℝ} {w dw : ℝ → ℝ} {p dp : ℝ → X → ℝ}, (∀ (θ : ℝ), 0 ≤ w θ) → (∀ (θ : ℝ) (x : X), 0 ≤ p θ x) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, w θ = 0 → dw θ = 0) → (∀ᵐ (z : ℝ × X) ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ, p z.1 z.2 = 0 → dp z.1 z.2 = 0) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, ∫ (x : X), p θ x ∂μ = 1) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, ∫ (x : X), Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x * p θ x ∂μ = 0) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField w dw p dp) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField w dw p dp) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.AEStronglyMeasurable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b) → MeasureTheory.Integrable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * (Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2)) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * (Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2)) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField w dw p dp z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ = Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation a b w dw + ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b","l":101,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a conditional likelihood density, its parameter derivative, a parameter value, and an observation, the guarded likelihood score is the derivative divided by the likelihood when that likelihood is positive, and is zero otherwise.","s":"{X : Type u_1} → (ℝ → X → ℝ) → (ℝ → X → ℝ) → ℝ → X → ℝ","l":43,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore_integral_eq_zero_of_normalization","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.GuardedInformation","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"At an interior parameter value, a normalized likelihood whose integral may be differentiated using the supplied derivative, with nonnegative density, zero-set derivative control, and both fields integrable has conditional mean-zero guarded likelihood score.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} {a b θ : ℝ} {p dp : ℝ → X → ℝ}, θ ∈ Set.Ioo a b → (∀ t ∈ Set.Icc a b, ∫ (x : X), p t x ∂μ = 1) → HasDerivAt (fun t => ∫ (x : X), p t x ∂μ) (∫ (x : X), dp θ x ∂μ) θ → (∀ (x : X), 0 ≤ p θ x) → (∀ᵐ (x : X) ∂μ, p θ x = 0 → dp θ x = 0) → MeasureTheory.Integrable (fun x => p θ x) μ → MeasureTheory.Integrable (fun x => dp θ x) μ → ∫ (x : X), Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp θ x * p θ x ∂μ = 0","l":77,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/GuardedInformation.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.observation_dependent_van_trees","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Main","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"A nondegenerate parameter interval with a continuously differentiable, compactly supported, normalized nonnegative prior; a normalized nonnegative dominated likelihood; the stated differentiation-under-the-integral, sectionwise absolute-continuity, derivative, boundary, product-measurability, and integrability conditions; and strictly positive finite total information ensures that Bayes mean-squared error for an estimator of an observation-dependent target is at least the squared joint mean of the target's parameter derivative divided by prior information plus average Fisher information.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.SigmaFinite μ] {ell u : ℝ}, ell < u → ∀ (w dw : ℝ → ℝ) (p dp g dg : ℝ → X → ℝ) (T : X → ℝ), ContDiff ℝ 1 w → Function.support w ⊆ Set.Icc ell u → (∀ (θ : ℝ), HasDerivAt w (dw θ) θ) → (∀ (θ : ℝ), 0 ≤ w θ) → ∫ (θ : ℝ), w θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u = 1 → (∀ (θ : ℝ) (x : X), 0 ≤ p θ x) → (∀ θ ∈ Set.Icc ell u, ∫ (x : X), p θ x ∂μ = 1) → (∀ θ ∈ Set.Icc ell u, MeasureTheory.Integrable (fun x => p θ x) μ) → (∀ θ ∈ Set.Icc ell u, MeasureTheory.Integrable (fun x => dp θ x) μ) → (∀ θ ∈ Set.Ioo ell u, HasDerivAt (fun t => ∫ (x : X), p t x ∂μ) (∫ (x : X), dp θ x ∂μ) θ) → (∀ᵐ (x : X) ∂μ, AbsolutelyContinuousOnInterval (fun θ => p θ x) ell u) → (∀ᵐ (x : X) ∂μ, AbsolutelyContinuousOnInterval (fun θ => g θ x) ell u) → (∀ᵐ (z : ℝ × X) ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ, HasDerivAt (fun t => p t z.2) (dp z.1 z.2) z.1) → (∀ᵐ (z : ℝ × X) ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ, HasDerivAt (fun t => g t z.2) (dg z.1 z.2) z.1) → (∀ᵐ (x : X) ∂μ, w u * p u x * (T x - g u x) = 0 ∧ w ell * p ell x * (T x - g ell x) = 0) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField w dw p dp g dg T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField w dw p dp g dg T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.errorScoreField w dw p dp g T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.errorScoreField w dw p dp g T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField w p dg) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField w p dg) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField w p g T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField w p g T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField w dw p dp) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField w dw p dp) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → MeasureTheory.Integrable (fun θ => w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2 ^ 2) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.AEStronglyMeasurable (fun z => w z.1 * p z.1 z.2 * (Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2)) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → MeasureTheory.Integrable (fun z => w z.1 * p z.1 z.2 * (Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore w dw z.1 * Causalean.Stat.Limit.ObservationDependentVanTrees.likelihoodScore p dp z.1 z.2)) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) → 0 < Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u w dw + ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u → (∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField w p dg z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ) ^ 2 / (Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u w dw + ∫ (θ : ℝ), w θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation μ p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u) ≤ ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField w p g T z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod μ","l":25,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Main.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a lower endpoint and an upper endpoint, the parameter reference measure is Lebesgue measure restricted to the closed interval from the lower to the upper endpoint.","s":"ℝ → ℝ → MeasureTheory.Measure ℝ","l":26,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a lower endpoint, an upper endpoint, a prior density, and its derivative, the prior Fisher information is the integral over the closed parameter interval of the prior density times the squared guarded prior score.","s":"ℝ → ℝ → (ℝ → ℝ) → (ℝ → ℝ) → ℝ","l":59,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation_smoothPrior","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie in an ambient interval, then the prior Fisher information equals ten divided by the squared bandwidth.","s":"∀ {ell u c a : ℝ}, 0 < a → ell ≤ c - a → c + a ≤ u → Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) = 10 / a ^ 2","l":508,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation_smoothPrior_le","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie in an ambient interval, then the prior Fisher information is at most forty divided by the squared bandwidth.","s":"∀ {ell u c a : ℝ}, 0 < a → ell ≤ c - a → c + a ≤ u → Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) ≤ 40 / a ^ 2","l":553,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, its parameter derivative, and a parameter value, the guarded prior score is the derivative divided by the density when the density is positive, and is zero otherwise.","s":"(ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ","l":37,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.product_integral_derivativeBalance_eq_zero","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.IntegrationByParts","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"In a σ-finite observation model, if the prior weight and almost every likelihood and target section are absolutely continuous with the stated derivative representatives, endpoint error products vanish, and the derivative-balance field is measurable and integrable, the joint integral of the derivative-balance field is zero.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.SigmaFinite μ] {a b : ℝ}, a ≤ b → ∀ {w dw : ℝ → ℝ} {p dp g dg : ℝ → X → ℝ} {T : X → ℝ}, AbsolutelyContinuousOnInterval w a b → (∀ᵐ (x : X) ∂μ, AbsolutelyContinuousOnInterval (fun θ => p θ x) a b) → (∀ᵐ (x : X) ∂μ, AbsolutelyContinuousOnInterval (fun θ => g θ x) a b) → (∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt w (dw θ) θ) → (∀ᵐ (x : X) ∂μ, ∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt (fun t => p t x) (dp θ x) θ) → (∀ᵐ (x : X) ∂μ, ∀ᵐ (θ : ℝ) ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b, HasDerivAt (fun t => g t x) (dg θ x) θ) → (∀ᵐ (x : X) ∂μ, w b * p b x * (T x - g b x) = 0 ∧ w a * p a x * (T x - g a x) = 0) → MeasureTheory.AEStronglyMeasurable (Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField w dw p dp g dg T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField w dw p dp g dg T) ((Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ) → ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.derivativeBalanceField w dw p dp g dg T z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure a b).prod μ = 0","l":104,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/IntegrationByParts.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.product_integral_eq_iterated","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.WeightedL2","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A real-valued product field that is almost-everywhere strongly measurable and integrable under two σ-finite measures has both iterated integrals equal to its product-measure integral.","s":"∀ {A : Type u_1} {B : Type u_2} [inst : MeasurableSpace A] [inst_1 : MeasurableSpace B] {μ : MeasureTheory.Measure A} {ν : MeasureTheory.Measure B} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] {F : A × B → ℝ}, MeasureTheory.AEStronglyMeasurable F (μ.prod ν) → MeasureTheory.Integrable F (μ.prod ν) → ∫ (z : A × B), F z ∂μ.prod ν = ∫ (a : A), ∫ (b : B), F (a, b) ∂ν ∂μ ∧ ∫ (z : A × B), F z ∂μ.prod ν = ∫ (b : B), ∫ (a : A), F (a, b) ∂μ ∂ν","l":21,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/WeightedL2.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.scoreSqField","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, its derivative, a conditional likelihood density, its derivative, and a parameter--observation pair, the squared-score field is the squared guarded joint score weighted by the joint density.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → ℝ × X → ℝ","l":103,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a prior density, a conditional likelihood density, the parameter derivative of a target function, and a parameter--observation pair, the target-sensitivity field is that derivative weighted by the joint density.","s":"{X : Type u_1} → (ℝ → ℝ) → (ℝ → X → ℝ) → (ℝ → X → ℝ) → ℝ × X → ℝ","l":80,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a center, a radius, and a parameter value, the smooth quartic prior is $(15/(16a))\\,[1-((θ-c)/a)^2]^2$ when $|θ-c|<a$, and is zero otherwise.","s":"ℝ → ℝ → ℝ → ℝ","l":124,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv","k":"def","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Given a center, a radius, and a parameter value, the smooth quartic prior derivative is $-(15/(4a^3))(θ-c)[1-((θ-c)/a)^2]$ when $|θ-c|<a$, and is zero otherwise.","s":"ℝ → ℝ → ℝ → ℝ","l":131,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_absolutelyContinuousOnInterval","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth and ordered endpoints ensure that the smooth quartic prior is absolutely continuous on the interval.","s":"∀ {ell u c a : ℝ}, 0 < a → ell ≤ u → AbsolutelyContinuousOnInterval (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) ell u","l":331,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_ambient_endpoints","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie strictly inside an ambient interval, then the prior vanishes at both ambient endpoints.","s":"∀ {ell u c a : ℝ}, 0 < a → ell < c - a → c + a < u → Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a ell = 0 ∧ Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a u = 0","l":206,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_contDiff","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth makes the smooth quartic prior continuously differentiable.","s":"∀ {c a : ℝ}, 0 < a → ContDiff ℝ 1 (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a)","l":311,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_finite_vanTrees_lower_bound","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Main","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"A positive bandwidth, strict containment of the left and right prior-support endpoints, finite-model regularity, a nonnegative sensitivity bound that lies below average sensitivity, a pointwise likelihood-information bound, and positive total information imply a native-real Bayes squared-risk lower bound equal to squared sensitivity divided by likelihood information plus forty over the squared bandwidth.","s":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSingletonClass X] {ell u c a s I : ℝ} {p dp g dg : ℝ → X → ℝ} {T : X → ℝ}, 0 < a → ell < c - a → c + a < u → Causalean.Stat.Limit.ObservationDependentVanTrees.FiniteVanTreesModelRegularity X ell u (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) p dp g dg T → 0 ≤ s → s ≤ ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.sensitivityField (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) p dg z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod MeasureTheory.Measure.count → (∀ (θ : ℝ), 0 < Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ → Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation MeasureTheory.Measure.count p dp θ ≤ I) → 0 < Causalean.Stat.Limit.ObservationDependentVanTrees.priorInformation ell u (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) + ∫ (θ : ℝ), Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ * Causalean.Stat.Limit.ObservationDependentVanTrees.fisherInformation MeasureTheory.Measure.count p dp θ ∂Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u → s ^ 2 / (I + 40 / a ^ 2) ≤ ∫ (z : ℝ × X), Causalean.Stat.Limit.ObservationDependentVanTrees.errorSqField (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) p g T z ∂(Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u).prod MeasureTheory.Measure.count","l":383,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Main.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_integrable_parameterMeasure","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth makes the smooth quartic prior integrable on every restricted parameter interval.","s":"∀ {ell u c a : ℝ}, 0 < a → MeasureTheory.Integrable (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u)","l":421,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_nonneg","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth makes the smooth quartic prior nonnegative at every parameter value.","s":"∀ {c a : ℝ}, 0 < a → ∀ (θ : ℝ), 0 ≤ Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ","l":138,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_pos_iff","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Under a positive bandwidth, the smooth quartic prior is positive exactly inside the open interval defined by its center and bandwidth.","s":"∀ {c a θ : ℝ}, 0 < a → (0 < Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ ↔ |θ - c| < a)","l":145,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_scoreSq_aestronglyMeasurable","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth makes the prior-weighted squared guarded score strongly measurable on every restricted parameter interval.","s":"∀ {ell u c a : ℝ}, 0 < a → MeasureTheory.AEStronglyMeasurable (fun θ => Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u)","l":490,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior_scoreSq_integrable","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"A positive bandwidth makes the prior-weighted squared guarded score integrable on every restricted parameter interval.","s":"∀ {ell u c a : ℝ}, 0 < a → MeasureTheory.Integrable (fun θ => Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a θ * Causalean.Stat.Limit.ObservationDependentVanTrees.priorScore (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPriorDeriv c a) θ ^ 2) (Causalean.Stat.Limit.ObservationDependentVanTrees.parameterMeasure ell u)","l":499,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.support_smoothPrior","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Under a positive bandwidth, the nonzero set of the smooth quartic prior is exactly its open support interval.","s":"∀ {c a : ℝ}, 0 < a → Function.support (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) = Set.Ioo (c - a) (c + a)","l":161,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.support_smoothPrior_subset_Icc","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie strictly inside an ambient interval, then every point where the prior is nonzero lies in the ambient closed interval.","s":"∀ {ell u c a : ℝ}, 0 < a → ell < c - a → c + a < u → Function.support (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) ⊆ Set.Icc ell u","l":186,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.tsupport_smoothPrior","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"Under a positive bandwidth, the topological support of the smooth quartic prior is exactly the corresponding closed interval.","s":"∀ {c a : ℝ}, 0 < a → tsupport (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) = Set.Icc (c - a) (c + a)","l":179,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.tsupport_smoothPrior_subset_Ioo","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.Basic","a":"Stat/Limit/ObservationDependentVanTrees","t":2,"d":"If the bandwidth is positive and the prior's left and right support endpoints lie strictly inside an ambient interval, then the prior's topological support lies in the ambient open interval.","s":"∀ {ell u c a : ℝ}, 0 < a → ell < c - a → c + a < u → tsupport (Causalean.Stat.Limit.ObservationDependentVanTrees.smoothPrior c a) ⊆ Set.Ioo ell u","l":196,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/Basic.lean"},{"n":"Causalean.Stat.Limit.ObservationDependentVanTrees.weighted_integral_mul_sq_le","k":"theorem","m":"Causalean.Stat.Limit.ObservationDependentVanTrees.WeightedL2","a":"Stat/Limit/ObservationDependentVanTrees","t":1,"d":"Under a nonnegative weight and measurable, integrable weighted square and cross-product fields, the squared weighted pairing is bounded by the product of the two weighted second moments.","s":"∀ {A : Type u_1} [inst : MeasurableSpace A] {μ : MeasureTheory.Measure A} {q f s : A → ℝ}, (∀ᵐ (a : A) ∂μ, 0 ≤ q a) → MeasureTheory.AEStronglyMeasurable (fun a => f a ^ 2 * q a) μ → MeasureTheory.AEStronglyMeasurable (fun a => s a ^ 2 * q a) μ → MeasureTheory.AEStronglyMeasurable (fun a => f a * s a * q a) μ → MeasureTheory.Integrable (fun a => f a ^ 2 * q a) μ → MeasureTheory.Integrable (fun a => s a ^ 2 * q a) μ → MeasureTheory.Integrable (fun a => f a * s a * q a) μ → (∫ (a : A), f a * s a * q a ∂μ) ^ 2 ≤ (∫ (a : A), f a ^ 2 * q a ∂μ) * ∫ (a : A), s a ^ 2 * q a ∂μ","l":32,"f":"Causalean/Stat/Limit/ObservationDependentVanTrees/WeightedL2.lean"},{"n":"Causalean.Stat.Limit.van_trees_inequality","k":"theorem","m":"Causalean.Stat.Limit.VanTreesInequality","a":"Stat/Limit","t":1,"d":"van Trees inequality (Bayesian Cramér–Rao bound), single parameter. Let a scalar parameter h range over an interval [a, b], drawn from a prior density q that is continuously differentiable on [a, b], nonnegative, positive on the open interior, and vanishing at both endpoints. Given h, data are drawn from a law P h whose score function S h has conditional mean zero and second moment equal to the Fisher information I h; let δ be an estimator of a target ψ that is differentiable with derivative dψ, whose posterior mean is differentiable with derivative equal to its covariance with the score. Assume also that δ, S, and their pointwise products are integrable, that the three quadratic building blocks (δ-ψ)², (δ-ψ)(S+dq/q), (S+dq/q)² are integrable under P h, that the posterior-mean derivative, dψ, dq, and their combination are interval-integrable, that ψ·dq, dq²/q, and I·q are interval-integrable, and that the three h-indexed second-moment integrands are interval-integrable — the standard integrability side conditions for the Bochner/interval-integral manipulations. Assume finally that the total information — prior information plus average experimental information — is positive. Then the Bayes mean-squared error ∫ E_h[(δ − ψ h)²] q(h) dh is at least (∫ ψ'(h) q(h) dh)² / (I_q + ∫ I(h) q(h) dh), where I_q = ∫ q'(h)²/q(h) dh.","s":"∀ {Z : Type u_1} [inst : MeasurableSpace Z] {a b : ℝ}, a ≤ b → ∀ (P : ℝ → MeasureTheory.Measure Z) [∀ (h : ℝ), MeasureTheory.IsProbabilityMeasure (P h)] (δ : Z → ℝ) (ψ dψ q dq I : ℝ → ℝ) (S : ℝ → Z → ℝ), (∀ h ∈ Set.Icc a b, HasDerivAt q (dq h) h) → (∀ h ∈ Set.Icc a b, 0 ≤ q h) → (∀ h ∈ Set.Ioo a b, 0 < q h) → q a = 0 → q b = 0 → (∀ h ∈ Set.Icc a b, HasDerivAt ψ (dψ h) h) → (∀ h ∈ Set.Icc a b, ∫ (z : Z), S h z ∂P h = 0) → (∀ h ∈ Set.Icc a b, ∫ (z : Z), S h z ^ 2 ∂P h = I h) → (∀ h ∈ Set.Icc a b, HasDerivAt (fun h' => ∫ (z : Z), δ z ∂P h') (∫ (z : Z), δ z * S h z ∂P h) h) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable δ (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (S h) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => δ z * S h z) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => S h z * S h z) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (δ z - ψ h) * (δ z - ψ h)) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (δ z - ψ h) * (S h z + dq h / q h)) (P h)) → (∀ h ∈ Set.Icc a b, MeasureTheory.Integrable (fun z => (S h z + dq h / q h) * (S h z + dq h / q h)) (P h)) → IntervalIntegrable (fun h => ∫ (z : Z), δ z * S h z ∂P h) MeasureTheory.volume a b → IntervalIntegrable dψ MeasureTheory.volume a b → IntervalIntegrable dq MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), δ z * S h z ∂P h) * q h + (∫ (z : Z), δ z ∂P h) * dq h) MeasureTheory.volume a b → IntervalIntegrable (fun h => ψ h * dq h) MeasureTheory.volume a b → IntervalIntegrable (fun h => dq h ^ 2 / q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => I h * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (δ z - ψ h) * (δ z - ψ h) ∂P h) * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (δ z - ψ h) * (S h z + dq h / q h) ∂P h) * q h) MeasureTheory.volume a b → IntervalIntegrable (fun h => (∫ (z : Z), (S h z + dq h / q h) * (S h z + dq h / q h) ∂P h) * q h) MeasureTheory.volume a b → 0 < (∫ (h : ℝ) in a..b, dq h ^ 2 / q h) + ∫ (h : ℝ) in a..b, I h * q h → (∫ (h : ℝ) in a..b, dψ h * q h) ^ 2 / ((∫ (h : ℝ) in a..b, dq h ^ 2 / q h) + ∫ (h : ℝ) in a..b, I h * q h) ≤ ∫ (h : ℝ) in a..b, (∫ (z : Z), (δ z - ψ h) ^ 2 ∂P h) * q h","l":48,"f":"Causalean/Stat/Limit/VanTreesInequality.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray","k":"structure","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"A martingale-difference triangular array consists of finite row lengths, real increments, and one filtration per row, such that each active increment is measurable at the next filtration time, is square integrable, and has conditional mean zero given the preceding filtration time; these data together form the array.","s":"(Ω : ℕ → Type u_2) → [mΩ : (n : ℕ) → MeasurableSpace (Ω n)] → ((n : ℕ) → MeasureTheory.Measure (Ω n)) → Type u_2","l":48,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.compensatedWeight","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":1,"d":"The compensated characteristic-function weight at time r is the ordinary characteristic-function weight multiplied by the Gaussian correction generated by the partial predictable quadratic variation.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → ℕ → ℝ → Ω n → ℂ","l":74,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.conditionalFourthMoment","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"The conditional fourth-moment sum for a row adds the conditional fourth moments of all increments, each conditioned on the preceding filtration time.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → Ω n → ℝ","l":100,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.conditionalLindeberg","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"The conditional Lindeberg sum for a row adds its conditional truncated second moments over all increments in that row.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℝ → (n : ℕ) → Ω n → ℝ","l":94,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.conditionalLindebergThrough","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":1,"d":"The conditional Lindeberg mass through index k is the sum of the conditional truncated second moments of the active row increments with indices at most k.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℝ → (n : ℕ) → ℕ → Ω n → ℝ","l":36,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.conditionalSecondMoment","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":1,"d":"The conditional second moment of one increment, given the filtration just before that increment, is its one-step predictable variance.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → ℕ → Ω n → ℝ","l":26,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.conditionalSecondMoment_le_sq_add_lindebergTerm","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.PredictableVarianceBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active increment and a positive threshold, its conditional second moment is at most the squared threshold plus its conditional Lindeberg term.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → ∀ (η : ℝ), 0 < η → (μ n)[fun ω => A.increment n k ω ^ 2 | ↑(A.filtration n) k] ≤ᵐ[μ n] fun ω => η ^ 2 + A.lindebergTerm η n k ω","l":26,"f":"Causalean/Stat/CLT/MartingaleArray/PredictableVarianceBounds.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.fourthMomentSum","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"The unconditional fourth-moment sum is the deterministic sum of the fourth moments of the increments in one row.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℕ → ℝ","l":107,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.integral_cexp_partialRowSum_succ","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTelescoping","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active increment, the integral characteristic function after that increment equals the preceding exponential weight times the exact conditional quadratic expansion.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → ∀ (t : ℝ), ∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.partialRowSum n (k + 1) ω)) ∂μ n = ∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.partialRowSum n k ω)) * (1 - ↑(t ^ 2 / 2) * ↑((μ n)[fun ω => A.increment n k ω ^ 2 | ↑(A.filtration n) k] ω) + (μ n)[fun ω => Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω) | ↑(A.filtration n) k] ω) ∂μ n","l":47,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.integral_compensatedWeight_succ","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active martingale increment, when the compensated predictable variance after that step is almost surely bounded, the integral of the next compensated weight equals the integral of the current weight times the exact conditionally centered quadratic update.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → ∀ (t K : ℝ), (A.partialPredictableQuadraticVariation n (k + 1) ≤ᵐ[μ n] fun x => K) → ∫ (ω : Ω n), A.compensatedWeight n (k + 1) t ω ∂μ n = ∫ (ω : Ω n), A.compensatedWeight n k t ω * (↑(Real.exp (t ^ 2 / 2 * A.conditionalSecondMoment n k ω)) * (1 - ↑(t ^ 2 / 2 * A.conditionalSecondMoment n k ω) + (μ n)[fun ω => Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω) | ↑(A.filtration n) k] ω)) ∂μ n","l":84,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.lindebergTerm","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"A conditional Lindeberg term is the conditional second moment of one increment after discarding values whose absolute size is at most the chosen threshold.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℝ → (n : ℕ) → ℕ → Ω n → ℝ","l":87,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.measure_stoppedArray_rowSum_ne_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a positive truncation threshold, the probability that stopping changes a row sum is bounded by the probability that the original row exceeds either stopping budget.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ), 0 < ε → ∀ (n : ℕ), (μ n) {ω | (A.stoppedArray ε K δ).rowSum n ω ≠ A.rowSum n ω} ≤ (μ n) {ω | K < A.predictableQuadraticVariation n ω ∨ δ < A.conditionalLindeberg ε n ω}","l":380,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.norm_integral_cexp_partialRowSum_succ_sub_quadratic_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTelescoping","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active increment, the error in replacing its conditional characteristic-function factor by the quadratic factor is bounded by the expected norm of its Taylor remainder.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → ∀ (t : ℝ), ‖∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.partialRowSum n (k + 1) ω)) ∂μ n - ∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.partialRowSum n k ω)) * (1 - ↑(t ^ 2 / 2) * ↑((μ n)[fun ω => A.increment n k ω ^ 2 | ↑(A.filtration n) k] ω)) ∂μ n‖ ≤ ∫ (ω : Ω n), ‖Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω)‖ ∂μ n","l":228,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.norm_integral_cexp_rowSum_sub_gaussian_le_of_budgets","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedTelescoping","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the truncation threshold is positive, the frequency-threshold product is at most one, both budgets are nonnegative, a row's predictable quadratic variation is at most K, and its conditional Lindeberg mass at threshold η is at most δ, then its ordinary characteristic function differs from the standard Gaussian characteristic function by the compensated budget plus a constant times the expected distance of predictable quadratic variation from one.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ) (t η K δ : ℝ), 0 < η → |t| * η ≤ 1 → 0 ≤ K → 0 ≤ δ → (A.predictableQuadraticVariation n ≤ᵐ[μ n] fun x => K) → (A.conditionalLindeberg η n ≤ᵐ[μ n] fun x => δ) → ‖∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.rowSum n ω)) ∂μ n - ↑(Real.exp (-(t ^ 2 / 2)))‖ ≤ Real.exp (t ^ 2 / 2 * K) * (|t| ^ 3 * η * K + (2 / η ^ 2 + |t| / η + t ^ 2 / 2) * δ + (t ^ 2 / 2) ^ 2 / 2 * ((η ^ 2 + δ) * K)) + Real.exp (t ^ 2 / 2 * max K 1) * (t ^ 2 / 2) * ∫ (ω : Ω n), |A.predictableQuadraticVariation n ω - 1| ∂μ n","l":183,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.norm_integral_cexp_rowSum_sub_quadraticTelescoping_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTelescoping","a":"Stat/CLT/MartingaleArray","t":2,"d":"For one finite martingale-difference row, the error in its full characteristic function after extracting all conditional quadratic terms is bounded by the sum of the expected one-increment Taylor remainders.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ) (t : ℝ), ‖∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.rowSum n ω)) ∂μ n - 1 + ↑(t ^ 2 / 2) * ∑ k ∈ Finset.range (A.rowLength n), ∫ (ω : Ω n), Complex.exp (Complex.I * ↑(t * A.partialRowSum n k ω)) * ↑((μ n)[fun ω => A.increment n k ω ^ 2 | ↑(A.filtration n) k] ω) ∂μ n‖ ≤ ∑ k ∈ Finset.range (A.rowLength n), ∫ (ω : Ω n), ‖Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω)‖ ∂μ n","l":332,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.norm_integral_compensatedWeight_rowSum_sub_gaussian_le_of_budgets","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedTelescoping","a":"Stat/CLT/MartingaleArray","t":1,"d":"If the truncation threshold is positive, the frequency-threshold product is at most one, both budgets are nonnegative, a row's predictable quadratic variation is at most K, and its conditional Lindeberg mass at threshold η is at most δ, then the integrated compensated row characteristic function differs from the standard Gaussian characteristic function by the displayed Taylor and predictable-mesh budget.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ) (t η K δ : ℝ), 0 < η → |t| * η ≤ 1 → 0 ≤ K → 0 ≤ δ → (A.predictableQuadraticVariation n ≤ᵐ[μ n] fun x => K) → (A.conditionalLindeberg η n ≤ᵐ[μ n] fun x => δ) → ‖∫ (ω : Ω n), A.compensatedWeight n (A.rowLength n) t ω ∂μ n - ↑(Real.exp (-(t ^ 2 / 2)))‖ ≤ Real.exp (t ^ 2 / 2 * K) * (|t| ^ 3 * η * K + (2 / η ^ 2 + |t| / η + t ^ 2 / 2) * δ + (t ^ 2 / 2) ^ 2 / 2 * ((η ^ 2 + δ) * K))","l":28,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.norm_integral_compensatedWeight_succ_sub_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":1,"d":"For an active martingale increment, if the variance budget is nonnegative and the compensated predictable variance after that step is at most K, then the change in the integrated compensated weight is bounded by the integrated Taylor remainder and the squared one-step predictable variance.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → ∀ (t K : ℝ), 0 ≤ K → (A.partialPredictableQuadraticVariation n (k + 1) ≤ᵐ[μ n] fun x => K) → ‖∫ (ω : Ω n), A.compensatedWeight n (k + 1) t ω ∂μ n - ∫ (ω : Ω n), A.compensatedWeight n k t ω ∂μ n‖ ≤ Real.exp (t ^ 2 / 2 * K) * (∫ (ω : Ω n), ‖Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω)‖ ∂μ n + (t ^ 2 / 2) ^ 2 / 2 * ∫ (ω : Ω n), A.conditionalSecondMoment n k ω ^ 2 ∂μ n)","l":214,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialPredictableQuadraticVariation","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":1,"d":"The partial predictable quadratic variation through time r sums the one-step predictable variances of the first r active increments.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → ℕ → Ω n → ℝ","l":32,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialPredictableQuadraticVariation_stronglyMeasurable","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":2,"d":"The partial predictable quadratic variation through time r is measurable with respect to the row filtration at time r.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n r : ℕ), MeasureTheory.StronglyMeasurable (A.partialPredictableQuadraticVariation n r)","l":59,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialPredictableQuadraticVariation_succ","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active increment, the next partial predictable quadratic variation is the current one plus that increment's conditional second moment.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n k : ℕ), k < A.rowLength n → A.partialPredictableQuadraticVariation n (k + 1) = A.partialPredictableQuadraticVariation n k + A.conditionalSecondMoment n k","l":44,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialPredictableQuadraticVariation_zero","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CompensatedStep","a":"Stat/CLT/MartingaleArray","t":2,"d":"The partial predictable quadratic variation at time zero is zero.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ), A.partialPredictableQuadraticVariation n 0 = 0","l":38,"f":"Causalean/Stat/CLT/MartingaleArray/CompensatedStep.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialRowSum","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTelescoping","a":"Stat/CLT/MartingaleArray","t":1,"d":"The partial row sum through time r adds the first r increments, capped at the row length.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → ℕ → Ω n → ℝ","l":28,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.partialRowSum_stronglyMeasurable","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTelescoping","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a martingale-difference triangular array, a row, and a time index, the partial row sum is measurable with respect to that row's filtration at that time.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n r : ℕ), MeasureTheory.StronglyMeasurable (A.partialRowSum n r)","l":34,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTelescoping.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.predictableQuadraticVariation","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"The predictable quadratic variation of a row is the sum of the conditional second moments of its increments, each conditioned on the preceding filtration time.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → Ω n → ℝ","l":80,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.predictableQuadraticVariationThrough","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":1,"d":"The predictable quadratic variation through index k is the sum of the conditional second moments of the active row increments with indices at most k.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → ℕ → Ω n → ℝ","l":28,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.rowSum","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"The row sum is the sum of the increments whose indices are below that row's length.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → (n : ℕ) → Ω n → ℝ","l":67,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.rowSum_aemeasurable","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a martingale-difference triangular array and one row, its row sum is almost-everywhere measurable under that row's probability measure.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ), AEMeasurable (A.rowSum n) (μ n)","l":71,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stopMultiplier","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":1,"d":"The predictable stopping multiplier for increment k is one exactly when the original row's predictable variance and conditional Lindeberg mass through k remain within the supplied budgets, and is zero otherwise.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℝ → ℝ → ℝ → (n : ℕ) → ℕ → Ω n → ℝ","l":43,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stopMultiplier_stronglyMeasurable","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"For an active row increment, its stopping multiplier is measurable at the preceding filtration time.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ) (n k : ℕ), k < A.rowLength n → MeasureTheory.StronglyMeasurable (A.stopMultiplier ε K δ n k)","l":51,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":1,"d":"Given a martingale-difference triangular array, a truncation threshold, a predictable-variance budget, and a conditional-Lindeberg budget, the predictably stopped array is another square-integrable martingale-difference array with the same row lengths and filtrations.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} → [∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] → Causalean.Stat.MartingaleDifferenceArray Ω μ → ℝ → ℝ → ℝ → Causalean.Stat.MartingaleDifferenceArray Ω μ","l":82,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray_conditionalLindeberg_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"With a positive truncation threshold and nonnegative variance and Lindeberg budgets, the stopped row's conditional Lindeberg sum at that threshold never exceeds its budget almost everywhere.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ), 0 < ε → 0 ≤ K → 0 ≤ δ → ∀ (n : ℕ), (A.stoppedArray ε K δ).conditionalLindeberg ε n ≤ᵐ[μ n] fun x => δ","l":284,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray_increment","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"A stopped increment is the original increment multiplied by the predictable zero-one stopping multiplier.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ) (n k : ℕ), (A.stoppedArray ε K δ).increment n k = fun ω => A.stopMultiplier ε K δ n k ω * A.increment n k ω","l":139,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray_predictableQuadraticVariation_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"With nonnegative variance and Lindeberg budgets, the stopped row's predictable quadratic variation never exceeds the variance budget almost everywhere.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ), 0 ≤ K → 0 ≤ δ → ∀ (n : ℕ), (A.stoppedArray ε K δ).predictableQuadraticVariation n ≤ᵐ[μ n] fun x => K","l":195,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray_rowLength","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"Stopping does not change the row length.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ) (n : ℕ), (A.stoppedArray ε K δ).rowLength n = A.rowLength n","l":134,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.stoppedArray_rowSum_ae_eq_of_bounds","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedArray","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the truncation threshold is positive and the original final predictable variance and conditional Lindeberg mass stay within their budgets, then predictable stopping leaves the row sum unchanged almost everywhere.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (ε K δ : ℝ), 0 < ε → ∀ (n : ℕ), ∀ᵐ (ω : Ω n) ∂μ n, A.predictableQuadraticVariation n ω ≤ K → A.conditionalLindeberg ε n ω ≤ δ → (A.stoppedArray ε K δ).rowSum n ω = A.rowSum n ω","l":146,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedArray.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.sum_integral_norm_expQuadraticRemainder_le_of_budgets","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.RemainderBudget","a":"Stat/CLT/MartingaleArray","t":1,"d":"If the truncation threshold is positive, the frequency-threshold product is at most one, both budgets are nonnegative, a row has predictable variance at most K, and conditional Lindeberg mass at most δ, then the sum of its integrated quadratic exponential remainders obeys the corresponding truncated Taylor bound.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ) (t η K δ : ℝ), 0 < η → |t| * η ≤ 1 → 0 ≤ K → 0 ≤ δ → (A.predictableQuadraticVariation n ≤ᵐ[μ n] fun x => K) → (A.conditionalLindeberg η n ≤ᵐ[μ n] fun x => δ) → ∑ k ∈ Finset.range (A.rowLength n), ∫ (ω : Ω n), ‖Causalean.Stat.expQuadraticRemainder (t * A.increment n k ω)‖ ∂μ n ≤ |t| ^ 3 * η * K + (2 / η ^ 2 + |t| / η + t ^ 2 / 2) * δ","l":26,"f":"Causalean/Stat/CLT/MartingaleArray/RemainderBudget.lean"},{"n":"Causalean.Stat.MartingaleDifferenceArray.sum_sq_conditionalSecondMoment_le_of_budgets","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.PredictableVarianceBounds","a":"Stat/CLT/MartingaleArray","t":1,"d":"If the truncation threshold is positive, the row predictable quadratic variation is bounded and its conditional Lindeberg sum is bounded, with nonnegative budgets, then the sum of squared conditional variance increments is at most (η² + δ)K.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (n : ℕ) (η K δ : ℝ), 0 < η → 0 ≤ K → 0 ≤ δ → (A.predictableQuadraticVariation n ≤ᵐ[μ n] fun x => K) → (A.conditionalLindeberg η n ≤ᵐ[μ n] fun x => δ) → (fun ω => ∑ k ∈ Finset.range (A.rowLength n), (μ n)[fun ω => A.increment n k ω ^ 2 | ↑(A.filtration n) k] ω ^ 2) ≤ᵐ[μ n] fun x => (η ^ 2 + δ) * K","l":87,"f":"Causalean/Stat/CLT/MartingaleArray/PredictableVarianceBounds.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.AmbientExactSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a label alphabet, a side alphabet, and action bounds, an ambient exact-table procedure is the function assigning a bounded action to each label and arbitrary real side-coordinate table.","s":"Type u_3 → Type u_4 → ℝ → ℝ → Type (max u_3 u_4)","l":75,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.BoundedDecision","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For a finite labeled alphabet and action bounds, a bounded decision assigns an action in the closed interval to every label.","s":"Type u_4 → ℝ → ℝ → Type u_4","l":24,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a labeled alphabet, a side alphabet, a sample size, and action bounds, an empirical-side procedure assigns a bounded action after observing a label and ordered side draws.","s":"Type u_4 → Type u_5 → ℕ → ℝ → ℝ → Type (max u_4 u_5)","l":33,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a labeled alphabet, a side alphabet, and action bounds, an exact-side procedure assigns a bounded action after observing a label and its exact finite side-law vector.","s":"Type u_4 → (C : Type u_5) → [Fintype C] → ℝ → ℝ → Type (max u_4 u_5)","l":27,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"A finite alphabet has as its coordinate probability simplex the subtype of real vectors with nonnegative coordinates summing to one.","s":"(C : Type u_1) → [Fintype C] → Type (max 0 u_1)","l":24,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf.nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"The mass assigned by a finite probability vector to every atom is nonnegative.","s":"∀ (C : Type u_1) [inst : Fintype C] (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) (c : C), 0 ≤ ↑w c","l":37,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf.sum_eq_one","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"The atom masses of a finite probability vector sum to one.","s":"∀ (C : Type u_1) [inst : Fintype C] (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), ∑ c, ↑w c = 1","l":42,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.IsContinuousAmbientExactSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given an ambient exact-table procedure, the continuity property requires every fixed-label section to vary continuously with its real coordinate table.","s":"{X : Type u_1} → {C : Type u_2} → {l u : ℝ} → Causalean.Stat.Minimax.FiniteSideInformation.AmbientExactSideProcedure X C l u → Prop","l":81,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.IsMeasurableAmbientExactSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given an ambient exact-table procedure, the measurability property requires every fixed-label section to be Borel measurable in its real coordinate table.","s":"{X : Type u_1} → {C : Type u_2} → {l u : ℝ} → Causalean.Stat.Minimax.FiniteSideInformation.AmbientExactSideProcedure X C l u → Prop","l":87,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.MeasurableAmbientExactSideProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a label alphabet, a finite side alphabet, and action bounds, a measurable ambient exact-table procedure is an ambient bounded rule equipped with Borel measurability of every label section.","s":"Type u_7 → (C : Type u_8) → [Fintype C] → ℝ → ℝ → Type (max 0 u_7 u_8)","l":218,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.closedSimplex_finiteSideInfo_minimax_tendsto","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.SimplexSpecialization","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For a nonempty closed model class inside a finite simplex, finite labeled and finite nonempty side alphabets, continuous simplex-valid label and side coordinates, and a continuous bounded target, the empirical-side minimax values converge to the exact-side benchmark.","s":"∀ {I : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype I] [inst_1 : Fintype X] [inst_2 : Fintype C] [DecidableEq C] [Nonempty C] {l u : ℝ} (K : Set (Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf I)), IsClosed K → K.Nonempty → ∀ (p : ↑K → X → ℝ) (q : ↑K → C → ℝ) (tau : ↑K → ℝ), (∀ (theta : ↑K), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : ↑K), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : ↑K), tau theta ∈ Set.Icc l u) → Filter.Tendsto (fun m => Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m) Filter.atTop (nhds (Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u))","l":37,"f":"Causalean/Stat/Minimax/FiniteSideInformation/SimplexSpecialization.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.conditionalAveragePolynomial","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a side-sample size, an empirical procedure, and a label, the conditional-average polynomial is the finite sum of action-weighted sample monomials.","s":"{X : Type u_1} → {C : Type u_2} → [Fintype C] → {l u : ℝ} → (m : ℕ) → Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u → X → MvPolynomial C ℝ","l":34,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.conditionalAverageProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Comparison","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given action bounds, a sample size, and an empirical-side procedure, the conditional-average exact-side procedure averages the action over an iid sample drawn from the supplied exact side law.","s":"{X : Type u_2} → {C : Type u_3} → [inst : Fintype C] → (l u : ℝ) → (m : ℕ) → Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u → Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure X C l u","l":26,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Comparison.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_atom","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Evaluation at an atom is a continuous coordinate on the finite probability simplex.","s":"∀ (C : Type u_1) [inst : Fintype C] (c : C), Continuous fun w => ↑w c","l":32,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_closed_finitePmf_coordinate","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.SimplexSpecialization","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Restricting an atom coordinate to a closed finite-simplex model class gives a continuous parameter coordinate.","s":"∀ {I : Type u_1} [inst : Fintype I] (K : Set (Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf I)) (i : I), Continuous fun w => ↑↑w i","l":30,"f":"Causalean/Stat/Minimax/FiniteSideInformation/SimplexSpecialization.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_conditionalAverageProcedure","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given action bounds, a side-sample size, an empirical procedure, and a label, the conditional-average procedure varies continuously with the finite side-law vector.","s":"∀ {X : Type u_1} {C : Type u_2} [Fintype X] [inst : Fintype C] (l u : ℝ) (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (x : X), Continuous (Causalean.Stat.Minimax.FiniteSideInformation.conditionalAverageProcedure l u m d x)","l":50,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_coordinateFamily_comp","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a parameter map, its continuity, a real coordinate family, and continuity of every coordinate, the pulled-back coordinate family is continuous.","s":"∀ {A : Type u_7} {B : Type u_8} {I : Type u_9} [inst : TopologicalSpace A] [inst_1 : TopologicalSpace B] (phi : A → B), Continuous phi → ∀ (f : B → I → ℝ), (∀ (i : I), Continuous fun b => f b i) → ∀ (i : I), Continuous fun a => f (phi a) i","l":268,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_finiteSquaredRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"When every label coordinate and the target are continuous, the finite squared risk of a fixed decision is continuous in the parameter.","s":"∀ {Theta : Type u_1} {X : Type u_2} [inst : TopologicalSpace Theta] [inst_1 : Fintype X] {l u : ℝ} (p : Theta → X → ℝ) (tau : Theta → ℝ), (∀ (x : X), Continuous fun theta => p theta x) → Continuous tau → ∀ (d : Causalean.Stat.Minimax.FiniteSideInformation.BoundedDecision X l u), Continuous fun theta => Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau d theta","l":66,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_finiteSquaredRisk_joint","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"When every label coordinate and the target are continuous, the finite squared risk is jointly continuous in the bounded decision and parameter.","s":"∀ {Theta : Type u_1} {X : Type u_2} [inst : TopologicalSpace Theta] [inst_1 : Fintype X] {l u : ℝ} (p : Theta → X → ℝ) (tau : Theta → ℝ), (∀ (x : X), Continuous fun theta => p theta x) → Continuous tau → Continuous fun z => Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau z.1 z.2","l":75,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_productProbability","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"For a fixed finite sample, its product probability varies continuously with the underlying probability vector.","s":"∀ (C : Type u_1) [inst : Fintype C] {m : ℕ} (z : Fin m → C), Continuous fun w => Causalean.Stat.Minimax.FiniteSideInformation.productProbability C w z","l":51,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.continuous_sidePmf","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"When side-probability coordinates are simplex-valid and continuous, the simplex-valued side law is continuous.","s":"∀ {Theta : Type u_1} {C : Type u_3} [inst : Fintype C] [inst_1 : TopologicalSpace Theta] (q : Theta → C → ℝ) (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (c : C), Continuous fun theta => q theta c) → Continuous (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq)","l":45,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.dist_empiricalPmf_le_empiricalL1","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"For a finite nonempty alphabet, a positive sample, and a probability vector, the distance from the empirical simplex point to the vector is at most their coordinatewise L1 error.","s":"∀ (C : Type u_1) [inst : Fintype C] [inst_1 : DecidableEq C] [inst_2 : Nonempty C] {m : ℕ} (hm : 0 < m) (z : Fin m → C) (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), dist (Causalean.Stat.Minimax.FiniteSideInformation.empiricalPmf C hm z) w ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalL1 C hm z w","l":67,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalFrequency","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a positive sample size, an ordered finite sample, and an atom, the empirical atom frequency is its sample count divided by the sample size.","s":"(C : Type u_1) → [DecidableEq C] → {m : ℕ} → 0 < m → (Fin m → C) → C → ℝ","l":27,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalFrequency_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Every empirical atom frequency is nonnegative.","s":"∀ (C : Type u_1) [Fintype C] [inst : DecidableEq C] {m : ℕ} (hm : 0 < m) (z : Fin m → C) (c : C), 0 ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalFrequency C hm z c","l":38,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalL1","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a positive sample size, a sample, and a side law, the empirical L1 error is the sum of the absolute atom-frequency errors.","s":"(C : Type u_1) → [inst : Fintype C] → [DecidableEq C] → {m : ℕ} → 0 < m → (Fin m → C) → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C → ℝ","l":33,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalL1_tail","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For a finite nonempty alphabet, a probability vector, a positive sample size, and a positive tolerance, the total product probability of samples whose empirical L1 error is at least the tolerance is at most the finite-union Hoeffding bound 2 |C| exp (-2 m (ε/|C|)^2).","s":"∀ (C : Type u_1) [inst : Fintype C] [inst_1 : DecidableEq C] [Nonempty C] {m : ℕ} (hm : 0 < m) (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) {ε : ℝ}, 0 < ε → (∑ z, if ε ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalL1 C hm z w then Causalean.Stat.Minimax.FiniteSideInformation.productProbability C w z else 0) ≤ 2 * ↑(Fintype.card C) * Real.exp (-2 * ↑m * (ε / ↑(Fintype.card C)) ^ 2)","l":112,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalL1_tail_bound_tendsto_zero","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"For every positive tolerance, the uniform finite-category empirical L1 tail bound tends to zero as the sample size tends to infinity.","s":"∀ (C : Type u_1) [inst : Fintype C] [DecidableEq C] [Nonempty C] {ε : ℝ}, 0 < ε → Filter.Tendsto (fun m => 2 * ↑(Fintype.card C) * Real.exp (-2 * ↑m * (ε / ↑(Fintype.card C)) ^ 2)) Filter.atTop (nhds 0)","l":221,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalPmf","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"A positive finite sample determines the probability-simplex point whose coordinates are its empirical atom frequencies.","s":"(C : Type u_1) → [inst : Fintype C] → [DecidableEq C] → [Nonempty C] → {m : ℕ} → 0 < m → (Fin m → C) → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C","l":52,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalPmf_apply","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a positive sample size, a sample, and an atom, the corresponding empirical probability-vector coordinate equals the empirical frequency.","s":"∀ (C : Type u_1) [inst : Fintype C] [inst_1 : DecidableEq C] [inst_2 : Nonempty C] {m : ℕ} (hm : 0 < m) (z : Fin m → C) (c : C), ↑(Causalean.Stat.Minimax.FiniteSideInformation.empiricalPmf C hm z) c = Causalean.Stat.Minimax.FiniteSideInformation.empiricalFrequency C hm z c","l":59,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label probabilities, side probabilities, a target, action bounds, and a sample size, the empirical-side minimax value is the infimum over bounded sample procedures of their worst-case squared risk over the parameter space.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → (Theta → X → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → (Theta → ℝ) → ℝ → ℝ → ℕ → ℝ","l":78,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue_comp_surjective","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given label probabilities, side-coordinate probabilities, simplex validity, a target, action bounds, a sample size, and a surjective parameter map, pulling all model coordinates back leaves the empirical-side minimax value unchanged.","s":"∀ {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {A : Type u_7} {B : Type u_8} (p : B → X → ℝ) (q : B → C → ℝ) (hq : ∀ (b : B), q b ∈ stdSimplex ℝ C) (tau : B → ℝ) (l u : ℝ) (m : ℕ) (phi : A → B), Function.Surjective phi → Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue (fun a => p (phi a)) (fun a => q (phi a)) (fun a => hq (phi a)) (fun a => tau (phi a)) l u m = Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m","l":302,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimax_limsup_le_exact","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Approximation","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Under the compact continuous finite-model assumptions, the limsup of empirical-side minimax values is at most the exact-side minimax value. The proof uses a measurable finite-cover selector and the uniform empirical L1 Hoeffding tail.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : TopologicalSpace Theta] [CompactSpace Theta] [Nonempty Theta] [inst_3 : Fintype X] [inst_4 : Fintype C] [DecidableEq C] [Nonempty C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Filter.limsup (fun m => Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m) Filter.atTop ≤ Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u","l":195,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Approximation.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label and side probabilities, a target, an empirical-side procedure, a sample size, and a parameter, the empirical-side squared risk averages squared loss over the label and all ordered iid side samples.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → {l u : ℝ} → (Theta → X → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → (Theta → ℝ) → (m : ℕ) → Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u → Theta → ℝ","l":61,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk_eq_integral","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given label probabilities, side probabilities, simplex validity, a target, sample size, an empirical procedure, and a parameter, the empirical finite-sum risk equals its iterated product-measure integral representation.","s":"∀ {X : Type u_1} {C : Type u_2} [inst : Fintype X] [inst_1 : Fintype C] {Theta : Type u_3} [inst_2 : MeasurableSpace X] [inst_3 : MeasurableSingletonClass X] [inst_4 : MeasurableSpace C] [inst_5 : MeasurableSingletonClass C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (hp : ∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) (tau : Theta → ℝ) (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk p q hq tau m d theta = ∫ (x : X), ∫ (z : Fin m → C), (↑(d x z) - tau theta) ^ 2 ∂MeasureTheory.Measure.pi fun x => Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq theta) ∂Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf p hp theta)","l":198,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Under probability-simplex label and side coordinates, with ordered action bounds containing the target, empirical-side squared risk is at most the squared action width.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype X] [inst_1 : Fintype C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) (tau : Theta → ℝ), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk p q hq tau m d theta ≤ (u - l) ^ 2","l":102,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Under probability-simplex label and side coordinates, empirical-side squared risk is nonnegative.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype X] [inst_1 : Fintype C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) (tau : Theta → ℝ) (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (theta : Theta), 0 ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk p q hq tau m d theta","l":87,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.eval_conditionalAveragePolynomial","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"For a side-sample size, empirical procedure, label, and real coordinate table, evaluating the conditional-average polynomial gives the corresponding product-weighted finite sum.","s":"∀ {X : Type u_1} {C : Type u_2} [Fintype X] [inst : Fintype C] {l u : ℝ} (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (x : X) (w : C → ℝ), (MvPolynomial.eval w) (Causalean.Stat.Minimax.FiniteSideInformation.conditionalAveragePolynomial m d x) = ∑ z, (∏ i, w (z i)) * ↑(d x z)","l":41,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label probabilities, side probabilities, a target, and action bounds, the exact-side minimax value is the infimum over bounded exact-law procedures of their worst-case squared risk over the parameter space.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → (Theta → X → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → (Theta → ℝ) → ℝ → ℝ → ℝ","l":70,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue_comp_surjective","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given label probabilities, side-coordinate probabilities, simplex validity, a target, action bounds, and a surjective parameter map, pulling all model coordinates back leaves the exact-side minimax value unchanged.","s":"∀ {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {A : Type u_7} {B : Type u_8} (p : B → X → ℝ) (q : B → C → ℝ) (hq : ∀ (b : B), q b ∈ stdSimplex ℝ C) (tau : B → ℝ) (l u : ℝ) (phi : A → B), Function.Surjective phi → Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue (fun a => p (phi a)) (fun a => q (phi a)) (fun a => hq (phi a)) (fun a => tau (phi a)) l u = Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u","l":289,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue_eq_iSup_fiber","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Comparison","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"On a compact parameter space, under continuous simplex-valid label and side coordinates and a continuous bounded target, the exact-side minimax value equals the supremum of its fiberwise minimax values.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype X] [inst_1 : Fintype C] {l u : ℝ} [inst_2 : TopologicalSpace Theta] [CompactSpace Theta] [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u = ⨆ theta0, Causalean.Stat.Minimax.FiniteSideInformation.fiberMinimaxValue p tau q hq l u theta0","l":112,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Comparison.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue_le_empiricalSideMinimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Comparison","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Under simplex-valid probabilities, ordered action bounds, and a bounded target, the exact-side minimax value is no larger than every empirical-side minimax value.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype X] [inst_1 : Fintype C] {l u : ℝ} [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ (m : ℕ), Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m","l":82,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Comparison.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue_le_measurableExactTableMinimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given simplex-valid label and side probabilities, a target, ordered action bounds, and target containment in those bounds, the unrestricted exact-side minimax value is at most the measurable ambient exact-table value.","s":"∀ {Theta : Type u_4} {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {l u : ℝ} [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u","l":316,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label and side probabilities, a target, an exact-side procedure, and a parameter, the exact-side squared risk averages only over the labeled outcome because the side-law vector is observed without noise.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → {l u : ℝ} → (Theta → X → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure X C l u → Theta → ℝ","l":53,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideRisk_conditionalAverage_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Comparison","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Under simplex-valid label and side coordinates and ordered action bounds, conditional averaging of an empirical procedure cannot increase squared risk at any parameter.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : Fintype X] [inst_1 : Fintype C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → ∀ (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSideInformation.exactSideRisk p q hq tau (Causalean.Stat.Minimax.FiniteSideInformation.conditionalAverageProcedure l u m d) theta ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideRisk p q hq tau m d theta","l":47,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Comparison.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exactSideRisk_eq_integral","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given label probabilities, side probabilities, simplex validity, a target, an exact-side procedure, and a parameter, the exact-side finite-sum risk equals its finite-measure integral representation.","s":"∀ {X : Type u_1} {C : Type u_2} [inst : Fintype X] [inst_1 : Fintype C] {Theta : Type u_3} [inst_2 : MeasurableSpace X] [inst_3 : MeasurableSingletonClass X] [inst_4 : MeasurableSpace C] [MeasurableSingletonClass C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (hp : ∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) (tau : Theta → ℝ) (d : Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure X C l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSideInformation.exactSideRisk p q hq tau d theta = ∫ (x : X), (↑(d x (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq theta)) - tau theta) ^ 2 ∂Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf p hp theta)","l":185,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exists_continuous_ambientExactSideProcedure","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given ordered action bounds, an exact-side procedure, and continuous label sections, an ambient procedure with continuous sections that agrees on simplex tables exists.","s":"∀ {X : Type u_1} {C : Type u_2} [Fintype X] [inst : Fintype C] [Nonempty C] {l u : ℝ}, l ≤ u → ∀ (d : Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure X C l u), (∀ (x : X), Continuous (d x)) → ∃ g, Causalean.Stat.Minimax.FiniteSideInformation.IsContinuousAmbientExactSideProcedure g ∧ ∀ (x : X) (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), g x ↑w = d x w","l":115,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exists_continuous_clippedExtension","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given ordered action bounds, an interval-valued simplex rule, and its continuity, a continuous interval-valued ambient extension agreeing on the simplex exists.","s":"∀ {C : Type u_2} [inst : Fintype C] [Nonempty C] {l u : ℝ}, l ≤ u → ∀ (f : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C → ↑(Set.Icc l u)), Continuous f → ∃ g, Continuous g ∧ ∀ (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), g ↑w = f w","l":93,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exists_measurable_clippedAmbientExactSideProcedure","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given ordered action bounds, an exact-side procedure, and continuous label sections, a Borel-measurable ambient procedure that agrees on simplex tables exists.","s":"∀ {X : Type u_1} {C : Type u_2} [Fintype X] [inst : Fintype C] [Nonempty C] {l u : ℝ}, l ≤ u → ∀ (d : Causalean.Stat.Minimax.FiniteSideInformation.ExactSideProcedure X C l u), (∀ (x : X), Continuous (d x)) → ∃ g, Causalean.Stat.Minimax.FiniteSideInformation.IsMeasurableAmbientExactSideProcedure g ∧ ∀ (x : X) (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), g x ↑w = d x w","l":127,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.exists_measurable_finiteCoverSelector","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Selector","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a number of centers with positive count, centers, radii that are positive, there is a measurable selector choosing a center within its radius at every point covered by the corresponding open balls.","s":"∀ {E : Type u_1} [inst : PseudoMetricSpace E] [inst_1 : MeasurableSpace E] [BorelSpace E] (k : ℕ), 0 < k → ∀ (center : Fin k → E) (radius : Fin k → ℝ), (∀ (i : Fin k), 0 < radius i) → ∃ select, Measurable select ∧ ∀ (y : E), (∃ i, dist y (center i) < radius i) → dist y (center (select y)) < radius (select y)","l":20,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Selector.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.fiberMinimaxValue","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Fiber","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For label probabilities, a target, side coordinates, action bounds, and a center parameter, the exact-fiber local minimax value minimizes finite squared risk over decisions and maximizes over parameters with the same side law as the center.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → (Theta → X → ℝ) → (Theta → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → ℝ → ℝ → Theta → ℝ","l":27,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Fiber.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finiteCover_approximateDecision","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Approximation","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"On a compact parameter space, under continuous simplex-valid label and side coordinates and a continuous bounded target, every positive tolerance admits a finite family of local decisions: smaller side-law balls cover the model with a uniform positive margin, and each decision has risk within the tolerance of the exact-side minimax value throughout its larger ball.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : TopologicalSpace Theta] [CompactSpace Theta] [Nonempty Theta] [inst_3 : Fintype X] [inst_4 : Fintype C] [DecidableEq C] [Nonempty C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ {ε : ℝ}, 0 < ε → ∃ k, ∃ (_ : 0 < k), ∃ center decision radius η, 0 < η ∧ (∀ (i : Fin k), η < radius i) ∧ (∀ (theta : Theta), ∃ i, dist (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq theta) (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq (center i)) < radius i - η) ∧ ∀ (i : Fin k) (theta : Theta), dist (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq theta) (Causalean.Stat.Minimax.FiniteSideInformation.sidePmf q hq (center i)) < radius i → Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau (decision i) theta ≤ Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u + ε","l":26,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Approximation.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finitePmfDesign","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a finite probability vector, the corresponding finite randomization design is the design with exactly those atom probabilities.","s":"{C : Type u_2} → [inst : Fintype C] → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C → Causalean.Experimentation.DesignBased.FiniteDesign C","l":140,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a finite probability vector, the finite-PMF measure is the atomic probability measure with those atom masses.","s":"{C : Type u_2} → [inst : Fintype C] → [inst_1 : MeasurableSpace C] → [MeasurableSingletonClass C] → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C → MeasureTheory.Measure C","l":147,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finiteSideInfo_minimax_tendsto","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Main","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For a compact parameter space, finite labeled and finite nonempty side alphabets, continuous simplex-valid label and side probabilities, and a continuous target contained in an ordered bounded action interval, the bounded squared-loss empirical-side minimax values converge to the exact-side minimax benchmark as the iid side sample size tends to infinity.","s":"∀ {l u : ℝ} {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : TopologicalSpace Theta] [CompactSpace Theta] [Nonempty Theta] [inst_3 : Fintype X] [inst_4 : Fintype C] [DecidableEq C] [Nonempty C] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Filter.Tendsto (fun m => Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m) Filter.atTop (nhds (Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u))","l":21,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Main.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label probabilities, a target, a bounded decision, and a parameter, the finite squared risk is the probability-weighted sum of squared errors over the labeled alphabet.","s":"{Theta : Type u_1} → {X : Type u_2} → [Fintype X] → {l u : ℝ} → (Theta → X → ℝ) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSideInformation.BoundedDecision X l u → Theta → ℝ","l":28,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk_bounds","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"If label probabilities are nonnegative and sum to one, and actions and targets lie in the same ordered interval, then the finite squared risk is bounded between zero and the squared interval width.","s":"∀ {Theta : Type u_1} {X : Type u_2} [TopologicalSpace Theta] [inst : Fintype X] {l u : ℝ} (p : Theta → X → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (x : X), 0 ≤ p theta x) → (∀ (theta : Theta), ∑ x, p theta x = 1) → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ (d : Causalean.Stat.Minimax.FiniteSideInformation.BoundedDecision X l u) (theta : Theta), 0 ≤ Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau d theta ∧ Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau d theta ≤ (u - l) ^ 2","l":42,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"If the label coordinates are nonnegative, every finite squared risk is nonnegative.","s":"∀ {Theta : Type u_1} {X : Type u_2} [TopologicalSpace Theta] [inst : Fintype X] {l u : ℝ} (p : Theta → X → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (x : X), 0 ≤ p theta x) → ∀ (d : Causalean.Stat.Minimax.FiniteSideInformation.BoundedDecision X l u) (theta : Theta), 0 ≤ Causalean.Stat.Minimax.FiniteSideInformation.finiteSquaredRisk p tau d theta","l":35,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.integral_finitePmfMeasure_eq_sum","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a finite probability vector and a real-valued function, integration against its atomic measure equals the probability-weighted finite sum.","s":"∀ {C : Type u_2} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : MeasurableSingletonClass C] (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) (f : C → ℝ), ∫ (c : C), f c ∂Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure w = ∑ c, ↑w c * f c","l":153,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.integral_pi_finitePmfMeasure_eq_sum_productProbability","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a finite probability vector, sample size, and real-valued sample function, integration against the iid product measure equals the exact product-probability sum.","s":"∀ {C : Type u_2} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : MeasurableSingletonClass C] (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) (m : ℕ) (f : (Fin m → C) → ℝ), (∫ (z : Fin m → C), f z ∂MeasureTheory.Measure.pi fun x => Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure w) = ∑ z, Causalean.Stat.Minimax.FiniteSideInformation.productProbability C w z * f z","l":170,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.isCompact_boundedDecision","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Risk","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"A finite labeled alphabet and ordered action bounds make the space of bounded decisions compact.","s":"∀ {X : Type u_2} [Fintype X] {l u : ℝ}, l ≤ u → IsCompact Set.univ","l":84,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Risk.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.isCompact_closed_finitePmf","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.SimplexSpecialization","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"A closed subset of a finite probability simplex is compact.","s":"∀ {I : Type u_1} [inst : Fintype I] {K : Set (Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf I)}, IsClosed K → IsCompact K","l":24,"f":"Causalean/Stat/Minimax/FiniteSideInformation/SimplexSpecialization.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.isCompact_finitePmf","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"The finite probability simplex is compact in its coordinate topology.","s":"∀ (C : Type u_1) [inst : Fintype C], IsCompact Set.univ","l":28,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.localMinimaxValue","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Fiber","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"For label probabilities, a target, side coordinates, action bounds, a radius, and a center parameter, the local minimax value minimizes finite squared risk over decisions and maximizes over parameters whose side law is within that radius of the center law.","s":"{Theta : Type u_1} → {X : Type u_2} → {C : Type u_3} → [Fintype X] → [inst : Fintype C] → (Theta → X → ℝ) → (Theta → ℝ) → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → ℝ → ℝ → ℝ → Theta → ℝ","l":38,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Fiber.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.localMinimaxValue_tendsto_fiber","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Fiber","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"On a compact parameter space, if label and side coordinates and the target are continuous, probabilities are simplex-valid, and the action interval contains the target, then local minimax values over radii 1/(n+1) converge to the exact-fiber minimax value.","s":"∀ {Theta : Type u_1} {X : Type u_2} {C : Type u_3} [inst : TopologicalSpace Theta] [CompactSpace Theta] [Nonempty Theta] [inst_3 : Fintype X] [inst_4 : Fintype C] {l u : ℝ} (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), (∀ (x : X), Continuous fun theta => p theta x) → (∀ (c : C), Continuous fun theta => q theta c) → Continuous tau → l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ (theta0 : Theta), Filter.Tendsto (fun n => Causalean.Stat.Minimax.FiniteSideInformation.localMinimaxValue p tau q hq l u (1 / (↑n + 1)) theta0) Filter.atTop (nhds (Causalean.Stat.Minimax.FiniteSideInformation.fiberMinimaxValue p tau q hq l u theta0))","l":49,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Fiber.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label probabilities, side-coordinate probabilities, a target, and action bounds, the measurable exact-table minimax value is the infimum of worst-case squared risks over globally Borel-measurable bounded ambient procedures.","s":"{Theta : Type u_4} → {X : Type u_5} → {C : Type u_6} → [Fintype X] → [Fintype C] → (Theta → X → ℝ) → (Theta → C → ℝ) → (Theta → ℝ) → ℝ → ℝ → ℝ","l":233,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue_comp_surjective","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given label probabilities, side-coordinate probabilities, a target, action bounds, a surjective parameter map, pulling all model coordinates back leaves the measurable exact-table minimax value unchanged.","s":"∀ {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {A : Type u_7} {B : Type u_8} (p : B → X → ℝ) (q : B → C → ℝ) (tau : B → ℝ) (l u : ℝ) (phi : A → B), Function.Surjective phi → Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue (fun a => p (phi a)) (fun a => q (phi a)) (fun a => tau (phi a)) l u = Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u","l":277,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue_eq_of_tendsto","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given simplex-valid label and side probabilities, a target, ordered action bounds, target containment, and convergence of the empirical-side minimax values to the exact-side value, the measurable exact-table minimax value equals that common limit.","s":"∀ {Theta : Type u_4} {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {l u : ℝ} [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Filter.Tendsto (fun m => Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m) Filter.atTop (nhds (Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u)) → Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u = Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u","l":426,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue_le_empiricalSideMinimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given simplex-valid label and side probabilities, a target, ordered action bounds, target containment in those bounds, and a sample size, the measurable ambient exact-table minimax value is at most the empirical-side minimax value.","s":"∀ {Theta : Type u_4} {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {l u : ℝ} [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → ∀ (m : ℕ), Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m","l":340,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimax_squeeze","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given simplex-valid label and side probabilities, a target, ordered action bounds, and target containment in those bounds, the measurable exact-table minimax value lies between the exact-side value and every empirical-side value.","s":"∀ {Theta : Type u_4} {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {l u : ℝ} [Nonempty Theta] (p : Theta → X → ℝ) (q : Theta → C → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta), p theta ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (theta : Theta), q theta ∈ stdSimplex ℝ C), l ≤ u → (∀ (theta : Theta), tau theta ∈ Set.Icc l u) → Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u ∧ ∀ (m : ℕ), Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue p q tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m","l":386,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimax_squeeze_continuousSurjection","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a continuous parameter map, its surjectivity, model coordinates and target, simplex validity, ordered action bounds, and target containment, the measurable exact-table value on the source parameterization is squeezed between the target exact-side value and every target empirical-side value.","s":"∀ {X : Type u_5} {C : Type u_6} [inst : Fintype X] [inst_1 : Fintype C] [Nonempty C] {l u : ℝ} {A : Type u_7} {B : Type u_8} [inst_3 : TopologicalSpace A] [inst_4 : TopologicalSpace B] [CompactSpace A] [CompactSpace B] [Nonempty A] [Nonempty B] (phi : A → B), Continuous phi → Function.Surjective phi → ∀ (p : B → X → ℝ) (q : B → C → ℝ) (tau : B → ℝ), (∀ (b : B), p b ∈ stdSimplex ℝ X) → ∀ (hq : ∀ (b : B), q b ∈ stdSimplex ℝ C), l ≤ u → (∀ (b : B), tau b ∈ Set.Icc l u) → Causalean.Stat.Minimax.FiniteSideInformation.exactSideMinimaxValue p q hq tau l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue (fun a => p (phi a)) (fun a => q (phi a)) (fun a => tau (phi a)) l u ∧ ∀ (m : ℕ), Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue (fun a => p (phi a)) (fun a => q (phi a)) (fun a => tau (phi a)) l u ≤ Causalean.Stat.Minimax.FiniteSideInformation.empiricalSideMinimaxValue p q hq tau l u m","l":404,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given label probabilities, side-coordinate probabilities, a target, a measurable ambient procedure, and a parameter, the measurable exact-table risk is the label-probability-weighted squared loss after evaluating the procedure at that parameter's raw side-coordinate table.","s":"{Theta : Type u_4} → {X : Type u_5} → {C : Type u_6} → [Fintype X] → [inst : Fintype C] → {l u : ℝ} → (Theta → X → ℝ) → (Theta → C → ℝ) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSideInformation.MeasurableAmbientExactSideProcedure X C l u → Theta → ℝ","l":224,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.measurable_conditionalAverageProcedure","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given action bounds, a side-sample size, an empirical procedure, and a label, the conditional-average procedure is Borel measurable in the finite side-law vector.","s":"∀ {X : Type u_1} {C : Type u_2} [Fintype X] [inst : Fintype C] (l u : ℝ) (m : ℕ) (d : Causalean.Stat.Minimax.FiniteSideInformation.EmpiricalSideProcedure X C m l u) (x : X), Measurable (Causalean.Stat.Minimax.FiniteSideInformation.conditionalAverageProcedure l u m d x)","l":63,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.minimaxValue_comp_surjective","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a risk, a parameter map, and surjectivity of that map, reindexing leaves the minimax value unchanged.","s":"∀ {E : Type u_7} {A : Type u_8} {B : Type u_9} (risk : E → B → ℝ) (phi : A → B), Function.Surjective phi → (Causalean.Stat.minimaxValue fun e a => risk e (phi a)) = Causalean.Stat.minimaxValue risk","l":258,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.pi_finitePmfMeasure_eq_productDesignMeasure","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a finite probability vector and sample size, the finite product of its atomic measures equals the product-design measure.","s":"∀ {C : Type u_2} [inst : Fintype C] [inst_1 : MeasurableSpace C] [inst_2 : MeasurableSingletonClass C] (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) (m : ℕ), (MeasureTheory.Measure.pi fun x => Causalean.Stat.Minimax.FiniteSideInformation.finitePmfMeasure w) = (Causalean.Experimentation.DesignBased.prodDesign fun x => Causalean.Stat.Minimax.FiniteSideInformation.finitePmfDesign w).toMeasure","l":161,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.productProbability","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given a finite probability vector and a finite sample, the product probability is the product of the sampled atom masses.","s":"(C : Type u_1) → [inst : Fintype C] → {m : ℕ} → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C → (Fin m → C) → ℝ","l":46,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.productProbability_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Every finite product probability is nonnegative.","s":"∀ (C : Type u_1) [inst : Fintype C] {m : ℕ} (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C) (z : Fin m → C), 0 ≤ Causalean.Stat.Minimax.FiniteSideInformation.productProbability C w z","l":59,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.sidePmf","k":"def","m":"Causalean.Stat.Minimax.FiniteSideInformation.Experiments","a":"Stat/Minimax/FiniteSideInformation","t":1,"d":"Given side-probability coordinates that are simplex-valid, the side law at a parameter is the corresponding finite probability vector.","s":"{Theta : Type u_1} → {C : Type u_3} → [inst : Fintype C] → (q : Theta → C → ℝ) → (∀ (theta : Theta), q theta ∈ stdSimplex ℝ C) → Theta → Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C","l":39,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Experiments.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.sum_empiricalFrequency","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Concentration","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Over a finite nonempty alphabet, the empirical atom frequencies of a positive sample sum to one.","s":"∀ (C : Type u_1) [inst : Fintype C] [inst_1 : DecidableEq C] [Nonempty C] {m : ℕ} (hm : 0 < m) (z : Fin m → C), ∑ c, Causalean.Stat.Minimax.FiniteSideInformation.empiricalFrequency C hm z c = 1","l":44,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Concentration.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.sum_productProbability","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Coordinates","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"The product probabilities of all length-m samples sum to one.","s":"∀ (C : Type u_1) [inst : Fintype C] {m : ℕ} (w : Causalean.Stat.Minimax.FiniteSideInformation.FinitePmf C), ∑ z, Causalean.Stat.Minimax.FiniteSideInformation.productProbability C w z = 1","l":64,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Coordinates.lean"},{"n":"Causalean.Stat.Minimax.FiniteSideInformation.worstCaseRisk_comp_surjective","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSideInformation.Measurable","a":"Stat/Minimax/FiniteSideInformation","t":2,"d":"Given a risk, a fixed procedure, a parameter map, and surjectivity of that map, reindexing leaves the worst-case risk unchanged.","s":"∀ {E : Type u_7} {A : Type u_8} {B : Type u_9} (risk : E → B → ℝ) (e : E) (phi : A → B), Function.Surjective phi → Causalean.Stat.worstCaseRisk (fun e a => risk e (phi a)) e = Causalean.Stat.worstCaseRisk risk e","l":242,"f":"Causalean/Stat/Minimax/FiniteSideInformation/Measurable.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a set of design indices and an observation space for each design index, the ambient coordinate space consists of one real design-weight vector and one real action coordinate for every design-index and observation pair.","s":"(R : Type u_4) → (R → Type u_5) → Type (max u_4 u_4 u_5)","l":75,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model","k":"structure","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"A dependent finite experiment assigns, at every state and design index, a nonnegative normalized mass function on that design's observation type, together with a real target.","s":"(Theta : Type u_1) → (B : Type u_2) → [Fintype Theta] → [Fintype B] → (X : B → Type u_3) → [(b : B) → Fintype (X b)] → Type (max (max u_1 u_2) u_3)","l":25,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.conditionalBarycenter_risk_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Nonnegative grid weights that sum to each design probability make the statewise risk of the guarded conditional barycenter no greater than the randomized grid-action risk.","s":"∀ {Theta : Type u_4} {B : Type u_5} {G : Type u_6} [inst : Fintype Theta] [inst_1 : Fintype B] [inst_2 : Fintype G] {X : B → Type u_7} [inst_3 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ) (gamma : G → ℝ) (d : ℝ), (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → ∀ (theta : Theta), ∑ b, pi.p b * ∑ x, M.P theta b x * (Causalean.Stat.Minimax.FiniteSquaredLoss.conditionalBarycenter pi w gamma d b x - M.tau theta) ^ 2 ≤ ∑ b, ∑ x, ∑ g, M.P theta b x * w b x g * (gamma g - M.tau theta) ^ 2","l":437,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.iInf_squaredRisk_eq_posteriorResidual","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The smallest prior squared-loss risk over all real decision rules for a fixed design equals its posterior residual risk.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B), ⨅ delta, ∑ theta, nu.p theta * ∑ x, M.P theta b x * (delta x - M.tau theta) ^ 2 = M.posteriorResidual nu b","l":231,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.joint_eq_zero_of_predictiveMass_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"A zero-mass predictive observation has zero joint probability in every state.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) {b : B} {x : X b}, M.predictiveMass nu b x = 0 → ∀ (theta : Theta), nu.p theta * M.P theta b x = 0","l":84,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.minimaxValue_le_of_randomizedGridRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Nonnegative grid weights that sum to each design probability, with the null-fiber default inside the action interval, all grid actions inside that interval, and randomized grid-action risk uniformly bounded across states, imply the dependent experiment's bounded squared-loss minimax value is no greater than that bound.","s":"∀ {Theta : Type u_4} {B : Type u_5} {G : Type u_6} [inst : Fintype Theta] [inst_1 : Fintype B] [inst_2 : Fintype G] {X : B → Type u_7} [Nonempty Theta] [inst_4 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ) (gamma : G → ℝ) (d : ℝ), (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → ∀ {l u c : ℝ}, d ∈ Set.Icc l u → (∀ (g : G), gamma g ∈ Set.Icc l u) → (∀ (theta : Theta), ∑ b, ∑ x, ∑ g, M.P theta b x * w b x g * (gamma g - M.tau theta) ^ 2 ≤ c) → Causalean.Stat.minimaxValue (Causalean.Stat.Minimax.FiniteSquaredLoss.risk M.P M.tau) ≤ c","l":513,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.posteriorMean","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the guarded posterior mean is the predictive target numerator divided by predictive mass when that mass is nonzero, and is zero when it is zero.","s":"{Theta : Type u_1} → {B : Type u_2} → [inst : Fintype Theta] → [inst_1 : Fintype B] → {X : B → Type u_3} → [inst_2 : (b : B) → Fintype (X b)] → Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X → Causalean.Experimentation.DesignBased.FiniteDesign Theta → (b : B) → X b → ℝ","l":75,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.posteriorResidual","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a dependent finite experiment, a prior probability design over states, and a design index, the posterior residual is the sum, over states and observations, of prior probability times observation likelihood times squared deviation of the guarded posterior mean from the state's target.","s":"{Theta : Type u_1} → {B : Type u_2} → [inst : Fintype Theta] → [inst_1 : Fintype B] → {X : B → Type u_3} → [inst_2 : (b : B) → Fintype (X b)] → Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X → Causalean.Experimentation.DesignBased.FiniteDesign Theta → B → ℝ","l":120,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.posteriorResidual_le_squaredRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The posterior residual risk is no greater than the prior squared-loss risk of any real decision rule for the same design.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B) (delta : X b → ℝ), M.posteriorResidual nu b ≤ ∑ theta, nu.p theta * ∑ x, M.P theta b x * (delta x - M.tau theta) ^ 2","l":220,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.posteriorResidual_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Every design-specific posterior residual risk is nonnegative.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B), 0 ≤ M.posteriorResidual nu b","l":129,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.predictiveMass","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the prior predictive mass is the sum over states of prior probability times the conditional probability of that observation.","s":"{Theta : Type u_1} → {B : Type u_2} → [inst : Fintype Theta] → [inst_1 : Fintype B] → {X : B → Type u_3} → [inst_2 : (b : B) → Fintype (X b)] → Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X → Causalean.Experimentation.DesignBased.FiniteDesign Theta → (b : B) → X b → ℝ","l":45,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.predictiveMass_mul_posteriorMean","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Predictive probability times the guarded posterior mean equals the target-weighted joint probability, including for null observations.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B) (x : X b), M.predictiveMass nu b x * M.posteriorMean nu b x = M.predictiveTarget nu b x","l":109,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.predictiveMass_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Every prior predictive probability is nonnegative.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B) (x : X b), 0 ≤ M.predictiveMass nu b x","l":62,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.predictiveTarget","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the predictive target numerator is the sum over states of prior probability times observation likelihood times the state's target.","s":"{Theta : Type u_1} → {B : Type u_2} → [inst : Fintype Theta] → [inst_1 : Fintype B] → {X : B → Type u_3} → [inst_2 : (b : B) → Fintype (X b)] → Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X → Causalean.Experimentation.DesignBased.FiniteDesign Theta → (b : B) → X b → ℝ","l":67,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.predictiveTarget_eq_zero_of_predictiveMass_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"A zero-mass predictive observation has zero target-weighted joint probability.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) {b : B} {x : X b}, M.predictiveMass nu b x = 0 → M.predictiveTarget nu b x = 0","l":97,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.risk_barycenterProcedure_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Nonnegative grid weights that sum to each design probability, with the null-fiber default inside the action interval and all grid actions inside that interval, make the bounded barycenter procedure's statewise risk no greater than the randomized grid-action risk.","s":"∀ {Theta : Type u_4} {B : Type u_5} {G : Type u_6} [inst : Fintype Theta] [inst_1 : Fintype B] [inst_2 : Fintype G] {X : B → Type u_7} [inst_3 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ) (gamma : G → ℝ) (d : ℝ) (hw : ∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) (hocc : ∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) {l u : ℝ} (hd : d ∈ Set.Icc l u) (hgamma : ∀ (g : G), gamma g ∈ Set.Icc l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.risk M.P M.tau (Causalean.Stat.Minimax.FiniteSquaredLoss.barycenterProcedure pi w gamma d hw hocc hd hgamma) theta ≤ ∑ b, ∑ x, ∑ g, M.P theta b x * w b x g * (gamma g - M.tau theta) ^ 2","l":495,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.sInf_posteriorResidual_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"An ordered action interval makes the least design-specific posterior residual a lower bound on the dependent experiment's bounded squared-loss minimax value.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] [Nonempty Theta] [Nonempty B] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) {l u : ℝ}, l ≤ u → sInf (Set.range (M.posteriorResidual nu)) ≤ Causalean.Stat.minimaxValue (Causalean.Stat.Minimax.FiniteSquaredLoss.risk M.P M.tau)","l":298,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.sInf_posteriorResidual_le_priorRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The least posterior residual across designs is no greater than the prior-average risk of any bounded dependent procedure.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] [Nonempty B] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) {l u : ℝ} (q : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u), sInf (Set.range (M.posteriorResidual nu)) ≤ ∑ theta, nu.p theta * Causalean.Stat.Minimax.FiniteSquaredLoss.risk M.P M.tau q theta","l":258,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.squaredRisk_eq_posteriorResidual_add","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The prior risk of any decision rule equals posterior residual risk plus its predictive-probability-weighted squared distance from the posterior mean.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B) (delta : X b → ℝ), ∑ theta, nu.p theta * ∑ x, M.P theta b x * (delta x - M.tau theta) ^ 2 = M.posteriorResidual nu b + ∑ x, M.predictiveMass nu b x * (delta x - M.posteriorMean nu b x) ^ 2","l":139,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.squaredRisk_posteriorMean","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The guarded posterior mean attains exactly the design-specific posterior residual risk.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B), ∑ theta, nu.p theta * ∑ x, M.P theta b x * (M.posteriorMean nu b x - M.tau theta) ^ 2 = M.posteriorResidual nu b","l":210,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Model.sum_predictiveMass","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The prior predictive probabilities within a fixed design sum to one.","s":"∀ {Theta : Type u_1} {B : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype B] {X : B → Type u_3} [inst_2 : (b : B) → Fintype (X b)] (M : Causalean.Stat.Minimax.FiniteSquaredLoss.Model Theta B X) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta) (b : B), ∑ x, M.predictiveMass nu b x = 1","l":53,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure","k":"structure","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"A procedure consists of a finite randomized design and, for every selected design and dependent observation, an action in the prescribed closed interval.","s":"{R : Type u_2} → [Fintype R] → (X : R → Type u_4) → [(r : R) → Fintype (X r)] → ℝ → ℝ → Type (max u_2 u_4)","l":31,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure.ofAmbient","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given an ambient coordinate point that belongs to the feasible procedure-coordinate set, the corresponding bounded finite procedure uses its design-weight coordinates as design probabilities and its action coordinates as bounded decisions.","s":"{R : Type u_2} → [inst : Fintype R] → {X : R → Type u_3} → [inst_1 : (r : R) → Fintype (X r)] → {l u : ℝ} → {z : Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X} → z ∈ Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u","l":100,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure.toAmbient","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a bounded finite randomized procedure, its ambient Euclidean coordinates are its design probabilities together with its action at every design-index and observation pair.","s":"{R : Type u_2} → [inst : Fintype R] → {X : R → Type u_3} → [inst_1 : (r : R) → Fintype (X r)] → {l u : ℝ} → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u → Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X","l":88,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure.toAmbient_mem","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The Euclidean coordinates of every bounded finite procedure belong to the feasible coordinate set.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] {l u : ℝ} (q : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u), q.toAmbient ∈ Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u","l":93,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure.toAmbient_ofAmbient","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Turning a feasible coordinate point into a procedure and back recovers that coordinate point.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] {l u : ℝ} {z : Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X} (hz : z ∈ Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u), (Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure.ofAmbient hz).toAmbient = z","l":111,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.barycenterProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a probability design over design indices, grid weights, real grid actions, a default action, nonnegative grid weights, weights that sum to the corresponding design probability, a default action in the closed interval from $l$ to $u$, and grid actions all in that interval, the barycenter procedure is the bounded finite procedure using that design and the guarded conditional barycenter as its decision rule.","s":"{B : Type u_5} → {G : Type u_6} → [inst : Fintype B] → [inst_1 : Fintype G] → {X : B → Type u_7} → (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) → [inst_2 : (b : B) → Fintype (X b)] → (w : (b : B) → X b → G → ℝ) → (gamma : G → ℝ) → (d : ℝ) → (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → {l u : ℝ} → d ∈ Set.Icc l u → (∀ (g : G), gamma g ∈ Set.Icc l u) → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u","l":477,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a real-valued risk vector over states and a vector of prior weights, the finite Bayes payoff is the sum over states of each prior weight times its risk-vector value.","s":"{Theta : Type u_1} → [Fintype Theta] → (Theta → ℝ) → (Theta → ℝ) → ℝ","l":225,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff_eq_E","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Pairing a risk vector with a finite prior equals that prior's existing finite-design expectation.","s":"∀ {Theta : Type u_1} [inst : Fintype Theta] (z : Theta → ℝ) (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff z nu.p = nu.E z","l":230,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.conditionalBarycenter","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a probability design over design indices, grid weights for each design index, observation, and grid action, real grid actions, a default action, a design index, and an observation in that design's observation space, the guarded conditional barycenter is the weighted average of grid actions divided by that design's probability when it is positive, and is the default action when it is zero.","s":"{B : Type u_5} → {G : Type u_6} → [inst : Fintype B] → [Fintype G] → {X : B → Type u_7} → Causalean.Experimentation.DesignBased.FiniteDesign B → ((b : B) → X b → G → ℝ) → (G → ℝ) → ℝ → (b : B) → X b → ℝ","l":328,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.conditionalBarycenter_mem_Icc","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Nonnegative grid weights that sum to the design probability, with the null-fiber default inside a closed interval and all grid actions inside that interval, produce a guarded conditional barycenter in the same interval.","s":"∀ {B : Type u_5} {G : Type u_6} [inst : Fintype B] [inst_1 : Fintype G] {X : B → Type u_7} (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ) (gamma : G → ℝ) (d : ℝ), (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → ∀ {l u : ℝ}, d ∈ Set.Icc l u → (∀ (g : G), gamma g ∈ Set.Icc l u) → ∀ (b : B) (x : X b), Causalean.Stat.Minimax.FiniteSquaredLoss.conditionalBarycenter pi w gamma d b x ∈ Set.Icc l u","l":351,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.continuous_bayesPayoff","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The finite Bayes payoff varies continuously with both the risk vector and the prior weights.","s":"∀ {Theta : Type u_1} [inst : Fintype Theta], Continuous fun p => Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff p.1 p.2","l":236,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.continuous_rawRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"For fixed likelihood coefficients and targets, the ambient squared-loss risk in any state varies continuously with the procedure coordinates.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) (theta : Theta), Continuous fun z => Causalean.Stat.Minimax.FiniteSquaredLoss.rawRisk P tau z theta","l":177,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.continuous_riskVector","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"For fixed likelihood coefficients, the vector collecting each state's ambient squared-loss risk varies continuously with the procedure coordinates.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), Continuous (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau)","l":193,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.convex_dominatedRiskVectors","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"With nonnegative likelihood coefficients and an ordered action interval, the set of risk vectors dominated by an ordinary finite procedure is convex.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∀ {l u : ℝ}, l ≤ u → Convex ℝ (Causalean.Stat.Minimax.FiniteSquaredLoss.dominatedRiskVectors P tau l u)","l":199,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.convex_procedureSet","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"An action interval with ordered endpoints makes the feasible Euclidean procedure set convex.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [(r : R) → Fintype (X r)] {l u : ℝ}, l ≤ u → Convex ℝ (Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u)","l":142,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.convexified_risk_game_has_saddle","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Saddle","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"For nonempty finite state and design spaces, given likelihood coefficients and an ordered action interval, the convexified risk game has an attaining saddle point consisting of a risk vector and a finite prior.","s":"∀ {Theta : Type u_1} {R : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype R] {X : R → Type u_3} [inst_2 : (r : R) → Fintype (X r)] [Nonempty Theta] [Nonempty R] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) {l u : ℝ}, l ≤ u → ∃ zstar ∈ (convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u), ∃ nustar ∈ stdSimplex ℝ Theta, IsSaddlePointOn ((convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u)) (stdSimplex ℝ Theta) Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff zstar nustar","l":29,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Saddle.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.designMass_mul_conditionalBarycenter_sq_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Nonnegative grid weights that sum to the design probability make design probability times the guarded barycenter's squared error no greater than the corresponding weighted grid-action squared error.","s":"∀ {B : Type u_5} {G : Type u_6} [inst : Fintype B] [inst_1 : Fintype G] {X : B → Type u_7} (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ) (gamma : G → ℝ) (d t : ℝ), (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → ∀ (b : B) (x : X b), pi.p b * (Causalean.Stat.Minimax.FiniteSquaredLoss.conditionalBarycenter pi w gamma d b x - t) ^ 2 ≤ ∑ g, w b x g * (gamma g - t) ^ 2","l":374,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.dominatedRiskVectors","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a parameter space, a finite set of design points, a finite observation set at each design point, likelihood coefficients, a target value for each parameter, and two action bounds, the set of dominated risk vectors consists of exactly those real-valued functions on the parameter space for which there exists a feasible finite randomized procedure whose squared-loss risk is no greater at every parameter.","s":"{Theta : Type u_1} → {R : Type u_2} → [Fintype R] → {X : R → Type u_3} → [(r : R) → Fintype (X r)] → (Theta → (r : R) → X r → ℝ) → (Theta → ℝ) → ℝ → ℝ → Set (Theta → ℝ)","l":186,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.exists_procedure_risk_le_of_mem_convexHull","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"With nonnegative likelihood coefficients and an ordered action interval, every convex combination of attainable risk vectors is coordinatewise dominated by the risk vector of an ordinary finite procedure.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∀ {l u : ℝ}, l ≤ u → ∀ {z : Theta → ℝ}, z ∈ (convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u) → ∃ q, ∀ (theta : Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau q theta ≤ z theta","l":237,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.finiteDesignOfSimplex","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a vector of weights over the finite state space that belongs to the standard probability simplex, the corresponding finite design is the probability design whose mass function is that vector.","s":"{Theta : Type u_1} → [inst : Fintype Theta] → {w : Theta → ℝ} → w ∈ stdSimplex ℝ Theta → Causalean.Experimentation.DesignBased.FiniteDesign Theta","l":216,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.finiteDesign_mem_stdSimplex","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Every finite randomized design has a mass function in the standard probability simplex.","s":"∀ {Theta : Type u_1} [inst : Fintype Theta] (nu : Causalean.Experimentation.DesignBased.FiniteDesign Theta), nu.p ∈ stdSimplex ℝ Theta","l":210,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.finite_bounded_squared_loss_has_saddle","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Saddle","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Finite bounded squared-loss minimax theorem. In nonempty finite state and design spaces, an ordered real action interval and nonnegative likelihood coefficients guarantee an ordinary randomized design, bounded decision rule, and finite least-favorable prior attaining both minimax saddle inequalities. No likelihood normalization, observation-space nonemptiness, or target-in-interval assumption is required.","s":"∀ {Theta : Type u_1} {R : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype R] {X : R → Type u_3} [inst_2 : (r : R) → Fintype (X r)] [Nonempty Theta] [Nonempty R] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) {l u : ℝ}, l ≤ u → (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∃ qstar deltastar nu, (∀ (theta : Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau { design := qstar, decision := deltastar } theta ≤ Causalean.Stat.minimaxValue (Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau)) ∧ ∀ (qprime : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u), Causalean.Stat.minimaxValue (Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau) ≤ nu.E (Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau qprime)","l":152,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Saddle.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.isCompact_convexHull_riskVectors","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Under given likelihood coefficients, the convex hull of the attainable finite risk vectors is compact.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) (l u : ℝ), IsCompact ((convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u))","l":251,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.isCompact_procedureSet","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"The feasible Euclidean procedure set is compact: it is the product of a finite probability simplex and finitely many closed action intervals.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [(r : R) → Fintype (X r)] (l u : ℝ), IsCompact (Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u)","l":151,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.isCompact_riskVector_image","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Under given likelihood coefficients, the set of risk vectors attainable by feasible finite procedures is compact.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) (l u : ℝ), IsCompact (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u)","l":202,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.mixProcedure","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a finite set of design points, a finite observation set at each design point, ordered action bounds, a mixing weight no smaller than zero, the same weight no larger than one, and two feasible randomized procedures, the mixed feasible randomized procedure assigns the convex combination of their design masses at each design point. At a point with positive mixed mass it averages their actions using the corresponding posterior mixture weights, and at a point with zero mixed mass it uses the lower action bound.","s":"{R : Type u_2} → [inst : Fintype R] → {X : R → Type u_3} → [inst_1 : (r : R) → Fintype (X r)] → {l u t : ℝ} → l ≤ u → 0 ≤ t → t ≤ 1 → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u","l":30,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given a family of finite observation spaces indexed by design, a lower action bound, and an upper action bound, the feasible procedure-coordinate set contains exactly those ambient coordinates whose design weights form a probability distribution and whose action coordinates all lie in the closed interval from the lower to the upper bound.","s":"{R : Type u_2} → [Fintype R] → (X : R → Type u_4) → ℝ → ℝ → Set (Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X)","l":81,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet_nonempty","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"A nonempty design space and a nonempty action interval make the feasible Euclidean procedure set nonempty.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [(r : R) → Fintype (X r)] [Nonempty R] {l u : ℝ}, l ≤ u → (Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u).Nonempty","l":134,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.procedure_nonempty","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"A nonempty design space and a nonempty action interval have a bounded finite randomized procedure, even when some observation spaces are empty.","s":"∀ {R : Type u_2} [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] [Nonempty R] {l u : ℝ}, l ≤ u → Nonempty (Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u)","l":61,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.rawRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given state-specific observation likelihoods, a real target for each state, an ambient coordinate point, and a state, the raw squared-loss risk is the design-coordinate-weighted sum of likelihood-weighted squared differences between action coordinates and that state's target.","s":"{Theta : Type u_1} → {R : Type u_2} → [Fintype R] → {X : R → Type u_3} → [(r : R) → Fintype (X r)] → (Theta → (r : R) → X r → ℝ) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X → Theta → ℝ","l":118,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.rawRisk_toAmbient","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"For given likelihood coefficients, the ambient polynomial representation of a procedure's risk equals its public squared-loss risk.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] {l u : ℝ} (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) (q : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.rawRisk P tau q.toAmbient theta = Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau q theta","l":126,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.risk","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given state-specific observation likelihoods, a real target for each state, a bounded randomized procedure, and a state, the statewise squared-loss risk is the procedure-design-weighted sum of likelihood-weighted squared differences between the selected action and that state's target.","s":"{Theta : Type u_1} → {R : Type u_2} → [inst : Fintype R] → {X : R → Type u_3} → [inst_1 : (r : R) → Fintype (X r)] → {l u : ℝ} → (Theta → (r : R) → X r → ℝ) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u → Theta → ℝ","l":39,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector","k":"def","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"Given state-specific observation likelihoods, a real target for each state, and an ambient coordinate point, the risk vector assigns to every state its raw squared-loss risk at that coordinate point.","s":"{Theta : Type u_1} → {R : Type u_2} → [Fintype R] → {X : R → Type u_3} → [(r : R) → Fintype (X r)] → (Theta → (r : R) → X r → ℝ) → (Theta → ℝ) → Causalean.Stat.Minimax.FiniteSquaredLoss.Ambient R X → Theta → ℝ","l":186,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector_image_subset_dominated","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Under given likelihood coefficients, every risk vector attained by feasible ambient coordinates is dominated by the risk of the associated ordinary finite procedure.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ) (l u : ℝ), Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u ⊆ Causalean.Stat.Minimax.FiniteSquaredLoss.dominatedRiskVectors P tau l u","l":224,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.risk_mixProcedure_le","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Mixing","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"With nonnegative likelihood coefficients, an ordered action interval, and a mixing weight between zero and one, conditional mixing produces a procedure whose statewise squared-loss risk is no greater than the corresponding mixture of the two original risks. No normalization of the coefficients is required.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∀ {l u t : ℝ} (hlu : l ≤ u) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (q₀ q₁ : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u) (theta : Theta), Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau (Causalean.Stat.Minimax.FiniteSquaredLoss.mixProcedure hlu ht0 ht1 q₀ q₁) theta ≤ t * Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau q₀ theta + (1 - t) * Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau q₁ theta","l":97,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Mixing.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.risk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Core","a":"Stat/Minimax/FiniteSquaredLoss","t":1,"d":"When the likelihood coefficients are nonnegative, every bounded finite procedure has nonnegative squared-loss risk in each state.","s":"∀ {Theta : Type u_1} {R : Type u_2} [Fintype Theta] [inst : Fintype R] {X : R → Type u_3} [inst_1 : (r : R) → Fintype (X r)] {l u : ℝ} (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∀ (q : Causalean.Stat.Minimax.FiniteSquaredLoss.Procedure X l u) (theta : Theta), 0 ≤ Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau q theta","l":47,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Core.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.saddle_value_eq_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.Saddle","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Given nonnegative likelihood coefficients, an ordered action interval, and a saddle point in the feasible convexified game, its Bayes payoff equals Causalean's minimax value over ordinary bounded finite procedures.","s":"∀ {Theta : Type u_1} {R : Type u_2} [inst : Fintype Theta] [inst_1 : Fintype R] {X : R → Type u_3} [inst_2 : (r : R) → Fintype (X r)] [Nonempty Theta] [Nonempty R] (P : Theta → (r : R) → X r → ℝ) (tau : Theta → ℝ), (∀ (theta : Theta) (r : R) (x : X r), 0 ≤ P theta r x) → ∀ {l u : ℝ}, l ≤ u → ∀ {zstar nustar : Theta → ℝ}, zstar ∈ (convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u) → nustar ∈ stdSimplex ℝ Theta → IsSaddlePointOn ((convexHull ℝ) (Causalean.Stat.Minimax.FiniteSquaredLoss.riskVector P tau '' Causalean.Stat.Minimax.FiniteSquaredLoss.procedureSet X l u)) (stdSimplex ℝ Theta) Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff zstar nustar → Causalean.Stat.Minimax.FiniteSquaredLoss.bayesPayoff zstar nustar = Causalean.Stat.minimaxValue (Causalean.Stat.Minimax.FiniteSquaredLoss.risk P tau)","l":86,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/Saddle.lean"},{"n":"Causalean.Stat.Minimax.FiniteSquaredLoss.weight_eq_zero_of_designMass_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.FiniteSquaredLoss.PosteriorBarycenter","a":"Stat/Minimax/FiniteSquaredLoss","t":2,"d":"Nonnegative grid weights that sum to the design probability are zero individually whenever that design has zero probability.","s":"∀ {B : Type u_5} {G : Type u_6} [inst : Fintype B] [inst_1 : Fintype G] {X : B → Type u_7} (pi : Causalean.Experimentation.DesignBased.FiniteDesign B) (w : (b : B) → X b → G → ℝ), (∀ (b : B) (x : X b) (g : G), 0 ≤ w b x g) → (∀ (b : B) (x : X b), ∑ g, w b x g = pi.p b) → ∀ {b : B}, pi.p b = 0 → ∀ (x : X b) (g : G), w b x g = 0","l":338,"f":"Causalean/Stat/Minimax/FiniteSquaredLoss/PosteriorBarycenter.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.bayesSquaredRisk","k":"def","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":1,"d":"Given a prior, an experiment kernel, a real-valued target, and an estimator, the Bayes squared risk averages parameterwise squared risk against the prior.","s":"{Θ : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace X] → MeasureTheory.Measure Θ → ProbabilityTheory.Kernel Θ X → (Θ → ℝ) → (X → ℝ) → ENNReal","l":28,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.bayesSquaredRisk_le_worstCase","k":"theorem","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":2,"d":"Given a probability prior, an experiment kernel, a real-valued target, and an estimator, Bayes squared risk is at most worst-case squared risk over the parameter class.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (target : Θ → ℝ) (estimator : X → ℝ) [MeasureTheory.IsProbabilityMeasure π], Causalean.Stat.Minimax.FuzzyHypotheses.bayesSquaredRisk π K target estimator ≤ Causalean.Stat.Minimax.FuzzyHypotheses.worstCaseSquaredRisk K target estimator","l":224,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.minimaxSquaredRisk","k":"def","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":1,"d":"Given an experiment kernel and a real-valued target, the measurable-estimator minimax squared risk is the infimum of worst-case risk over all measurable real-valued estimators.","s":"{Θ : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace X] → ProbabilityTheory.Kernel Θ X → (Θ → ℝ) → ENNReal","l":42,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.squaredRisk","k":"def","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":1,"d":"Given an experiment kernel, a real-valued target, an estimator, and a parameter value, the squared risk is the expected squared estimation error under the experiment law at that parameter, represented as a nonnegative lower integral.","s":"{Θ : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace X] → ProbabilityTheory.Kernel Θ X → (Θ → ℝ) → (X → ℝ) → Θ → ENNReal","l":21,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.twoFuzzyHypotheses_bayesRisk_lower","k":"theorem","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":2,"d":"Given two probability priors, a probability experiment kernel, a measurable real-valued target, a measurable estimator, two target centers, a nonnegative center separation, a nonnegative concentration radius smaller than half the separation, the stated separation of the centers, nonnegative prior-tail bounds, and a nonnegative predictive-distance bound, if the first prior concentrates around its center, the second prior concentrates around its center, and the predictive mixtures satisfy the distance bound, then the larger Bayes squared risk is at least half the squared residual separation times one minus the three error terms.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (target : Θ → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1], (∀ (θ : Θ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (estimator : X → ℝ), Measurable estimator → Measurable target → ∀ (center0 center1 Delta r alpha0 alpha1 beta : ℝ), 0 ≤ Delta → 0 ≤ r → r < Delta / 2 → Delta ≤ center1 - center0 → 0 ≤ alpha0 → 0 ≤ alpha1 → 0 ≤ beta → π0.real {θ | r < |target θ - center0|} ≤ alpha0 → π1.real {θ | r < |target θ - center1|} ≤ alpha1 → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π1 K) ≤ beta → ENNReal.ofReal ((Delta / 2 - r) ^ 2 * (1 - beta - alpha0 - alpha1) / 2) ≤ max (Causalean.Stat.Minimax.FuzzyHypotheses.bayesSquaredRisk π0 K target estimator) (Causalean.Stat.Minimax.FuzzyHypotheses.bayesSquaredRisk π1 K target estimator)","l":48,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.twoFuzzyHypotheses_minimax_lower_standard","k":"theorem","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":1,"d":"Given two probability priors, a probability experiment kernel, a measurable real-valued target, two target centers, and a positive center separation, if the centers have at least that separation, each prior puts at most one eighth of its mass beyond one quarter of the separation from its center, and the predictive mixtures are within one sixteenth in total variation, then the minimax squared risk over all measurable estimators is at least eleven times the squared separation divided by 512.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (target : Θ → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1], (∀ (θ : Θ), MeasureTheory.IsProbabilityMeasure (K θ)) → Measurable target → ∀ (center0 center1 Delta : ℝ), 0 < Delta → Delta ≤ center1 - center0 → π0.real {θ | Delta / 4 < |target θ - center0|} ≤ 1 / 8 → π1.real {θ | Delta / 4 < |target θ - center1|} ≤ 1 / 8 → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π1 K) ≤ 1 / 16 → ENNReal.ofReal (11 * Delta ^ 2 / 512) ≤ Causalean.Stat.Minimax.FuzzyHypotheses.minimaxSquaredRisk K target","l":267,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.twoFuzzyHypotheses_worstCase_lower_standard","k":"theorem","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":2,"d":"Given two probability priors, a probability experiment kernel, a measurable real-valued target, a measurable estimator, two target centers, and a positive center separation, if the centers have at least that separation, each prior puts at most one eighth of its mass beyond one quarter of the separation from its center, and the predictive mixtures are within one sixteenth in total variation, then the estimator's worst-case squared risk is at least eleven times the squared separation divided by 512.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (target : Θ → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1], (∀ (θ : Θ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (estimator : X → ℝ), Measurable estimator → Measurable target → ∀ (center0 center1 Delta : ℝ), 0 < Delta → Delta ≤ center1 - center0 → π0.real {θ | Delta / 4 < |target θ - center0|} ≤ 1 / 8 → π1.real {θ | Delta / 4 < |target θ - center1|} ≤ 1 / 8 → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π1 K) ≤ 1 / 16 → ENNReal.ofReal (11 * Delta ^ 2 / 512) ≤ Causalean.Stat.Minimax.FuzzyHypotheses.worstCaseSquaredRisk K target estimator","l":236,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.FuzzyHypotheses.worstCaseSquaredRisk","k":"def","m":"Causalean.Stat.Minimax.FuzzyHypotheses","a":"Stat/Minimax","t":1,"d":"Given an experiment kernel, a real-valued target, and an estimator, the worst-case squared risk is the supremum of parameterwise risk over the model's parameter class.","s":"{Θ : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace X] → ProbabilityTheory.Kernel Θ X → (Θ → ℝ) → (X → ℝ) → ENNReal","l":35,"f":"Causalean/Stat/Minimax/FuzzyHypotheses.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoissonObservation","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the displayed assumptions and is given by the following defining expression. The two-count observation consisting of aggregate treated and aggregate control counts.","s":"Type","l":22,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the displayed assumptions and is given by the following defining expression. One coordinate of the marked-Poisson experiment: labeled treated counts with marks one and zero, followed by auxiliary-treated and aggregate-control counts.","s":"Type","l":23,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate","k":"structure","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The certificate structure is specified using the finite node index set, the moment-matching degree. A normalized finite signed moment certificate consists of distinct real nodes, signed weights of total absolute mass one, and vanishing algebraic moments through degree L.","s":"(ι : Type u_1) → [Fintype ι] → ℕ → Type u_1","l":25,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.abs_polarSign_ae","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The atomic polar sign has absolute value one almost everywhere for the variation measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), ∀ᵐ (x : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, |C.polarSign x| = 1","l":197,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.absoluteMeasure","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The atomic absolute-weight measure associated with a finite signed certificate.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → MeasureTheory.Measure ℝ","l":45,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.exists_aggregatePoisson_jordan_sqrt_tail_bound","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the overlap fraction, the support ratio, positive overlap fraction, the overlap upper bound, the support-ratio identity. For fixed overlap geometry, the aggregate affine-Poisson mixtures of the positive and negative Jordan priors satisfy a square-root exponential-tail TV bound with argument proportional to t * B.","s":"∀ (ε κ : ℝ), 0 < ε → ε < 1 / 2 → κ = (1 - 2 * ε) / ε → ∃ A, 0 < A ∧ ∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a B t : ℝ), 0 < a → 0 < B → 0 < t → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.positivePrior ε a t) (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.negativePrior ε a t) ≤ √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail L (A * (t * B)))","l":568,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.exists_geometric_aggregatePoisson_jordan_tv_bound","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The stated conclusion follows from the overlap fraction, the support ratio, positive overlap fraction, the overlap upper bound, the support-ratio identity. For fixed overlap geometry, moment matching makes the two aggregate Jordan-prior predictive laws geometrically close whenever t * B is at most a sufficiently small multiple of the matching degree.","s":"∀ (ε κ : ℝ), 0 < ε → ε < 1 / 2 → κ = (1 - 2 * ε) / ε → ∃ b D ρ, 0 < b ∧ 0 < D ∧ ρ ∈ Set.Ioo 0 1 ∧ ∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a B t : ℝ), 0 < a → 0 < B → 0 < t → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → t * B ≤ b * ↑L → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.positivePrior ε a t) (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.negativePrior ε a t) ≤ D * ρ ^ L","l":754,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.exists_geometric_markedPoisson_tv_bound","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The stated conclusion follows from the overlap fraction, the support ratio, positive overlap fraction, the overlap upper bound, the support-ratio identity. For fixed overlap and support-ratio parameters there are positive constants b,C₀ and a geometric factor ρ < 1 such that moment matching through L bounds one-coordinate marked-Poisson TV by C₀ * u * a * ρ^L whenever (u+v)B ≤ bL. The statement includes the edge cases u = 0 and v = 0 as long as total intensity is positive.","s":"∀ (ε κ : ℝ), 0 < ε → ε < 1 / 2 → κ = (1 - 2 * ε) / ε → ∃ b C₀ ρ, 0 < b ∧ 0 < C₀ ∧ ρ ∈ Set.Ioo 0 1 ∧ ∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a B u v : ℝ), 0 < a → 0 < B → 0 ≤ u → 0 ≤ v → 0 < u + v → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → (u + v) * B ≤ b * ↑L → Causalean.Stat.tvDist (C.markedPoissonPredictive ε a u v false) (C.markedPoissonPredictive ε a u v true) ≤ C₀ * u * a * ρ ^ L","l":1279,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integrable_polarSign","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The atomic polar sign is integrable against the variation measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), MeasureTheory.Integrable C.polarSign (MeasureTheory.VectorMeasure.variation C.signedMeasure)","l":205,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_coordinate_sq_zeroInflatedProductPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, the coordinate index, positive shift, the support-ratio identity, the compact-support condition. Each coordinate of the finite i.i.d. prior has the scalar zero-inflated second moment.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ) (i : Fin k), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∫ (w : Fin k → ℝ), w i ^ 2 ∂C.zeroInflatedProductPrior a k = ∫ (p : ℝ), p ^ 2 ∂C.zeroInflatedPrior a","l":239,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_coordinate_zeroInflatedProductPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, the coordinate index, positive shift, the support-ratio identity, the compact-support condition. Each coordinate of the finite i.i.d. prior has the scalar zero-inflated first moment.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ) (i : Fin k), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∫ (w : Fin k → ℝ), w i ∂C.zeroInflatedProductPrior a k = a * ∫ (p : ℝ), p / (p + a) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure","l":224,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_id_zeroInflatedPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, positive shift, the support-ratio identity, the compact-support condition. The first moment of the zero-inflated prior is a times the variation expectation of p / (p + a).","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∫ (p : ℝ), p ∂C.zeroInflatedPrior a = a * ∫ (p : ℝ), p / (p + a) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure","l":134,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_pow_signedMeasure_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the requested moment degree, the degree condition. Integration against the atomic signed measure recovers every vanishing certificate moment through degree L.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), ∀ j ≤ L, ∫ᵛ (x : ℝ), x ^ j ∂<•C.signedMeasure = 0","l":159,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_sq_zeroInflatedPrior_le","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, positive shift, the support-ratio identity, the compact-support condition. The second moment of the zero-inflated prior is at most B times its first moment.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∫ (p : ℝ), p ^ 2 ∂C.zeroInflatedPrior a ≤ B * ∫ (p : ℝ), p ∂C.zeroInflatedPrior a","l":175,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.integral_sum_zeroInflatedProductPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, positive shift, the support-ratio identity, the compact-support condition. The sum of all i.i.d. coordinates has expectation k times the scalar first moment.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∫ (w : Fin k → ℝ), ∑ i, w i ∂C.zeroInflatedProductPrior a k = ↑k * (a * ∫ (p : ℝ), p / (p + a) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure)","l":273,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.jordanDecomposition_negPart_eq","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The negative part of the signed measure's Jordan decomposition is one half of the normalized negative Jordan prior.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), C.signedMeasure.toJordanDecomposition.negPart = 2⁻¹ • C.negativePrior","l":440,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.jordanDecomposition_posPart_eq","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The positive part of the signed measure's Jordan decomposition is one half of the normalized positive Jordan prior.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), C.signedMeasure.toJordanDecomposition.posPart = 2⁻¹ • C.positivePrior","l":420,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.jordanPriors_ae_mem_range","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The positive and negative priors are concentrated on the finite set of certificate nodes.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), (∀ᵐ (x : ℝ) ∂C.positivePrior, x ∈ Set.range C.node) ∧ ∀ᵐ (x : ℝ) ∂C.negativePrior, x ∈ Set.range C.node","l":469,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.jordanPriors_moments_eq","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the requested moment degree, the degree condition. The two normalized Jordan priors have equal moments through degree L.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), ∀ j ≤ L, ∫ (x : ℝ), x ^ j ∂C.positivePrior = ∫ (x : ℝ), x ^ j ∂C.negativePrior","l":460,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.markedPoissonPredictive","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite node index set, the moment-matching degree, the finite signed certificate, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark branch and is given by the following defining expression. The one-coordinate prior-predictive marked-Poisson law obtained by mixing the zero-inflated variation prior over the latent mass.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → ℝ → ℝ → ℝ → ℝ → Bool → MeasureTheory.Measure Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation","l":113,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.markedPoissonPredictive_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite node index set, the moment-matching degree, the finite signed certificate, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the support ratio, the support upper bound, the outcome-mark branch, positive shift, the support-ratio identity, the compact-support condition. The one-coordinate marked-Poisson predictive law is a probability measure under the support assumptions that normalize the zero-inflated prior.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (ε a u v κ B : ℝ) (branch : Bool), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → MeasureTheory.IsProbabilityMeasure (C.markedPoissonPredictive ε a u v branch)","l":123,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.markedPoissonProductPredictive","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Product","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the number of independent coordinates, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark branch and is given by the following defining expression. The k-fold i.i.d. product of a one-coordinate marked-Poisson predictive law.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → (k : ℕ) → ℝ → ℝ → ℝ → ℝ → Bool → MeasureTheory.Measure (Fin k → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation)","l":20,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.markedPoissonProductPredictive_geometric_tv_le","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Product","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The stated conclusion follows from the finite signed certificate, the number of independent coordinates, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the support ratio, the support upper bound, the geometric-bound constant, the geometric decay factor, positive shift, the support-ratio identity, the compact-support condition, the one-coordinate TV bound. The geometric one-coordinate estimate composes with finite i.i.d. products, giving k times the one-coordinate bound.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (k : ℕ) (ε a u v κ B C₀ ρ : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → Causalean.Stat.tvDist (C.markedPoissonPredictive ε a u v false) (C.markedPoissonPredictive ε a u v true) ≤ C₀ * u * a * ρ ^ L → Causalean.Stat.tvDist (C.markedPoissonProductPredictive k ε a u v false) (C.markedPoissonProductPredictive k ε a u v true) ≤ C₀ * u * ↑k * a * ρ ^ L","l":71,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.markedPoissonProductPredictive_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Product","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the number of independent coordinates, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the support ratio, the support upper bound, the outcome-mark branch, positive shift, the support-ratio identity, the compact-support condition. The i.i.d. product marked-Poisson predictive law is a probability measure under the scalar prior's support assumptions.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (k : ℕ) (ε a u v κ B : ℝ) (branch : Bool), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → MeasureTheory.IsProbabilityMeasure (C.markedPoissonProductPredictive k ε a u v branch)","l":28,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.measurableSet_noLabeledTreated","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the displayed assumptions. The no-labeled-treated event is measurable.","s":"MeasurableSet Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.noLabeledTreated","l":146,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.measurable_polarSign","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The atomic polar sign is Borel measurable.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), Measurable C.polarSign","l":191,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.negativePrior","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The normalized negative Jordan prior, obtained by doubling the negative atomic mass of the signed certificate.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → MeasureTheory.Measure ℝ","l":226,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.negativePrior_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The negative Jordan prior is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), MeasureTheory.IsProbabilityMeasure C.negativePrior","l":407,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.noLabeledTreated","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the displayed assumptions and is given by the following defining expression. The event that no labeled treated observation appears in either outcome mark.","s":"Set Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation","l":141,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.ofFiniteMomentDual","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the target function, the lower endpoint, the upper endpoint, the alternation dual certificate and is given by the following defining expression. An existing finite alternation moment dual canonically supplies a normalized finite signed moment certificate.","s":"{L : ℕ} → {f : ℝ → ℝ} → {r s : ℝ} → Causalean.Mathlib.Analysis.FinitePolynomialAlternationDuality.FiniteMomentDual f r s L → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate (Fin (L + 2)) L","l":538,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPrior0","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the target function and is given by the following defining expression. The null prior oriented for a target f; the naming is chosen so that the target mean under prior one is at least that under prior zero.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → (ℝ → ℝ) → MeasureTheory.Measure ℝ","l":486,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPrior0_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the finite signed certificate, the target function and is given by the following defining expression. The target-oriented null prior is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (f : ℝ → ℝ), MeasureTheory.IsProbabilityMeasure (C.orientedPrior0 f)","l":498,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPrior1","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the target function and is given by the following defining expression. The alternative prior oriented for a target f; it swaps the Jordan priors exactly when the certificate evaluates f negatively.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → (ℝ → ℝ) → MeasureTheory.Measure ℝ","l":492,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPrior1_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the finite signed certificate, the target function and is given by the following defining expression. The target-oriented alternative prior is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (f : ℝ → ℝ), MeasureTheory.IsProbabilityMeasure (C.orientedPrior1 f)","l":505,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPrior_target_separation","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The stated conclusion follows from the finite signed certificate, the target function. Target orientation turns the signed evaluation into the exact nonnegative difference of target expectations under the two Jordan priors.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (f : ℝ → ℝ), ∫ (x : ℝ), f x ∂C.orientedPrior1 f - ∫ (x : ℝ), f x ∂C.orientedPrior0 f = 2 * |∑ i, C.weight i * f (C.node i)|","l":512,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.orientedPriors_moments_eq","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the target function, the requested moment degree, the degree condition. The target-oriented Jordan priors retain equal moments through degree L.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (f : ℝ → ℝ), ∀ j ≤ L, ∫ (x : ℝ), x ^ j ∂C.orientedPrior0 f = ∫ (x : ℝ), x ^ j ∂C.orientedPrior1 f","l":526,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.polarSign","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the evaluation point and is given by the following defining expression. The measurable atomic polar sign: it equals the sign of the unique weight at a certificate node and is zero away from all nodes.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → ℝ → ℝ","l":185,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.positivePrior","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The normalized positive Jordan prior, obtained by doubling the positive atomic mass of the signed certificate.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → MeasureTheory.Measure ℝ","l":220,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.positivePrior_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The positive Jordan prior is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), MeasureTheory.IsProbabilityMeasure C.positivePrior","l":394,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.restrict_markedPoissonPredictive_noLabeledTreated_eq","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite node index set, the moment-matching degree, the finite signed certificate, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the support ratio, the support upper bound, positive shift, the support-ratio identity, the compact-support condition. On the zero-labeled-count event, the two prior-predictive laws agree exactly; the auxiliary treated and aggregate control counts are branch invariant.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (ε a u v κ B : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → (C.markedPoissonPredictive ε a u v false).restrict Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.noLabeledTreated = (C.markedPoissonPredictive ε a u v true).restrict Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.noLabeledTreated","l":166,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.signedMeasure","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The atomic signed measure represented by a finite signed certificate.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → MeasureTheory.SignedMeasure ℝ","l":40,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.signedMeasure_eq_withDensity_polarSign","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. Polar decomposition of the atomic signed measure: the signed certificate is its variation weighted by the measurable polar sign.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), C.signedMeasure = (MeasureTheory.VectorMeasure.variation C.signedMeasure).withDensityᵥ C.polarSign","l":212,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.signedMeasure_univ_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The total signed mass of the certificate is zero, as the degree-zero moment constraint requires.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), C.signedMeasure Set.univ = 0","l":150,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.tvDist_markedPoissonPredictive_le_palm_aggregate","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite node index set, the moment-matching degree, the finite signed certificate, the overlap fraction, the support ratio, the positive shift, the support upper bound, the labeled treated intensity, the auxiliary treated intensity, positive overlap fraction, the overlap upper bound, the support-ratio identity, positive shift, positive upper bound, nonnegative labeled intensity, nonnegative auxiliary intensity, positive total intensity, the compact-support condition. The marked-law discrepancy is at most the labeled Palm intensity u * ε * a times the discrepancy between aggregate treated/control mixtures of the two Jordan priors. The result includes u = 0 and v = 0; only the total intensity must be positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (ε κ a B u v : ℝ), 0 < ε → ε < 1 / 2 → κ = (1 - 2 * ε) / ε → 0 < a → 0 < B → 0 ≤ u → 0 ≤ v → 0 < u + v → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → Causalean.Stat.tvDist (C.markedPoissonPredictive ε a u v false) (C.markedPoissonPredictive ε a u v true) ≤ u * ε * a * Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.positivePrior ε a (u + v)) (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive C.negativePrior ε a (u + v))","l":1080,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.tvDist_markedPoissonProductPredictive_le","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Product","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the number of independent coordinates, the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the support ratio, the support upper bound, the one-coordinate distance bound, positive shift, the support-ratio identity, the compact-support condition, the one-coordinate TV bound. A one-coordinate TV estimate tensorizes to k independent coordinates by tvDist_pi_iid_le.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (k : ℕ) (ε a u v κ B δ : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → Causalean.Stat.tvDist (C.markedPoissonPredictive ε a u v false) (C.markedPoissonPredictive ε a u v true) ≤ δ → Causalean.Stat.tvDist (C.markedPoissonProductPredictive k ε a u v false) (C.markedPoissonProductPredictive k ε a u v true) ≤ ↑k * δ","l":43,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.variance_coordinate_zeroInflatedProductPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, the coordinate index, positive shift, the support-ratio identity, the compact-support condition. Each product coordinate has variance equal to the scalar zero-inflated variance.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ) (i : Fin k), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ProbabilityTheory.variance (fun w => w i) (C.zeroInflatedProductPrior a k) = ProbabilityTheory.variance id (C.zeroInflatedPrior a)","l":253,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.variance_sum_zeroInflatedProductPrior","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, positive shift, the support-ratio identity, the compact-support condition. The variance of the sum of the finite i.i.d. coordinates is k times the scalar variance.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ProbabilityTheory.variance (fun w => ∑ i, w i) (C.zeroInflatedProductPrior a k) = ↑k * ProbabilityTheory.variance id (C.zeroInflatedPrior a)","l":298,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.variation_eq_absoluteMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate. The signed certificate's vector-measure variation is exactly its atomic absolute-weight measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), MeasureTheory.VectorMeasure.variation C.signedMeasure = C.absoluteMeasure","l":134,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.variation_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.Certificate","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the finite signed certificate and is given by the following defining expression. The variation of a normalized finite signed certificate is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L), MeasureTheory.IsProbabilityMeasure (MeasureTheory.VectorMeasure.variation C.signedMeasure)","l":173,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/Certificate.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedPrior","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the positive shift and is given by the following defining expression. The zero-inflated prior obtained by weighting the variation probability by a / (p + a) and assigning the residual probability to zero.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → ℝ → MeasureTheory.Measure ℝ","l":56,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedPrior_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, positive shift, the support-ratio identity, the compact-support condition. Under positive shift and positive compact support, the zero-inflated construction is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → MeasureTheory.IsProbabilityMeasure (C.zeroInflatedPrior a)","l":65,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedPrior_support","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, positive shift, the support-ratio identity, the compact-support condition. The zero-inflated prior is supported on zero together with the original compact support interval.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∀ᵐ (p : ℝ) ∂C.zeroInflatedPrior a, p = 0 ∨ p ∈ Set.Icc (a / κ) B","l":87,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedProductPrior","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the finite signed certificate, the positive shift, the number of independent coordinates and is given by the following defining expression. The finite i.i.d. product of the zero-inflated scalar prior.","s":"{ι : Type u_1} → [inst : Fintype ι] → {L : ℕ} → Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L → ℝ → (k : ℕ) → MeasureTheory.Measure (Fin k → ℝ)","l":192,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedProductPrior_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, positive shift, the support-ratio identity, the compact-support condition. The finite i.i.d. zero-inflated product is a probability measure.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → MeasureTheory.IsProbabilityMeasure (C.zeroInflatedProductPrior a k)","l":198,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate.zeroInflatedProductPrior_support","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.ZeroInflated","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the finite signed certificate, the positive shift, the support ratio, the support upper bound, the number of independent coordinates, positive shift, the support-ratio identity, the compact-support condition. Every coordinate of the finite i.i.d. prior lies almost surely at zero or in the original compact interval.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {L : ℕ} (C : Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.NormalizedFiniteSignedMomentCertificate ι L) (a κ B : ℝ) (k : ℕ), 0 < a → 0 < κ → (∀ᵐ (p : ℝ) ∂MeasureTheory.VectorMeasure.variation C.signedMeasure, p ∈ Set.Icc (a / κ) B) → ∀ᵐ (w : Fin k → ℝ) ∂C.zeroInflatedProductPrior a k, ∀ (i : Fin k), w i = 0 ∨ w i ∈ Set.Icc (a / κ) B","l":209,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/ZeroInflated.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregateControlRate","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the aggregate intensity, the latent mass and is given by the following defining expression.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":29,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonKernel","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the aggregate intensity and is given by the following defining expression. The aggregate affine-Poisson experiment as a kernel from latent mass to the treated/control count pair.","s":"ℝ → ℝ → ℝ → ProbabilityTheory.Kernel ℝ Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoissonObservation","l":70,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonKernel_apply","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the overlap fraction, the positive shift, the aggregate intensity, the latent mass. The aggregate kernel fibre is the explicit product-Poisson law.","s":"∀ (ε a t p : ℝ), (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonKernel ε a t) p = Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonLaw ε a t p","l":77,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonLaw","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the aggregate intensity, the latent mass and is given by the following defining expression. The explicit pair of independent Poisson laws with aggregate treated and control rates.","s":"ℝ → ℝ → ℝ → ℝ → MeasureTheory.Measure Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoissonObservation","l":33,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the overlap fraction, the positive shift, the aggregate intensity, the latent mass and is given by the following defining expression. Every aggregate two-count law is a probability measure.","s":"∀ (ε a t p : ℝ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonLaw ε a t p)","l":40,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregatePoissonPredictive","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the latent prior, the overlap fraction, the positive shift, the aggregate intensity and is given by the following defining expression. Mixing the aggregate affine-Poisson kernel against a latent prior gives its prior-predictive treated/control count law.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ → MeasureTheory.Measure Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoissonObservation","l":247,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.aggregateTreatedRate","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the aggregate intensity, the latent mass and is given by the following defining expression.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":26,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.branchMark","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the outcome-mark branch, the outcome-mark function and is given by the following defining expression. The signed outcome mark for a branch: the alternative uses h, while the null uses -h.","s":"Bool → ℝ → ℝ","l":36,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.controlMass","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the latent mass and is given by the following defining expression. The aggregate control mass left after subtracting the treated mass from the latent cell mass.","s":"ℝ → ℝ → ℝ → ℝ","l":32,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.exists_geometric_sqrt_exponentialSeriesTail_bound","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.AggregatePoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the tail-scale constant, positive tail-scale constant. A square root of an exponential-series tail is uniformly geometric in the matching degree when its argument is at most a sufficiently small fixed multiple of that degree.","s":"∀ (A : ℝ), 0 < A → ∃ b D ρ, 0 < b ∧ 0 < D ∧ ρ ∈ Set.Ioo 0 1 ∧ ∀ (L : ℕ) (x : ℝ), 0 ≤ x → x ≤ b * ↑L → √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail L (A * x)) ≤ D * ρ ^ L","l":325,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/AggregatePoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonKernel","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark function, measurability of the mark function, the outcome-mark branch and is given by the following defining expression. The four-count experiment as a probability kernel from the latent real mass to marked-Poisson observations.","s":"ℝ → ℝ → ℝ → ℝ → (h : ℝ → ℝ) → Measurable h → Bool → ProbabilityTheory.Kernel ℝ Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation","l":84,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonKernel_apply","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The stated conclusion follows from the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark function, measurability of the mark function, the outcome-mark branch, the latent mass. The marked-Poisson kernel's fibre is the explicit four-count law.","s":"∀ (ε a u v : ℝ) (h : ℝ → ℝ) (hh : Measurable h) (branch : Bool) (p : ℝ), (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonKernel ε a u v h hh branch) p = Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonLaw ε a u v h branch p","l":96,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonLaw","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark function, the outcome-mark branch, the latent mass and is given by the following defining expression. The explicit law of the four independent Poisson counts at latent p. Real rates are converted to nonnegative rates by Real.toNNReal; the main theorems' support assumptions ensure the intended rates are already nonnegative.","s":"ℝ → ℝ → ℝ → ℝ → (ℝ → ℝ) → Bool → ℝ → MeasureTheory.Measure Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoissonObservation","l":40,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonLaw_isProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":2,"d":"The defined object is determined by the overlap fraction, the positive shift, the labeled treated intensity, the auxiliary treated intensity, the outcome-mark function, the outcome-mark branch, the latent mass and is given by the following defining expression. The marked-Poisson law is a probability measure for every parameter.","s":"∀ (ε a u v : ℝ) (h : ℝ → ℝ) (branch : Bool) (p : ℝ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.markedPoissonLaw ε a u v h branch p)","l":54,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.treatedMass","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.FiniteSignedMomentMarkedPoissonMixture.MarkedPoisson","a":"Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture","t":1,"d":"The defined object is determined by the overlap fraction, the positive shift, the latent mass and is given by the following defining expression. The treated mass attached to latent mass p, overlap level ε, and shift a.","s":"ℝ → ℝ → ℝ → ℝ","l":28,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/FiniteSignedMomentMarkedPoissonMixture/MarkedPoisson.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.abs_priorMoment_le_pow_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The absolute raw moment of order n of a probability prior is at most the support radius raised to that order when the radius is nonnegative and the prior is supported within that radius.","s":"∀ (π : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure π] (a : ℝ) (n : ℕ), 0 ≤ a → π {θ | |θ| ≤ a} = 1 → |Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment π n| ≤ a ^ n","l":126,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.ae_abs_le_of_measure_interval_eq_one","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"A probability prior that assigns mass one to the interval of parameters with absolute value at most the chosen radius is almost surely concentrated on that interval.","s":"∀ (π : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure π] (a : ℝ), π {θ | |θ| ≤ a} = 1 → ∀ᵐ (θ : ℝ) ∂π, |θ| ≤ a","l":19,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given two real-valued priors and an interaction scale, their exponential prior energy averages the exponential of the scaled product over independent draws from the priors, as a nested Bochner integral: an inner or outer integrand that is not integrable contributes zero, so this is the exponential prior energy only when those integrals exist (for example for compactly supported probability priors).","s":"MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → ℝ → ℝ","l":31,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy_quadratic_eq_tsum","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given two probability priors, a nonnegative interaction scale, and a nonnegative common support radius, if both priors are supported within that radius, then the alternating sum of their four exponential energies equals the convergent exponential series of squared moment differences.","s":"∀ (π0 π1 : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] (lambda a : ℝ), 0 ≤ lambda → 0 ≤ a → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π0 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π1 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π0 lambda + Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π1 lambda = ∑' (n : ℕ), lambda ^ n / ↑n.factorial * (Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment π0 n - Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment π1 n) ^ 2","l":164,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy_quadratic_le_tail","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given two probability priors, a nonnegative interaction scale, a nonnegative common support radius, and a matching degree, if both priors are supported within that radius and their moments agree through the matching degree, then their alternating exponential energy is at most four times the unmatched exponential-series tail.","s":"∀ (π0 π1 : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment π0 n = Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment π1 n) → Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π0 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π1 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π0 lambda + Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π1 lambda ≤ 4 * Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2)","l":190,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a matched degree and a real series argument, the exponential-series tail is the sum of the exponential-series terms whose degree is strictly larger than the matched degree.","s":"ℕ → ℝ → ℝ","l":18,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integrable_mixtureLikelihood_mul_of_supported_exponentialGram","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The product of likelihood averages under two probability priors is integrable under the observation law when the priors are supported on the interval of parameters with absolute value at most the chosen radius, the observation law is a probability law, the likelihood is jointly measurable and nonnegative on that interval, and its supported fibres have exponential inner products at a nonnegative interaction scale, with a nonnegative support radius.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π ρ : MeasureTheory.Measure ℝ) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] [MeasureTheory.IsProbabilityMeasure ρ] [MeasureTheory.IsProbabilityMeasure Q] (lambda a : ℝ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (θ' : ℝ), |θ'| ≤ a → ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π {θ | |θ| ≤ a} = 1 → ρ {θ | |θ| ≤ a} = 1 → MeasureTheory.Integrable (fun x => Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood x * Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood ρ likelihood x) Q","l":274,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integrable_mixtureLikelihood_of_supported_density","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The prior average of a likelihood family is integrable under a common dominating measure when the experiment kernel has probability fibres, the dominating measure is sigma-finite, the prior is supported on the interval of parameters with absolute value at most the chosen radius, the likelihood is jointly measurable and nonnegative on that interval, and supported kernel fibres have the stated likelihood densities.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] [MeasureTheory.SigmaFinite Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (a : ℝ), π {θ | |θ| ≤ a} = 1 → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → MeasureTheory.Integrable (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood) Q","l":51,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integral_priorPredictive","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"The expectation of an integrable real-valued statistic under the mixture from a prior and experiment kernel equals the prior average of its conditional expectations.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (f : X → ℝ), MeasureTheory.Integrable f (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K) → ∫ (x : X), f x ∂Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K = ∫ (θ : Θ), ∫ (x : X), f x ∂K θ ∂π","l":147,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integral_sq_mixtureLikelihood_sub_eq_exponentialPriorEnergy","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given two probability priors, a probability dominating measure, a jointly measurable nonnegative likelihood family, a nonnegative interaction scale, and a nonnegative support radius, if likelihood inner products have the exponential product form and both priors are supported within the radius, then the squared distance between their mixture likelihoods equals the alternating sum of the four exponential prior energies.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q] (lambda a : ℝ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ θ' : ℝ), ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → ∫ (x : X), (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π0 likelihood x - Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π1 likelihood x) ^ 2 ∂Q = Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π0 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π1 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π0 lambda + Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π1 lambda","l":85,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integral_sq_mixtureLikelihood_sub_eq_exponentialPriorEnergy_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The integrated squared difference between likelihood averages under two probability priors equals the alternating sum of their four exponential prior energies when the priors are supported on the interval of parameters with absolute value at most the chosen radius, the observation law is a probability law, the likelihood is jointly measurable and nonnegative on that interval, and supported fibres have exponential inner products at a nonnegative interaction scale, with a nonnegative support radius.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q] (lambda a : ℝ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (θ' : ℝ), |θ'| ≤ a → ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → ∫ (x : X), (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π0 likelihood x - Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π1 likelihood x) ^ 2 ∂Q = Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π0 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π0 π1 lambda - Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π0 lambda + Causalean.Stat.Minimax.MomentMatchedMixture.exponentialPriorEnergy π1 π1 lambda","l":295,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integral_sq_mixtureLikelihood_sub_le_tail","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given two probability priors, a probability dominating measure, a jointly measurable nonnegative likelihood family, a nonnegative interaction scale, a nonnegative support radius, and a matching degree, if likelihood inner products have the exponential product form, both priors are supported within the radius, and their moments agree through that degree, then the squared distance between their mixture likelihoods is at most four times the unmatched exponential-series tail.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q] (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ θ' : ℝ), ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → ∫ (x : X), (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π0 likelihood x - Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π1 likelihood x) ^ 2 ∂Q ≤ 4 * Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2)","l":230,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.integral_sq_mixtureLikelihood_sub_le_tail_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The integrated squared difference between likelihood averages under two probability priors is at most four times the unmatched exponential-series tail when their moments agree through the chosen degree, the priors are supported on the interval of parameters with absolute value at most the chosen radius, the observation law is a probability law, the likelihood is jointly measurable and nonnegative on that interval, and supported fibres have exponential inner products at a nonnegative interaction scale, with a nonnegative support radius.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q] (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (θ' : ℝ), |θ'| ≤ a → ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → ∫ (x : X), (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π0 likelihood x - Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π1 likelihood x) ^ 2 ∂Q ≤ 4 * Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2)","l":355,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.lintegral_priorPredictive","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"The lower integral of a nonnegative measurable statistic under the mixture from a prior and experiment kernel equals the iterated prior-then-experiment lower integral.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) (f : X → ENNReal), Measurable f → ∫⁻ (x : X), f x ∂Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K = ∫⁻ (θ : Θ), ∫⁻ (x : X), f x ∂K θ ∂π","l":139,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.measurable_mixtureLikelihood","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The average under a sigma-finite prior of a real-valued likelihood family that is jointly measurable in parameter and observation is measurable as a function of the observation.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) [MeasureTheory.SFinite π] (likelihood : ℝ → X → ℝ), (Measurable fun p => likelihood p.1 p.2) → Measurable (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood)","l":29,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a real-valued prior, a parameter-indexed likelihood family, and an observation, the mixture likelihood is the prior average of the component likelihoods at that observation, taken as a Bochner integral: it is zero when the likelihood is not integrable against the prior, so it is the mixture likelihood for a probability prior and an integrable likelihood.","s":"{X : Type u_1} → MeasureTheory.Measure ℝ → (ℝ → X → ℝ) → X → ℝ","l":20,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood_nonnegative_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The average under a probability prior of a real-valued likelihood family at an observation is nonnegative when the prior is supported on the interval of parameters with absolute value at most the chosen radius and the likelihood is nonnegative throughout that interval, regardless of its values elsewhere.","s":"∀ {X : Type u_1} [MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] (a : ℝ), π {θ | |θ| ≤ a} = 1 → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → ∀ (x : X), 0 ≤ Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood x","l":38,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.momentMatchedMixture_tv_le_sqrt_tail","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given two probability priors, a probability experiment kernel, a probability dominating law, a jointly measurable nonnegative likelihood family, a nonnegative interaction scale, a nonnegative support radius, and a matching degree, if each experiment law has the stated density, likelihood inner products have the exponential product form, both priors are supported within the radius, and their moments agree through that degree, then the two one-coordinate prior-predictive mixtures are within the square root of the explicit unmatched series tail in total variation.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (∀ (θ θ' : ℝ), ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π1 K) ≤ √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2))","l":368,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.momentMatchedMixture_tv_le_sqrt_tail_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The predictive laws formed from two probability priors and an experiment kernel with probability fibres are within the square root of the unmatched exponential-series tail in total variation when their moments agree through the chosen degree, the priors are supported on the interval of parameters with absolute value at most the chosen radius, the common dominating observation law is a probability law, and the likelihood family is jointly measurable while only on that interval it is nonnegative, represents the kernel densities, and has exponential inner products at a nonnegative interaction scale, with a nonnegative support radius.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π0 π1 : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (θ' : ℝ), |θ'| ≤ a → ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π1 K) ≤ √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2))","l":380,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.momentMatchedProductMixture_tv_le","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a dimension, two probability priors, a probability experiment kernel, a probability dominating law, a jointly measurable nonnegative likelihood family, a nonnegative interaction scale, a nonnegative support radius, and a matching degree, if each experiment law has the stated density, likelihood inner products have the exponential product form, both priors are supported within the radius, and their moments agree through that degree, then the product prior-predictive mixtures are within the dimension times the square root of the explicit unmatched series tail in total variation.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (π0 π1 : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (∀ (θ θ' : ℝ), ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π1 K) ≤ ↑d * √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2))","l":262,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.momentMatchedProductMixture_tv_le_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"The finite products over the chosen number of independent coordinates of predictive laws formed from two probability priors and an experiment kernel with probability fibres are within the dimension times the square root of the unmatched exponential-series tail in total variation when their moments agree through the chosen degree, the priors are supported on the interval of parameters with absolute value at most the chosen radius, the common dominating observation law is a probability law, and the likelihood family is jointly measurable while only on that interval it is nonnegative, represents the kernel densities, and has exponential inner products at a nonnegative interaction scale, with a nonnegative support radius.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (π0 π1 : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π0] [MeasureTheory.IsProbabilityMeasure π1] [MeasureTheory.IsProbabilityMeasure Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (lambda a : ℝ) (degree : ℕ), 0 ≤ lambda → 0 ≤ a → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (θ' : ℝ), |θ'| ≤ a → ∫ (x : X), likelihood θ x * likelihood θ' x ∂Q = Real.exp (lambda * θ * θ')) → π0 {θ | |θ| ≤ a} = 1 → π1 {θ | |θ| ≤ a} = 1 → (∀ n ≤ degree, ∫ (θ : ℝ), θ ^ n ∂π0 = ∫ (θ : ℝ), θ ^ n ∂π1) → Causalean.Stat.tvDist (Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π0 K) (Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π1 K) ≤ ↑d * √(Causalean.Stat.Minimax.MomentMatchedMixture.exponentialSeriesTail degree (lambda * a ^ 2))","l":473,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorMoment","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.ExponentialEnergy","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a real-valued prior and a nonnegative integer order, its raw moment of that order is the prior average of the corresponding power, taken as a Bochner integral, which is zero when the power is not integrable. It coincides with Causalean.Stat.MomentProblems.rawMoment.","s":"MeasureTheory.Measure ℝ → ℕ → ℝ","l":24,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/ExponentialEnergy.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive","k":"def","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":1,"d":"Given a prior measure and a measurable experiment kernel, the prior-predictive law first draws a parameter from the prior and then draws an observation from the experiment at that parameter.","s":"{Θ : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace X] → MeasureTheory.Measure Θ → ProbabilityTheory.Kernel Θ X → MeasureTheory.Measure X","l":114,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_absolutelyContinuous","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a probability prior, an experiment kernel, a dominating measure, and a jointly measurable nonnegative likelihood family, if each experiment law has the stated density, then the prior-predictive mixture is absolutely continuous with respect to the dominating measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π], (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K).AbsolutelyContinuous Q","l":255,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_absolutelyContinuous_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The predictive law obtained by averaging a probability prior through an experiment kernel is absolutely continuous with respect to a common measure when the prior is supported on the interval of parameters with absolute value at most the chosen radius and, on that interval, the likelihood family represents every kernel fibre as a density relative to the common measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] (a : ℝ), π {θ | |θ| ≤ a} = 1 → (∀ (θ : ℝ), |θ| ≤ a → K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K).AbsolutelyContinuous Q","l":155,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_apply","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"The mass that a prior and experiment kernel assign to a measurable observation event equals the prior average of its conditional event probabilities.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) {A : Set X}, MeasurableSet A → (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K) A = ∫⁻ (θ : Θ), (K θ) A ∂π","l":131,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_eq_withDensity_mixtureLikelihood","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a probability prior, an experiment kernel, a σ-finite dominating measure, and a jointly measurable nonnegative likelihood family, if every experiment law is a probability law and has the stated density against the dominating measure, then the prior-predictive law has density equal to the prior average of the component likelihoods.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] [MeasureTheory.SigmaFinite Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ) (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K = Q.withDensity fun x => ENNReal.ofReal (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood x)","l":28,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_eq_withDensity_mixtureLikelihood_of_supported","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.SupportLocalized","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"The predictive law obtained by averaging a probability prior through an experiment kernel with probability fibres equals the common dominating measure weighted by the prior-averaged likelihood when the dominating measure is sigma-finite, the likelihood family is jointly measurable, the prior is supported on the interval of parameters with absolute value at most the chosen radius, and only on that interval the likelihood is nonnegative and represents each kernel fibre's density.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (Q : MeasureTheory.Measure X) (likelihood : ℝ → X → ℝ) [MeasureTheory.IsProbabilityMeasure π] [MeasureTheory.SigmaFinite Q], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → ∀ (a : ℝ), π {θ | |θ| ≤ a} = 1 → (Measurable fun p => likelihood p.1 p.2) → (∀ (θ : ℝ), |θ| ≤ a → ∀ (x : X), 0 ≤ likelihood θ x) → (∀ (θ : ℝ), |θ| ≤ a → K θ = Q.withDensity fun x => ENNReal.ofReal (likelihood θ x)) → Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K = Q.withDensity fun x => ENNReal.ofReal (Causalean.Stat.Minimax.MomentMatchedMixture.mixtureLikelihood π likelihood x)","l":91,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/SupportLocalized.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_isProbability","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"Mixing probability experiment laws from a measurable kernel against a probability prior produces a probability law on observations.","s":"∀ {Θ : Type u_1} {X : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : MeasurableSpace X] (π : MeasureTheory.Measure Θ) (K : ProbabilityTheory.Kernel Θ X) [MeasureTheory.IsProbabilityMeasure π], (∀ (θ : Θ), MeasureTheory.IsProbabilityMeasure (K θ)) → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K)","l":120,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_productPrior_apply_pi","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a dimension, a probability prior, a one-coordinate experiment kernel with probability component laws, and a product experiment kernel whose fibres are the coordinatewise product laws, the mixture mass of a coordinate rectangle factors into the product of the one-coordinate predictive masses.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (productKernel : ProbabilityTheory.Kernel (Fin d → ℝ) (Fin d → X)) [MeasureTheory.IsProbabilityMeasure π], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → (∀ (θ : Fin d → ℝ), productKernel θ = MeasureTheory.Measure.pi fun i => K (θ i)) → ∀ (s : Fin d → Set X), (∀ (i : Fin d), MeasurableSet (s i)) → (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive (Causalean.Stat.Minimax.MomentMatchedMixture.productPrior d π) productKernel) (Set.univ.pi s) = ∏ i, (Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive π K) (s i)","l":41,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive_productPrior_eq_productPriorPredictive","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a dimension, a probability prior, a one-coordinate experiment kernel with probability component laws, and a product experiment kernel whose fibres are the coordinatewise product laws, mixing the product experiment against the product prior equals the product of the one-coordinate prior-predictive mixtures.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) (productKernel : ProbabilityTheory.Kernel (Fin d → ℝ) (Fin d → X)) [MeasureTheory.IsProbabilityMeasure π], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → (∀ (θ : Fin d → ℝ), productKernel θ = MeasureTheory.Measure.pi fun i => K (θ i)) → Causalean.Stat.Minimax.MomentMatchedMixture.priorPredictive (Causalean.Stat.Minimax.MomentMatchedMixture.productPrior d π) productKernel = Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π K","l":102,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.productPrior","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a dimension and a scalar prior, the independent coordinate prior is the finite product of that prior across all coordinates.","s":"(d : ℕ) → MeasureTheory.Measure ℝ → MeasureTheory.Measure (Fin d → ℝ)","l":19,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive","k":"def","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":1,"d":"Given a dimension, a scalar prior, and a one-coordinate experiment kernel, the independent coordinate prior-predictive law is the finite product of the one-coordinate mixture law.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (d : ℕ) → MeasureTheory.Measure ℝ → ProbabilityTheory.Kernel ℝ X → MeasureTheory.Measure (Fin d → X)","l":24,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive_isProbability","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a dimension, a probability prior, and an experiment kernel whose component laws are probability laws, the independent coordinate prior-predictive law is a probability law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (π : MeasureTheory.Measure ℝ) (K : ProbabilityTheory.Kernel ℝ X) [MeasureTheory.IsProbabilityMeasure π], (∀ (θ : ℝ), MeasureTheory.IsProbabilityMeasure (K θ)) → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Minimax.MomentMatchedMixture.productPriorPredictive d π K)","l":31,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.tvDist_le_half_sqrt_integral_sq_density_sub","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Analytic","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given two probability laws, a probability dominating law, and two measurable nonnegative densities, if the laws have those densities and their squared difference is integrable, then their total variation distance is at most half the square root of the integrated squared density difference.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (μ ν Q : MeasureTheory.Measure X) (p q : X → ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [MeasureTheory.IsProbabilityMeasure Q], Measurable p → Measurable q → (∀ (x : X), 0 ≤ p x) → (∀ (x : X), 0 ≤ q x) → (μ = Q.withDensity fun x => ENNReal.ofReal (p x)) → (ν = Q.withDensity fun x => ENNReal.ofReal (q x)) → MeasureTheory.Integrable (fun x => (p x - q x) ^ 2) Q → Causalean.Stat.tvDist μ ν ≤ 1 / 2 * √(∫ (x : X), (p x - q x) ^ 2 ∂Q)","l":274,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Analytic.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.tvDist_pi_iid_le","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given a number of coordinates and two one-coordinate probability laws, the total variation distance between their finite independent product laws is at most the number of coordinates times their one-coordinate distance.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (d : ℕ) (μ ν : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.tvDist (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) ≤ ↑d * Causalean.Stat.tvDist μ ν","l":187,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.MomentMatchedMixture.tvDist_prod_le_add","k":"theorem","m":"Causalean.Stat.Minimax.MomentMatchedMixture.Product","a":"Stat/Minimax/MomentMatchedMixture","t":2,"d":"Given two probability laws on one coordinate and two probability laws on a second coordinate, the total variation distance between the binary product laws is at most the sum of the two coordinatewise distances.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {Y : Type u_2} [inst_1 : MeasurableSpace Y] (μ0 μ1 : MeasureTheory.Measure X) (ν0 ν1 : MeasureTheory.Measure Y) [MeasureTheory.IsProbabilityMeasure μ0] [MeasureTheory.IsProbabilityMeasure μ1] [MeasureTheory.IsProbabilityMeasure ν0] [MeasureTheory.IsProbabilityMeasure ν1], Causalean.Stat.tvDist (μ0.prod ν0) (μ1.prod ν1) ≤ Causalean.Stat.tvDist μ0 μ1 + Causalean.Stat.tvDist ν0 ν1","l":123,"f":"Causalean/Stat/Minimax/MomentMatchedMixture/Product.lean"},{"n":"Causalean.Stat.Minimax.leCam_two_point_L1_lower","k":"theorem","m":"Causalean.Stat.Minimax.LeCamTwoPoint","a":"Stat/Minimax","t":1,"d":"Le Cam two-point lower bound on finite Bochner L¹ risk. Fix a strictly positive KL budget C with C at most 1/2. Then there is a universal positive constant c₀ such that, for every sample size, every pair of single-observation laws whose scaled KL divergence is bounded by C and whose n-fold product KL divergence is controlled by a supplied product-KL comparison and a supplied product-form Pinsker bound, every pair of separated real targets, and every measurable estimator with integrable absolute loss under both n-fold product laws, the worst-case Bochner L¹ risk is at least c₀ times the target separation.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (C : ℝ), 0 < C → C ≤ 1 / 2 → ∃ c₀, 0 < c₀ ∧ ∀ (n : ℕ), 1 ≤ n → ∀ (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q] (θP θQ δ : ℝ), ↑n * (InformationTheory.klDiv P Q).toReal ≤ C → (InformationTheory.klDiv (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q)).toReal ≤ ↑n * (InformationTheory.klDiv P Q).toReal → Causalean.Stat.PinskerBound (MeasureTheory.Measure.pi fun x => P) (MeasureTheory.Measure.pi fun x => Q) → 0 ≤ δ → δ ≤ |θP - θQ| → ∀ (T : (Fin n → Ω) → ℝ), Measurable T → MeasureTheory.Integrable (fun ω => |T ω - θP|) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (fun ω => |T ω - θQ|) (MeasureTheory.Measure.pi fun x => Q) → c₀ * δ ≤ max (∫ (ω : Fin n → Ω), |T ω - θP| ∂MeasureTheory.Measure.pi fun x => P) (∫ (ω : Fin n → Ω), |T ω - θQ| ∂MeasureTheory.Measure.pi fun x => Q)","l":67,"f":"Causalean/Stat/Minimax/LeCamTwoPoint.lean"},{"n":"Causalean.Stat.Minimax.le_cam_two_point_mse","k":"theorem","m":"Causalean.Stat.Minimax.LeCamTwoPoint","a":"Stat/Minimax","t":1,"d":"Le Cam two-point reduction in mean-squared-error form, uniform over a finite KL budget. For every Kullback–Leibler budget K, there is a single positive constant c_K (here exp(−K)/32), chosen before the laws, such that for every pair of probability laws Q₀, Q₁ whose divergence obeys KL(Q₀, Q₁) ≤ K, any measurable estimator with integrable squared loss under both laws has worst-case mean-squared error at least c_K times the squared separation of the two candidate parameter values. The finite-budget hypothesis is encoded as the ℝ≥0∞ inequality klDiv Q₀ Q₁ ≤ ENNReal.ofReal K, which forces a finite divergence (so it cannot be met vacuously by an infinite divergence collapsing under .toReal) and pins c_K to K alone.","s":"∀ (K : ℝ), ∃ cK, 0 < cK ∧ ∀ {S : Type u_1} [inst : MeasurableSpace S] (Q0 Q1 : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure Q0] [MeasureTheory.IsProbabilityMeasure Q1] (theta0 theta1 : ℝ), InformationTheory.klDiv Q0 Q1 ≤ ENNReal.ofReal K → ∀ (T : S → ℝ), Measurable T → MeasureTheory.Integrable (fun s => (T s - theta0) ^ 2) Q0 → MeasureTheory.Integrable (fun s => (T s - theta1) ^ 2) Q1 → cK * (theta1 - theta0) ^ 2 ≤ max (∫ (s : S), (T s - theta0) ^ 2 ∂Q0) (∫ (s : S), (T s - theta1) ^ 2 ∂Q1)","l":213,"f":"Causalean/Stat/Minimax/LeCamTwoPoint.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"μ is admissible for the envelope at level v: a probability measure a.e. supported in [0,1] with second moment ∫ y² ∂μ = v². These are exactly the laws over which the residual l2ResidualQuadratic is maximized to give ρ(v).","s":"ℝ → MeasureTheory.Measure ℝ → Prop","l":77,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible.moment2_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Second moment of an admissible law in rawMoment-form: rawMoment μ 2 = v².","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.rawMoment μ 2 = v ^ 2","l":98,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.continuous_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The continuous polynomial t ↦ envelopeQuartic t q.","s":"∀ (q : ℝ), Continuous fun t => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q","l":63,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.crossMoment_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"0 ≤ crossMoment m q m₃ m₄. The certificate cross moment equals the integral ∫ y (1 − y) (y − xᵥ)² ∂μ (expand the degree-4 polynomial and integrate term by term, matching the definition of crossMoment); on [0,1] the integrand is ≥ 0, so the integral is ≥ 0.","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment (Causalean.Stat.MomentProblems.rawMoment μ 1) (Causalean.Stat.MomentProblems.rawMoment μ 2) (Causalean.Stat.MomentProblems.rawMoment μ 3) (Causalean.Stat.MomentProblems.rawMoment μ 4)","l":82,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_deriv_neg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Strict negativity of the derivative on (v², v). For v ∈ (0,1) and t ∈ (v², v), 4 t³ − 6 t² + 4 v² t − 2 v⁴ < 0. (This is the derivative of envelopeQuartic · (v²).) Proof: nlinarith from v² < t, t < v, 0 < v, v < 1.","s":"∀ (v t : ℝ), 0 < v → v < 1 → v ^ 2 < t → t < v → 4 * t ^ 3 - 6 * t ^ 2 + 4 * v ^ 2 * t - 2 * (v ^ 2) ^ 2 < 0","l":105,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_neg_at_v","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"envelopeQuartic v (v²) = −2 v³ (v − 1)² < 0 for v ∈ (0,1). This is the negative endpoint of the sign change (at t = v = √q) that locates the interior root.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic v (v ^ 2) < 0","l":51,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.envelopeQuartic_pos_at_q","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"envelopeQuartic q q = q² (1 − q)² > 0 for q ∈ (0,1). This is the positive endpoint of the sign change that locates the interior root.","s":"∀ (q : ℝ), 0 < q → q < 1 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic q q","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the extremal three-point measure places the nonnegative parts of the three prescribed weights at $0$, at the associated interior support point, and at $1$, respectively.","s":"ℝ → MeasureTheory.Measure ℝ","l":63,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_admissible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The extremal law is admissible.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v)","l":329,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_isProb","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"extremalMeasure v is a probability measure (total mass w₀ + w₁ + w₂ = 1, weights ≥ 0).","s":"∀ (v : ℝ), 0 < v → v < 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v)","l":205,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment1","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"First moment: ∫ y ∂(extremalMeasure v) = μᵥ. Algebraic: w₁ xᵥ + w₂ = μᵥ.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.rawMoment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 1 = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v","l":234,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Second moment: ∫ y² ∂(extremalMeasure v) = v². Algebraic: w₁ xᵥ² + w₂ = v².","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.rawMoment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 2 = v ^ 2","l":246,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment3","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Third moment matches the extremal M₃: w₁ xᵥ³ + w₂ = extremalM3 μᵥ (v²). Pure algebra (field_simp/ring) using xᵥ = extremalMid μᵥ (v²) and the weight formulas.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.rawMoment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 3 = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3 (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2)","l":258,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment4","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Fourth moment matches the extremal M₄: w₁ xᵥ⁴ + w₂ = extremalM4 μᵥ (v²). Pure algebra.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.rawMoment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) 4 = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4 (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2)","l":294,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_moment_pow","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Raw moments of the extremal law. For k ≥ 1, ∫ yᵏ ∂(extremalMeasure v) = w₁ xᵥᵏ + w₂ (the mass at 0 contributes nothing, the mass at 1 contributes w₂). Proof: expand via integral_add_measure, integral_smul_measure, integral_dirac, using w_i ≥ 0 to convert ENNReal.ofReal wᵢ back to wᵢ, and (0:ℝ)^k = 0, (1:ℝ)^k = 1.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∀ {k : ℕ}, 1 ≤ k → Causalean.Stat.MomentProblems.rawMoment (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) k = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1 v * Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp v ^ k + Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2 v","l":174,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_residual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The extremal law realizes ρ(v). Its residual equals the envelope value: l2ResidualQuadratic (extremalMeasure v) = rhoEnvelope v.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v) = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":336,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure_supp","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"extremalMeasure v is a.e. supported in [0,1] (its atoms 0, xᵥ, 1 all lie in [0,1]).","s":"∀ (v : ℝ), 0 < v → v < 1 → ∀ᵐ (y : ℝ) ∂Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalMeasure v, y ∈ Set.Icc 0 1","l":223,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the interior support point of the extremal three-point law is $(\\mu_v^2-2\\mu_vv^2+v^2)/(2\\mu_v(1-\\mu_v))$, where $\\mu_v$ is the selected maximizing first moment at second moment $v^2$.","s":"ℝ → ℝ","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The interior support point lies in (0,1): xᵥ = (μᵥ² − 2 μᵥ v² + v²)/(2 μᵥ (1 − μᵥ)) ∈ (0,1) for μᵥ ∈ (v², v). Proof: nlinarith from v² < μᵥ < v < 1.","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalSupp v ∈ Set.Ioo 0 1","l":73,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW0","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the weight on the support point $0$ is one minus the weights assigned to the interior support point and to $1$.","s":"ℝ → ℝ","l":57,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the weight on the interior support point is $ (\\mu_v-v^2)/(x_v(1-x_v)) $, where $\\mu_v$ is the selected maximizing first moment and $x_v$ is the interior support point.","s":"ℝ → ℝ","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the weight on the support point $1$ is $ (v^2-\\mu_v x_v)/(1-x_v) $, where $\\mu_v$ is the selected maximizing first moment and $x_v$ is the interior support point.","s":"ℝ → ℝ","l":49,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"The three weights are nonnegative for v ∈ (0,1). Proof: w₁ > 0 since μᵥ > v² and xᵥ ∈ (0,1); w₂ ≥ 0 since v² ≥ μᵥ xᵥ; w₀ ≥ 0 since w₁ + w₂ ≤ 1. All via nlinarith from v² < μᵥ < v and extremalSupp_mem.","s":"∀ (v : ℝ), 0 < v → v < 1 → 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW0 v ∧ 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW1 v ∧ 0 ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.extremalW2 v","l":100,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.finiteMoment4_of_admissible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Every power y ↦ yᵏ (k ≤ 4) is integrable against an admissible law: on [0,1] we have |yᵏ| ≤ 1, and μ is a finite measure, so Integrable.mono' against the constant 1 applies.","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ","l":38,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.hasDerivAt_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Derivative of the quartic. d/dt envelopeQuartic t q = 4 t³ − 6 t² + 4 q t − 2 q².","s":"∀ (t q : ℝ), HasDerivAt (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic s q) (4 * t ^ 3 - 6 * t ^ 2 + 4 * q * t - 2 * q ^ 2) t","l":69,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.interior_quartic_exists","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"Interior existence. For v ∈ (0,1) and q = v², the quartic has a root strictly inside (v², v). Proof: the sign change envelopeQuartic (v²) (v²) > 0, envelopeQuartic v (v²) < 0 plus continuity, via the intermediate value theorem (intermediate_value_Ioo').","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃ u ∈ Set.Ioo (v ^ 2) v, Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u (v ^ 2) = 0","l":88,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.interior_quartic_unique_root","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Unique interior root of the FOC quartic. For v strictly between 0 and 1 (write q = v²), there is a unique μᵥ ∈ (v², v) with envelopeQuartic μᵥ (v²) = 0. This μᵥ is the envelope maximizer selecting ρ(v). Existence is interior_quartic_exists; uniqueness follows from strict antitonicity (strictAntiOn_envelopeQuartic), whose InjOn forces two roots to coincide.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃! u, u ∈ Set.Ioo (v ^ 2) v ∧ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u (v ^ 2) = 0","l":138,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.l2ResidualQuadratic_eq_zero_of_degenerate","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"In the degenerate case m² = q the closed-form residual is 0 (the Hankel ratio's denominator m₁² − m₂ vanishes, so momentResidual _ _ _ _ = _ / 0 = 0).","s":"∀ {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 = Causalean.Stat.MomentProblems.rawMoment μ 2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ = 0","l":124,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.l2ResidualQuadratic_le_rho","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Measure-level sharp upper bound. For v strictly between 0 and 1 and every admissible probability law μ on [0,1] with second moment v², the closed-form residual is at most the envelope value: l2ResidualQuadratic μ ≤ rhoEnvelope v.","s":"∀ (v : ℝ) (μ : MeasureTheory.Measure ℝ), Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":132,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the selected maximizing root is the unique root in $(v^2,v)$ of the envelope first-order quartic when $0<v<1$, and is $0$ otherwise.","s":"ℝ → ℝ","l":32,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"maximizingRoot v lies in the open interval (v², v).","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v ∈ Set.Ioo (v ^ 2) v","l":41,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot_quartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"maximizingRoot v is a root of the FOC quartic at q = v².","s":"∀ (v : ℝ), 0 < v → v < 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.maximizingRoot v) (v ^ 2) = 0","l":47,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.moment1_sq_le_moment2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"m² ≤ q: the variance is nonnegative. For a probability measure, (∫ y)² ≤ ∫ y² (Cauchy–Schwarz / Jensen).","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 ≤ Causalean.Stat.MomentProblems.rawMoment μ 2","l":62,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.moment2_le_moment1","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Bounds","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"q ≤ m: the second moment is at most the first, because y² ≤ y a.e. on [0,1]. Uses integral_mono_ae on the a.e. support bound plus integrability of y, y².","s":"∀ {v : ℝ} {μ : MeasureTheory.Measure ℝ}, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ → Causalean.Stat.MomentProblems.rawMoment μ 2 ≤ Causalean.Stat.MomentProblems.rawMoment μ 1","l":51,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Bounds.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.residualSet","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the set of residual values realized by admissible laws contains exactly the quadratic least-squares residuals of probability measures that are almost surely supported on $[0,1]$ and have second moment $v^2$.","s":"ℝ → Set ℝ","l":88,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"For a real number $v$, the measure-level residual envelope is $(\\mu_v-v^2)(v^2-\\mu_v^2)/(4\\mu_v(1-\\mu_v))$, where $\\mu_v$ is the selected maximizing root at second moment $v^2$.","s":"ℝ → ℝ","l":53,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Defs","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"The envelope value ρ(v) is strictly positive for v strictly between 0 and 1: momentEnvelope μᵥ (v²) = (μᵥ − v²)(v² − μᵥ²) / (4 μᵥ (1 − μᵥ)) has all four factors positive when μᵥ ∈ (v², v).","s":"∀ (v : ℝ), 0 < v → v < 1 → 0 < Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":61,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Defs.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rho_envelope_attained","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Attainment","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Envelope attainment. For v strictly between 0 and 1, there is an admissible law whose residual is exactly ρ(v). Together with the upper bound this makes ρ(v) the supremum.","s":"∀ (v : ℝ), 0 < v → v < 1 → ∃ μ, Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.Admissible v μ ∧ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ = Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v","l":364,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Attainment.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rho_envelope_isLUB","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.Envelope","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":1,"d":"Measure-level sharp envelope (IsLUB). For v strictly between 0 and 1, rhoEnvelope v is the least upper bound of the set of residuals l2ResidualQuadratic μ over admissible laws μ. Equivalently: the sup over all probability measures on [0,1] with ∫ y² ∂μ = v² of the L² residual of y² on span{1, y} equals the closed form ρ(v), and is attained (by the extremal three-point law).","s":"∀ (v : ℝ), 0 < v → v < 1 → IsLUB (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.residualSet v) (Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.rhoEnvelope v)","l":31,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/Envelope.lean"},{"n":"Causalean.Stat.MomentProblems.BoundedOutcomeEnvelope.strictAntiOn_envelopeQuartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.BoundedOutcomeEnvelope.QuarticRoot","a":"Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope","t":2,"d":"t ↦ envelopeQuartic t (v²) is strictly antitone on Icc (v²) v.","s":"∀ (v : ℝ), 0 < v → v < 1 → StrictAntiOn (fun t => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t (v ^ 2)) (Set.Icc (v ^ 2) v)","l":127,"f":"Causalean/Stat/Nonparametric/MomentProblems/BoundedOutcomeEnvelope/QuarticRoot.lean"},{"n":"Causalean.Stat.MomentProblems.IsGaussianLaw","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a measure on the real line, the Gaussian-law property holds exactly when there exist a real mean and a nonnegative variance parameter such that the measure is the associated normal distribution. The variance parameter may be zero, so point masses count as degenerate Gaussian laws.","s":"MeasureTheory.Measure ℝ → Prop","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Finite-fourth-moment hypothesis bundle: integrability of y, of y², of y³, and of y⁴ against μ. For a probability measure with ∫ y⁴ ∂μ < ∞ all four hold, so this is exactly \"μ has a finite fourth moment\".","s":"MeasureTheory.Measure ℝ → Prop","l":56,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.iInf_residualQuad","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Headline infimum identity. For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), the infimum, over every intercept and slope, of the mean squared residual of fitting b₀ + b₁y to y² equals the closed-form Hankel-determinant residual l2ResidualQuadratic μ, attained at the least-squares-optimal coefficients (optIntercept μ, optSlope μ):","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ⨅ b₀, ⨅ b₁, Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":177,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a measure on the real line, the moment-defined quadratic residual is the closed-form residual-variance expression calculated from its first four raw moments.","s":"MeasureTheory.Measure ℝ → ℝ","l":80,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic_le","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Lower bound (infimum). The closed-form residual is a lower bound of the integral objective at every choice of coefficients, when the design is non-degenerate. Together with residualQuad_opt_eq this identifies l2ResidualQuadratic μ as the minimum ⨅ b₀ b₁, ∫ (y² − b₀ − b₁ y)² ∂μ.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ∀ (b₀ b₁ : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁","l":158,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic_nonneg","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Non-negativity. The closed-form residual is non-negative (it is the value at the optimum of an integral of a square), for a non-degenerate design.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":169,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optIntercept","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a measure on the real line, the least-squares optimal intercept is the moment-based coefficient $(m_1m_3-m_2^2)/(m_1^2-m_2)$ for fitting $y^2$ by an affine function.","s":"MeasureTheory.Measure ℝ → ℝ","l":89,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optSlope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a measure on the real line, the least-squares optimal slope is the moment-based coefficient $(m_3-m_1m_2)/(m_2-m_1^2)$ for fitting $y^2$ by an affine function.","s":"MeasureTheory.Measure ℝ → ℝ","l":95,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a measure on the real line, an intercept, and a slope, the quadratic regression objective is the integral of the squared residual from fitting $y^2$ by the affine function $b_0+b_1y$.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":74,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Bridge (integral → moments). For a probability measure with finite fourth moment, the integral objective equals the moment polynomial: expanding (y² − b₀ − b₁ y)² = y⁴ − 2b₁ y³ + (b₁² − 2b₀) y² + 2b₀b₁ y + b₀² and integrating term by term (using ∫ 1 ∂μ = 1),","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → ∀ (b₀ b₁ : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ b₀ b₁ = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad (Causalean.Stat.MomentProblems.rawMoment μ 1) (Causalean.Stat.MomentProblems.rawMoment μ 2) (Causalean.Stat.MomentProblems.rawMoment μ 3) (Causalean.Stat.MomentProblems.rawMoment μ 4) b₀ b₁","l":101,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad_opt_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MeasureBridge","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Attainment. At the optimal coefficients (optIntercept μ, optSlope μ) the integral objective attains the closed-form residual l2ResidualQuadratic μ, whenever the design is non-degenerate (m₁² < m₂).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.residualQuad μ (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optIntercept μ) (Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.optSlope μ) = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":148,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MeasureBridge.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given real numbers representing the first through fourth raw moments, the dual-certificate cross moment is $-m_4+(1+2x)m_3-(2x+x^2)q+x^2m$, where $x=(m^2-2mq+q)/(2m(1-m))$.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":242,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real support parameter and a real second-moment value, the envelope first-order quartic is $t^4-2t^3+2qt^2-2q^2t+q^2$.","s":"ℝ → ℝ → ℝ","l":141,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff0","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real first-moment value and a real second-moment value, the intercept of the dual-certificate linear fit is $(mM_3-q^2)/(m^2-q)$, where $M_3$ is the extremal third moment at first moment $m$ and second moment $q$.","s":"ℝ → ℝ → ℝ","l":232,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real first-moment value and a real second-moment value, the slope of the dual-certificate linear fit is $(mq-M_3)/(m^2-q)$, where $M_3$ is the extremal third moment at first moment $m$ and second moment $q$.","s":"ℝ → ℝ → ℝ","l":237,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM1","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real support parameter, the first moment of the extremal three-point law is that support parameter itself.","s":"ℝ → ℝ","l":123,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real support parameter and a real second-moment value, the third moment of the extremal three-point law is $(2q^2t-q^2-qt^2-qt+t^3)/(2t(t-1))$.","s":"ℝ → ℝ → ℝ","l":127,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real support parameter and a real second-moment value, the fourth moment of the extremal three-point law is $(4q^3t^2-4q^3t+q^3-4q^2t^3+q^2t+3qt^4-2qt^3+2qt^2+t^5-2t^4)/(4t^2(t-1)^2)$.","s":"ℝ → ℝ → ℝ","l":133,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalMid","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real first-moment value and a real second-moment value, the interior support point of the extremal three-point law is $(m^2-2mq+q)/(2m(1-m))$.","s":"ℝ → ℝ → ℝ","l":228,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalResidual_eq_envelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Family residual identity. For every nondegenerate support parameter t (with t ≠ 0, t ≠ 1, and second moment q ≠ t²), the residual variance of regressing y² on {1, y} under the three-point law with moments (t, q, M₃, M₄) equals the closed form momentEnvelope t q. In particular ρ(v) is attained by an admissible law at second moment q = v².","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → t ^ 2 ≠ q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM1 t) q (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM3 t q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalM4 t q) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope t q","l":145,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given a real support parameter and a real second-moment value, the envelope value is $(t-q)(q-t^2)/(4t(1-t))$.","s":"ℝ → ℝ → ℝ","l":119,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_hasDerivAt","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Stationarity / envelope FOC. As a function of the support parameter t, the family residual momentEnvelope t q has derivative envelopeQuartic t q / (4 t² (t − 1)²). Hence the quartic is exactly the numerator of dρ/dt, and the envelope's stationary points are its roots.","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → HasDerivAt (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope s q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q / (4 * t ^ 2 * (t - 1) ^ 2)) t","l":169,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_le_root","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Envelope maximality. Among admissible first moments the envelope momentEnvelope · q is maximized at a quartic root u = μᵥ: if q < u, u² < q, and envelopeQuartic u q = 0, then for every admissible first moment m with q ≤ m, m² < q, one has momentEnvelope m q ≤ momentEnvelope u q.","s":"∀ (m q u : ℝ), 0 < q → q < 1 → q ≤ m → m ^ 2 < q → q < u → u ^ 2 < q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u q = 0 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope u q","l":289,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope_stationary_of_quartic","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"At a root t of the quartic, the family residual momentEnvelope · q is stationary. The maximizing root t = μᵥ ∈ (q, √q) therefore realizes the envelope value ρ(v).","s":"∀ (t q : ℝ), t ≠ 0 → t ≠ 1 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic t q = 0 → deriv (fun s => Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope s q) t = 0","l":207,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given four real numbers representing the first through fourth raw moments, the closed-form residual variance is the ratio of the leading third- and second-order Hankel determinants formed from those moments.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":63,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual_le_envelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Conditional moment-level envelope bound. Consider raw moments m, q, m3, m4 of a law and a candidate root u. If the second moment q lies strictly between 0 and 1, if the first moment m is at least q while m² still lies below q, i.e. the design has positive variance, if a supplied cross-moment combination of m, q, m3, m4 is nonnegative, and if u satisfies q < u, u² < q, and solves the envelope's stationarity quartic exactly, then the moment-level residual variance of regressing y² on {1, y} under moments (m, q, m3, m4) is at most the envelope value momentEnvelope u q.","s":"∀ (m q m3 m4 u : ℝ), 0 < q → q < 1 → q ≤ m → m ^ 2 < q → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4 → q < u → u ^ 2 < q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.envelopeQuartic u q = 0 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m q m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope u q","l":346,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual_le_momentEnvelope","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Sharp per-instance bound. For an admissible moment tuple (1, m, q, m₃, m₄) of a law on [0,1] with positive variance (m² < q) and nonnegative cross moment (0 ≤ crossMoment m q m₃ m₄, automatic for any law on [0,1]), the residual variance of regressing y² on {1, y} is at most the envelope value at the law's own first moment: momentResidual m q m₃ m₄ ≤ momentEnvelope m q.","s":"∀ (m q m3 m4 : ℝ), 0 < m → m < 1 → m ^ 2 < q → 0 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m q m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q","l":274,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optIntercept","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given three real numbers representing the first three raw moments, the optimal regression intercept is $(m_1m_3-m_2^2)/(m_1^2-m_2)$.","s":"ℝ → ℝ → ℝ → ℝ","l":73,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optSlope","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given three real numbers representing the first three raw moments, the optimal regression slope is $(m_1m_2-m_3)/(m_1^2-m_2)$.","s":"ℝ → ℝ → ℝ → ℝ","l":77,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Given four real numbers representing the first through fourth raw moments and an intercept and slope, the moment-coordinate regression objective is the polynomial obtained by expanding the squared residual from fitting $y^2$ by $b_0+b_1y$.","s":"ℝ → ℝ → ℝ → ℝ → ℝ → ℝ → ℝ","l":57,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_extremalCoeff_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Dual-certificate identity (integrated). For the certificate's linear-fit coefficients, the regression objective equals the envelope value minus the cross moment: residualQuad m q m₃ m₄ b₀ b₁ = momentEnvelope m q − crossMoment m q m₃ m₄. This is the moment-level form of the pointwise SOS certificate, a pure algebraic identity (ring) valid for m ≠ 0, m ≠ 1, m² ≠ q.","s":"∀ (m q m3 m4 : ℝ), m ≠ 0 → m ≠ 1 → m ^ 2 ≠ q → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m q m3 m4 (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff0 m q) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.extremalCoeff1 m q) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentEnvelope m q - Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.crossMoment m q m3 m4","l":254,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_ge_momentResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"The closed-form residual momentResidual is a lower bound for the regression objective at every choice of coefficients, when the design is nondegenerate (m₁² < m₂, positive variance). Together with residualQuad_optimalCoeff this shows momentResidual is the minimum min_{b₀,b₁} ∫ (y² − b₀ − b₁ y)² dμ.","s":"∀ (m1 m2 m3 m4 b0 b1 : ℝ), m1 ^ 2 < m2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m1 m2 m3 m4 ≤ Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m1 m2 m3 m4 b0 b1","l":94,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad_optimalCoeff","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.MomentAlgebra","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Attainment at the optimal coefficients. For raw moments m1, m2, m3, m4 of a law with first moment squared strictly below the second moment (positive variance), the regression objective, evaluated at the optimal intercept and slope (optIntercept m1 m2 m3, optSlope m1 m2 m3), equals the closed-form residual momentResidual m1 m2 m3 m4.","s":"∀ (m1 m2 m3 m4 : ℝ), m1 ^ 2 < m2 → Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.residualQuad m1 m2 m3 m4 (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optIntercept m1 m2 m3) (Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.optSlope m1 m2 m3) = Causalean.Stat.MomentProblems.ResidualQuadratic.MomentAlgebra.momentResidual m1 m2 m3 m4","l":81,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/MomentAlgebra.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integrable_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"q is integrable (finite measure + q ∈ L²).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.Integrable (Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ) μ","l":78,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_id_mul_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Orthogonality to y. ∫ y·q dμ = 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ∫ (y : ℝ), y * Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = 0","l":116,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"Orthogonality to the constant. ∫ q dμ = 0.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ∫ (y : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = 0","l":84,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_sq_mul_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y²-moment. ∫ y²·q dμ = r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ∫ (y : ℝ), y ^ 2 * Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ∂μ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":168,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.integral_sq_projResidual","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"Self L²-norm of the projection residual. For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), the squared L² norm of the projection residual q(y) = y² − (optIntercept μ + optSlope μ · y) equals the closed-form residual l2ResidualQuadratic μ: ∫ q² dμ = l2ResidualQuadratic μ.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → ∫ (y : ℝ), Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ y ^ 2 ∂μ = Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":150,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.memL2_id","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y ↦ y is in L²(μ) when the second moment is finite.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (fun y => y) 2 μ","l":53,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.memL2_sq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"y ↦ y² is in L²(μ) when the fourth moment is finite.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (fun y => y ^ 2) 2 μ","l":59,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":1,"d":"For a measure on the real line, let $m_j$ denote its $j$-th raw moment. The projection-residual function assigns to each real value $y$ the value $y^2-a-by$, where $a=(m_1m_3-m_2^2)/(m_1^2-m_2)$ and $b=(m_1m_2-m_3)/(m_1^2-m_2)$.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":43,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual_memL2","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ResidualQuadratic.ProjectionResidual","a":"Stat/Nonparametric/MomentProblems/ResidualQuadratic","t":2,"d":"The residual function q is square-integrable (q ∈ L²(μ)): it is a degree-2 polynomial in y and μ has a finite fourth moment.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → MeasureTheory.MemLp (Causalean.Stat.MomentProblems.ResidualQuadratic.ProjectionResidual.projResidual μ) 2 μ","l":65,"f":"Causalean/Stat/Nonparametric/MomentProblems/ResidualQuadratic/ProjectionResidual.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore","k":"structure","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"Feasibility of a score. s : ℝ → ℝ is feasible for target x if it lies in L²(μ) and is moment-orthogonal to 1 and y while carrying prescribed y²-moment x: s ∈ L²(μ), ∫ s dμ = 0, ∫ y·s dμ = 0, ∫ y²·s dμ = x.","s":"MeasureTheory.Measure ℝ → ℝ → (ℝ → ℝ) → Prop","l":53,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.feasibleScore_cost_lower_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Lower bound (duality ≤). Every feasible score has cost at least x² / r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ) {s : ℝ → ℝ}, Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore μ x s → x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ ≤ ∫ (y : ℝ), s y ^ 2 ∂μ","l":133,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"For a measure on the real line and a real target moment, the optimal score assigns to each real value the projection residual of its square multiplied by the target moment divided by the closed-form residual scalar calculated from the first four raw moments of the measure.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":70,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore_cost","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Attainment. The optimal score s* achieves cost exactly x² / r: ∫ (s*)² dμ = ∫ (x/r)²·q² dμ = (x/r)²·r = x²/r.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), ∫ (y : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.optScore μ x y ^ 2 ∂μ = x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":118,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.optScore_feasible","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":2,"d":"Feasibility of the optimal score. s* = (x/r)·q is a feasible score for target x: it lies in L²(μ) and satisfies the three moment constraints. The y²-moment constraint uses ∫ y²·q = r and (x/r)·r = x (needs r ≠ 0).","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.FeasibleScore μ x (Causalean.Stat.MomentProblems.ScoreProgram.optScore μ x)","l":89,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.scoreCost","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"For a measure on the real line and a real target moment, the score-program value is the infimum, over all square-integrable real-valued functions whose mean and first moment-weighted mean are zero and whose second-moment-weighted mean equals the target, of their integrated square.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":80,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.ScoreProgram.scoreCost_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.ScoreProgram.ScoreProgram","a":"Stat/Nonparametric/MomentProblems/ScoreProgram","t":1,"d":"The score-program duality (headline). For a probability measure μ on ℝ with a finite fourth moment and first moment squared strictly below the second moment (positive variance), if moreover the closed-form residual l2ResidualQuadratic μ is positive, then for every target y²-moment x, the value of the constrained minimum-norm score program equals x² / l2ResidualQuadratic μ:","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.FiniteMoment4 μ → Causalean.Stat.MomentProblems.rawMoment μ 1 ^ 2 < Causalean.Stat.MomentProblems.rawMoment μ 2 → 0 < Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ → ∀ (x : ℝ), Causalean.Stat.MomentProblems.ScoreProgram.scoreCost μ x = x ^ 2 / Causalean.Stat.MomentProblems.ResidualQuadratic.MeasureBridge.l2ResidualQuadratic μ","l":215,"f":"Causalean/Stat/Nonparametric/MomentProblems/ScoreProgram/ScoreProgram.lean"},{"n":"Causalean.Stat.MomentProblems.TruncatedMomentInterior","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.TruncatedCumulantInterior","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative truncation order, the interior-of-the-truncated-cumulant-range condition says that there exist a real cumulant sequence and a strictly positive real radius such that every real sequence whose entries of every order from two through the truncation order differ from the corresponding target entries by less than that radius is the cumulant sequence, at each of those orders, of some centered non-Gaussian probability law on the real line with finite moment of the truncation order.","s":"ℕ → Prop","l":40,"f":"Causalean/Stat/Nonparametric/MomentProblems/TruncatedCumulantInterior.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L and a prescribed vector of real raw moments of orders zero through L, the moment-matching weight vector is the result of applying the inverse transpose of the Vandermonde matrix of the symmetric atoms to that vector.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → Fin (L + 1) → ℝ","l":136,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_mulVec","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The solve is a left inverse of taking moments: solving for the weights from the moments that a given weight vector produces returns that same weight vector.","s":"∀ (L : ℕ) (q : Fin (L + 1) → ℝ), Causalean.Stat.MomentProblems.atomSolve L ((Causalean.Stat.MomentProblems.atomVandermonde L).transpose.mulVec q) = q","l":162,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_spec","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The solved weights do reproduce the target: the weighted sum of the k-th powers of the atoms is the k-th target moment, for every order k from 0 to L.","s":"∀ (L : ℕ) (b : Fin (L + 1) → ℝ) (k : Fin (L + 1)), ∑ i, Causalean.Stat.MomentProblems.atomSolve L b i * Causalean.Stat.MomentProblems.symmetricAtoms L i ^ ↑k = b k","l":145,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomSolve_uniformMoments","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Solving at the moments of the uniform-weight law returns the uniform weights.","s":"∀ (L : ℕ), (Causalean.Stat.MomentProblems.atomSolve L fun k => Causalean.Stat.MomentProblems.uniformMoments L ↑k) = Causalean.Stat.MomentProblems.uniformWeights L","l":172,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomVandermonde","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the Vandermonde matrix of the symmetric atoms is the square real matrix whose row for each atom lists successive powers of that atom, from power zero through power L.","s":"(L : ℕ) → Matrix (Fin (L + 1)) (Fin (L + 1)) ℝ","l":122,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomVandermonde_transpose_isUnit","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The transposed Vandermonde matrix of the symmetric atoms is invertible, because the atoms are pairwise distinct.","s":"∀ (L : ℕ), IsUnit (Causalean.Stat.MomentProblems.atomVandermonde L).transpose.det","l":129,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.atomicLaw","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a natural number of atoms, a list of their real locations, and a matching list of real weights, the finite atomic law is the measure that puts each weight as point mass at its corresponding location.","s":"(n : ℕ) → (Fin n → ℝ) → (Fin n → ℝ) → MeasureTheory.Measure ℝ","l":36,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicLaw_singleton","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"A finite atomic law with distinct locations and nonnegative weights puts exactly its own weight of mass on each single atom.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∀ (j : Fin n), (Causalean.Stat.MomentProblems.atomicLaw n x p) {x j} = ENNReal.ofReal (p j)","l":83,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicWeight","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a natural number of atoms, a list of their real locations, and a matching list of real weights, the weight assigned to a real point is the sum of the weights of precisely those atoms located at that point.","s":"(n : ℕ) → (Fin n → ℝ) → (Fin n → ℝ) → ℝ → ℝ","l":30,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.atomicWeight_apply","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"When the atom locations are pairwise distinct, the total weight sitting at one of them is exactly that atom's own weight — no two indices collide.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → ∀ (j : Fin n), Causalean.Stat.MomentProblems.atomicWeight n x p (x j) = p j","l":43,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_atomSolve","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The moment-matching solve depends continuously on the target moments — it is linear.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.atomSolve L)","l":181,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_cumulantToWeights","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The cumulant-to-weights map is continuous. For any truncation order L, the map sending a truncated cumulant target to the recovered symmetric-atom weights is continuous, since both the moment↔cumulant inversion and the Vandermonde solve it composes are continuous.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.cumulantToWeights L)","l":217,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_momFromCum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Continuity of the inversion. At any fixed order r, the reconstructed moment of that order is a continuous function of the prescribed cumulant sequence, so small perturbations of the target cumulants move the moments only slightly — the key to the openness arguments that use this inversion.","s":"∀ (r : ℕ), Continuous fun c => Causalean.Stat.MomentProblems.momFromCum c r","l":221,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.continuous_padCumulants","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Zero-extension of a finite cumulant target is continuous in the target.","s":"∀ (L : ℕ), Continuous (Causalean.Stat.MomentProblems.padCumulants L)","l":199,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative order and a real moment sequence, the cumulant read from that moment sequence is the signed, factorial-weighted sum over all partitions of a set of that many elements, with each partition contributing the product of the moments indexed by its block sizes.","s":"ℕ → (ℕ → ℝ) → ℝ","l":31,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The order-r cumulant depends only on the moments up to order r: changing higher moments leaves it unchanged.","s":"∀ (r : ℕ) {m m' : ℕ → ℝ}, (∀ (k : ℕ), 1 ≤ k → k ≤ r → m k = m' k) → Causalean.Stat.MomentProblems.cumFromMom r m = Causalean.Stat.MomentProblems.cumFromMom r m'","l":131,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_congr_up_to","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.CumulantTransfer","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":2,"d":"When two abstract moment sequences agree through a cutoff and the requested cumulant order lies below that cutoff, their combinatorial cumulants at that order agree.","s":"∀ {m₁ m₂ : ℕ → ℝ} {K k : ℕ}, (∀ j ≤ K, m₁ j = m₂ j) → k ≤ K → Causalean.Stat.MomentProblems.cumFromMom k m₁ = Causalean.Stat.MomentProblems.cumFromMom k m₂","l":20,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/CumulantTransfer.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Triangularity. At any positive order the cumulant equals the moment of that same order plus a remainder assembled only from strictly lower moments. This is what makes the moment-to-cumulant map invertible by recursion.","s":"∀ (r : ℕ), 1 ≤ r → ∀ (m : ℕ → ℝ), Causalean.Stat.MomentProblems.cumFromMom r m = m r + Causalean.Stat.MomentProblems.restFromMom r m","l":99,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumFromMom_momFromCum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Correctness of the inversion. Reading the cumulants back off the reconstructed moment sequence returns the prescribed cumulants, at every order r at least two.","s":"∀ (c : ℕ → ℝ) (r : ℕ), 2 ≤ r → Causalean.Stat.MomentProblems.cumFromMom r (Causalean.Stat.MomentProblems.momFromCum c) = c r","l":198,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.cumulantToWeights","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L and a prescribed vector of cumulants of orders zero through L, the cumulant-to-weights map first reconstructs the corresponding raw moments and then returns the symmetric-atom weights obtained by solving their Vandermonde moment-matching system.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → Fin (L + 1) → ℝ","l":209,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.cumulantToWeights_uniformCumulantPoint","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The base point maps to the uniform weights. For any truncation order L, feeding the cumulants of the uniform-weight law on the symmetric atoms into the cumulant-to-weights map returns exactly the uniform weights — all strictly positive, hence strictly inside the positivity constraints.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.cumulantToWeights L (Causalean.Stat.MomentProblems.uniformCumulantPoint L) = Causalean.Stat.MomentProblems.uniformWeights L","l":241,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.exists_bounded_gaussian_orthogonal_perturbation","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.OrthogonalPerturbation","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"For a finite degree cutoff, there is a measurable profile bounded by one, nonzero under the standard Gaussian law, and orthogonal to every monomial through that cutoff.","s":"∀ (K : ℕ), ∃ h, Measurable h ∧ (∀ (x : ℝ), |h x| ≤ 1) ∧ 0 < ∫ (x : ℝ), |h x| ∂ProbabilityTheory.gaussianReal 0 1 ∧ ∀ k ≤ K, MeasureTheory.Integrable (fun x => x ^ k * h x) (ProbabilityTheory.gaussianReal 0 1) ∧ ∫ (x : ℝ), x ^ k * h x ∂ProbabilityTheory.gaussianReal 0 1 = 0","l":73,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/OrthogonalPerturbation.lean"},{"n":"Causalean.Stat.MomentProblems.exists_finiteMoment_near_gaussian_perturbation","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.Main","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"Given a finite moment cutoff of at least three and a strictly positive testing-distance radius, there is a non-Gaussian probability law within that radius of the standard Gaussian, with matching raw moments and source cumulants through the cutoff, all absolute moments, and a divergent Hamburger--Carleman series.","s":"∀ (K : ℕ) (rho : ℝ), 3 ≤ K → 0 < rho → ∃ F, MeasureTheory.IsProbabilityMeasure F ∧ ∫ (x : ℝ), x ∂F = 0 ∧ ProbabilityTheory.variance id F = 1 ∧ ¬Causalean.Stat.MomentProblems.IsGaussianLaw F ∧ Causalean.Stat.tvDist F (ProbabilityTheory.gaussianReal 0 1) < rho ∧ (∀ k ≤ K, ∫ (x : ℝ), x ^ k ∂F = ∫ (x : ℝ), x ^ k ∂ProbabilityTheory.gaussianReal 0 1) ∧ (∀ (k : ℕ), MeasureTheory.Integrable (fun x => |x| ^ k) F) ∧ ∑' (s : ℕ), (ENNReal.ofReal |∫ (x : ℝ), x ^ (2 * (s + 1)) ∂F|).rpow (-1 / (2 * (↑s + 1))) = ⊤ ∧ ∀ k ≤ K, Causalean.Stat.MomentProblems.sourceCumulant F id k = Causalean.Stat.MomentProblems.sourceCumulant (ProbabilityTheory.gaussianReal 0 1) id k","l":25,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/Main.lean"},{"n":"Causalean.Stat.MomentProblems.gaussianPerturbation","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.DensityPerturbation","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"Given a signed profile and a real amplitude, the Gaussian density perturbation is given by weighting the standard Gaussian law by the positive part of one plus the scaled profile. Where one plus the scaled profile is negative the weight is zero, so this is the signed tilt only when that quantity is nonnegative.","s":"(ℝ → ℝ) → ℝ → MeasureTheory.Measure ℝ","l":25,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/DensityPerturbation.lean"},{"n":"Causalean.Stat.MomentProblems.gaussianPerturbation_spec","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.DensityPerturbation","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"If a profile is measurable, bounded by one, nonzero under the standard Gaussian law, and orthogonal to the required monomials, while its amplitude is positive, below one, and below the requested distance radius, then the resulting density perturbation is a distinct nearby probability law with the specified raw moments, all absolute moments, and a Gaussian-scale even-moment bound.","s":"∀ {K : ℕ} {h : ℝ → ℝ} {ε rho : ℝ}, Measurable h → (∀ (x : ℝ), |h x| ≤ 1) → 0 < ∫ (x : ℝ), |h x| ∂ProbabilityTheory.gaussianReal 0 1 → (∀ k ≤ K, MeasureTheory.Integrable (fun x => x ^ k * h x) (ProbabilityTheory.gaussianReal 0 1) ∧ ∫ (x : ℝ), x ^ k * h x ∂ProbabilityTheory.gaussianReal 0 1 = 0) → 0 < ε → ε < 1 → ε < rho → have F := Causalean.Stat.MomentProblems.gaussianPerturbation h ε; MeasureTheory.IsProbabilityMeasure F ∧ F ≠ ProbabilityTheory.gaussianReal 0 1 ∧ Causalean.Stat.tvDist F (ProbabilityTheory.gaussianReal 0 1) < rho ∧ (∀ k ≤ K, Causalean.Stat.MomentProblems.rawMoment F k = Causalean.Stat.MomentProblems.rawMoment (ProbabilityTheory.gaussianReal 0 1) k) ∧ (∀ (k : ℕ), MeasureTheory.Integrable (fun x => |x| ^ k) F) ∧ ∀ (n : ℕ), 0 < n → |Causalean.Stat.MomentProblems.rawMoment F (2 * n)| ≤ 2 * (2 * ↑n) ^ n","l":90,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/DensityPerturbation.lean"},{"n":"Causalean.Stat.MomentProblems.hamburgerCarlemanSeries","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.Basic","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"Given a real measure, the explicit Hamburger--Carleman series is given by the sum of inverse roots of its positive even raw moments. It is the Carleman series only for a measure whose even moments exist: a non-integrable even power has raw moment zero by the integral convention, and the zero base with a negative exponent makes that term infinite.","s":"MeasureTheory.Measure ℝ → ENNReal","l":24,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/Basic.lean"},{"n":"Causalean.Stat.MomentProblems.hamburgerCarlemanSeries_eq","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.Basic","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"For a real measure, the Hamburger--Carleman notation equals its explicit integral-series formula.","s":"∀ (ν : MeasureTheory.Measure ℝ), Causalean.Stat.MomentProblems.hamburgerCarlemanSeries ν = ∑' (s : ℕ), (ENNReal.ofReal |∫ (x : ℝ), x ^ (2 * (s + 1)) ∂ν|).rpow (-1 / (2 * (↑s + 1)))","l":33,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/Basic.lean"},{"n":"Causalean.Stat.MomentProblems.hamburgerCarlemanSeries_eq_top_of_evenMoment_le","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.Carleman","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"If a real measure has positive even raw moments bounded at the Gaussian scale, then its explicit Hamburger--Carleman series diverges. The hypothesis bounds the real-valued raw moments, which are zero for non-integrable powers, so the result carries Carleman content only for a measure whose even moments exist, as in its use for Gaussian perturbations.","s":"∀ (ν : MeasureTheory.Measure ℝ), (∀ (n : ℕ), 0 < n → |Causalean.Stat.MomentProblems.rawMoment ν (2 * n)| ≤ 2 * (2 * ↑n) ^ n) → Causalean.Stat.MomentProblems.hamburgerCarlemanSeries ν = ⊤","l":57,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/Carleman.lean"},{"n":"Causalean.Stat.MomentProblems.integral_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Integrating a function against a finite atomic law with distinct locations and nonnegative weights gives the weighted sum of the function's values at the atoms.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∀ (f : ℝ → ℝ), ∫ (t : ℝ), f t ∂Causalean.Stat.MomentProblems.atomicLaw n x p = ∑ i, p i * f (x i)","l":51,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.isProbabilityMeasure_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"A finite atomic law on the real line, built from n pairwise distinct atom locations carrying nonnegative weights that sum to one, is a probability measure.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∑ i, p i = 1 → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":68,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.jointCumulant","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Given a sample space equipped with a σ-algebra, a measure on that space, two real-valued random variables, and two nonnegative integer orders, the joint cumulant at the specified bidegree is the sum over every partition of the combined slots, with the first $p$ slots assigned to the first variable and the remaining $q$ slots assigned to the second, of a signed factorial weight times the product, across its blocks, of the mixed moment whose exponents equal that block's counts of first- and second-variable slots.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → ℕ → ℕ → ℝ","l":45,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.memLp_id_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"A finite atomic probability law has finite moments of every order: its support is a bounded finite set, so the identity function is bounded almost everywhere and hence in every Lᵖ.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 ≤ p i) → ∑ i, p i = 1 → ∀ (q : ENNReal), MeasureTheory.MemLp id q (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":94,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a real sequence of prescribed cumulants, the reconstructed moment sequence is defined by giving its zeroth moment the value one, giving its first moment the value zero, and at every order at least two, subtracting from the prescribed cumulant the lower-order remainder computed from the already reconstructed moments.","s":"(ℕ → ℝ) → ℕ → ℝ","l":158,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Locality of the inversion. The reconstructed moment of order r depends only on the prescribed cumulants of orders two through r, so truncating the cumulant target beyond r is harmless.","s":"∀ {c c' : ℕ → ℝ} (r : ℕ), (∀ (k : ℕ), 2 ≤ k → k ≤ r → c k = c' k) → Causalean.Stat.MomentProblems.momFromCum c r = Causalean.Stat.MomentProblems.momFromCum c' r","l":205,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_eq_of_cum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Uniqueness of the inversion. Any moment sequence with total mass one and mean zero whose cumulants are the prescribed ones is exactly the sequence produced by the recursion.","s":"∀ (m : ℕ → ℝ), m 0 = 1 → m 1 = 0 → ∀ (c : ℕ → ℝ), (∀ (r : ℕ), 2 ≤ r → c r = Causalean.Stat.MomentProblems.cumFromMom r m) → ∀ (r : ℕ), Causalean.Stat.MomentProblems.momFromCum c r = m r","l":247,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_one","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The reconstructed moment sequence is centered: its mean is zero.","s":"∀ (c : ℕ → ℝ), Causalean.Stat.MomentProblems.momFromCum c 1 = 0","l":178,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_padCumulants_uniform","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Inverting the base cumulant target order by order returns the moments of the uniform-weight law.","s":"∀ (L : ℕ) (k : Fin (L + 1)), Causalean.Stat.MomentProblems.momFromCum (Causalean.Stat.MomentProblems.padCumulants L (Causalean.Stat.MomentProblems.uniformCumulantPoint L)) ↑k = Causalean.Stat.MomentProblems.uniformMoments L ↑k","l":232,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_succ","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The defining recursion: at every order at least two, the reconstructed moment is the target cumulant of that order minus the remainder built from the lower reconstructed moments.","s":"∀ (c : ℕ → ℝ) (r : ℕ), 2 ≤ r → Causalean.Stat.MomentProblems.momFromCum c r = c r - Causalean.Stat.MomentProblems.restFromMom r (Causalean.Stat.MomentProblems.momFromCum c)","l":182,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_uniformCumulants","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Inverting the cumulants of the uniform-weight law returns its moments: the base point is a fixed point of the moment↔cumulant round trip.","s":"∀ (L k : ℕ), Causalean.Stat.MomentProblems.momFromCum (Causalean.Stat.MomentProblems.uniformCumulants L) k = Causalean.Stat.MomentProblems.uniformMoments L k","l":112,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.momFromCum_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The reconstructed moment sequence has total mass one.","s":"∀ (c : ℕ → ℝ), Causalean.Stat.MomentProblems.momFromCum c 0 = 1","l":174,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.not_isGaussianLaw_atomicLaw","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.AtomicLaw","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"A finite atomic probability law with at least two locations that are pairwise distinct and each carry strictly positive mass is not a Gaussian law: a normal law with positive variance has no point masses at all, and one with zero variance is a single point mass.","s":"∀ {n : ℕ} {x p : Fin n → ℝ}, Function.Injective x → (∀ (i : Fin n), 0 < p i) → 2 ≤ n → ¬Causalean.Stat.MomentProblems.IsGaussianLaw (Causalean.Stat.MomentProblems.atomicLaw n x p)","l":115,"f":"Causalean/Stat/Nonparametric/MomentProblems/AtomicLaw.lean"},{"n":"Causalean.Stat.MomentProblems.padCumulants","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L and a prescribed vector of cumulants of orders zero through L, the zero-padded cumulant sequence agrees with that vector through order L and equals zero at every higher order.","s":"(L : ℕ) → (Fin (L + 1) → ℝ) → ℕ → ℝ","l":193,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.rawMoment","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.RawMoment","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Given a real measure and a nonnegative integer order, the raw moment is given by integrating the corresponding power. This is the real-valued Bochner integral, which is zero when the power is not integrable, so it is the moment only when the moment exists.","s":"MeasureTheory.Measure ℝ → ℕ → ℝ","l":21,"f":"Causalean/Stat/Nonparametric/MomentProblems/RawMoment.lean"},{"n":"Causalean.Stat.MomentProblems.rawMoment_eq_integral","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.RawMoment","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a real measure at a nonnegative integer order, the raw-moment notation equals the integral of the corresponding power.","s":"∀ (ν : MeasureTheory.Measure ℝ) (k : ℕ), Causalean.Stat.MomentProblems.rawMoment ν k = ∫ (x : ℝ), x ^ k ∂ν","l":28,"f":"Causalean/Stat/Nonparametric/MomentProblems/RawMoment.lean"},{"n":"Causalean.Stat.MomentProblems.restFromMom","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative order and a real moment sequence, the lower-order remainder of the cumulant formula is the same partition sum with the sole one-block partition omitted; hence it uses only moments of order strictly below that order.","s":"ℕ → (ℕ → ℝ) → ℝ","l":40,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.restFromMom_congr","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The lower-order remainder at order r depends only on the moments strictly below r: changing the order-r moment, or any higher one, leaves it unchanged.","s":"∀ (r : ℕ) {m m' : ℕ → ℝ}, (∀ (k : ℕ), 1 ≤ k → k < r → m k = m' k) → Causalean.Stat.MomentProblems.restFromMom r m = Causalean.Stat.MomentProblems.restFromMom r m'","l":144,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.sourceCumulant","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.Cumulant","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"Given a sample space equipped with a σ-algebra, a measure on that space, a real-valued random variable, and a nonnegative integer order, the cumulant of that order is the joint cumulant obtained by placing that same variable in every slot.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℕ → ℝ","l":67,"f":"Causalean/Stat/Nonparametric/MomentProblems/Cumulant.lean"},{"n":"Causalean.Stat.MomentProblems.sourceCumulant_eq_cumFromMom","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.MomentCumulantInversion","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a real random variable with law ν and any order r, the order-r cumulant of ν equals the abstract combinatorial cumulant formula evaluated at ν's own raw-moment sequence: the measure-theoretic and combinatorial definitions agree.","s":"∀ (ν : MeasureTheory.Measure ℝ) (r : ℕ), Causalean.Stat.MomentProblems.sourceCumulant ν id r = Causalean.Stat.MomentProblems.cumFromMom r fun k => ∫ (t : ℝ), t ^ k ∂ν","l":80,"f":"Causalean/Stat/Nonparametric/MomentProblems/MomentCumulantInversion.lean"},{"n":"Causalean.Stat.MomentProblems.sourceCumulant_eq_of_rawMoment_eq_up_to","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.FiniteMomentNearGaussianPerturbation.CumulantTransfer","a":"Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation","t":1,"d":"Given two real measures, a finite cutoff, and equality of their raw moments through that cutoff, the source cumulants of the identity statistic agree at every order through the cutoff.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) (K : ℕ), (∀ j ≤ K, Causalean.Stat.MomentProblems.rawMoment μ j = Causalean.Stat.MomentProblems.rawMoment ν j) → ∀ k ≤ K, Causalean.Stat.MomentProblems.sourceCumulant μ id k = Causalean.Stat.MomentProblems.sourceCumulant ν id k","l":35,"f":"Causalean/Stat/Nonparametric/MomentProblems/FiniteMomentNearGaussianPerturbation/CumulantTransfer.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the symmetric atom locations are the L + 1 real numbers −L, −L + 2, …, L, indexed in increasing order; equivalently, the location with index i is twice i minus L.","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":34,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms_injective","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The symmetric atoms are pairwise distinct.","s":"∀ (L : ℕ), Function.Injective (Causalean.Stat.MomentProblems.symmetricAtoms L)","l":42,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.symmetricAtoms_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The symmetric atoms sum to zero — they are balanced about the origin.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.symmetricAtoms L i = 0","l":50,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.truncatedMomentInterior","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.TruncatedCumulantInterior","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"The truncated cumulant range has nonempty interior. For every truncation order L, there is a cumulant vector, of orders two through L, together with a strictly positive radius, such that every cumulant vector within that radius is the truncated cumulant vector of some centered, non-Gaussian probability law on the real line with finite L-th moment.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.TruncatedMomentInterior L","l":84,"f":"Causalean/Stat/Nonparametric/MomentProblems/TruncatedCumulantInterior.lean"},{"n":"Causalean.Stat.MomentProblems.uniformCumulantPoint","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the base cumulant target is the vector of cumulants of orders zero through L of the uniform probability law on the symmetric atoms.","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":227,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformCumulants","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the cumulant sequence of the uniform law on the symmetric atoms is the cumulant sequence calculated from that law's raw moments.","s":"ℕ → ℕ → ℝ","l":105,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the raw-moment sequence of the uniform law on the symmetric atoms assigns to every nonnegative order the equally weighted average of that power of the L + 1 symmetric atom locations.","s":"ℕ → ℕ → ℝ","l":91,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments_one","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law is centered.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.uniformMoments L 1 = 0","l":101,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformMoments_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law has total mass one.","s":"∀ (L : ℕ), Causalean.Stat.MomentProblems.uniformMoments L 0 = 1","l":97,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights","k":"def","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":1,"d":"For a nonnegative integer L, the uniform weight vector assigns mass $1/(L+1)$ to each of the L + 1 symmetric atoms.","s":"(L : ℕ) → Fin (L + 1) → ℝ","l":68,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_mean","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform-weight law on the symmetric atoms has mean zero.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.uniformWeights L i * Causalean.Stat.MomentProblems.symmetricAtoms L i = 0","l":84,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"Every uniform weight is strictly positive.","s":"∀ (L : ℕ) (i : Fin (L + 1)), 0 < Causalean.Stat.MomentProblems.uniformWeights L i","l":73,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.MomentProblems.uniformWeights_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.MomentProblems.SymmetricAtomSolve","a":"Stat/Nonparametric/MomentProblems","t":2,"d":"The uniform weights sum to one, so the uniform-weight atomic law is a probability law.","s":"∀ (L : ℕ), ∑ i, Causalean.Stat.MomentProblems.uniformWeights L i = 1","l":79,"f":"Causalean/Stat/Nonparametric/MomentProblems/SymmetricAtomSolve.lean"},{"n":"Causalean.Stat.NeymanOrthogonal","k":"def","m":"Causalean.Stat.Orthogonality.Orthogonality","a":"Stat/Orthogonality","t":1,"d":"Given an abstract real vector space of nuisance values, a measurable sample space, a real-valued moment function of a nuisance value, an observation, and a target value, a reference nuisance value, a target value, and a population measure on the sample space, Neyman orthogonality means that the population moment at the reference nuisance and target values is zero and that, for every nuisance direction, the population moment along the line from the reference nuisance toward that direction, divided by the nonzero line parameter, converges to zero as the parameter tends to zero.","s":"{H : Type u_1} → {X : Type u_2} → [inst : AddCommGroup H] → [Module ℝ H] → [inst : MeasurableSpace X] → (H → X → ℝ → ℝ) → H → ℝ → MeasureTheory.Measure X → Prop","l":35,"f":"Causalean/Stat/Orthogonality/Orthogonality.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.degenerate_uStatistic_variance","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":1,"d":"Exact variance of the degenerate order-2 U-statistic. For an i.i.d. sample S, if the kernel g is symmetric, square-integrable, and doubly degenerate — its conditional expectation given either argument vanishes — and the sample size n is at least 2, then the order-2 U-statistic built from g on n observations has variance Var[Uₙ] = 2ζ / (n(n−1)), where ζ = ∬ g² dP dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ = 2 * Causalean.Stat.IIDSample.zeta P g / (↑n * (↑n - 1))","l":66,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.degenerate_uStatistic_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":1,"d":"HOIF degenerate U-statistic variance bound O(J/(nh)²). For an i.i.d. sample S, if the kernel g is symmetric, square-integrable, and doubly degenerate, the sample size n is at least 2, the bandwidth h is positive, the projection dimension J is nonnegative, the trace constant C is nonnegative, and the localized, J-dimensional projection kernel g has L²-energy ζ = ∬ g² dP dP bounded by C·J/h², then the degenerate order-2 U-statistic's variance satisfies Var[Uₙ] ≤ 4C·J / (nh)².","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {C J h : ℝ} {n : ℕ}, 2 ≤ n → 0 < h → 0 ≤ J → 0 ≤ C → Causalean.Stat.IIDSample.zeta P g ≤ C * J / h ^ 2 → ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ ≤ 4 * C * J / (↑n * h) ^ 2","l":99,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.gram","k":"def","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"For a measurable sample space, a nonnegative integer giving the number of basis functions, a real-valued basis evaluated at each sample point, and a measure on that sample space, the second-moment, or Gram, matrix of the basis has entry $k,l$ equal to $\\int c_k(x)c_l(x)\\,dP(x)$.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {J : ℕ} → (X → Fin J → ℝ) → MeasureTheory.Measure X → Matrix (Fin J) (Fin J) ℝ","l":49,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.gram_symm","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"The Gram matrix is symmetric.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {J : ℕ} (c : X → Fin J → ℝ), (Causalean.Stat.Nonparametric.HOIF.gram c P).transpose = Causalean.Stat.Nonparametric.HOIF.gram c P","l":66,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_order_choice_negligible","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":1,"d":"Rate selection: a high-enough HOIF order makes the squared remainder o(ρ_n). For a squared product-remainder bound of the power-law form K·n^{-a} and a target risk rate ρ_n = n^{-κ}, if the remainder's decay exponent a exceeds the target rate's exponent κ, then the ratio (K·n^{-a}) / n^{-κ} = K·n^{κ-a} tends to 0 as n → ∞: the HOIF estimation bias is asymptotically negligible relative to the target rate.","s":"∀ {K a κ : ℝ}, κ < a → Filter.Tendsto (fun n => K * ↑n ^ (κ - a)) Filter.atTop (nhds 0)","l":82,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_projection_risk_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectionRisk","a":"Stat/Nonparametric/HOIF","t":1,"d":"Order-m HOIF projection-risk bound. For an i.i.d. sample S and a projection kernel g, suppose g is symmetric, square-integrable, and doubly degenerate, the sample size n is at least 2, and the bandwidth h is positive, the projection dimension J and trace constant C are nonnegative, and the kernel's L²-energy ζ obeys the trace bound ζ ≤ C·J/h². Suppose further that the first-order variance V1 obeys the rate V1 ≤ Cv1/(nh) and the projection bias² Bsq obeys the rate Bsq ≤ Cb·J^{-4s/d}, and that the order-m estimation remainder R is controlled, over a finite index set T, by a sum of products of m+1 nuisance-error factors: every such factor is nonnegative, every factor is at most the largest nuisance error δ, and the absolute value of R is bounded by that sum of products. If, finally, the estimator's risk decomposes as risk ≤ V1 + Bsq + Var[Uₙ] + R², then the risk is bounded by the explicit sum Cv1/(nh) + Cb·J^{-4s/d} + 4C·J/(nh)² + |T|²·δ^{2(m+1)}.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {ι : Type u_3} (T : Finset ι) (e : ι → ℕ → ℝ) (R : ℝ) {risk V1 Bsq Cv1 Cb C J h s d δ : ℝ} {n m : ℕ}, 2 ≤ n → 0 < h → 0 ≤ J → 0 ≤ C → Causalean.Stat.IIDSample.zeta P g ≤ C * J / h ^ 2 → V1 ≤ Cv1 / (↑n * h) → Bsq ≤ Cb * J ^ (-(4 * s / d)) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), 0 ≤ e t k) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), e t k ≤ δ) → |R| ≤ ∑ t ∈ T, ∏ k ∈ Finset.range (m + 1), e t k → risk ≤ V1 + Bsq + ProbabilityTheory.variance (Causalean.Stat.uStatistic S g n) μ + R ^ 2 → risk ≤ Cv1 / (↑n * h) + Cb * J ^ (-(4 * s / d)) + 4 * C * J / (↑n * h) ^ 2 + ↑T.card ^ 2 * δ ^ (2 * (m + 1))","l":46,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectionRisk.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.hoif_remainder_sq_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":2,"d":"Squared HOIF product-remainder bound. The order-m HOIF estimation-bias remainder R is dominated by a sum, over a finite index set T of terms, of products of m+1 nuisance-error factors e t k; every factor lies in [0, δ], where δ is the largest nuisance L²-error. Then the remainder's squared contribution obeys R² ≤ |T|²·δ^{2(m+1)}. This is the order-m doubly-robust bound: the higher the order, the higher the power of δ.","s":"∀ {ι : Type u_1} (T : Finset ι) (e : ι → ℕ → ℝ) (m : ℕ) (δ R : ℝ), (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), 0 ≤ e t k) → (∀ t ∈ T, ∀ k ∈ Finset.range (m + 1), e t k ≤ δ) → |R| ≤ ∑ t ∈ T, ∏ k ∈ Finset.range (m + 1), e t k → R ^ 2 ≤ ↑T.card ^ 2 * δ ^ (2 * (m + 1))","l":45,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.integral_uStatistic_eq_zero","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":2,"d":"Mean of the degenerate U-statistic is zero.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → ∫ (ω : Ω), Causalean.Stat.uStatistic S g n ω ∂μ = 0","l":56,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.memLp_uStatistic","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.DegenerateUStatVariance","a":"Stat/Nonparametric/HOIF","t":2,"d":"The degenerate order-2 U-statistic is in L² (rescaling the √n-version).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {g : X → X → ℝ} (S : Causalean.Stat.IIDSample Ω X μ P), Causalean.Stat.DegenKernel P g → ∀ {n : ℕ}, 2 ≤ n → MeasureTheory.MemLp (Causalean.Stat.uStatistic S g n) 2 μ","l":42,"f":"Causalean/Stat/Nonparametric/HOIF/DegenerateUStatVariance.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.prod_le_pow_of_factors_le","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProductRemainder","a":"Stat/Nonparametric/HOIF","t":2,"d":"A product of m+1 factors each in [0, δ] is at most δ^{m+1}. Each nuisance-error factor of a single HOIF remainder term is nonnegative and bounded by the largest error δ; multiplying m+1 of them keeps the product below δ^{m+1}.","s":"∀ {ι : Type u_1} (e : ι → ℝ) (m : ℕ) (δ : ℝ) (s : Finset ℕ) (idx : ℕ → ι), s.card = m + 1 → (∀ k ∈ s, 0 ≤ e (idx k)) → (∀ k ∈ s, e (idx k) ≤ δ) → ∏ k ∈ s, e (idx k) ≤ δ ^ (m + 1)","l":33,"f":"Causalean/Stat/Nonparametric/HOIF/ProductRemainder.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel","k":"def","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"For a sample space, a nonnegative integer giving the number of basis functions, a real-valued basis evaluated at each sample point, and a real square weighting matrix indexed by those basis functions, the projected higher-order influence-function kernel assigns to two sample points $x,y$ the value $\\sum_{k,l} c_k(x)M_{kl}c_l(y)$.","s":"{X : Type u_1} → {J : ℕ} → (X → Fin J → ℝ) → Matrix (Fin J) (Fin J) ℝ → X → X → ℝ","l":56,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_L2_eq_dim","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":1,"d":"Projected degenerate-kernel L²-energy equals the dimension: ζ = J. For a feature map c into J-dimensional space, if every coordinate of c is square-integrable under P, the Gram matrix of c under P is invertible, and M is the inverse of that Gram matrix, then the squared L²(P⊗P)-norm of the HOIF projection kernel g(x,y) = ⟨c(x), M·c(y)⟩ equals the projection dimension J.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (k : Fin J), MeasureTheory.MemLp (fun x => c x k) 2 P) → IsUnit (Causalean.Stat.Nonparametric.HOIF.gram c P).det → M = (Causalean.Stat.Nonparametric.HOIF.gram c P)⁻¹ → ∫ (p : X × X), Causalean.Stat.Nonparametric.HOIF.projKernel c M p.1 p.2 ^ 2 ∂P.prod P = ↑J","l":222,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_L2_eq_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"L²-energy of the projected kernel (expanded form). By independence of the two sample points, the double integral of the squared projected kernel factors into Gram entries:","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (k : Fin J), MeasureTheory.MemLp (fun x => c x k) 2 P) → ∫ (p : X × X), Causalean.Stat.Nonparametric.HOIF.projKernel c M p.1 p.2 ^ 2 ∂P.prod P = ∑ k, ∑ l, ∑ k', ∑ l', M k l * M k' l' * Causalean.Stat.Nonparametric.HOIF.gram c P k k' * Causalean.Stat.Nonparametric.HOIF.gram c P l l'","l":97,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.projKernel_degen","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"One-sided degeneracy of the projected kernel. If the basis coordinates are centered (∫ c_l dP = 0 for every l), then ∫ g(x, ·) dP = 0 for every x. This records the zero-integral field only; measurability, symmetry, and square-integrability are separate requirements for packaging the kernel as a degenerate U-statistic kernel.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {J : ℕ} (c : X → Fin J → ℝ) (M : Matrix (Fin J) (Fin J) ℝ), (∀ (l : Fin J), MeasureTheory.Integrable (fun x => c x l) P) → (∀ (l : Fin J), ∫ (x : X), c x l ∂P = 0) → ∀ (x : X), ∫ (y : X), Causalean.Stat.Nonparametric.HOIF.projKernel c M x y ∂P = 0","l":73,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HOIF.sum_collapse_dim","k":"theorem","m":"Causalean.Stat.Nonparametric.HOIF.ProjectedKernelTrace","a":"Stat/Nonparametric/HOIF","t":2,"d":"The 4-fold Gram sum collapses to the projection dimension. With the HOIF weighting M = Σ⁻¹ (Σ = gram c P invertible), the expanded L²-energy equals the projection dimension J, because Σ Σ⁻¹ = Σ⁻¹ Σ = 1.","s":"∀ {J : ℕ} (M S : Matrix (Fin J) (Fin J) ℝ), IsUnit S.det → M = S⁻¹ → ∑ k, ∑ l, ∑ k', ∑ l', M k l * M k' l' * S k k' * S l l' = ↑J","l":191,"f":"Causalean/Stat/Nonparametric/HOIF/ProjectedKernelTrace.lean"},{"n":"Causalean.Stat.Nonparametric.HolderBallStd","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"For a nonnegative integer dimension, a real-valued function on that-dimensional real space, a real smoothness order, a real radius, and a region of that space, the standard multivariate Hölder ball condition requires that (1) the function has continuous derivatives through order $\\lceil\\mathrm{order}\\rceil-1$ on the region, (2) every derivative of order at most $\\lceil\\mathrm{order}\\rceil-1$ has norm at most the radius there, and (3) the derivative of order $\\lceil\\mathrm{order}\\rceil-1$ changes between any two points in the region by at most the radius times their distance to the power $\\mathrm{order}-(\\lceil\\mathrm{order}\\rceil-1)$.","s":"{d : ℕ} → ((Fin d → ℝ) → ℝ) → ℝ → ℝ → Set (Fin d → ℝ) → Prop","l":98,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.KernelOrder","k":"structure","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"A kernel of order at least p: supported in [-1,1], integrable, with unit mass ∫ K = 1 and vanishing moments ∫ uʲ K(u) du = 0 for 1 ≤ j ≤ p. These are the inputs of the classical interior local-polynomial / kernel bias theorem.","s":"(ℝ → ℝ) → ℕ → Prop","l":35,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinateDirections","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a bivariate multi-index and a position in a list whose length is its total order, the associated ordered standard-coordinate direction is the first standard coordinate direction when the position is smaller than the first coordinate order, and the second standard coordinate direction otherwise.","s":"(alpha : Fin 2 → ℕ) → Fin (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) → EuclideanSpace ℝ (Fin 2)","l":26,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a bivariate multi-index, the total order of that multi-index is the sum of its two coordinate orders.","s":"(Fin 2 → ℕ) → ℕ","l":22,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a real-valued function on two-dimensional Euclidean space, a bivariate multi-index, and a point in that space, the scalar coordinate partial derivative is the iterated derivative of total order given by the multi-index, evaluated at the point along the associated ordered standard-coordinate directions.","s":"(EuclideanSpace ℝ (Fin 2) → ℝ) → (Fin 2 → ℕ) → EuclideanSpace ℝ (Fin 2) → ℝ","l":35,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial_abs_le_iteratedFDeriv_norm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"Evaluating the iterated Fréchet derivative of a function f along the standard coordinate directions given by a bivariate multi-index alpha at a point x, cannot increase its operator norm — the resulting scalar coordinate partial is bounded in absolute value by the operator norm of the full iterated derivative.","s":"∀ (f : EuclideanSpace ℝ (Fin 2) → ℝ) (alpha : Fin 2 → ℕ) (x : EuclideanSpace ℝ (Fin 2)), |Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha x| ≤ ‖iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f x‖","l":42,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial_sub_abs_le_iteratedFDeriv_sub_norm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.CoordinateDerivative","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Differences of scalar coordinate partials are bounded by the operator norm of the corresponding Fréchet-derivative difference.","s":"∀ (f : EuclideanSpace ℝ (Fin 2) → ℝ) (alpha : Fin 2 → ℕ) (x z : EuclideanSpace ℝ (Fin 2)), |Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha x - Causalean.Stat.Nonparametric.LocalPolynomial.coordinatePartial f alpha z| ≤ ‖iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f x - iteratedFDeriv ℝ (Causalean.Stat.Nonparametric.LocalPolynomial.coordinateMultiOrder alpha) f z‖","l":62,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/CoordinateDerivative.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a nonnegative polynomial degree and real coefficients indexed from zero through that degree, the local-polynomial coefficient polynomial is $\\sum_i v_i u^i$.","s":"(p : ℕ) → (Fin (p + 1) → ℝ) → Polynomial ℝ","l":22,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial_eq_zero_iff","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The coefficient polynomial vanishes only when every coefficient does.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial p v = 0 ↔ v = 0","l":35,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial_eval","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Evaluation of the coefficient polynomial is the dot product with the monomial basis.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ) (u : ℝ), Polynomial.eval u (Causalean.Stat.Nonparametric.LocalPolynomial.localPolynomial p v) = ∑ i, v i * u ^ ↑i","l":28,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy","k":"def","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a nonnegative polynomial degree and real coefficients indexed from zero through that degree, the radial polynomial energy is the double sum of $v_i v_j$ divided by $i+j+2$, over all coefficient indices $i,j$ from zero through that degree.","s":"(p : ℕ) → (Fin (p + 1) → ℝ) → ℝ","l":55,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_coercive","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":1,"d":"For a polynomial degree bound p, there is a positive constant such that the radial polynomial energy of any coefficient vector is bounded below by that constant times the sum of the squared coefficients: on the unit sphere of the sup norm on coefficient vectors, radial polynomial energy has a positive minimum, and homogeneity packages this as a coercive lower bound for all coefficient vectors.","s":"∀ (p : ℕ), ∃ c, 0 < c ∧ ∀ (v : Fin (p + 1) → ℝ), c * ∑ i, v i ^ 2 ≤ Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":148,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_continuous","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The radial energy is a continuous quadratic function of its coefficient vector.","s":"∀ (p : ℕ), Continuous (Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p)","l":126,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_eq_integral","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The explicit moment matrix is exactly the weighted squared-polynomial integral on the unit interval.","s":"∀ (p : ℕ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v = ∫ (u : ℝ) in 0..1, (∑ i, v i * u ^ ↑i) ^ 2 * u","l":63,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"A nonzero coefficient vector has strictly positive radial energy.","s":"∀ (p : ℕ) {v : Fin (p + 1) → ℝ}, v ≠ 0 → 0 < Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":89,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy_smul","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"Radial energy is homogeneous of degree two in the coefficient vector.","s":"∀ (p : ℕ) (a : ℝ) (v : Fin (p + 1) → ℝ), Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p (a • v) = a ^ 2 * Causalean.Stat.Nonparametric.LocalPolynomial.radialPolynomialEnergy p v","l":134,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.LocalPolynomial.signedRadialPolynomialEnergy_coercive","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPolynomial.GramCoercivity","a":"Stat/Nonparametric/LocalPolynomial","t":2,"d":"The same coercivity constant works in both signed-distance orientations. The negative orientation merely changes coefficient i by the sign (-1)^i, which preserves the sum of coefficient squares.","s":"∀ (p : ℕ), ∃ c, 0 < c ∧ ∀ (t : Bool) (v : Fin (p + 1) → ℝ), c * ∑ i, v i ^ 2 ≤ ∫ (u : ℝ) in 0..1, (∑ i, v i * (if t = true then u else -u) ^ ↑i) ^ 2 * u","l":212,"f":"Causalean/Stat/Nonparametric/LocalPolynomial/GramCoercivity.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.jacksonSeriesBias_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Jackson bias rate for the series least-squares objective. If the noise-free least-squares objective A := lstsqObjective Φ w f c0 obeys the squared Jackson best-approximation bound A ≤ (∑ᵢ wᵢ)·(C_J·J^{−s/d})² — the shape produced by seriesApprox_le_of_sup with sup-error δ = C_J·J^{−s/d} — then the same objective satisfies the explicit doubled-exponent bias rate A ≤ ((∑ᵢ wᵢ)·C_J²) · J^{−2s/d}:","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {s d C_J : ℝ} {J : ℕ}, Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * (C_J * ↑J ^ (-(s / d))) ^ 2 → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * C_J ^ 2 * ↑J ^ (-(2 * s / d))","l":44,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.seriesEffectiveDoF_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Effective degrees of freedom is controlled by J/N. If the sample size N is positive, the least-squares weights are normalized as wᵢ = 1/N, and the hat map a obeys the Frobenius/trace bound ∑ᵢ ∑ₖ aᵢₖ² ≤ Cvar·J, then the weighted coefficient sum — the effective degrees of freedom V of the oracle inequality — obeys the bound Cvar · J / N:","s":"∀ {N : ℕ} {a : Fin N → Fin N → ℝ} {w : Fin N → ℝ} {Cvar : ℝ} {J : ℕ}, 0 < N → (∀ (i : Fin N), w i = 1 / ↑N) → ∑ i, ∑ k, a i k ^ 2 ≤ Cvar * ↑J → ∑ i, w i * ∑ k, a i k ^ 2 ≤ Cvar * ↑J / ↑N","l":72,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.SeriesSieve.seriesLS_prediction_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.PredictionRate","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Series / sieve least-squares prediction rate O(J^{−2s/d} + J/N). Assume a positive sample size; the noise-free projection c0 has residual orthogonal to every design column; the data fit chat ω differs from c0 by a deterministic linear image a of a noise family ε that is square-integrable, mean zero, and spherical with scale σ; the noise-free objective obeys the squared Jackson best-approximation bound at rate s/d with constant C_J; and the weights are normalized wᵢ = 1/N with the hat map obeying the Frobenius/trace bound ∑ᵢ ∑ₖ aᵢₖ² ≤ Cvar·J. Then the expected weighted quadratic prediction error of the fitted series coefficients obeys the standard series/sieve rate:","s":"∀ {Ω : Type u_1} {N : ℕ} {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {chat : Ω → ι → ℝ} {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {s d C_J Cvar σ : ℝ} {J : ℕ}, 0 < N → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → (∀ (ω : Ω) (i : Fin N), ∑ j, (c0 j - chat ω j) * Φ i j = ∑ k, a i k * ε k ω) → (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ (∑ i, w i) * (C_J * ↑J ^ (-(s / d))) ^ 2 → (∀ (i : Fin N), w i = 1 / ↑N) → ∑ i, ∑ k, a i k ^ 2 ≤ Cvar * ↑J → ∫ (ω : Ω), Causalean.Stat.Nonparametric.lstsqObjective Φ w f (chat ω) ∂μ ≤ (∑ i, w i) * C_J ^ 2 * ↑J ^ (-(2 * s / d)) + σ ^ 2 * (Cvar * ↑J / ↑N)","l":98,"f":"Causalean/Stat/Nonparametric/SeriesSieve/PredictionRate.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"For a real left endpoint, a real right endpoint, a nonnegative integer number of cells, and a real point, the base point of the point's cell is the left endpoint plus the cell index times the uniform cell width. The piecewise-Taylor approximant expands its function around this point.","s":"ℝ → ℝ → ℕ → ℝ → ℝ","l":50,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase_dist","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"x is within one cell width to the right of its base point: 0 ≤ x − base ≤ δ.","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → 0 ≤ x - Causalean.Stat.Nonparametric.cellBase lo hi J x ∧ x - Causalean.Stat.Nonparametric.cellBase lo hi J x ≤ Causalean.Stat.Nonparametric.cellWidth lo hi J","l":106,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellBase_mem","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"The cell base point lies inside the window [lo, hi].","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → ∀ {x : ℝ}, Causalean.Stat.Nonparametric.cellBase lo hi J x ∈ Set.Icc lo hi","l":73,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellIdx","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"For a real left endpoint, a real right endpoint, a nonnegative integer number of cells, and a real point, the uniform-partition cell index is the nonnegative floor of the point's displacement from the left endpoint divided by the cell width, capped at $J-1$ so that the right endpoint belongs to the final cell.","s":"ℝ → ℝ → ℕ → ℝ → ℕ","l":43,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellWidth","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"For a real left endpoint, a real right endpoint, and a nonnegative integer number of cells, the width of each cell in the uniform partition is $(\\mathrm{hi}-\\mathrm{lo})/J$.","s":"ℝ → ℝ → ℕ → ℝ","l":38,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.cellWidth_pos","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"The cell width is positive on a nondegenerate window.","s":"∀ {lo hi : ℝ} {J : ℕ}, lo < hi → 0 < J → 0 < Causalean.Stat.Nonparametric.cellWidth lo hi J","l":65,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Given a nonnegative polynomial degree, a sample indexed by a nonnegative number of observations, real-valued centered design coordinates for those observations, and real-valued observation weights, the weighted design moment matrix is the matrix whose $(j,k)$ entry is $\\sum_i w_i x_i^j x_i^k$.","s":"(p : ℕ) → {N : ℕ} → (Fin N → ℝ) → (Fin N → ℝ) → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":39,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The design moment matrix is symmetric.","s":"∀ {N p : ℕ} (x w : Fin N → ℝ), (Causalean.Stat.Nonparametric.designMatrix p x w).IsHermitian","l":62,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_isUnit_det","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Invertibility of the design moment matrix from design non-degeneracy. If the design weights w are nonnegative and the design is non-degenerate — no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ vanishing at every positively weighted design point, then the design moment matrix's determinant is a unit, i.e. the matrix is invertible: a positive definite matrix has a unit determinant, discharging the IsUnit (designMatrix p x w).det hypothesis used throughout the local-polynomial analysis.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0) → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det","l":102,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_posDef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"The design moment matrix is positive definite under a non-degeneracy condition on the design. If the design weights w are nonnegative and no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ that vanishes at every positively weighted design point, then the design moment matrix is positive definite. (Implied by the existence of p+1 distinct design points with positive weight, via Vandermonde.)","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0) → (Causalean.Stat.Nonparametric.designMatrix p x w).PosDef","l":81,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_posSemidef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The design moment matrix is positive semidefinite when the weights are nonnegative (a sum of weighted rank-one squares).","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (Causalean.Stat.Nonparametric.designMatrix p x w).PosSemidef","l":69,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.designMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Gram quadratic form of the design moment matrix. vᵀ M v = ∑ᵢ wᵢ (∑ⱼ vⱼ xᵢʲ)² — the design moment matrix is the weighted Gram matrix of the monomial feature vectors.","s":"∀ {N p : ℕ} (x w : Fin N → ℝ) (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.designMatrix p x w).mulVec v = ∑ i, w i * (∑ j, v j * x i ^ ↑j) ^ 2","l":31,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.entry00_eq_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The (0,0) entry of a matrix as a quadratic form at the first basis vector: A₀₀ = e₀ᵀ A e₀ with e₀ = Pi.single 0 1.","s":"∀ {p : ℕ} (A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), A 0 0 = Pi.single 0 1 ⬝ᵥ A.mulVec (Pi.single 0 1)","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.entry00_le_of_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich bound on the (0,0) entry. If A ⪯ C·B in the Loewner order on quadratic forms (wᵀ A w ≤ C·wᵀ B w for all w), then the top weight entries satisfy A₀₀ ≤ C·B₀₀. (Just evaluate the quadratic forms at the first basis vector.)","s":"∀ {p : ℕ} {A B : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ} {C : ℝ}, (∀ (w : Fin (p + 1) → ℝ), w ⬝ᵥ A.mulVec w ≤ C * w ⬝ᵥ B.mulVec w) → A 0 0 ≤ C * B 0 0","l":137,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Given a nonnegative polynomial degree, a sample indexed by a nonnegative number of observations, real-valued centered design coordinates, real-valued observation weights, and an observation in that sample, the local-polynomial equivalent-kernel weight for that observation is $\\sum_k (M^{-1})_{0k}w_i x_i^k$, where $M$ is the corresponding weighted design moment matrix. This weight extracts the fitted intercept: the degree-$p$ weighted least-squares intercept is the outcome-weighted sum of these weights.","s":"ℕ → {N : ℕ} → (Fin N → ℝ) → (Fin N → ℝ) → Fin N → ℝ","l":44,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_abs_sum_sq_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Cauchy–Schwarz leverage bound for the equivalent kernel. The ℓ¹ leverage of the equivalent-kernel weights is controlled by the product of the (0,0) entries of the design moment matrix and its inverse: (∑ᵢ |Sᵢ|)² ≤ M₀₀ · (M⁻¹)₀₀, where M₀₀ = ∑ᵢ wᵢ is the total weight. (Cauchy–Schwarz on Sᵢ = wᵢ gᵢ split as √wᵢ · √wᵢ gᵢ, using ∑ᵢ wᵢ gᵢ² = (M⁻¹)₀₀.) Together with equivKernelWeight_sq_sum_le this reduces *both* the bias leverage ∑ᵢ|Sᵢ| and the variance leverage ∑ᵢ Sᵢ² to the design quantities M₀₀ and (M⁻¹)₀₀.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∑ i, |Causalean.Stat.Nonparametric.equivKernelWeight p x w i|) ^ 2 ≤ Causalean.Stat.Nonparametric.designMatrix p x w 0 0 * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":316,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_reproduces","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Polynomial reproduction of the equivalent kernel. If the weighted design moment matrix is invertible, then the local-polynomial equivalent-kernel weights reproduce polynomials up to degree p: ∑ᵢ Sᵢ xᵢᵐ = [m = 0] for every m ≤ p. This discharges the reproduction hypothesis of linearSmoother_bias_of_reproduces.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → ∀ m ≤ p, ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * x i ^ m = if m = 0 then 1 else 0","l":48,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernelWeight_sq_sum_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Leverage bound for the equivalent kernel. With nonnegative weights bounded by W, the local-polynomial equivalent-kernel weights satisfy ∑ᵢ Sᵢ² ≤ W · (M⁻¹)₀₀. Combined with linearSmoother_variance_le, this reduces the interior O((Nh)^{−1/2}) stochastic-error rate to the single concentration bound (M⁻¹)₀₀ = O(1/(Nh)).","s":"∀ {N p : ℕ} {x w : Fin N → ℝ} {W : ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i ^ 2 ≤ W * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":264,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.equivKernel_weighted_sq_sum","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"Leverage identity for the equivalent kernel. Writing the equivalent-kernel weight as Sᵢ = wᵢ · gᵢ with gᵢ = ∑ₖ (M⁻¹)₀ₖ xᵢᵏ the unweighted kernel, the weighted leverage equals the (0,0) entry of the inverse moment matrix: ∑ᵢ wᵢ gᵢ² = (M⁻¹)₀₀. (Algebraically e₀ᵀ M⁻¹ M M⁻¹ e₀ = e₀ᵀ M⁻¹ e₀.)","s":"∀ {N p : ℕ} {x w : Fin N → ℝ}, IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → ∑ i, w i * (∑ k, (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 k * x i ^ ↑k) ^ 2 = (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0","l":185,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorBias_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional bias of a truncated estimator. Let est be an integrable statistic bounded by M in absolute value estimating a target θ with |θ| ≤ M, relative to a sub-σ-algebra m of the ambient σ-algebra. If G is an m-measurable good-design event on which, almost everywhere, the conditional bias |𝔼[est | m] − θ| is at most a nonnegative constant B, then the full-sample-law bias obeys |𝔼[est] − θ| ≤ B + 2M·μ(Gᶜ), the conditional bias B plus a truncation tail proportional to the bad-design probability.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ B M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.Integrable est μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ B → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → |μ[est | m] ω - θ| ≤ B) → |∫ (ω : Ω), est ω ∂μ - θ| ≤ B + 2 * M * (μ Gᶜ).toReal","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorStochL2_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional stochastic L² error of a truncated estimator. The √· form of estimatorVariance_unconditional: for an L² statistic est bounded by M in absolute value estimating θ with |θ| ≤ M relative to a sub-σ-algebra m of the ambient σ-algebra, and an m-measurable good-design event G on which, almost everywhere, the conditional variance is at most a nonnegative rate Vrate and the squared conditional bias is at most a nonnegative constant Bsq, the full-sample-law stochastic L² error obeys √Var(est) ≤ √(Vrate + Bsq + 5M²·μ(Gᶜ)). With Vrate = Cvar²·(Nh)⁻¹, Bsq = Cbias²·h^{2β} and a negligible truncation tail (5M²·μ(Gᶜ) driven to o((Nh)⁻¹) by designMatrix_inv_concentration), this gives the full-sample stochastic L² rate up to the bias and the negligible tail.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ Bsq Vrate M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.MemLp est 2 μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ Vrate → 0 ≤ Bsq → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → ProbabilityTheory.condVar m est μ ω ≤ Vrate) → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → (μ[est | m] ω - θ) ^ 2 ≤ Bsq) → √(ProbabilityTheory.variance est μ) ≤ √(Vrate + Bsq + 5 * M ^ 2 * (μ Gᶜ).toReal)","l":189,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.estimatorVariance_unconditional","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Unconditional","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Unconditional variance of a truncated estimator (law of total variance). Let est be an L² statistic bounded by M in absolute value estimating a target θ with |θ| ≤ M, relative to a sub-σ-algebra m of the ambient σ-algebra. Let G be an m-measurable good-design event on which, almost everywhere, the conditional variance Var[est | m] is at most a nonnegative rate Vrate and the squared conditional bias (𝔼[est | m] − θ)² is at most a nonnegative constant Bsq. Then the full-sample-law variance obeys Var(est) ≤ Vrate + Bsq + 5M²·μ(Gᶜ), the within-design variance rate plus the squared bias plus a truncation tail.","s":"∀ {Ω : Type u_1} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {est : Ω → ℝ} {θ Bsq Vrate M : ℝ} {G : Set Ω}, m ≤ m0 → MeasureTheory.MemLp est 2 μ → (∀ (ω : Ω), |est ω| ≤ M) → |θ| ≤ M → 0 ≤ Vrate → 0 ≤ Bsq → MeasurableSet G → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → ProbabilityTheory.condVar m est μ ω ≤ Vrate) → (∀ᵐ (ω : Ω) ∂μ, ω ∈ G → (μ[est | m] ω - θ) ^ 2 ≤ Bsq) → ProbabilityTheory.variance est μ ≤ Vrate + Bsq + 5 * M ^ 2 * (μ Gᶜ).toReal","l":117,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Unconditional.lean"},{"n":"Causalean.Stat.Nonparametric.exists_moment_cancelling_kernel_1d","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"For any prescribed finite order m, a compactly supported one-dimensional kernel can be chosen to have unit total mass while cancelling every polynomial moment through that order.","s":"∀ (m : ℕ), ∃ k, Continuous k ∧ (∀ (u : ℝ), 1 < |u| → k u = 0) ∧ ∫ (u : ℝ) in Set.Icc (-1) 1, k u = 1 ∧ ∀ (j : ℕ), 1 ≤ j → j ≤ m → ∫ (u : ℝ) in Set.Icc (-1) 1, u ^ j * k u = 0","l":161,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.expected_weighted_sq_image_spherical","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Expected weighted quadratic form of a deterministic linear image of spherical mean-zero noise. If ε is a spherical mean-zero square-integrable family with scale σ, and a is a deterministic matrix of coefficients, then the linear image Dᵢ(ω) = ∑ₖ aᵢₖ εₖ(ω) has expected weighted sum of squares","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {w : Fin N → ℝ} {σ : ℝ}, (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → ∫ (ω : Ω), ∑ i, w i * (∑ k, a i k * ε k ω) ^ 2 ∂μ = σ ^ 2 * ∑ i, w i * ∑ k, a i k ^ 2","l":87,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.holderDerivOrder","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"Given a real smoothness index, the Hölder derivative order is the nonnegative integer obtained by subtracting one, with truncation at zero, from the least nonnegative integer that is at least the smoothness index. For positive noninteger $\\beta$ this equals $\\lfloor\\beta\\rfloor$, and for positive integer $\\beta=m$ it equals $m-1$.","s":"ℝ → ℕ","l":36,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holderDerivOrder_lt","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"","s":"∀ {β : ℝ}, 0 < β → ↑(Causalean.Stat.Nonparametric.holderDerivOrder β) < β","l":39,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holder_point_l1_interpolation","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"Hölder pointwise ⟹ local L¹ mass interpolation. Fix a point x0 in d-dimensional Euclidean space. If the Hölder exponent γ is positive, the Hölder constant M is positive, the neighbourhood radius r is positive, and the sup-norm ball of radius r around x0 is contained in the domain S, then there is a constant c_H > 0, depending only on γ, d, M, r and uniform over the Hölder ball, such that every function g in the standard Hölder ball of exponent γ, constant M, and domain S satisfies c_H · |g(x0)|^{1 + d/γ} ≤ ∫_{supBall x0 r} |g|.","s":"∀ {d : ℕ} {γ M r : ℝ} {x0 : Fin d → ℝ} {S : Set (Fin d → ℝ)}, 0 < γ → 0 < M → 0 < r → Causalean.Stat.Nonparametric.supBall x0 r ⊆ S → ∃ cH, 0 < cH ∧ ∀ (g : (Fin d → ℝ) → ℝ), Causalean.Stat.Nonparametric.HolderBallStd g γ M S → cH * |g x0| ^ (1 + ↑d / γ) ≤ ∫ (x : Fin d → ℝ) in Causalean.Stat.Nonparametric.supBall x0 r, |g x|","l":796,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_monomial_approx","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylorMonomial","a":"Stat/Nonparametric/Approximation","t":1,"d":"Local monomial approximation in a Hölder ball. Fix a centre x0 in d-dimensional Euclidean space and a finite family expo of exponent multi-indices. If the smoothness index β is positive, the Hölder constant L is positive, the neighbourhood radius r is positive, the closed sup-norm cube of radius r around x0 is contained in the domain S, and expo lists every exponent multi-index of total degree up to the Taylor order, then there is a constant C_b ≥ 0, depending only on β and d, such that every function f in the standard Hölder ball of exponent β, constant L, and domain S is approximated near x0, at any bandwidth h ∈ (0, r), by a monomial combination in the expo basis with error at most C_b · L · h^β, uniformly over the unit cube of rescaled directions.","s":"∀ {d p : ℕ} {β L r : ℝ} {x0 : Fin d → ℝ} {S : Set (Fin d → ℝ)}, 0 < β → 0 < L → 0 < r → {x | ∀ (i : Fin d), |x i - x0 i| ≤ r} ⊆ S → ∀ (expo : Fin p → Fin d → ℕ), (∀ (e : Fin d → ℕ), ∑ j, e j ≤ ⌈β⌉₊ - 1 → ∃ k, expo k = e) → ∃ Cb, 0 ≤ Cb ∧ ∀ (f : (Fin d → ℝ) → ℝ), Causalean.Stat.Nonparametric.HolderBallStd f β L S → ∀ (h : ℝ), 0 < h → h < r → ∃ θ, ∀ (u : Fin d → ℝ), (∀ (j : Fin d), |u j| ≤ 1) → |f (x0 + h • u) - ∑ k, θ k * ∏ j, u j ^ expo k j| ≤ Cb * L * h ^ β","l":306,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylorMonomial.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_remainder","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"Hölder–Taylor remainder bound under the standard Hölder convention. Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the expansion point t lies in a window [lo,hi], the evaluation point a lies in the same window, f is p times continuously differentiable, and its p-th derivative is (β − p)-Hölder with constant M on the window, then the error of the degree-p Taylor approximation of f at t, evaluated at a, is bounded by (M / p!) · |a − t|^β. For positive integer β = m, this uses derivative order m - 1 and Hölder exponent 1.","s":"∀ {f : ℝ → ℝ} {M β lo hi t a : ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → a ∈ Set.Icc lo hi → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → |f a - Causalean.Stat.Nonparametric.taylorPoly (Causalean.Stat.Nonparametric.holderDerivOrder β) f t a| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * |a - t| ^ β","l":187,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.holder_taylor_remainder_of_lt","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Hölder-Taylor remainder when the evaluation point lies to the right of the base.","s":"∀ {f : ℝ → ℝ} {M β lo hi t a : ℝ} {p : ℕ}, 0 ≤ M → t ∈ Set.Icc lo hi → a ∈ Set.Icc lo hi → ContDiff ℝ (↑p) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv p f x - iteratedDeriv p f y| ≤ M * |x - y| ^ (β - ↑p)) → 0 < p → ↑p ≤ β → t < a → |f a - Causalean.Stat.Nonparametric.taylorPoly p f t a| ≤ M / ↑p.factorial * |a - t| ^ β","l":114,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.hpkPoly_continuous","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The coefficient polynomial of degree at most m built from a coefficient vector is a continuous function on the real line.","s":"∀ (m : ℕ) (c : Fin (m + 1) → ℝ), Continuous (Causalean.Stat.Nonparametric.hpkPoly✝ m c)","l":50,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Given a nonnegative polynomial degree, a measure on the real line, and a real target point, the integral population moment matrix is the matrix whose $(j,k)$ entry is $\\int (a-t)^j(a-t)^k\\,d\\nu(a)$.","s":"(p : ℕ) → MeasureTheory.Measure ℝ → ℝ → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":36,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"The integral moment matrix is symmetric.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).IsHermitian","l":97,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_isUnit_det","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Invertibility of the integral moment matrix from design non-degeneracy: a positive definite matrix has a unit determinant, discharging the IsUnit S.det hypothesis for the population shape matrix.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ¬∀ᵐ (a : ℝ) ∂ν, ∑ j, v j * (a - t) ^ ↑j = 0) → IsUnit (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).det","l":156,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_posDef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"The integral moment matrix is positive definite whenever every centered-monomial product (a−t)^j·(a−t)^k is integrable against the design measure ν and the design law is non-degenerate: no nonzero coefficient vector v makes the centered polynomial ∑ⱼ vⱼ (a−t)^j vanish ν-almost everywhere. Then the integral moment matrix intMomentMatrix p ν t is positive definite.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ¬∀ᵐ (a : ℝ) ∂ν, ∑ j, v j * (a - t) ^ ↑j = 0) → (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).PosDef","l":114,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_posSemidef","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"The integral moment matrix is positive semidefinite (a Gram matrix of an L² family).","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).PosSemidef","l":104,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.intMomentMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.IntegralMoment","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Gram quadratic form of the integral moment matrix. vᵀ S v = ∫ (∑ⱼ vⱼ (a−t)^j)² dν — the population moment matrix is the integral Gram matrix of the centered monomial features.","s":"∀ {p : ℕ} {ν : MeasureTheory.Measure ℝ} {t : ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun a => (a - t) ^ ↑j * (a - t) ^ ↑k) ν) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.intMomentMatrix p ν t).mulVec v = ∫ (a : ℝ), (∑ j, v j * (a - t) ^ ↑j) ^ 2 ∂ν","l":41,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/IntegralMoment.lean"},{"n":"Causalean.Stat.Nonparametric.integrable_of_abs_le_const_mul_kernel","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":2,"d":"If a function F is dominated in absolute value by a constant multiple C·|K| of an integrable kernel K, then F is itself integrable. This is the integrability workhorse for kernel-smoothing arguments: every factor multiplying K (a monomial, a Taylor remainder) is bounded by |K| on the kernel's support and vanishes off it.","s":"∀ {K F : ℝ → ℝ} {C : ℝ}, MeasureTheory.Integrable K MeasureTheory.volume → MeasureTheory.AEStronglyMeasurable F MeasureTheory.volume → (∀ (v : ℝ), |F v| ≤ C * |K v|) → MeasureTheory.Integrable F MeasureTheory.volume","l":53,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Leverage entry under diagonal-conjugation scaling. If S factors as κ • (D · T · D) for a nonzero scalar κ with D = diagonal d an invertible diagonal matrix (every d i nonzero) whose first entry is 1, and T invertible, then S is invertible and its (0,0) inverse entry collapses to (S⁻¹)₀₀ = κ⁻¹ · (T⁻¹)₀₀: the diagonal bandwidth powers cancel at the intercept because D₀₀ = 1.","s":"∀ {p : ℕ} {κ : ℝ}, κ ≠ 0 → ∀ {d : Fin (p + 1) → ℝ}, (∀ (i : Fin (p + 1)), d i ≠ 0) → d 0 = 1 → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → IsUnit S.det ∧ S⁻¹ 0 0 = κ⁻¹ * T⁻¹ 0 0","l":80,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_eq_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The (0,0) entry of an inverse as an inverse quadratic form at the first basis vector: (A⁻¹)₀₀ = e₀ᵀ A⁻¹ e₀ with e₀ = Pi.single 0 1.","s":"∀ {p : ℕ} (A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ), A⁻¹ 0 0 = Pi.single 0 1 ⬝ᵥ A⁻¹.mulVec (Pi.single 0 1)","l":76,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.inv00_le_of_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich bound on the (0,0) inverse entry. If A and B are positive definite and A dominates c·B in the Loewner order on quadratic forms (c·(wᵀ B w) ≤ wᵀ A w for all w) with c > 0, then the intercept leverage entries satisfy","s":"∀ {p : ℕ} {A B : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, A.PosDef → B.PosDef → ∀ {c : ℝ}, 0 < c → (∀ (w : Fin (p + 1) → ℝ), c * w ⬝ᵥ B.mulVec w ≤ w ⬝ᵥ A.mulVec w) → A⁻¹ 0 0 ≤ B⁻¹ 0 0 / c","l":88,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.inv_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Diagonal-conjugation inverse formula. If S = κ • (D · T · D) with κ ≠ 0 and D = diagonal d an invertible diagonal matrix (d i ≠ 0), T invertible, then S is invertible with S⁻¹ = κ⁻¹ • (D⁻¹ · T⁻¹ · D⁻¹) where D⁻¹ = diagonal (fun i => (d i)⁻¹).","s":"∀ {p : ℕ} {κ : ℝ}, κ ≠ 0 → ∀ {d : Fin (p + 1) → ℝ}, (∀ (i : Fin (p + 1)), d i ≠ 0) → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → IsUnit S.det ∧ S⁻¹ = κ⁻¹ • ((Matrix.diagonal fun i => (d i)⁻¹) * T⁻¹ * Matrix.diagonal fun i => (d i)⁻¹)","l":36,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.isCompact_cube","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"An axis-aligned box over a finite index type is compact: it is a finite product of closed bounded intervals.","s":"∀ {ι : Type u_1} [Finite ι] (a b : ℝ), IsCompact (Set.univ.pi fun x => Set.Icc a b)","l":91,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.isCompact_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"A closed sup-norm ball — the axis-aligned cube, or box, of half-width r centred at a point x0 in d-dimensional space — is compact, being a finite product of closed bounded intervals.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), IsCompact (Causalean.Stat.Nonparametric.supBall x0 r)","l":57,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.kernelDensity_lower_dom","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Pointwise weight domination from the design-density window bound. If the kernel K is nonnegative and supported in [-1,1] and the design density p obeys cDesign ≤ p(a) for every a within bandwidth h of t, then cDesign · K(u) ≤ K(u) · p(t + h·u) for every u: on the support |u| ≤ 1 the argument t + h·u lies in the window, off the support both sides vanish.","s":"∀ {K pdens : ℝ → ℝ} {t h cDesign : ℝ}, 0 < h → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → ∀ (u : ℝ), cDesign * K u ≤ K u * pdens (t + h * u)","l":225,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.kernelDensity_upper_dom","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Pointwise weight domination, upper side. Same hypotheses with an upper density bound p(a) ≤ CDesign on the window give K(u) · p(t + h·u) ≤ CDesign · K(u) for every u.","s":"∀ {K pdens : ℝ → ℝ} {t h CDesign : ℝ}, 0 < h → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (a : ℝ), |a - t| ≤ h → pdens a ≤ CDesign) → ∀ (u : ℝ), K u * pdens (t + h * u) ≤ CDesign * K u","l":248,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"Given a real-valued regression function, a real-valued kernel function, a real target point, and a real bandwidth, the population kernel-smoothing bias is $\\int h^{-1}K((u-t)/h)\\{f(u)-f(t)\\}\\,du$. It is the bias of estimating the regression function's value at the target point by kernel smoothing with that bandwidth.","s":"(ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ → ℝ","l":49,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":1,"d":"Interior kernel smoothing bias is O(h^β). Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the bandwidth h is positive, the kernel K has order p, the regression function f is p times continuously differentiable, and its p-th derivative is (β−p)-Hölder with constant M on the window [t−h, t+h], then the kernel smoothing bias of f at t with bandwidth h is bounded by (M/p! · ∫|K|)·h^β. For positive integer β = m, p = m - 1 and the Hölder exponent is 1. (Fan–Gijbels 1996 §3.1–3.3; Tsybakov 2009 Ch. 1.)","s":"∀ {f K : ℝ → ℝ} {β M t h : ℝ}, 0 < β → 0 ≤ M → 0 < h → Causalean.Stat.Nonparametric.KernelOrder K (Causalean.Stat.Nonparametric.holderDerivOrder β) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc (t - h) (t + h), ∀ y ∈ Set.Icc (t - h) (t + h), |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → |Causalean.Stat.Nonparametric.kernelSmoothingBias f K t h| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∫ (u : ℝ), |K u|) * h ^ β","l":89,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.kernelSmoothingBias_changeOfVar","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.Kernel","a":"Stat/Nonparametric/Approximation","t":2,"d":"Change of variables for the kernel smoothing bias. Substituting u = t + h v (h > 0) turns the bias integral into ∫ K(v) (f(t+hv) − f t) dv: the h⁻¹ prefactor cancels the Jacobian h, and (u−t)/h = v.","s":"∀ (f K : ℝ → ℝ) (t h : ℝ), 0 < h → Causalean.Stat.Nonparametric.kernelSmoothingBias f K t h = ∫ (v : ℝ), K v * (f (t + h * v) - f t)","l":64,"f":"Causalean/Stat/Nonparametric/Approximation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.l1_lower_of_bias_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The bandwidth optimization converts a lower bound on a kernel-smoothed signal into the Hölder interpolation rate: local absolute mass grows at least as the pointwise signal to the power one plus dimension divided by smoothness.","s":"∀ {d : ℕ} {γ Ksup Δ Ival cstar h : ℝ}, 0 < γ → 0 < Ksup → 0 < Δ → 0 < cstar → h = cstar * Δ ^ (1 / γ) → 3 * Δ / 4 ≤ Ksup * h⁻¹ ^ d * Ival → 3 / (4 * Ksup) * cstar ^ d * Δ ^ (1 + ↑d / γ) ≤ Ival","l":47,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_bias_of_reproduces","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Bias of a polynomial-reproducing linear smoother. If the weights Sᵢ reproduce polynomials up to degree p = holderDerivOrder β at t (∑ᵢ Sᵢ (aᵢ − t)ᵏ = [k = 0] for k ≤ p), the design points aᵢ and t lie in a window [lo, hi], and f is β-Hölder there (p-times continuously differentiable with (β−p)-Hölder top derivative, constant M), then the smoother's bias is controlled by the weighted spread of the design: |∑ᵢ Sᵢ f(aᵢ) − f t| ≤ (M/p!) · ∑ᵢ |Sᵢ| · |aᵢ − t|^β. (Tsybakov 2009, Ch. 1.)","s":"∀ {f : ℝ → ℝ} {β M lo hi t : ℝ} {N : ℕ} {a S : Fin N → ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → (∀ k ≤ Causalean.Stat.Nonparametric.holderDerivOrder β, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → |∑ i, S i * f (a i) - f t| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |S i| * |a i - t| ^ β","l":65,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_bias_window","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Interior O(h^β) bias of a polynomial-reproducing linear smoother. Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the target point t lies in a window [lo,hi], every design point aᵢ lies in the same window, every design point is within bandwidth h of t, f is p times continuously differentiable, its p-th derivative is (β−p)-Hölder with constant M on the window, and the smoother weights S reproduce polynomials of degree up to p at t, then the smoother's bias collapses to the standard rate |∑ᵢ Sᵢ f(aᵢ) − f t| ≤ (M/p!) · (∑ᵢ |Sᵢ|) · h^β.","s":"∀ {f : ℝ → ℝ} {β M lo hi t h : ℝ} {N : ℕ} {a S : Fin N → ℝ}, 0 < β → 0 ≤ M → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → (∀ k ≤ Causalean.Stat.Nonparametric.holderDerivOrder β, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → |∑ i, S i * f (a i) - f t| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |S i|) * h ^ β","l":105,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_reproduces_taylorPoly","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherBias","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"A linear smoother whose weights reproduce polynomials up to degree p (∑ᵢ Sᵢ (aᵢ − t)ᵏ = [k = 0] for k ≤ p) reproduces the degree-p Taylor polynomial of f at t exactly: ∑ᵢ Sᵢ · T_p(aᵢ; t) = f t. Only the constant term of the Taylor polynomial survives the reproduction identities.","s":"∀ {f : ℝ → ℝ} {t : ℝ} {N : ℕ} {a S : Fin N → ℝ} (p : ℕ), (∀ k ≤ p, ∑ i, S i * (a i - t) ^ k = if k = 0 then 1 else 0) → ∑ i, S i * Causalean.Stat.Nonparametric.taylorPoly p f t (a i) = f t","l":37,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherBias.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherVariance","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Stochastic-error bound for a fixed-weight linear smoother. If each response Yᵢ is square-integrable, the responses form a spherical family with common scale σ — each has variance σ², and distinct responses are uncorrelated, and the sum of squared smoother weights, ∑ᵢ Sᵢ², is bounded by V, then the variance of the linear smoother ∑ᵢ Sᵢ Yᵢ is at most σ² V.","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin N → Ω → ℝ} {S : Fin N → ℝ} {σ V : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → ∑ i, S i ^ 2 ≤ V → ProbabilityTheory.variance (fun ω => ∑ i, S i * Y i ω) μ ≤ σ ^ 2 * V","l":49,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.linearSmoother_variance_spherical","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.SmootherVariance","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Variance of a fixed-weight linear smoother under spherical errors. If Y is a spherical random family with scale σ (each cell has variance σ², distinct cells are uncorrelated) and each Yᵢ is L², then the linear smoother with deterministic weights S has variance Var[∑ᵢ Sᵢ Yᵢ] = σ² · ∑ᵢ Sᵢ².","s":"∀ {Ω : Type u_1} {N : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin N → Ω → ℝ} {S : Fin N → ℝ} {σ : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → ProbabilityTheory.variance (fun ω => ∑ i, S i * Y i ω) μ = σ ^ 2 * ∑ i, S i ^ 2","l":32,"f":"Causalean/Stat/Nonparametric/LeastSquares/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_density_inv00_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityLeverage","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Explicit Θ(1/(Nh)) leverage rate from density + kernel-moment constants. Fix a bandwidth h > 0, a sample size N with N > 0, and a design-density lower bound cDesign > 0. Let the kernel K be nonnegative and supported in [-1,1], with both centered-monomial integrands — against the shape weight K·p(t+h·) and against the pure kernel K — integrable, the pure kernel-moment matrix G positive definite, and the kernel shape matrix T positive definite; suppose the design density obeys cDesign ≤ p on the window |a − t| ≤ h. On the good design event where the population matrix's inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative scale η of the population matrix S = popDesignMatrix p N K pdens t h, and where the regime constants are small enough (c·(p+1)·η ≤ 1/2 and 2c²(p+1)η is at most the explicit density + kernel-moment constant cInv/(Nh)), then the empirical moment matrix M is invertible and its intercept leverage obeys the explicit interior rate (M⁻¹)₀₀ ≤ 2·cInv/(Nh), with cInv = (G⁻¹)₀₀/cDesign an explicit density + kernel-moment constant — no S-level invertibility or leverage assumption remains.","s":"∀ {p N : ℕ} {h cDesign η c t : ℝ} {K pdens : ℝ → ℝ} {M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < h → 0 < ↑N → 0 < cDesign → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * pdens (t + h * u) * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (Causalean.Stat.Nonparametric.weightMomentMatrix p K).PosDef → (Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)).PosDef → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |(Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h)⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → 2 * c ^ 2 * (↑(p + 1) * η) ≤ (Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h) → IsUnit M.det ∧ M⁻¹ 0 0 ≤ 2 * ((Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h))","l":44,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityLeverage.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_density_leverage_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityLeverage","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Bandwidth-free density bound on the local-polynomial leverage product. Fix a bandwidth h > 0, a sample size N > 0, and density-window bounds 0 < cDesign with cDesign ≤ CDesign. Let the kernel K be nonnegative and supported in [-1,1], with both centered-monomial integrands integrable, the pure kernel-moment matrix G positive definite, and the kernel shape matrix T positive definite; suppose the design density obeys the lower window bound cDesign ≤ p and the upper window bound p ≤ CDesign on |a − t| ≤ h. On the good design event — the population matrix's inverse row sums bounded by a nonnegative c, the empirical moment matrix M entrywise within a nonnegative scale η of the population matrix, with the regime constants small (c·(p+1)·η ≤ 1/2), the perturbation bound 2c²(p+1)η ≤ cInv/(Nh), and η at most Nh — and given the total weight M₀₀ is nonnegative and the inverse leverage (M⁻¹)₀₀ is nonnegative, then the geometric mean of the total weight and the inverse leverage is bounded by the bandwidth-free density constant √(M₀₀·(M⁻¹)₀₀) ≤ √(2·cInv·(cTop+1)), with cInv = (G⁻¹)₀₀/cDesign and cTop = CDesign·G₀₀. Via equivKernelWeight_abs_sum_sq_le this controls the ℓ¹ bias leverage ∑ᵢ|Sᵢ| by a bandwidth-free constant.","s":"∀ {p N : ℕ} {h cDesign CDesign η c t : ℝ} {K pdens : ℝ → ℝ} {M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < h → 0 < ↑N → 0 < cDesign → cDesign ≤ CDesign → (∀ (u : ℝ), 0 ≤ K u) → (∀ (u : ℝ), 1 < |u| → K u = 0) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * pdens (t + h * u) * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => K u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (Causalean.Stat.Nonparametric.weightMomentMatrix p K).PosDef → (Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)).PosDef → (∀ (a : ℝ), |a - t| ≤ h → cDesign ≤ pdens a) → (∀ (a : ℝ), |a - t| ≤ h → pdens a ≤ CDesign) → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |(Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h)⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → 2 * c ^ 2 * (↑(p + 1) * η) ≤ (Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign / (↑N * h) → η ≤ ↑N * h → 0 ≤ M 0 0 → 0 ≤ M⁻¹ 0 0 → √(M 0 0 * M⁻¹ 0 0) ≤ √(2 * ((Causalean.Stat.Nonparametric.weightMomentMatrix p K)⁻¹ 0 0 / cDesign) * (CDesign * Causalean.Stat.Nonparametric.weightMomentMatrix p K 0 0 + 1))","l":111,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityLeverage.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_estimatorBias_window","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.EstimatorRisk","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Interior local-polynomial estimator bias ≤ Cbias · h^β. Let ĉ₀ = c 0 be the degree- p = holderDerivOrder β weighted least-squares intercept fit — the coefficient vector c minimizing the weighted sum of squared residuals — of a β-Hölder function f (exponent β > 0) with Hölder constant MH ≥ 0 bounding the highest derivative's modulus of continuity at noise-free responses, where the weights are nonnegative, the target point t and every design point a i lie in a common interval [lo, hi], every design point is within bandwidth h ≥ 0 of t, and the design moment matrix is invertible. If the bandwidth-free leverage bound √(M₀₀·(M⁻¹)₀₀) ≤ L holds for some L ≥ 0 — supplied by localPoly_density_leverage_bound — then the conditional bias obeys |ĉ₀ − f(t)| ≤ (MH/p!) · L · h^β, the standard local-polynomial bias rate with constant Cbias = (MH/p!) · L.","s":"∀ {N : ℕ} {β MH lo hi t h L : ℝ} {a w : Fin N → ℝ} {f : ℝ → ℝ} {c : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ}, 0 < β → 0 ≤ MH → 0 ≤ h → (∀ (i : Fin N), 0 ≤ w i) → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ MH * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → IsUnit (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w).det → (∀ (c' : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ), ∑ i, w i * (f (a i) - ∑ j, c j * (a i - t) ^ ↑j) ^ 2 ≤ ∑ i, w i * (f (a i) - ∑ j, c' j * (a i - t) ^ ↑j) ^ 2) → √(Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w 0 0 * (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w)⁻¹ 0 0) ≤ L → |c 0 - f t| ≤ MH / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * L * h ^ β","l":42,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/EstimatorRisk.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_estimatorStochL2","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.EstimatorRisk","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Interior local-polynomial estimator stochastic L² error ≤ Cvar · (Nh)^{-1/2}. The degree-p local-polynomial equivalent-kernel smoother ĉ₀ = ∑ᵢ Sᵢ Yᵢ, applied to a family Y of square-integrable responses that is spherical with common scale σ, with an invertible design moment matrix and nonnegative weights bounded by a constant W ≥ 0, satisfies, on the good design event where the intercept leverage obeys the rate (M⁻¹)₀₀ ≤ 2·cInv/Nh for a nonnegative constant cInv and a positive scale Nh, the conditional stochastic L² error bound √Var(ĉ₀) ≤ √(2σ²·W·cInv) · (Nh)^{-1/2}, the standard interior stochastic rate with constant Cvar = √(2σ²·W·cInv).","s":"∀ {Ω : Type u_1} {N p : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {x w : Fin N → ℝ} {Y : Fin N → Ω → ℝ} {σ W cInv Nh : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → 0 ≤ W → 0 ≤ cInv → 0 < Nh → (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0 ≤ 2 * (cInv / Nh) → √(ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i ω) μ) ≤ √(2 * σ ^ 2 * W * cInv) * √Nh⁻¹","l":86,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/EstimatorRisk.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_intercept_bias","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Bias","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Interior local-polynomial bias is O(h^β). Let p denote the largest natural number strictly below the smoothness index β. If β is positive, the Hölder constant M is nonnegative, the design weights w are nonnegative, the target point t lies in a window [lo,hi], every design point aᵢ lies in the same window, every design point is within bandwidth h of t, the regression function f is p times continuously differentiable, its p-th derivative is (β−p)-Hölder with constant M on the window, the weighted design moment matrix is invertible, and c globally minimizes the weighted degree-p least-squares objective at noise-free responses f(aᵢ), then the fitted intercept c 0 estimates f t with bias |c 0 − f t| ≤ (M/p!)·(∑ᵢ |Sᵢ|)·h^β, where Sᵢ are the local-polynomial equivalent-kernel weights.","s":"∀ {N : ℕ} {β M lo hi t h : ℝ} {a w : Fin N → ℝ} {f : ℝ → ℝ} {c : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ}, 0 < β → 0 ≤ M → (∀ (i : Fin N), 0 ≤ w i) → t ∈ Set.Icc lo hi → (∀ (i : Fin N), a i ∈ Set.Icc lo hi) → (∀ (i : Fin N), |a i - t| ≤ h) → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → IsUnit (Causalean.Stat.Nonparametric.designMatrix (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w).det → (∀ (c' : Fin (Causalean.Stat.Nonparametric.holderDerivOrder β + 1) → ℝ), ∑ i, w i * (f (a i) - ∑ j, c j * (a i - t) ^ ↑j) ^ 2 ≤ ∑ i, w i * (f (a i) - ∑ j, c' j * (a i - t) ^ ↑j) ^ 2) → |c 0 - f t| ≤ (M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ∑ i, |Causalean.Stat.Nonparametric.equivKernelWeight (Causalean.Stat.Nonparametric.holderDerivOrder β) (fun i => a i - t) w i|) * h ^ β","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/Bias.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_intercept_variance_le","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.SmootherVariance","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Variance of the interior local-polynomial estimator. The degree-p local-polynomial equivalent-kernel smoother ∑ᵢ Sᵢ Yᵢ, applied to a family Y of square-integrable responses that is spherical with common scale σ, with an invertible design moment matrix and nonnegative weights bounded above by a constant W, has variance Var[∑ᵢ Sᵢ Yᵢ] ≤ σ² · W · (M⁻¹)₀₀. This reduces the interior O((Nh)^{−1/2}) stochastic-error rate to the single design-concentration bound (M⁻¹)₀₀ = O(1/(Nh)).","s":"∀ {Ω : Type u_1} {N p : ℕ} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {x w : Fin N → ℝ} {Y : Fin N → Ω → ℝ} {σ W : ℝ}, (∀ (i : Fin N), MeasureTheory.MemLp (Y i) 2 μ) → Causalean.GaussMarkov.SphericalFamily Y μ σ → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (i : Fin N), 0 ≤ w i) → (∀ (i : Fin N), w i ≤ W) → ProbabilityTheory.variance (fun ω => ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i ω) μ ≤ σ ^ 2 * (W * (Causalean.Stat.Nonparametric.designMatrix p x w)⁻¹ 0 0)","l":25,"f":"Causalean/Stat/Nonparametric/LocalPoly/SmootherVariance.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_inv00_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Explicit Θ(1/(Nh)) rate for the local-polynomial leverage. On a good design event with positive scale Nh where the population moment matrix S is invertible, its inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative perturbation scale η of S, the perturbation is small relative to the dimension: c·(p+1)·η ≤ 1/2, and the population intercept leverage sits at the Θ(Nh) scale: (S⁻¹)₀₀ ≤ cInv/(Nh) and 2c²(p+1)η ≤ cInv/(Nh), then the empirical moment matrix M is invertible and its intercept leverage obeys the explicit interior rate (M⁻¹)₀₀ ≤ 2·cInv/(Nh). This is the variance-rate capstone for the local-polynomial upper bound: combined with localPoly_intercept_variance_le it yields the O((Nh)^{-1/2}) stochastic error.","s":"∀ {p N : ℕ} {h c cInv η : ℝ} {S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < ↑N * h → IsUnit S.det → 0 ≤ c → 0 ≤ η → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → S⁻¹ 0 0 ≤ cInv / (↑N * h) → 2 * c ^ 2 * (↑(p + 1) * η) ≤ cInv / (↑N * h) → IsUnit M.det ∧ M⁻¹ 0 0 ≤ 2 * (cInv / (↑N * h))","l":77,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.localPoly_leverage_bound","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":1,"d":"Bandwidth-free bound on the local-polynomial leverage product. On the same good design event with positive scale Nh, where the population moment matrix S is invertible, its inverse row sums are bounded by a nonnegative constant c, the empirical moment matrix M lies entrywise within a nonnegative perturbation scale η that is small relative to the dimension (c·(p+1)·η ≤ 1/2) and at most Nh, with nonnegative density constants cInv and cTop such that the population intercept leverage and perturbation obey (S⁻¹)₀₀ ≤ cInv/(Nh) and 2c²(p+1)η ≤ cInv/(Nh), the population top weight obeys S₀₀ ≤ cTop·(Nh), and the empirical top weight M₀₀ and inverse leverage (M⁻¹)₀₀ are both nonnegative, the geometric mean of the total weight and the inverse leverage is bounded by the bandwidth-free constant √(M₀₀·(M⁻¹)₀₀) ≤ √(2·cInv·(cTop+1)). The Θ(Nh) growth of M₀₀ ≤ (cTop+1)·(Nh) exactly cancels the Θ(1/(Nh)) decay of (M⁻¹)₀₀. Via equivKernelWeight_abs_sum_sq_le ((∑ᵢ|Sᵢ|)² ≤ M₀₀·(M⁻¹)₀₀) this controls the ℓ¹ bias leverage ∑ᵢ|Sᵢ| by a bandwidth-free constant, the second leverage capstone used by the upper-bound analysis.","s":"∀ {p N : ℕ} {h c cInv cTop η : ℝ} {S M : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, 0 < ↑N * h → IsUnit S.det → 0 ≤ c → 0 ≤ η → 0 ≤ cInv → 0 ≤ cTop → (∀ (i : Fin (p + 1)), ∑ j, |S⁻¹ i j| ≤ c) → (∀ (j k : Fin (p + 1)), |M j k - S j k| ≤ η) → c * (↑(p + 1) * η) ≤ 1 / 2 → S⁻¹ 0 0 ≤ cInv / (↑N * h) → 2 * c ^ 2 * (↑(p + 1) * η) ≤ cInv / (↑N * h) → S 0 0 ≤ cTop * (↑N * h) → η ≤ ↑N * h → 0 ≤ M 0 0 → 0 ≤ M⁻¹ 0 0 → √(M 0 0 * M⁻¹ 0 0) ≤ √(2 * cInv * (cTop + 1))","l":102,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.lstsqObjective","k":"def","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Given a sample indexed by a nonnegative number of observations and a finite collection of basis indices, a real-valued design array, real-valued observation weights, real-valued outcomes, and real-valued coefficients indexed by the basis, the weighted least-squares objective is $\\sum_i w_i r_i(c)^2$, where $r_i(c)$ is the residual of observation $i$ under those coefficients.","s":"{N : ℕ} → {ι : Type u_1} → [Fintype ι] → (Fin N → ι → ℝ) → (Fin N → ℝ) → (Fin N → ℝ) → (ι → ℝ) → ℝ","l":43,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsqResidual","k":"def","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Given a sample indexed by a nonnegative number of observations and a finite collection of basis indices, a real-valued design array, real-valued outcomes, real-valued coefficients indexed by the basis, and an observation in the sample, the least-squares residual for that observation is its outcome minus the fitted value $\\sum_j c_j\\Phi_{ij}$.","s":"{N : ℕ} → {ι : Type u_1} → [Fintype ι] → (Fin N → ι → ℝ) → (Fin N → ℝ) → (ι → ℝ) → Fin N → ℝ","l":38,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_normal_equations","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":1,"d":"Normal equations for weighted linear least squares. If the weights w are nonnegative and the coefficient vector c globally minimizes the weighted sum of squares ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ Φᵢⱼ)² over all coefficient vectors, then the weighted residual is orthogonal to every design column: ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ Φᵢⱼ) Φᵢₖ = 0 for each basis index k.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c ≤ Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c') → ∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0","l":76,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_objective_le_of_orthogonal","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Optimality of the orthogonal least-squares fit. With nonnegative weights, a residual orthogonal to every design column attains the minimal weighted sum of squares: SSE(c) ≤ SSE(c') for every c'. (Immediate from lstsq_pythagoras, the excess term being a nonnegative weighted sum of squares.)","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0) → ∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c ≤ Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c'","l":229,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.lstsq_pythagoras","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Pythagorean decomposition for weighted least squares. If the residual at c is orthogonal to every design column (∑ᵢ wᵢ rᵢ(c) Φᵢₖ = 0 for all k — e.g. c is the least-squares minimizer, by lstsq_normal_equations), then for any coefficient vector c' the weighted sum of squares splits exactly as SSE(c') = SSE(c) + ∑ᵢ wᵢ (∑ⱼ (cⱼ − c'ⱼ) Φᵢⱼ)²: the orthogonality kills the cross term, so the excess error is the weighted norm of the fitted-value difference.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w Y : Fin N → ℝ} {c : ι → ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ Y c i * Φ i k = 0) → ∀ (c' : ι → ℝ), Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c' = Causalean.Stat.Nonparametric.lstsqObjective Φ w Y c + ∑ i, w i * (∑ j, (c j - c' j) * Φ i j) ^ 2","l":159,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.Nonparametric.measurableSet_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball (an axis-aligned cube, or box) is a measurable set.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), MeasurableSet (Causalean.Stat.Nonparametric.supBall x0 r)","l":64,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.mem_cube","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"Membership in an axis-aligned box: a point lies in it exactly when each of its coordinates lies between a and b.","s":"∀ {ι : Type u_1} {a b : ℝ} {x : ι → ℝ}, (x ∈ Set.univ.pi fun x => Set.Icc a b) ↔ ∀ (i : ι), a ≤ x i ∧ x i ≤ b","l":85,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.mem_supBall_self","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball (axis-aligned cube) of nonnegative half-width contains its own centre.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) {r : ℝ}, 0 ≤ r → x0 ∈ Causalean.Stat.Nonparametric.supBall x0 r","l":69,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.nondegenerate_of_distinct_points","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.DesignMatrixPosDef","a":"Stat/Nonparametric/LocalPoly","t":1,"d":"Design non-degeneracy from distinct positively-weighted points (Vandermonde). For a set S of design indices, if every index in S carries a positive weight and the design points xᵢ take at least p+1 distinct values on S, then the design non-degeneracy condition holds: no nonzero coefficient vector v yields a degree-p polynomial ∑ⱼ vⱼ xᵢʲ that vanishes at every positively weighted design point.","s":"∀ {N p : ℕ} {x w : Fin N → ℝ} (S : Finset (Fin N)), (∀ i ∈ S, 0 < w i) → p + 1 ≤ (Finset.image x S).card → ∀ (v : Fin (p + 1) → ℝ), v ≠ 0 → ∃ i, 0 < w i ∧ ∑ j, v j * x i ^ ↑j ≠ 0","l":115,"f":"Causalean/Stat/Nonparametric/LocalPoly/DesignMatrixPosDef.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylorApprox","k":"def","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"For a nonnegative Taylor degree, a real-valued function, a real left endpoint, a real right endpoint, a nonnegative integer number of cells, and a real evaluation point, the piecewise-Taylor sieve approximant is the degree-$p$ Taylor polynomial of the function, expanded at the left endpoint of the uniform-partition cell containing the evaluation point and evaluated at that point.","s":"ℕ → (ℝ → ℝ) → ℝ → ℝ → ℕ → ℝ → ℝ","l":57,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylor_sup_approx","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Piecewise-Taylor approximation bound under the standard Hölder convention. If f has p = holderDerivOrder β continuous derivatives and its p-th derivative obeys a Hölder bound with exponent β - p on the window, then the corresponding piecewise-Taylor approximant on a uniform J-cell partition has pointwise error at most (M / p!)·((hi − lo)/J)^β. For positive integer β = m, this uses derivative order m - 1 and Hölder exponent 1.","s":"∀ {f : ℝ → ℝ} {M β lo hi : ℝ} {J : ℕ}, 0 < β → 0 ≤ M → lo < hi → 0 < J → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → |f x - Causalean.Stat.Nonparametric.piecewiseTaylorApprox (Causalean.Stat.Nonparametric.holderDerivOrder β) f lo hi J x| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * ((hi - lo) / ↑J) ^ β","l":174,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.piecewiseTaylor_sup_approx_rate","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Jackson","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Jackson rate, J^{−β} form. Fix a Hölder exponent β > 0, a nonnegative Hölder constant M, and a nondegenerate window [lo, hi] with lo < hi subdivided into a positive number J of uniform cells. If f has p = holderDerivOrder β continuous derivatives on the window and its p-th derivative obeys the Hölder bound |f^(p) x − f^(p) y| ≤ M · |x − y|^(β − p) for all x, y in the window, then at any evaluation point x in the window, the piecewise-Taylor approximant on the uniform J-cell partition has pointwise error |f x − g x| ≤ C · J^{−β}, with constant C = (M / p!) · (hi − lo)^β independent of J.","s":"∀ {f : ℝ → ℝ} {M β lo hi : ℝ} {J : ℕ}, 0 < β → 0 ≤ M → lo < hi → 0 < J → ContDiff ℝ (↑(Causalean.Stat.Nonparametric.holderDerivOrder β)) f → (∀ x ∈ Set.Icc lo hi, ∀ y ∈ Set.Icc lo hi, |iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f x - iteratedDeriv (Causalean.Stat.Nonparametric.holderDerivOrder β) f y| ≤ M * |x - y| ^ (β - ↑(Causalean.Stat.Nonparametric.holderDerivOrder β))) → ∀ {x : ℝ}, x ∈ Set.Icc lo hi → |f x - Causalean.Stat.Nonparametric.piecewiseTaylorApprox (Causalean.Stat.Nonparametric.holderDerivOrder β) f lo hi J x| ≤ M / ↑(Causalean.Stat.Nonparametric.holderDerivOrder β).factorial * (hi - lo) ^ β * ↑J ^ (-β)","l":210,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Jackson.lean"},{"n":"Causalean.Stat.Nonparametric.popDesignMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Given a nonnegative polynomial degree, a nonnegative sample size, a real-valued kernel function, a real-valued design-weight function, a real target point, and a real bandwidth, the population design moment matrix is the matrix whose $(j,k)$ entry is $N\\int K((a-t)/h)(a-t)^{j+k}p(a)\\,da$.","s":"(p : ℕ) → ℕ → (ℝ → ℝ) → (ℝ → ℝ) → ℝ → ℝ → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":82,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.popDesignMatrix_factor","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Diagonal-conjugation factorization of the population moment matrix. For a positive bandwidth h > 0, writing T for the kernel shape matrix weightMomentMatrix p (fun u => K u · p(t+h·u)) and D for the diagonal matrix diagonal (fun j => h^j), the population design moment matrix factors as popDesignMatrix p N K pdens t h = (N·h) • (D · T · D). This is the literal S = (Nh)·(D T D) hypothesis of population_scaling_of_conj, proved by the single-entry change of variables popMomentEntry_changeOfVar.","s":"∀ {p : ℕ} (N : ℕ) (K pdens : ℝ → ℝ) (t h : ℝ), 0 < h → Causalean.Stat.Nonparametric.popDesignMatrix p N K pdens t h = (↑N * h) • (((Matrix.diagonal fun j => h ^ ↑j) * Causalean.Stat.Nonparametric.weightMomentMatrix p fun u => K u * pdens (t + h * u)) * Matrix.diagonal fun j => h ^ ↑j)","l":88,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.popMomentEntry_changeOfVar","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.Factorization","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Change of variables for a single moment entry. For h > 0, ∫ a, K((a−t)/h) · (a−t)^m · p(a) da = h^{m+1} · ∫ u, K(u) · u^m · p(t+h·u) du: the substitution a = t + h·u contributes the Jacobian h and turns (a−t)^m into (h·u)^m = h^m u^m.","s":"∀ (K pdens : ℝ → ℝ) (t h : ℝ), 0 < h → ∀ (m : ℕ), ∫ (a : ℝ), K ((a - t) / h) * (a - t) ^ m * pdens a = h ^ (m + 1) * ∫ (u : ℝ), K u * u ^ m * pdens (t + h * u)","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/Factorization.lean"},{"n":"Causalean.Stat.Nonparametric.population_scaling_of_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Θ(Nh) leverage scaling from the change-of-variables factorization. If the population moment matrix factors as S = (Nh)·D·T·D with D = diagonal (fun j => h^j) (so D₀₀ = 1) and a bandwidth-free shape matrix T that is invertible with (T⁻¹)₀₀ ≤ cInv and T₀₀ ≤ cTop, then S is invertible and its intercept leverage scales as (S⁻¹)₀₀ ≤ cInv/(Nh) while its top weight scales as S₀₀ ≤ cTop·(Nh). This discharges the leverage-scaling hypotheses of the rate capstones from the bandwidth-free shape matrix T.","s":"∀ {p N : ℕ} {h cInv cTop : ℝ}, 0 < h → 0 < ↑N → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, IsUnit T.det → T⁻¹ 0 0 ≤ cInv → T 0 0 ≤ cTop → S = (↑N * h) • ((Matrix.diagonal fun j => h ^ ↑j) * T * Matrix.diagonal fun j => h ^ ↑j) → IsUnit S.det ∧ S⁻¹ 0 0 ≤ cInv / (↑N * h) ∧ S 0 0 ≤ cTop * (↑N * h)","l":41,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"For a one-dimensional real-valued kernel and a nonnegative integer dimension, the multivariate product kernel assigns to each point the product of the one-dimensional kernel evaluated at all of its coordinates.","s":"(ℝ → ℝ) → (d : ℕ) → (Fin d → ℝ) → ℝ","l":114,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_abs_le","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A product kernel is uniformly bounded by the coordinatewise bound raised to the dimension, whenever its one-dimensional factor has that bound.","s":"∀ {d : ℕ} {k : ℝ → ℝ} {B : ℝ}, (∀ (u : ℝ), |k u| ≤ B) → ∀ (u : Fin d → ℝ), |Causalean.Stat.Nonparametric.prodKernel k d u| ≤ B ^ d","l":214,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_continuous","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The tensor kernel is continuous when the 1-D kernel is.","s":"∀ {d : ℕ} {k : ℝ → ℝ}, Continuous k → Continuous (Causalean.Stat.Nonparametric.prodKernel k d)","l":282,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_integral","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"The total mass of a product kernel factorizes into the product of the one-dimensional masses, so a unit-mass factor yields a unit-mass multivariate kernel.","s":"∀ {d : ℕ} (k : ℝ → ℝ), ∫ (u : Fin d → ℝ), Causalean.Stat.Nonparametric.prodKernel k d u = (∫ (t : ℝ), k t) ^ d","l":227,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.prodKernel_moment","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Kernel","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"Each polynomial moment of a product kernel factorizes into the corresponding one-dimensional moments, allowing any cancelled coordinate moment to cancel the entire multivariate moment.","s":"∀ {d : ℕ} (k : ℝ → ℝ) (ν : Fin d → ℕ), ∫ (u : Fin d → ℝ), (∏ i, u i ^ ν i) * Causalean.Stat.Nonparametric.prodKernel k d u = ∏ i, ∫ (t : ℝ), t ^ ν i * k t","l":235,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Kernel.lean"},{"n":"Causalean.Stat.Nonparametric.seriesApprox_le_of_sup","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Empirical approximation error is controlled by the best uniform approximant. If the least-squares fit c to target values f has residual orthogonal to every design column (lstsq_normal_equations) and some comparator cstar approximates f uniformly within δ at the design points (|fᵢ − ∑ⱼ cstarⱼ Φᵢⱼ| ≤ δ), then the empirical fit error is at most (∑ᵢ wᵢ)·δ². With the Jackson rate δ = C·J^{−s/d} this is the empirical best-approximation half of the series prediction rate.","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c cstar : ι → ℝ} {δ : ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c i * Φ i k = 0) → (∀ (i : Fin N), |f i - ∑ j, cstar j * Φ i j| ≤ δ) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c ≤ (∑ i, w i) * δ ^ 2","l":41,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.seriesLS_expected_prediction_le","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":1,"d":"Conditional oracle inequality for series least-squares prediction. Suppose the noise-free projection coefficients c0 have residual orthogonal to every design column, so that the noise-free least-squares objective is bounded by A, and suppose the data fit chat ω differs from c0, at every design row i, by a deterministic linear image ∑ₖ a i k · ε k ω of a noise family ε that is square-integrable, mean zero, and spherical with scale σ, with the weighted trace sum ∑ᵢ wᵢ ∑ₖ aᵢₖ² bounded by V. Then the expected weighted quadratic prediction error of the fitted series coefficients against the target values f is at most A + σ² V:","s":"∀ {Ω : Type u_1} {N : ℕ} {ι : Type u_2} [inst : Fintype ι] [inst_1 : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 : ι → ℝ} {chat : Ω → ι → ℝ} {ε : Fin N → Ω → ℝ} {a : Fin N → Fin N → ℝ} {σ A V : ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 ≤ A → (∀ (ω : Ω) (i : Fin N), ∑ j, (c0 j - chat ω j) * Φ i j = ∑ k, a i k * ε k ω) → (∀ (k : Fin N), MeasureTheory.MemLp (ε k) 2 μ) → (∀ (k : Fin N), ∫ (ω : Ω), ε k ω ∂μ = 0) → Causalean.GaussMarkov.SphericalFamily ε μ σ → ∑ i, w i * ∑ k, a i k ^ 2 ≤ V → ∫ (ω : Ω), Causalean.Stat.Nonparametric.lstsqObjective Φ w f (chat ω) ∂μ ≤ A + σ ^ 2 * V","l":171,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.seriesLS_prediction_decomp","k":"theorem","m":"Causalean.Stat.Nonparametric.SeriesSieve.Prediction","a":"Stat/Nonparametric/SeriesSieve","t":2,"d":"Pythagorean decomposition of the series prediction error. If the noise-free projection coefficient c0 (least-squares fit to the true values f) has residual orthogonal to every design column, then for *any* coefficient vector chat the prediction error of Φ·chat against the truth f splits exactly into the approximation error of the projection plus the squared fitted-value gap:","s":"∀ {N : ℕ} {ι : Type u_1} [inst : Fintype ι] {Φ : Fin N → ι → ℝ} {w f : Fin N → ℝ} {c0 chat : ι → ℝ}, (∀ (k : ι), ∑ i, w i * Causalean.Stat.Nonparametric.lstsqResidual Φ f c0 i * Φ i k = 0) → Causalean.Stat.Nonparametric.lstsqObjective Φ w f chat = Causalean.Stat.Nonparametric.lstsqObjective Φ w f c0 + ∑ i, w i * (∑ j, (c0 j - chat j) * Φ i j) ^ 2","l":70,"f":"Causalean/Stat/Nonparametric/SeriesSieve/Prediction.lean"},{"n":"Causalean.Stat.Nonparametric.supBall","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"For a nonnegative integer dimension, a centre in that-dimensional real space, and a real radius, the closed coordinatewise neighbourhood is the set of points whose every coordinate differs from the corresponding coordinate of the centre by at most the radius.","s":"{d : ℕ} → (Fin d → ℝ) → ℝ → Set (Fin d → ℝ)","l":40,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.supBall_eq_pi","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":2,"d":"A closed sup-norm ball — the axis-aligned cube of half-width r centred at a point — is exactly the product of the coordinate intervals of radius r around the centre's coordinates.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), Causalean.Stat.Nonparametric.supBall x0 r = Set.univ.pi fun i => Set.Icc (x0 i - r) (x0 i + r)","l":47,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly","k":"def","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":1,"d":"Given a nonnegative degree, a real-valued function, a real base point, and a real evaluation point, the degree-$p$ Taylor polynomial evaluated at the evaluation point is $\\sum_{k=0}^{p} f^{(k)}(t)(a-t)^k/k!$.","s":"ℕ → (ℝ → ℝ) → ℝ → ℝ → ℝ","l":46,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_eval_base","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"The Taylor polynomial of f at base point t, evaluated at t itself, is f t (every positive-degree term carries a (t − t)^k = 0 factor).","s":"∀ (p : ℕ) (f : ℝ → ℝ) (t : ℝ), Causalean.Stat.Nonparametric.taylorPoly p f t t = f t","l":50,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_reflect","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Reflecting across the base point preserves the Taylor polynomial value at the reflected evaluation point.","s":"∀ (p : ℕ) (f : ℝ → ℝ) (t a : ℝ), Causalean.Stat.Nonparametric.taylorPoly p (fun x => f (2 * t - x)) t (2 * t - a) = Causalean.Stat.Nonparametric.taylorPoly p f t a","l":89,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorPoly_succ","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"Split the last term off the Taylor polynomial.","s":"∀ (n : ℕ) (f : ℝ → ℝ) (t a : ℝ), Causalean.Stat.Nonparametric.taylorPoly (n + 1) f t a = Causalean.Stat.Nonparametric.taylorPoly n f t a + iteratedDeriv (n + 1) f t / ↑(n + 1).factorial * (a - t) ^ (n + 1)","l":81,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.taylorWithinEval_eq_taylorPoly","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderTaylor","a":"Stat/Nonparametric/Approximation","t":2,"d":"On an interval with distinct endpoints, the Taylor polynomial computed from derivatives restricted to the interval equals the usual Taylor polynomial computed from ordinary derivatives.","s":"∀ {f : ℝ → ℝ} {p n : ℕ} {x₀ x : ℝ}, ContDiff ℝ (↑p) f → n ≤ p → x₀ < x → taylorWithinEval f n (Set.Icc x₀ x) x₀ x = Causalean.Stat.Nonparametric.taylorPoly n f x₀ x","l":61,"f":"Causalean/Stat/Nonparametric/Approximation/HolderTaylor.lean"},{"n":"Causalean.Stat.Nonparametric.top00_diag_conj","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Rate.Conjugation","a":"Stat/Nonparametric/LocalPoly/Rate","t":2,"d":"Top entry under diagonal-conjugation scaling. Under the same factorization S = κ • (D · T · D) with d 0 = 1, the (0,0) entry of S itself is S₀₀ = κ · T₀₀.","s":"∀ {p : ℕ} {κ : ℝ} {d : Fin (p + 1) → ℝ}, d 0 = 1 → ∀ {T S : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, S = κ • (Matrix.diagonal d * T * Matrix.diagonal d) → S 0 0 = κ * T 0 0","l":95,"f":"Causalean/Stat/Nonparametric/LocalPoly/Rate/Conjugation.lean"},{"n":"Causalean.Stat.Nonparametric.two_dotProduct_sub_quadForm_le_inv","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.SquareCompletion","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Completing-the-square inequality for a positive definite quadratic form. For a positive definite matrix A and any vectors e, w, the affine functional 2 eᵀw − wᵀ A w is bounded above by the inverse quadratic form eᵀ A⁻¹ e, with equality at w = A⁻¹ e. This is the elementary identity eᵀ A⁻¹ e − (2 eᵀw − wᵀ A w) = (w − A⁻¹ e)ᵀ A (w − A⁻¹ e) ≥ 0.","s":"∀ {p : ℕ} {A : Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ}, A.PosDef → ∀ (e w : Fin (p + 1) → ℝ), 2 * e ⬝ᵥ w - w ⬝ᵥ A.mulVec w ≤ e ⬝ᵥ A⁻¹.mulVec e","l":38,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/SquareCompletion.lean"},{"n":"Causalean.Stat.Nonparametric.volume_supBall","k":"theorem","m":"Causalean.Stat.Nonparametric.Approximation.HolderInterpolation.Defs","a":"Stat/Nonparametric/Approximation/HolderInterpolation","t":1,"d":"The Lebesgue volume of a closed sup-norm ball — the axis-aligned cube of side 2r centred at x0 in d dimensions — equals the d-th power of the side length, (2r)^d. No sign restriction on r is needed: a negative half-width gives an empty cube and a zero right-hand side.","s":"∀ {d : ℕ} (x0 : Fin d → ℝ) (r : ℝ), MeasureTheory.volume (Causalean.Stat.Nonparametric.supBall x0 r) = ENNReal.ofReal (2 * r) ^ d","l":75,"f":"Causalean/Stat/Nonparametric/Approximation/HolderInterpolation/Defs.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix","k":"def","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"For a nonnegative integer degree and a real-valued weight function on the real line, the weight moment matrix is the square matrix whose row-$j$, column-$k$ entry is the Lebesgue integral $\\int W(u)u^j u^k\\,du$ for $j,k=0,\\ldots,p$.","s":"(p : ℕ) → (ℝ → ℝ) → Matrix (Fin (p + 1)) (Fin (p + 1)) ℝ","l":39,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_isHermitian","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"The weight moment matrix is symmetric.","s":"∀ {p : ℕ} {W : ℝ → ℝ}, (Causalean.Stat.Nonparametric.weightMomentMatrix p W).IsHermitian","l":107,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Gram quadratic form of the weight moment matrix. vᵀ (weightMomentMatrix p W) v = ∫ W(u) (∑ⱼ vⱼ uʲ)² du.","s":"∀ {p : ℕ} {W : ℝ → ℝ}, (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => W u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p W).mulVec v = ∫ (u : ℝ), W u * (∑ j, v j * u ^ ↑j) ^ 2","l":49,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm_sandwich","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":1,"d":"Loewner sandwich on quadratic forms from pointwise weight domination. Let WT and WG be weight functions and c a nonnegative scalar. If every centered-monomial integrand built from WT is integrable and the same holds for WG, and if the weights obey the pointwise domination c · WG(u) ≤ WT(u) for every u, then for every coefficient vector v the shape-matrix quadratic form dominates c times the pure-matrix quadratic form: c · (vᵀ G v) ≤ vᵀ T v, where T = weightMomentMatrix p WT and G = weightMomentMatrix p WG.","s":"∀ {p : ℕ} {WT WG : ℝ → ℝ} {c : ℝ}, 0 ≤ c → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WT u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WG u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (u : ℝ), c * WG u ≤ WT u) → ∀ (v : Fin (p + 1) → ℝ), c * v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WG).mulVec v ≤ v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WT).mulVec v","l":114,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.weightMomentMatrix_quadForm_sandwich_upper","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.EstimatorRisk.DensityConstants","a":"Stat/Nonparametric/LocalPoly/EstimatorRisk","t":2,"d":"Loewner sandwich on quadratic forms, upper direction. If W_T(u) ≤ C · W_G(u) pointwise (C ≥ 0) and both monomial integrands are integrable, then vᵀ T v ≤ C · (vᵀ G v) for every v, where T = weightMomentMatrix p W_T, G = weightMomentMatrix p W_G.","s":"∀ {p : ℕ} {WT WG : ℝ → ℝ} {C : ℝ}, 0 ≤ C → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WT u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (j k : Fin (p + 1)), MeasureTheory.Integrable (fun u => WG u * (u ^ ↑j * u ^ ↑k)) MeasureTheory.volume) → (∀ (u : ℝ), WT u ≤ C * WG u) → ∀ (v : Fin (p + 1) → ℝ), v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WT).mulVec v ≤ C * v ⬝ᵥ (Causalean.Stat.Nonparametric.weightMomentMatrix p WG).mulVec v","l":171,"f":"Causalean/Stat/Nonparametric/LocalPoly/EstimatorRisk/DensityConstants.lean"},{"n":"Causalean.Stat.Nonparametric.wls_intercept_eq_equivKernelSmoother","k":"theorem","m":"Causalean.Stat.Nonparametric.LocalPoly.Weights","a":"Stat/Nonparametric/LocalPoly","t":2,"d":"The local-polynomial WLS intercept is the equivalent-kernel linear smoother. If the weighted design moment matrix is invertible and c minimizes the weighted sum of squares, then the fitted intercept c 0 equals the linear smoother ∑ᵢ Sᵢ Yᵢ with the equivalent-kernel weights Sᵢ = equivKernelWeight p x w i. Combined with equivKernelWeight_reproduces and linearSmoother_bias_of_reproduces, this yields the interior local-polynomial bias estimate.","s":"∀ {N p : ℕ} {x w Y : Fin N → ℝ} {c : Fin (p + 1) → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → IsUnit (Causalean.Stat.Nonparametric.designMatrix p x w).det → (∀ (c' : Fin (p + 1) → ℝ), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) ^ 2 ≤ ∑ i, w i * (Y i - ∑ j, c' j * x i ^ ↑j) ^ 2) → c 0 = ∑ i, Causalean.Stat.Nonparametric.equivKernelWeight p x w i * Y i","l":97,"f":"Causalean/Stat/Nonparametric/LocalPoly/Weights.lean"},{"n":"Causalean.Stat.Nonparametric.wls_normal_equations","k":"theorem","m":"Causalean.Stat.Nonparametric.LeastSquares.NormalEquations","a":"Stat/Nonparametric/LeastSquares","t":2,"d":"Normal equations for weighted polynomial least squares (local-polynomial design Φᵢⱼ = (xᵢ)ʲ). The weighted least-squares minimizer of ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ xᵢʲ)² has residual orthogonal to every design monomial: ∑ᵢ wᵢ (Yᵢ − ∑ⱼ cⱼ xᵢʲ) xᵢᵏ = 0.","s":"∀ {N p : ℕ} {x w Y : Fin N → ℝ} {c : Fin (p + 1) → ℝ}, (∀ (i : Fin N), 0 ≤ w i) → (∀ (c' : Fin (p + 1) → ℝ), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) ^ 2 ≤ ∑ i, w i * (Y i - ∑ j, c' j * x i ^ ↑j) ^ 2) → ∀ (k : Fin (p + 1)), ∑ i, w i * (Y i - ∑ j, c j * x i ^ ↑j) * x i ^ ↑k = 0","l":139,"f":"Causalean/Stat/Nonparametric/LeastSquares/NormalEquations.lean"},{"n":"Causalean.Stat.OneShotSplit","k":"structure","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"One-shot sample split. For an i.i.d. sample S, a fold-size schedule n₁ : ℕ → ℕ splitting the sample of size n into a nuisance fold $A(n) = \\{0,\\dots,n_1(n)-1\\}$ and an estimation fold $B(n) = \\{n_1(n),\\dots,n-1\\}$, where the nuisance fold never exceeds the full sample, $n_1(n) \\le n$, the nuisance fold grows without bound, and the estimation fold also grows without bound, $n - n_1(n) \\to \\infty$.","s":"{Ω : Type u_3} → {X : Type u_4} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → Type","l":37,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.card_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"The size of the estimation fold is the sample size minus the split point.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), (split.foldB n).card = n - split.n₁ n","l":73,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.OneShotSplit.foldA","k":"def","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a two-fold split of that sample, and a sample size, the nuisance-fold index set consists of the nonnegative indices strictly smaller than the split's nuisance-fold size at that sample size.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → Causalean.Stat.OneShotSplit S → ℕ → Finset ℕ","l":55,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldA_disjoint_foldB","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"Fold-A and fold-B index sets are disjoint.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), Disjoint (split.foldA n) (split.foldB n)","l":86,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB","k":"def","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a two-fold split of that sample, and a sample size, the estimation-fold index set consists of the indices from the split's nuisance-fold size through one less than that sample size.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → {S : Causalean.Stat.IIDSample Ω X μ P} → Causalean.Stat.OneShotSplit S → ℕ → Finset ℕ","l":64,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_card","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"Cardinality of the estimation fold: |B(n)| = n - n₁(n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), (split.foldB n).card = n - split.n₁ n","l":73,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_card_tendsto","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":2,"d":"The estimation-fold cardinality grows to infinity.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S), Filter.Tendsto (fun n => (split.foldB n).card) Filter.atTop Filter.atTop","l":81,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_eq_Ico","k":"theorem","m":"Causalean.Stat.SampleSplit.PartialFoldCLT","a":"Stat/SampleSplit","t":2,"d":"The estimation fold of a one-shot split is the interval from the split point to the sample size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), split.foldB n = Finset.Ico (split.n₁ n) n","l":65,"f":"Causalean/Stat/SampleSplit/PartialFoldCLT.lean"},{"n":"Causalean.Stat.OneShotSplit.foldB_sampleMean_tendsto_inProb","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBWLLN","a":"Stat/SampleSplit","t":1,"d":"Fold-B weak law of large numbers. For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a measurable statistic g that is square-integrable under the population measure, the estimation-fold sample average of g converges in probability to the population integral $\\int g\\,dP$ as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {g : X → ℝ}, Measurable g → MeasureTheory.MemLp g 2 P → Causalean.Stat.Tendsto_inProb (fun n ω => (↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, g (S.Z i ω)) (fun x => ∫ (x : X), g x ∂P) μ","l":85,"f":"Causalean/Stat/SampleSplit/FoldBWLLN.lean"},{"n":"Causalean.Stat.OneShotSplit.folds_indep","k":"theorem","m":"Causalean.Stat.SampleSplit.OneShot","a":"Stat/SampleSplit","t":1,"d":"Independence of folds. For a fixed sample size n, the sample sub-tuple indexed by the nuisance fold A(n) is independent, under μ, of the sub-tuple indexed by the estimation fold B(n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {S : Causalean.Stat.IIDSample Ω X μ P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ), ProbabilityTheory.IndepFun (fun ω i => S.Z (↑i) ω) (fun ω i => S.Z (↑i) ω) μ","l":96,"f":"Causalean/Stat/SampleSplit/OneShot.lean"},{"n":"Causalean.Stat.OneShotSplit.integral_finiteSelector_sq_le","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"Given an iid one-shot sample split, a sample horizon, a measurable finite branch selector on the pilot fold, a measurable good pilot event, a nonnegative common risk bound, eligibility of every branch selected on the good event, integrability of every eligible branch's squared error under the tail-fold law, and the common fixed-branch risk bound, the selected squared risk on the good event is at most that common bound.","s":"∀ {Omega : Type u_1} {Iota : Type u_4} {X : Type u_5} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Iota] [inst_2 : MeasurableSpace X] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [Fintype Iota] [MeasurableSingletonClass Iota] {S : Causalean.Stat.IIDSample Omega X mu P} (split : Causalean.Stat.OneShotSplit S) (n : ℕ) {select : (↥(split.foldA n) → X) → Iota} {err : Iota → (↥(split.foldB n) → X) → ℝ} {Eligible : Iota → Prop} {V : ℝ}, Measurable select → ∀ {good : Set (↥(split.foldA n) → X)}, MeasurableSet good → 0 ≤ V → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map (fun omega j => S.Z (↑j) omega) mu)) → (∀ (i : Iota), Eligible i → ∫ (b : ↥(split.foldB n) → X), err i b ^ 2 ∂MeasureTheory.Measure.map (fun omega j => S.Z (↑j) omega) mu ≤ V) → ∫ (omega : Omega) in (fun omega j => S.Z (↑j) omega) ⁻¹' good, (err (select fun j => S.Z (↑j) omega) fun j => S.Z (↑j) omega) ^ 2 ∂mu ≤ V","l":423,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.OrderDegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Fully degenerate order-m kernel. A kernel g on m-tuples over X is completely degenerate under the product measure $P^{\\otimes m}$ when g is measurable, g is invariant under permuting its m coordinates, integrating g over any one coordinate against P gives zero, whichever coordinate and values are held fixed for the rest, and g is square-integrable under $P^{\\otimes m}$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":69,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.OrderDegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A square-integrable order-m degenerate kernel is integrable under the product law.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderDegenKernel P g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P)","l":82,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.OrderDegenKernel.integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"A fully degenerate order-m kernel has zero product-law mean.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_2 : NeZero m] {g : (Fin m → X) → ℝ}, Causalean.Stat.OrderDegenKernel P g → Causalean.Stat.uMeanOrder g P = 0","l":90,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.OrderDegenerateNegligible","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a positive integer order, and a real-valued kernel of that order, the order-$m$ degeneracy-negligibility condition asserts that the kernel's higher-order Hájek remainder, multiplied by $\\sqrt n$, converges in probability to zero as the sample size tends to infinity.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":95,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":1,"d":"First-order degenerate order-m kernel. A kernel g on m-tuples over X, together with the population measure P, is first-order degenerate when g is measurable, its first Hoeffding projection vanishes in every coordinate — integrating g over the other $m-1$ coordinates against the product measure leaves zero, whichever coordinate and value are held fixed, and g is square-integrable under the product measure $P^{\\otimes m}$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → Prop","l":44,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel.integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":2,"d":"A first-order degenerate square-integrable kernel is integrable under the product law.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → MeasureTheory.Integrable g (MeasureTheory.Measure.pi fun x => P)","l":70,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.OrderFirstDegenKernel.integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.FirstDegenKernel","a":"Stat/UStatistic/OrderM","t":1,"d":"Population mean of a first-order degenerate kernel is zero. If the order-m kernel g is first-order degenerate: measurable, square-integrable under the m-fold product law, and with zero mean after integrating out all but any single coordinate, then the population mean of g under the m-fold product law is zero.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} [inst_1 : NeZero m] {g : (Fin m → X) → ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.OrderFirstDegenKernel P g → Causalean.Stat.uMeanOrder g P = 0","l":79,"f":"Causalean/Stat/UStatistic/OrderM/FirstDegenKernel.lean"},{"n":"Causalean.Stat.PartialMatching","k":"structure","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"A partial matching between ordered coordinate sets of sizes r and s selects a subset from each side and pairs the selected coordinates bijectively.","s":"ℕ → ℕ → Type","l":25,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.MergedIndex","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For a partial matching between coordinate sets of sizes $r$ and $s$, the merged coordinate set contains every left coordinate together with precisely those right coordinates that are not selected by the matching.","s":"{r s : ℕ} → Causalean.Stat.PartialMatching r s → Type","l":80,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.empty","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For two nonnegative coordinate-set sizes, the empty partial matching selects no coordinate on either side and therefore contains no pairs.","s":"(r s : ℕ) → Causalean.Stat.PartialMatching r s","l":48,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.empty_size","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"For coordinate-set sizes r and s, the empty partial matching has no pairs.","s":"∀ (r s : ℕ), (Causalean.Stat.PartialMatching.empty r s).size = 0","l":59,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.eq_empty_of_size_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching with no pairs is the empty partial matching.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.size = 0 → M = Causalean.Stat.PartialMatching.empty r s","l":64,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.leftInjection","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For a partial matching and a left coordinate, the left-coordinate injection assigns that coordinate its own position in the merged coordinate set.","s":"{r s : ℕ} → (M : Causalean.Stat.PartialMatching r s) → Fin r → M.MergedIndex","l":86,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.mergedIndex_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"For a partial matching between coordinate sets of sizes r and s, the merged coordinate set contains r + s minus the matching size coordinates.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), Fintype.card M.MergedIndex = r + s - M.size","l":100,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.rightInjection","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For a partial matching and a right coordinate, the right-coordinate injection assigns the coordinate to its matched left-coordinate position when it is matched, and otherwise to its own separate position in the merged coordinate set.","s":"{r s : ℕ} → (M : Causalean.Stat.PartialMatching r s) → Fin s → M.MergedIndex","l":91,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.right_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"Given a partial matching, its selected right subset has the same number of coordinates as the matching.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.right.card = M.size","l":43,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.size","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For a partial matching, its matching size is the number of selected left coordinates, equivalently the number of paired coordinates.","s":"{r s : ℕ} → Causalean.Stat.PartialMatching r s → ℕ","l":39,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PartialMatching.size_le_min","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching has at most the smaller coordinate-set size many pairs.","s":"∀ {r s : ℕ} (M : Causalean.Stat.PartialMatching r s), M.size ≤ min r s","l":208,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.PinskerBound","k":"def","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"For a measurable sample space and two measures on that space, the Pinsker bound is the proposition that their total-variation distance is at most the square root of one half of the real-valued Kullback--Leibler divergence from the first measure to the second.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → Prop","l":243,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail","k":"structure","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Polynomial lower tail. For a measurable [0,1]-valued function U, says that the lower CDF t ↦ P{U ≤ t} is squeezed between cm·tᵏ and cp·tᵏ on the window (0, t₀], with a positive exponent κ, a window endpoint t₀ strictly between 0 and 1, and constants with 0 < cm < cp. This is the sole distributional input to the inverse-moment asymptotics.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ → ℝ → ℝ → Prop","l":58,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail.cp_pos","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The upper constant is positive.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < cp","l":84,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.PolyTail.t0_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The window endpoint is positive and < 1 packaged together.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < t₀ ∧ t₀ < 1","l":87,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.TailSetup","k":"structure","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Tail setup. Bundles the structural hypotheses on U: measurability and 0 < U ≤ 1 almost surely. IsProbabilityMeasure P is required separately as a typeclass on the theorems that need it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → Prop","l":95,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrableOn_rpow_neg_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"t^{-κ} is integrable on the window (t₀⁻¹, λ⁻¹].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → MeasureTheory.IntegrableOn (fun t => t ^ (-κ)) (Set.Ioc t₀⁻¹ lam⁻¹) MeasureTheory.volume","l":70,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.integrableOn_tailIntegrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The tail integrand is integrable on Ioc a b.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P] (a b : ℝ), MeasureTheory.IntegrableOn (fun t => P.real {ω | U ω ≤ t⁻¹}) (Set.Ioc a b) MeasureTheory.volume","l":80,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_invMomentI_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The I-integrand is integrable (bounded a.e. on a probability space).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (fun ω => U ω / max (U ω) lam ^ 2) P","l":224,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_invMomentJ_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The J-integrand is integrable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (fun ω => (max (U ω) lam)⁻¹) P","l":234,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integrable_trimWeight","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover weight is integrable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → MeasureTheory.Integrable (Causalean.Stat.PolynomialTail.trimWeight U lam) P","l":244,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.integral_inv_neg_Ioc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":2,"d":"Power integral, κ = 1. ∫ t in (a,b], t^{-1} = log b − log a.","s":"∀ {a b : ℝ}, 0 < a → a ≤ b → ∫ (t : ℝ) in Set.Ioc a b, t ^ (-1) = Real.log b - Real.log a","l":54,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.PolynomialTail.integral_rpow_neg_Ioc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":1,"d":"Power integral, κ ≠ 1. If the exponent κ is not equal to 1, a is positive, and a is at most b, then the interval integral of t ↦ t^{-κ} over (a, b] equals (b^{1-κ} − a^{1-κ})/(1−κ).","s":"∀ {κ : ℝ}, κ ≠ 1 → ∀ {a b : ℝ}, 0 < a → a ≤ b → ∫ (t : ℝ) in Set.Ioc a b, t ^ (-κ) = (b ^ (1 - κ) - a ^ (1 - κ)) / (1 - κ)","l":45,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued variable, and a real threshold, the truncated inverse second moment is the integral of the variable divided by the square of the larger of its value and the threshold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":117,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_antitone","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"I is antitone in λ. As λ decreases, max U λ decreases, so the integrand U/(max U λ)² increases. Hence I P U λ₂ ≤ I P U λ₁ when λ₁ ≤ λ₂.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → ∀ {l1 l2 : ℝ}, 0 < l1 → l1 ≤ l2 → Causalean.Stat.PolynomialTail.invMomentI P U l2 ≤ Causalean.Stat.PolynomialTail.invMomentI P U l1","l":103,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_eq","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ = 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent exactly 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is squeezed between a·log(1/λ) and A·log(1/λ) + A, uniformly for every λ ∈ (0, t₀]. Upper: I ≤ J; lower: I = J − (J−I) ≥ J − cp, uniformized with the constant shell floor.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ = 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * Real.log (1 / lam) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A * Real.log (1 / lam) + A","l":214,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ > 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly above 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is bounded between a and A, uniformly for every λ ∈ (0, t₀]: the inverse second moment does not blow up. Lower: I is antitone, so I(λ) ≥ I(t₀) ≥ (shell at t₀), a positive constant; upper: I ≤ J.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A","l":272,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_bounds_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime 0 < κ < 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, there exist constants 0 < a ≤ A such that the truncated inverse second moment I P U λ is squeezed between a·λ^{κ-1} and A·λ^{κ-1}, uniformly for every λ ∈ (0, t₀]. Lower: shell; upper: I ≤ J.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam ∧ Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ A * lam ^ (κ - 1)","l":190,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_ge_shell","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"Shell lower bound on I. (ρ·cm/2)·λ^{κ-1} ≤ I P U λ, with ρ = (cm/(2cp))^{1/κ}. Valid for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → cm / 2 * (cm / (2 * cp)) ^ (1 / κ) * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentI P U lam","l":118,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_integrand_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"On {0 < U ≤ 1} the I-integrand is squeezed in [0, 1/λ²].","s":"∀ {lam : ℝ}, 0 < lam → ∀ {x : ℝ}, 0 < x → x ≤ 1 → 0 ≤ x / max x lam ^ 2 ∧ x / max x lam ^ 2 ≤ (lam ^ 2)⁻¹","l":197,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_isTheta_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for I (κ > 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly above 1, the truncated inverse second moment I P U λ is Θ(1) — bounded — as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → (fun lam => Causalean.Stat.PolynomialTail.invMomentI P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun x => 1","l":355,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_isTheta_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for I (κ < 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, the truncated inverse second moment I P U λ is Θ(λ^{κ-1}) as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → (fun lam => Causalean.Stat.PolynomialTail.invMomentI P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun lam => lam ^ (κ - 1)","l":331,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentI_le_invMomentJ","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"I ≤ J. Pointwise U/(max U λ)² ≤ (max U λ)⁻¹ (since U ≤ max U λ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam","l":45,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"For a measurable sample space, a measure on that space, a real-valued variable, and a real threshold, the truncated inverse first moment is the integral of the reciprocal of the larger of the variable's value and the threshold.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":125,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_eq","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ = 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent exactly 1, there exist constants 0 < a ≤ A such that the truncated inverse first moment J P U λ is squeezed between a·log(1/λ) and A·log(1/λ) + A, uniformly for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ = 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * Real.log (1 / lam) ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A * Real.log (1 / lam) + A","l":126,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime κ > 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly above 1, there exist constants 0 < a ≤ A such that the truncated inverse first moment J P U λ is bounded between a and A, uniformly for every λ ∈ (0, t₀]: the inverse first moment does not blow up.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A","l":173,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_bounds_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":1,"d":"Regime 0 < κ < 1. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, there exist constants 0 < a ≤ A, depending only on κ, cm, cp, t₀, such that the truncated inverse first moment J P U λ is squeezed between a·λ^{κ-1} and A·λ^{κ-1}, uniformly for every λ ∈ (0, t₀].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → ∃ a A, 0 < a ∧ a ≤ A ∧ ∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * lam ^ (κ - 1) ≤ Causalean.Stat.PolynomialTail.invMomentJ P U lam ∧ Causalean.Stat.PolynomialTail.invMomentJ P U lam ≤ A * lam ^ (κ - 1)","l":65,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_eq_one_add_tailIntegral","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":1,"d":"Master layer-cake reduction. For a variable U that is measurable and almost surely strictly positive and at most 1, and a threshold λ that is positive and strictly below 1, the truncated inverse first moment J P U λ equals 1 plus the threshold tail integral tailIntegral P U λ:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam < 1 → Causalean.Stat.PolynomialTail.invMomentJ P U lam = 1 + Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":91,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_integrand_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The J-integrand is squeezed in [0, 1/λ].","s":"∀ {lam : ℝ}, 0 < lam → ∀ (x : ℝ), 0 ≤ (max x lam)⁻¹ ∧ (max x lam)⁻¹ ≤ lam⁻¹","l":207,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_isTheta_gt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for J (κ > 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly above 1, the truncated inverse first moment J P U λ is Θ(1) — bounded — as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 1 < κ → (fun lam => Causalean.Stat.PolynomialTail.invMomentJ P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun x => 1","l":343,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_isTheta_lt","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":1,"d":"IsTheta for J (κ < 1). Under the polynomial lower-tail hypothesis with U measurable and almost surely in (0,1] and tail exponent strictly between 0 and 1, the truncated inverse first moment J P U λ is Θ(λ^{κ-1}) as λ → 0⁺.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → κ < 1 → (fun lam => Causalean.Stat.PolynomialTail.invMomentJ P U lam) =Θ[nhdsWithin 0 (Set.Ioi 0)] fun lam => lam ^ (κ - 1)","l":319,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.invMomentJ_sub_invMomentI_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"J − I is controlled by the tail mass. J P U λ − I P U λ ≤ cp·λ^{κ-1}. Pointwise the gap equals wλ/(max U λ) ≤ λ⁻¹·wλ, and ∫ wλ ≤ P{U<λ} ≤ cp·λᵏ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → Causalean.Stat.PolynomialTail.invMomentJ P U lam - Causalean.Stat.PolynomialTail.invMomentI P U lam ≤ cp * lam ^ (κ - 1)","l":61,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.inv_rpow_eq_rpow_neg","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"(t⁻¹)^κ = t^{-κ} for t > 0.","s":"∀ {t : ℝ}, 0 < t → ∀ (κ : ℝ), t⁻¹ ^ κ = t ^ (-κ)","l":48,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.isTheta_of_two_sided","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentIBounds","a":"Stat/PolynomialTail","t":2,"d":"Generic two-sided-bound ⟹ IsTheta packager on 𝓝[>] 0.","s":"∀ {f g : ℝ → ℝ} {t₀ a A : ℝ}, 0 < t₀ → 0 < a → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → 0 < g lam) → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → 0 ≤ f lam) → (∀ (lam : ℝ), 0 < lam → lam ≤ t₀ → a * g lam ≤ f lam ∧ f lam ≤ A * g lam) → f =Θ[nhdsWithin 0 (Set.Ioi 0)] g","l":298,"f":"Causalean/Stat/PolynomialTail/MomentIBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.le_truncDen","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"max U λ ≥ λ (pointwise, no hypotheses).","s":"∀ {Ω : Type u_1} {U : Ω → ℝ} {lam : ℝ} (ω : Ω), lam ≤ Causalean.Stat.PolynomialTail.truncDen U lam ω","l":172,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_invMomentI_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The inverse second-moment integrand is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable fun ω => U ω / max (U ω) lam ^ 2","l":151,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_invMomentJ_integrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The inverse first-moment integrand is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable fun ω => (max (U ω) lam)⁻¹","l":157,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_tailIntegrand","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand t ↦ P{U ≤ t⁻¹} is measurable (monotone CDF ∘ inversion).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], Measurable fun t => P.real {ω | U ω ≤ t⁻¹}","l":63,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_trimWeight","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover trimming weight is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable (Causalean.Stat.PolynomialTail.trimWeight U lam)","l":163,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measurable_truncDen","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The truncated denominator is measurable when U is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {U : Ω → ℝ} {lam : ℝ}, Measurable U → Measurable (Causalean.Stat.PolynomialTail.truncDen U lam)","l":145,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.measureReal_lt_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"Mass below λ. Under the polynomial lower-tail hypothesis packaged in h, for a threshold λ that is positive and at most the window endpoint t₀, the open lower level set {U < λ} has P-mass at most cp · λᵏ. Immediate from monotonicity and the upper tail bound at t = λ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam κ t₀ cm cp : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → P.real {ω | U ω < lam} ≤ cp * lam ^ κ","l":184,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.monotone_cdf","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The lower CDF s ↦ P{U ≤ s} is monotone.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsFiniteMeasure P], Monotone fun s => P.real {ω | U ω ≤ s}","l":58,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.powInt_eq_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":2,"d":"PowInt = log(1/λ) − log(1/t₀) when κ = 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → κ = 1 → 0 < lam → lam ≤ t₀ → ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) = Real.log (1 / lam) - Real.log (1 / t₀)","l":55,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.powInt_ne_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.MomentJBounds","a":"Stat/PolynomialTail","t":2,"d":"PowInt = (λ^{κ-1} − t₀^{κ-1})/(1−κ) when κ ≠ 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ}, Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → κ ≠ 1 → 0 < lam → lam ≤ t₀ → ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) = (lam ^ (κ - 1) - t₀ ^ (κ - 1)) / (1 - κ)","l":46,"f":"Causalean/Stat/PolynomialTail/MomentJBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral","k":"def","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":1,"d":"For a measurable sample space, a measure on that sample space, a real-valued random variable on that sample space, and a real threshold, the threshold tail integral is the integral over $1<t\\leq\\lambda^{-1}$ of the probability that the random variable is at most $t^{-1}$.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → ℝ → ℝ","l":50,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_ge","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":1,"d":"Lower bound on the tail integral. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp, for any threshold λ that is positive and at most the window endpoint t₀, cm times the power comparison integral over (t₀⁻¹, λ⁻¹] is a lower bound for the layer-cake tail integral tailIntegral P U λ:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → cm * ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ) ≤ Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":122,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_le","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":1,"d":"Upper bound on the tail integral. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp, for any threshold λ that is positive and at most the window endpoint t₀, the layer-cake tail integral tailIntegral P U λ is bounded by a constant pre-window contribution t₀⁻¹ − 1 plus cp times the power comparison integral over (t₀⁻¹, λ⁻¹]:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → 0 < lam → lam ≤ t₀ → Causalean.Stat.PolynomialTail.tailIntegral P U lam ≤ t₀⁻¹ - 1 + cp * ∫ (t : ℝ) in Set.Ioc t₀⁻¹ lam⁻¹, t ^ (-κ)","l":83,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegral_nonneg","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"tailIntegral is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {lam : ℝ} [MeasureTheory.IsProbabilityMeasure P], 0 ≤ Causalean.Stat.PolynomialTail.tailIntegral P U lam","l":150,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_le_one","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand is bounded by 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} [MeasureTheory.IsProbabilityMeasure P] (t : ℝ), P.real {ω | U ω ≤ t⁻¹} ≤ 1","l":73,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_lower_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"Lower sandwich of the tail integrand on the polynomial window.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → ∀ {t : ℝ}, t ∈ Set.Ioc t₀⁻¹ lam⁻¹ → cm * t ^ (-κ) ≤ P.real {ω | U ω ≤ t⁻¹}","l":61,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_nonneg","k":"theorem","m":"Causalean.Stat.PolynomialTail.LayerCakeReduction","a":"Stat/PolynomialTail","t":2,"d":"The integrand is nonnegative.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} (t : ℝ), 0 ≤ P.real {ω | U ω ≤ t⁻¹}","l":69,"f":"Causalean/Stat/PolynomialTail/LayerCakeReduction.lean"},{"n":"Causalean.Stat.PolynomialTail.tailIntegrand_upper_window","k":"theorem","m":"Causalean.Stat.PolynomialTail.TailIntegralBounds","a":"Stat/PolynomialTail","t":2,"d":"Upper sandwich of the tail integrand on the polynomial window.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam : ℝ} [MeasureTheory.IsFiniteMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → ∀ {t : ℝ}, t ∈ Set.Ioc t₀⁻¹ lam⁻¹ → P.real {ω | U ω ≤ t⁻¹} ≤ cp * t ^ (-κ)","l":52,"f":"Causalean/Stat/PolynomialTail/TailIntegralBounds.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"For a sample space, a real-valued variable on that space, a real threshold, and a sample point, the trimming weight is one minus the variable's value divided by the larger of that value and the threshold.","s":"{Ω : Type u_1} → (Ω → ℝ) → ℝ → Ω → ℝ","l":133,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight_bias_bound","k":"theorem","m":"Causalean.Stat.PolynomialTail.TrimBias","a":"Stat/PolynomialTail","t":1,"d":"General trimming-bias bound. Under the polynomial lower-tail hypothesis PolyTail P U κ t₀ cm cp with U measurable and almost surely in (0,1], fix a threshold λ that is positive and at most the window endpoint t₀. If the envelope constant Cβ is nonnegative, g is measurable, and g obeys the Hölder envelope |g| ≤ Cβ · λ^β on the event {U < λ}, then the integrated bias |∫ wλ · g dP| is at most Cβ · cp · λ^{κ+β}:","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} {U : Ω → ℝ} {κ t₀ cm cp lam β : ℝ} [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.PolynomialTail.PolyTail P U κ t₀ cm cp → Causalean.Stat.PolynomialTail.TailSetup P U → 0 < lam → lam ≤ t₀ → ∀ {g : Ω → ℝ} {Cβ : ℝ}, 0 ≤ Cβ → Measurable g → (∀ (ω : Ω), U ω < lam → |g ω| ≤ Cβ * lam ^ β) → |∫ (ω : Ω), Causalean.Stat.PolynomialTail.trimWeight U lam ω * g ω ∂P| ≤ Cβ * cp * lam ^ (κ + β)","l":43,"f":"Causalean/Stat/PolynomialTail/TrimBias.lean"},{"n":"Causalean.Stat.PolynomialTail.trimWeight_mem","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"The leftover weight lies in [0,1] whenever U > 0.","s":"∀ {lam : ℝ}, 0 < lam → ∀ {x : ℝ}, 0 < x → 0 ≤ 1 - x / max x lam ∧ 1 - x / max x lam ≤ 1","l":215,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.truncDen","k":"def","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":1,"d":"For a sample space, a real-valued variable on that space, a real threshold, and a sample point, the truncated denominator is the larger of the variable's value at that point and the threshold.","s":"{Ω : Type u_1} → (Ω → ℝ) → ℝ → Ω → ℝ","l":112,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.truncDen_pos","k":"theorem","m":"Causalean.Stat.PolynomialTail.Basic","a":"Stat/PolynomialTail","t":2,"d":"For 0 < λ, the denominator is positive.","s":"∀ {Ω : Type u_1} {U : Ω → ℝ} {lam : ℝ}, 0 < lam → ∀ (ω : Ω), 0 < Causalean.Stat.PolynomialTail.truncDen U lam ω","l":176,"f":"Causalean/Stat/PolynomialTail/Basic.lean"},{"n":"Causalean.Stat.PolynomialTail.zero_notMem_uIcc","k":"theorem","m":"Causalean.Stat.PolynomialTail.PowerIntegral","a":"Stat/PolynomialTail","t":2,"d":"0 ∉ [[a,b]] when 0 < a ≤ b.","s":"∀ {a b : ℝ}, 0 < a → a ≤ b → 0 ∉ Set.uIcc a b","l":39,"f":"Causalean/Stat/PolynomialTail/PowerIntegral.lean"},{"n":"Causalean.Stat.PositiveCovariance","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For a finite coordinate type, the positive covariance matrices are real square matrices equipped with a proof of positive definiteness.","s":"(V : Type u_2) → [Fintype V] → Type (max 0 u_2)","l":23,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.Privacy.approx_dp_postprocess","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Post-processing preserves approximate differential privacy. Given a vector-valued mechanism M obeying an $(\\varepsilon,\\delta)$-approximate DP bound between the releases M d and M d' for every measurable event, with a nonnegative failure allowance $\\delta$, post-processing the release by any measurable scalar summary f again satisfies the same $(\\varepsilon,\\delta)$-approximate DP bound, now between the f-summaries of M d and M d'.","s":"∀ {D : Type u_1} {ι : Type u_2} (M : D → MeasureTheory.Measure (ι → ℝ)) (d d' : D) (ε δ : ℝ), (∀ (s : Set (ι → ℝ)), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s + δ) → 0 ≤ δ → ∀ (f : (ι → ℝ) → ℝ), Measurable f → ∀ (s : Set ℝ), MeasurableSet s → (MeasureTheory.Measure.map f (M d)).real s ≤ Real.exp ε * (MeasureTheory.Measure.map f (M d')).real s + δ","l":474,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"For a real scale parameter, the centered Laplace measure is Lebesgue measure weighted by the nonnegative part of the centered Laplace density at that scale.","s":"ℝ → MeasureTheory.Measure ℝ","l":33,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_integrable_abs","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The absolute value of a centered Laplace draw has a finite expectation at every positive scale.","s":"∀ (b : ℝ), 0 < b → MeasureTheory.Integrable (fun x => |x|) (Causalean.Stat.Privacy.laplaceMeasure b)","l":533,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_integral_abs","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The expected absolute value of a centered Laplace draw equals its positive scale.","s":"∀ (b : ℝ), 0 < b → ∫ (x : ℝ), |x| ∂Causalean.Stat.Privacy.laplaceMeasure b = b","l":543,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMeasure_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"At every positive scale, the centered Laplace measure has total mass one and therefore defines a probability distribution.","s":"∀ (b : ℝ), 0 < b → MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMeasure b)","l":107,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"For a dataset domain, a real noise scale, and a real-valued query on that domain, the scalar Laplace mechanism assigns to each dataset the distribution of the query evaluated at that dataset plus an independent centered Laplace draw with that scale.","s":"{D : Type u_1} → ℝ → (D → ℝ) → D → MeasureTheory.Measure ℝ","l":151,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"For a dataset domain, a finite coordinate index set, a real noise scale, and a coordinate-indexed real-valued query on the dataset domain, the finite-dimensional Laplace mechanism assigns to each dataset the distribution obtained by adding independent centered Laplace draws with that scale to all query coordinates.","s":"{D : Type u_1} → {ι : Type u_2} → [Fintype ι] → ℝ → (D → ι → ℝ) → D → MeasureTheory.Measure (ι → ℝ)","l":250,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Adding independent positive-scale Laplace noise in finitely many coordinates produces a probability distribution for the vector release under every dataset.","s":"∀ {D : Type u_1} {ι : Type u_2} [inst : Fintype ι] (b : ℝ), 0 < b → ∀ (q : D → ι → ℝ) (d : D), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMechPi b q d)","l":258,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMechPi_pure_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Vector Laplace mechanism is purely differentially private. Given a positive $\\ell^1$-sensitivity bound $\\Delta$ and a positive privacy level $\\varepsilon$ such that the coordinatewise absolute differences of the query q sum to at most $\\Delta$ between any pair of adjacent datasets, releasing q after adding independent Laplace noise of scale $\\Delta/\\varepsilon$ to each coordinate satisfies pure $\\varepsilon$-differential privacy: for every adjacent pair, the probability of any measurable event under one release is at most $e^\\varepsilon$ times its probability under the other.","s":"∀ {D : Type u_1} {ι : Type u_2} [inst : Fintype ι] (Adj : D → D → Prop) (q : D → ι → ℝ) {Δ ε : ℝ}, 0 < Δ → 0 < ε → (∀ (d d' : D), Adj d d' → ∑ i, |q d i - q d' i| ≤ Δ) → ∀ (d d' : D), Adj d d' → ∀ (s : Set (ι → ℝ)), MeasurableSet s → (Causalean.Stat.Privacy.laplaceMechPi (Δ / ε) q d).real s ≤ Real.exp ε * (Causalean.Stat.Privacy.laplaceMechPi (Δ / ε) q d').real s","l":378,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Adding positive-scale Laplace noise to a scalar query produces a probability distribution for the release under every dataset.","s":"∀ {D : Type u_1} (b : ℝ), 0 < b → ∀ (q : D → ℝ) (d : D), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.Privacy.laplaceMech b q d)","l":183,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplaceMech_pure_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Scalar Laplace mechanism is purely differentially private. Given a positive sensitivity bound $\\Delta$ and a positive privacy level $\\varepsilon$ such that the query q changes by at most $\\Delta$ between any pair of adjacent datasets, releasing q after adding independent Laplace noise of scale $\\Delta/\\varepsilon$ satisfies pure $\\varepsilon$-differential privacy: for every adjacent pair, the probability of any measurable event under one release is at most $e^\\varepsilon$ times its probability under the other.","s":"∀ {D : Type u_1} (Adj : D → D → Prop) (q : D → ℝ) {Δ ε : ℝ}, 0 < Δ → 0 < ε → (∀ (d d' : D), Adj d d' → |q d - q d'| ≤ Δ) → ∀ (d d' : D), Adj d d' → ∀ (s : Set ℝ), MeasurableSet s → (Causalean.Stat.Privacy.laplaceMech (Δ / ε) q d).real s ≤ Real.exp ε * (Causalean.Stat.Privacy.laplaceMech (Δ / ε) q d').real s","l":192,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePDF","k":"def","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"For a real scale parameter and a real evaluation point, the centered Laplace density is $(2b)^{-1}\\exp(-|x|/b)$.","s":"ℝ → ℝ → ℝ","l":29,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePDF_shift_le","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"Moving the center of a positive-scale Laplace density changes its value at any observation by at most an exponential factor determined by the distance between the old and new centers divided by the scale.","s":"∀ (b u v z : ℝ), 0 < b → Causalean.Stat.Privacy.laplacePDF b (z - u) ≤ Real.exp (|u - v| / b) * Causalean.Stat.Privacy.laplacePDF b (z - v)","l":126,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePi_integrable_euclidean_norm","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The Euclidean norm of finitely many independent centered Laplace draws has a finite expectation whenever their common scale is positive.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (b : ℝ), 0 < b → MeasureTheory.Integrable (fun w => √(∑ i, w i ^ 2)) (MeasureTheory.Measure.pi fun x => Causalean.Stat.Privacy.laplaceMeasure b)","l":621,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.laplacePi_integral_euclidean_norm_le","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The expected Euclidean norm of a finite vector of independent centered Laplace draws is at most the number of coordinates times their common positive scale.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (b : ℝ), 0 < b → (∫ (w : ι → ℝ), √(∑ i, w i ^ 2) ∂MeasureTheory.Measure.pi fun x => Causalean.Stat.Privacy.laplaceMeasure b) ≤ ↑(Fintype.card ι) * b","l":584,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.measurable_laplacePDF","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":2,"d":"The Laplace density with scale b is a measurable function on the real line.","s":"∀ (b : ℝ), Measurable (Causalean.Stat.Privacy.laplacePDF b)","l":43,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.pure_dp_implies_approx_dp","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Pure DP implies approximate DP. Given a mechanism M satisfying pure $\\varepsilon$-differential privacy between releases M d and M d', i.e. the probability of every measurable event under one release is at most $e^\\varepsilon$ times its probability under the other, adding any nonnegative failure allowance $\\delta$ to the bound still yields a valid $(\\varepsilon,\\delta)$-approximate differential-privacy guarantee between M d and M d'.","s":"∀ {D : Sort u_2} {α : Type u_1} [inst : MeasurableSpace α] (M : D → MeasureTheory.Measure α) (d d' : D) (ε δ : ℝ), (∀ (s : Set α), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s) → 0 ≤ δ → ∀ (s : Set α), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s + δ","l":442,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Privacy.pure_dp_postprocess","k":"theorem","m":"Causalean.Stat.Privacy.LaplaceMechanism","a":"Stat/Privacy","t":1,"d":"Post-processing preserves pure differential privacy. Given a vector-valued mechanism M satisfying pure $\\varepsilon$-differential privacy between the releases M d and M d', post-processing the release by any measurable scalar summary f again satisfies pure $\\varepsilon$-differential privacy, now between the f-summaries of M d and M d'.","s":"∀ {D : Type u_1} {ι : Type u_2} (M : D → MeasureTheory.Measure (ι → ℝ)) (d d' : D) (ε : ℝ), (∀ (s : Set (ι → ℝ)), MeasurableSet s → (M d).real s ≤ Real.exp ε * (M d').real s) → ∀ (f : (ι → ℝ) → ℝ), Measurable f → ∀ (s : Set ℝ), MeasurableSet s → (MeasureTheory.Measure.map f (M d)).real s ≤ Real.exp ε * (MeasureTheory.Measure.map f (M d')).real s","l":457,"f":"Causalean/Stat/Privacy/LaplaceMechanism.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization","k":"structure","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"A Boolean mark factorization packages a positive requested-mark mass and the identity expressing each marked outcome-set mass as that mass times the outcome law.","s":"MeasureTheory.Measure (Bool × ℝ) → Bool → ENNReal → MeasureTheory.Measure ℝ → Prop","l":27,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID","k":"structure","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"A finite marked iid experiment packages the marked observation family, its coordinatewise measurability, mutual independence, and the common marginal law.","s":"(Ω : Type u_1) → [inst : MeasurableSpace Ω] → ℕ → MeasureTheory.Measure Ω → MeasureTheory.Measure (Bool × ℝ) → Type u_1","l":18,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.map_snd_cond_mark_eq","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, its mass, a probability law for selected outcomes, a Boolean mark factorization, and a coordinate, conditioning that coordinate's outcome on its requested mark gives the selected-outcome law.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (e : ENNReal) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization ν a e ρ → ∀ (i : Fin n), MeasureTheory.Measure.map (fun ω => (S.Z i ω).2) μ[|{ω | (S.Z i ω).1 = a}] = ρ","l":57,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.measure_wordEvent_eq_prod","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment and a Boolean word, the probability of its exact-word event equals the product of its coordinatewise mark probabilities.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (w : Fin n → Bool), μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w) = ∏ i, ν {z | z.1 = w i}","l":35,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.outcomes_iIndep_cond_word","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a Boolean word, and positive probability for its exact-word event, all outcome coordinates remain mutually independent after conditioning on that event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (w : Fin n → Bool), μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w) ≠ 0 → ProbabilityTheory.iIndepFun (fun i ω => (S.Z i ω).2) μ[|Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w]","l":98,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.selectedOutcome_map_cond_word_eq","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, its mass, a selected-outcome law, a Boolean mark factorization, a Boolean word, positive probability for that word, and a selected-coordinate index, that reindexed selected outcome has law ρ after conditioning on the complete word.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (e : ENNReal) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization ν a e ρ → ∀ (w : Fin n → Bool), μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w) ≠ 0 → ∀ (j : Fin (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a)), MeasureTheory.Measure.map (fun ω => Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes S.Z w a ω j) μ[|Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w] = ρ","l":135,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.selectedOutcomes_conditionalLaw","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"Given a finite marked iid experiment, a requested mark, its mass, a selected-outcome law, a Boolean mark factorization, a Boolean word, and positive probability for that word, the reindexed selected outcome vector conditioned on the word has the finite iid product law with marginal ρ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (e : ENNReal) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization ν a e ρ → ∀ (w : Fin n → Bool), μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w) ≠ 0 → MeasureTheory.Measure.map (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes S.Z w a) μ[|Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w] = MeasureTheory.Measure.pi fun x => ρ","l":222,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID.selectedOutcomes_iIndep_cond_word","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, a Boolean word, and positive probability for its exact-word event, the reindexed selected outcomes remain mutually independent after conditioning on that event.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (w : Fin n → Bool), μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w) ≠ 0 → ProbabilityTheory.iIndepFun (fun j ω => Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes S.Z w a ω j) μ[|Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent S.Z w]","l":199,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.conditionalMarkedSubsample_dkwRadius","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.TailLift","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, its mass, a selected-outcome law, a Boolean mark factorization, a confidence parameter between zero and one, and the fixed-size DKW--Massart bound at every positive sample size, the selected-subsample empirical-CDF deviation at the DKW radius has tail probability at most α/2.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (e : ENNReal) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization ν a e ρ → ∀ (α : ℝ), 0 < α → α ≤ 1 → (∀ (m : ℕ), 0 < m → (MeasureTheory.Measure.pi fun x => ρ) (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.fixedCDFBadSet ρ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.dkwRadius α m)) ≤ ENNReal.ofReal (α / 2)) → μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent S.Z a ρ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.dkwRadius α)) ≤ ENNReal.ofReal (α / 2)","l":115,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/TailLift.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.conditionalMarkedSubsample_empiricalCDF_tail","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.TailLift","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"Given a finite marked iid experiment, a requested mark, its mass, a selected-outcome law, a Boolean mark factorization, a count-indexed radius, a probability bound, and the corresponding fixed-size iid empirical-CDF bound at every positive sample size, the probability that the selected arm is nonempty and its uniform empirical-CDF deviation exceeds its count-indexed radius is at most β.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (e : ENNReal) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.BooleanMarkFactorization ν a e ρ → ∀ (radius : ℕ → ℝ) (β : ENNReal), (∀ (m : ℕ), 0 < m → (MeasureTheory.Measure.pi fun x => ρ) (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.fixedCDFBadSet ρ (radius m)) ≤ β) → μ (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent S.Z a ρ radius) ≤ β","l":58,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/TailLift.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.disjoint_wordEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a marked sample and two distinct Boolean words, their exact-word events are disjoint.","s":"∀ {Ω : Type u_1} {n : ℕ} (Z : Fin n → Ω → Bool × ℝ) {w v : Fin n → Bool}, w ≠ v → Disjoint (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w) (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z v)","l":85,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.dkwRadius","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.TailLift","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a confidence parameter and a selected count, the DKW confidence radius is the square root of log(4/α) divided by twice that count. It is a DKW radius for a confidence parameter that is positive and at most one and a positive count, as assumed by the tail theorems; outside that domain the logarithm, division and square-root conventions give values without that meaning (for example zero at count zero).","s":"ℝ → ℕ → ℝ","l":107,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/TailLift.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.empiricalCDFVec","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a finite real vector and a threshold, the vector empirical CDF is the normalized sum of lower-ray indicators. For the empty vector the normalizing inverse is zero by convention, so the value is zero.","s":"{m : ℕ} → (Fin m → ℝ) → ℝ → ℝ","l":133,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.empiricalCDFVec_continuousWithinAt_Ici","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Measurability","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"For a finite real vector and a threshold, its empirical CDF is continuous from the right at that threshold.","s":"∀ {m : ℕ} (x : Fin m → ℝ) (y : ℝ), ContinuousWithinAt (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.empiricalCDFVec x) (Set.Ici y) y","l":17,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Measurability.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.empiricalCDFVec_selectedOutcomes_eq","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a marked sample, a requested mark, a Boolean word, an outcome in its exact-word event, and a threshold, the reindexed-vector empirical CDF equals the directly selected empirical CDF.","s":"∀ {Ω : Type u_1} {n : ℕ} (Z : Fin n → Ω → Bool × ℝ) (a : Bool) (w : Fin n → Bool) {ω : Ω}, ω ∈ Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w → ∀ (y : ℝ), Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.empiricalCDFVec (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes Z w a ω) y = Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedEmpiricalCDF Z a ω y","l":150,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.fixedCDFBadSet","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a measure on the real line (a population probability law in the results using it) and a deviation radius, the fixed-size bad set is the set of vectors whose uniform empirical-CDF deviation exceeds that radius.","s":"{m : ℕ} → MeasureTheory.Measure ℝ → ℝ → Set (Fin m → ℝ)","l":174,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.iUnion_wordEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"For a marked sample, the union of all exact-word events is the whole sample space.","s":"∀ {Ω : Type u_1} {n : ℕ} (Z : Fin n → Ω → Bool × ℝ), ⋃ w, Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w = Set.univ","l":97,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.measurableSet_fixedCDFBadSet","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Measurability","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a probability law and a deviation radius, the fixed-size uniform empirical-CDF bad set is measurable.","s":"∀ {m : ℕ} (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ] (radius : ℝ), MeasurableSet (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.fixedCDFBadSet ρ radius)","l":128,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Measurability.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.measurableSet_selectedCDFBadEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.TailLift","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, a selected-outcome probability law, and a count-indexed radius, the selected-subsample empirical-CDF bad event is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ] (radius : ℕ → ℝ), MeasurableSet (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent S.Z a ρ radius)","l":21,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/TailLift.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.measurableSet_wordEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a marked sample, measurability of every marked observation, and a Boolean word, the corresponding exact-word event is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} (Z : Fin n → Ω → Bool × ℝ), (∀ (i : Fin n), Measurable (Z i)) → ∀ (w : Fin n → Bool), MeasurableSet (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w)","l":73,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.measurable_selectedOutcomes","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.ConditionalLaw","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a finite marked iid experiment, a requested mark, and a Boolean word, the map from a sample outcome to its reindexed selected outcome vector is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {n : ℕ} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure (Bool × ℝ)} (S : Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.MarkedIID Ω n μ ν) (a : Bool) (w : Fin n → Bool), Measurable (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes S.Z w a)","l":213,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/ConditionalLaw.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.measurable_uniformCDFDeviation","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Measurability","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"For a probability law, the uniform empirical-CDF deviation of a finite real vector from that law is measurable.","s":"∀ {m : ℕ} (ρ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure ρ], Measurable (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.uniformCDFDeviation ρ)","l":54,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Measurability.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a marked sample, a requested mark, a measure on the real line (a population probability law in the results using it), and a count-indexed radius, the selected-sample bad event is the event that the selected arm is nonempty and its uniform deviation exceeds the radius at its observed size.","s":"{Ω : Type u_1} → {n : ℕ} → (Fin n → Ω → Bool × ℝ) → Bool → MeasureTheory.Measure ℝ → (ℕ → ℝ) → Set Ω","l":180,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent_inter_wordEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"Given a marked sample, a requested mark, a population law, a count-indexed radius, a Boolean word, and a positive selected count for that word, intersecting the selected bad event with its exact-word event equals the reindexed fixed-size bad event intersected with that word event.","s":"∀ {Ω : Type u_1} {n : ℕ} (Z : Fin n → Ω → Bool × ℝ) (a : Bool) (ρ : MeasureTheory.Measure ℝ) (radius : ℕ → ℝ) (w : Fin n → Bool), 0 < Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a → Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCDFBadEvent Z a ρ radius ∩ Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w = Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes Z w a ⁻¹' Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.fixedCDFBadSet ρ (radius (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a)) ∩ Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w","l":189,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCount","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a marked sample, a requested mark, and a sample outcome, the selected count is the number of observed coordinates carrying that mark.","s":"{Ω : Type u_1} → {n : ℕ} → (Fin n → Ω → Bool × ℝ) → Bool → Ω → ℕ","l":105,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCount_eq_on_wordEvent","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"Given a marked sample, a requested mark, a Boolean word, and an outcome in that word's event, the observed selected count equals the word's deterministic selected count.","s":"∀ {Ω : Type u_1} {n : ℕ} (Z : Fin n → Ω → Bool × ℝ) (a : Bool) (w : Fin n → Bool) {ω : Ω}, ω ∈ Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent Z w → Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedCount Z a ω = Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a","l":112,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedEmpiricalCDF","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a marked sample, a requested mark, a sample outcome, and a threshold, the selected empirical CDF is the lower-ray count in the selected arm divided by its observed size. When the selected arm is empty the inverse size is zero by convention and the value is zero; the bad event below requires a nonempty arm.","s":"{Ω : Type u_1} → {n : ℕ} → (Fin n → Ω → Bool × ℝ) → Bool → Ω → ℝ → ℝ","l":139,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndex","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a Boolean word and a requested mark, the selected-coordinate embedding enumerates the selected positions in their original order, using the ordered finite-set enumeration.","s":"{n : ℕ} → (w : Fin n → Bool) → (a : Bool) → Fin (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a) ↪o Fin n","l":29,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndex_mark","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"For a Boolean word, a requested mark, and a selected-coordinate index, the coordinate selected by the embedding carries the requested mark.","s":"∀ {n : ℕ} (w : Fin n → Bool) (a : Bool) (j : Fin (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a)), w ((Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndex w a) j) = a","l":36,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndices","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a Boolean word and a requested mark, the selected positions are the finite set of coordinates carrying that mark, as given by filtering all coordinates.","s":"{n : ℕ} → (Fin n → Bool) → Bool → Finset (Fin n)","l":18,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedOutcomes","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a marked sample, a Boolean word, and a requested mark, the selected outcome vector takes the outcome component at each reindexed selected coordinate.","s":"{Ω : Type u_1} → {n : ℕ} → (Fin n → Ω → Bool × ℝ) → (w : Fin n → Bool) → (a : Bool) → Ω → Fin (Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount w a) → ℝ","l":126,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedWordCount","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a Boolean word and a requested mark, the selected count is the number of its selected positions, given by their cardinality.","s":"{n : ℕ} → (Fin n → Bool) → Bool → ℕ","l":24,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.sum_selectedIndex","k":"theorem","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":2,"d":"For a Boolean word, a requested mark, and a summand function, summing over the reindexed selected coordinates equals summing over all requested-mark positions.","s":"∀ {n : ℕ} (w : Fin n → Bool) (a : Bool) {R : Type u_1} [inst : AddCommMonoid R] (f : Fin n → R), ∑ j, f ((Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndex w a) j) = ∑ i ∈ Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.selectedIndices w a, f i","l":48,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.uniformCDFDeviation","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a measure on the real line and a finite real vector, the uniform empirical-CDF deviation is the supremum over thresholds of the absolute empirical-minus-population CDF difference. No probability assumption is made here; for a probability law, as assumed by the measurability and tail results, the differences are bounded and this is the usual uniform deviation, while an unbounded family has real supremum zero by convention.","s":"{m : ℕ} → MeasureTheory.Measure ℝ → (Fin m → ℝ) → ℝ","l":166,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.wordEvent","k":"def","m":"Causalean.Stat.Quantile.ConditionalMarkedSubsampleDkw.Basic","a":"Stat/Quantile/ConditionalMarkedSubsampleDkw","t":1,"d":"For a marked sample and a Boolean word, the exact-word event is the set of outcomes whose complete mark vector equals that word.","s":"{Ω : Type u_1} → {n : ℕ} → (Fin n → Ω → Bool × ℝ) → (Fin n → Bool) → Set Ω","l":67,"f":"Causalean/Stat/Quantile/ConditionalMarkedSubsampleDkw/Basic.lean"},{"n":"Causalean.Stat.QuantileRegularity","k":"structure","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Quantile-estimator regularity. Bundles the analytic and empirical-process hypotheses under which a quantile-estimator sequence $q̂_n$ is $\\sqrt n$-asymptotically linear for the $\\tau$-quantile $q_0$ of $P$ with density $f_0$: the level lies in the open unit interval $\\tau \\in (0,1)$, the density at the quantile is positive, $q_0$ is indeed the population $\\tau$-quantile, i.e. the cdf satisfies $F(q_0) = \\tau$, the cdf is differentiable at $q_0$ with derivative $f_0$, so that $\\tau(1-\\tau)/f_0^2$ is the genuine asymptotic variance, and the rescaled estimator matches the normalized influence-function sum up to a term vanishing in probability — the Bahadur remainder.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure ℝ} → Causalean.Stat.IIDSample Ω ℝ μ P → (ℕ → Ω → ℝ) → ℝ → ℝ → ℝ → Prop","l":143,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.QuantileRegularity.isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Regularity implies asymptotic linearity. Given a QuantileRegularity witness h for the estimator sequence qn, the estimator is asymptotically linear at the quantile q₀ with influence function ψ_τ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {S : Causalean.Stat.IIDSample Ω ℝ μ P} {qn : ℕ → Ω → ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.QuantileRegularity S qn τ q₀ f₀ → Causalean.Stat.IsAsymLinear qn q₀ (Causalean.Stat.quantileIF τ q₀ f₀) S fun m => Finset.range m","l":173,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.QuantileRegularity.tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Sample-quantile asymptotic normality. Given a QuantileRegularity witness h for the estimator sequence qn — interior level $\\tau\\in(0,1)$, positive density $f_0$ at the population quantile $q_0$, cdf identification, and the exposed Bahadur/Donsker remainder — provided the rescaled estimator is almost-everywhere measurable at each sample size and the normalized influence-function sum is almost-everywhere measurable at each sample size, then $\\sqrt n\\,(\\hat q_n-q_0)$ converges in distribution to the centered Gaussian law with variance $\\tau(1-\\tau)/f_0^2$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {S : Causalean.Stat.IIDSample Ω ℝ μ P} {qn : ℕ → Ω → ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.QuantileRegularity S qn τ q₀ f₀ → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator qn q₀ (fun m => Finset.range m) n) μ), (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.normalizedSum S (Causalean.Stat.quantileIF τ q₀ f₀) (fun m => Finset.range m) n) μ) → Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator qn q₀ fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (τ * (1 - τ) / f₀ ^ 2)) μ hθn_meas","l":183,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.SampleQuantileReg","k":"structure","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":1,"d":"Regularity for sample-quantile asymptotics. Bundles the hypotheses on the population cdf $F$ of $P$ under which the empirical sample quantile admits a Bahadur representation at level $\\tau$: the level lies in the open unit interval $\\tau \\in (0,1)$, the density $f_0$ at the quantile is positive, $q_0$ is the population $\\tau$-quantile, i.e. $F(q_0) = \\tau$, $F$ is differentiable at $q_0$ with derivative $f_0$, and $F$ is continuous, so the population has no atoms.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ → Prop","l":70,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.StochEquicontAt","k":"def","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.StochEquicont","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"On a measurable observation space and a normed real vector space, for a score function taking a parameter and an observation to a vector, a distinguished parameter value, an observation-space measure, a measurable sample space, a measure on that sample space, an independent identically distributed sample with that observation law, and a sequence of sample-dependent parameter estimates, the stochastic equicontinuity property at the distinguished parameter holds when, for every positive tolerance, there is a positive radius such that the probability of the stated large empirical-process gap while the estimate lies within that radius converges to zero as the sample size tends to infinity.","s":"{X : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace X] → [inst_1 : NormedAddCommGroup E] → [NormedSpace ℝ E] → (E → X → E) → E → (P : MeasureTheory.Measure X) → {Ω : Type u_3} → [inst_3 : MeasurableSpace Ω] → (μ : MeasureTheory.Measure Ω) → Causalean.Stat.IIDSample Ω X μ P → (ℕ → Ω → E) → Prop","l":37,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/StochEquicont.lean"},{"n":"Causalean.Stat.TendstoInDistribution","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"Given one probability space for every row, almost-everywhere measurable row variables, and a target probability law, convergence in distribution to that law means weak convergence of the row pushforward laws.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) → [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] → (Y : (n : ℕ) → Ω n → ℝ) → (Q : MeasureTheory.Measure ℝ) → [MeasureTheory.IsProbabilityMeasure Q] → (∀ (n : ℕ), AEMeasurable (Y n) (μ n)) → Prop","l":37,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.TendstoInProbability","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.Basic","a":"Stat/CLT/MartingaleArray","t":1,"d":"Given one measure space for every row, a row-indexed family of real random variables converges in probability to the target constant when, at every positive tolerance, the row probability of exceeding that tolerance tends to zero.","s":"{Ω : ℕ → Type u_1} → {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} → ((n : ℕ) → MeasureTheory.Measure (Ω n)) → ((n : ℕ) → Ω n → ℝ) → ℝ → Prop","l":29,"f":"Causalean/Stat/CLT/MartingaleArray/Basic.lean"},{"n":"Causalean.Stat.Tendsto_L2","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Given a sequence of real-valued random variables, a real-valued limiting random variable, and a measure on a measurable sample space, $L^2$ convergence means that the $L^2$ norm of the difference between the indexed random variable and the limit tends to zero as the index tends to infinity.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Prop","l":69,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Given a sequence of almost-everywhere measurable real-valued random variables, a probability law on the real line, and a probability measure on a measurable sample space, convergence in distribution means that the sequence of induced laws converges weakly to the specified real-line probability law.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (Xn : ℕ → Ω → ℝ) → (Q : MeasureTheory.Measure ℝ) → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → [MeasureTheory.IsProbabilityMeasure Q] → (∀ (n : ℕ), AEMeasurable (Xn n) μ) → Prop","l":83,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.add_isLittleOp_one","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"If Xn ⇒ Q in distribution and Yn − Xn = o_p(1), then Yn ⇒ Q.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn Yn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Causalean.Stat.IsLittleOp (fun n ω => Yn n ω - Xn n ω) (fun x => 1) μ → Causalean.Stat.Tendsto_dist Yn Q μ hYn","l":237,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.congr_ae","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"Convergence in distribution is invariant under eventual a.e. equality of the random variables.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn Yn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → (∀ᶠ (n : ℕ) in Filter.atTop, Xn n =ᵐ[μ] Yn n) → Causalean.Stat.Tendsto_dist Yn Q μ hYn","l":345,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.const_mul_tendsto","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Deterministic-scalar Slutsky for the project's measure-level Tendsto_dist wrapper.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] {a : ℕ → ℝ} {a₀ : ℝ} (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Filter.Tendsto a Filter.atTop (nhds a₀) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => a n * Xn n ω) μ, MeasureTheory.Measure.isProbabilityMeasure_map (AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id)) (hXn n))⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map (fun x => a₀ * x) Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))⟩)","l":112,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.const_mul_tendsto_gaussian","k":"theorem","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"Deterministic-scalar Slutsky for Gaussian limits, phrased for the project's measure-level Tendsto_dist wrapper.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {a : ℕ → ℝ} {a₀ v : ℝ} (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn (Causalean.Stat.gaussianMeasure 0 v) μ hXn → Filter.Tendsto a Filter.atTop (nhds a₀) → Causalean.Stat.Tendsto_dist (fun n ω => a n * Xn n ω) (Causalean.Stat.gaussianMeasure 0 (a₀ ^ 2 * v)) μ fun n => AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id)) (hXn n)","l":361,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.Tendsto_dist.div_tendsto_inProb_gaussian","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":1,"d":"Generic studentized convergence. Let Xn be a real-valued rescaled-estimator sequence and σ_hat a variance-estimator sequence, with σ₀ a positive scale. Suppose Xn is measurable at every sample size and it converges in distribution to the project's Gaussian law with mean zero and variance σ₀², that σ_hat converges in probability to σ₀, and the studentized ratio Xn / σ_hat is measurable at every sample size. Then the studentized ratio Xn / σ_hat converges in distribution to the standard Gaussian law.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn σ_hat : ℕ → Ω → ℝ} {σ₀ : ℝ}, 0 < σ₀ → ∀ (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn (Causalean.Stat.gaussianMeasure 0 (σ₀ ^ 2)) μ hXn → Causalean.Stat.Tendsto_inProb σ_hat (fun x => σ₀) μ → ∀ (hdiv : ∀ (n : ℕ), AEMeasurable (fun ω => Xn n ω / σ_hat n ω) μ), Causalean.Stat.Tendsto_dist (fun n ω => Xn n ω / σ_hat n ω) (Causalean.Stat.gaussianMeasure 0 1) μ hdiv","l":101,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.limsup_measure_closed_le","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Rate","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Portmanteau (closed-set limsup). If Xn ⇒ Q in distribution, then for every closed F, limsup μ{ω | Xn n ω ∈ F} ≤ Q F. Mirrors the closed-set half of Tendsto_dist.tightness.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → ∀ {F : Set ℝ}, IsClosed F → Filter.limsup (fun n => μ {ω | Xn n ω ∈ F}) Filter.atTop ≤ Q F","l":103,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Rate.lean"},{"n":"Causalean.Stat.Tendsto_dist.tendsto_measure_of_null_frontier","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"Continuity-set form of portmanteau for the project's Tendsto_dist wrapper.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → ∀ {E : Set ℝ}, Q (frontier E) = 0 → Filter.Tendsto (fun n => ((MeasureTheory.Measure.map (Xn n) μ) E).toReal) Filter.atTop (nhds (Q E).toReal)","l":53,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.tightness","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Tightness from convergence in distribution. Suppose a real-valued sequence Xn is measurable at every sample size and it converges in distribution under μ to a probability measure Q on ℝ. Then Xn is bounded in probability, O_p(1). Standard fact: any single tight limit gives a tight sequence (Prokhorov).","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {Q : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":186,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":1,"d":"Generic Wald asymptotic coverage. Suppose a studentized statistic sequence Sn is measurable at every sample size and converges in distribution to the standard Gaussian law, and fix a positive half-width z. If a coverage-probability sequence coverProb is asymptotically equivalent to the studentized-interval event {Sn ∈ [-z, z]}, then coverProb converges to the standard-Gaussian probability of [-z, z].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Sn : ℕ → Ω → ℝ} (hSn : ∀ (n : ℕ), AEMeasurable (Sn n) μ), Causalean.Stat.Tendsto_dist Sn (Causalean.Stat.gaussianMeasure 0 1) μ hSn → ∀ {z : ℝ}, 0 < z → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Sn n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":159,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_Iic","k":"theorem","m":"Causalean.Stat.Inference.WaldVec","a":"Stat/Inference","t":1,"d":"Wald / confidence-ellipsoid asymptotic coverage. Suppose the scalar Wald statistic sequence Wₙ is measurable at every sample size and converges in distribution to a limit law χ, and that χ gives zero mass to the boundary frontier (Iic c) = {c}. If a real sequence coverProb is asymptotically equivalent to the ellipsoid event {ω | Wₙ ω ≤ c}, then coverProb converges to χ(Iic c).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) μ) {χ : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure χ], Causalean.Stat.Tendsto_dist Wn χ μ hWn → ∀ {c : ℝ}, χ (frontier (Set.Iic c)) = 0 → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds (χ (Set.Iic c)).toReal)","l":70,"f":"Causalean/Stat/Inference/WaldVec.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_Iic_of_noAtoms","k":"theorem","m":"Causalean.Stat.Inference.WaldVec","a":"Stat/Inference","t":1,"d":"Wald / ellipsoid coverage with an atomless limit. Suppose the scalar Wald statistic sequence Wₙ is measurable at every sample size and converges in distribution to a limit law χ with no atoms (which holds for any non-degenerate χ²_d, and more generally for any continuous limit). If a real sequence coverProb is asymptotically equivalent to the ellipsoid event {ω | Wₙ ω ≤ c}, then coverProb converges to χ(Iic c). The boundary-null hypothesis of wald_coverage_Iic is automatic here: frontier (Iic c) = {c} and χ {c} = 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) μ) {χ : MeasureTheory.Measure ℝ} [inst_2 : MeasureTheory.IsProbabilityMeasure χ] [MeasureTheory.NullSingletonClass χ], Causalean.Stat.Tendsto_dist Wn χ μ hWn → ∀ (c : ℝ) (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds (χ (Set.Iic c)).toReal)","l":110,"f":"Causalean/Stat/Inference/WaldVec.lean"},{"n":"Causalean.Stat.Tendsto_dist.wald_coverage_chiSq","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"χ²-coverage of the Wald confidence ellipsoid. Suppose the Wald statistic sequence Wₙ is measurable at every sample size, d is a positive-integer degrees-of-freedom parameter, and Wₙ converges in distribution to the χ²_d law. If a coverage-probability sequence coverProb is asymptotically equivalent to the ellipsoid event {Wₙ ≤ c}, then coverProb converges to the χ²_d probability of (-∞, c].","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {ν : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Wn : ℕ → Ω → ℝ} (hWn : ∀ (n : ℕ), AEMeasurable (Wn n) ν) {d : ℕ}, 1 ≤ d → Causalean.Stat.Tendsto_dist Wn (Causalean.Stat.chiSqDist d) ν hWn → ∀ (c : ℝ) (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (ν {ω | Wn n ω ≤ c}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.chiSqDist d) (Set.Iic c)).toReal)","l":174,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.Tendsto_dist_iff","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Convergence in distribution of a sequence of real random variables to a probability law on the real line is exactly convergence, in the space of probability measures on the real line, of the pushforward laws of the random variables to that law.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Xn : ℕ → Ω → ℝ) (Q : MeasureTheory.Measure ℝ) (μ : MeasureTheory.Measure Ω) [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist Xn Q μ hXn ↔ Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Xn n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hXn n)⟩) Filter.atTop (nhds ⟨Q, inst_2⟩)","l":98,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec","k":"def","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"For a measurable sample space and a pseudo-metric outcome space whose open sets are measurable, a sequence of random elements in the outcome space, a probability measure on that outcome space, a probability measure on the sample space, and almost-everywhere measurability of every random element in the sequence, convergence in distribution of the sequence to the outcome-space probability measure means weak convergence of its induced distributions.","s":"{Ω : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : PseudoMetricSpace E] → [inst_2 : MeasurableSpace E] → [OpensMeasurableSpace E] → (Xn : ℕ → Ω → E) → (Q : MeasureTheory.Measure E) → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsProbabilityMeasure μ] → [MeasureTheory.IsProbabilityMeasure Q] → (∀ (n : ℕ), AEMeasurable (Xn n) μ) → Prop","l":49,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.add_isLittleOp_one","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"Vector Slutsky absorption. Suppose Xn and Yn are E-valued sequences that are each measurable at every sample size, Xn converges in distribution to a probability measure Q on E, and the norm of the perturbation ‖Yn − Xn‖ is o_p(1). Then Yn also converges in distribution to Q.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : NormedAddCommGroup E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] {Xn Yn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_5 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → Causalean.Stat.IsLittleOp (fun n ω => ‖Yn n ω - Xn n ω‖) (fun x => 1) μ → Causalean.Stat.Tendsto_dist_vec Yn Q μ hYn","l":76,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.congr_ae","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":2,"d":"Vector convergence in distribution is invariant under eventual a.e. equality of the random variables. Vector analogue of Tendsto_dist.congr_ae.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : PseudoMetricSpace E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] {Xn Yn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_5 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ) (hYn : ∀ (n : ℕ), AEMeasurable (Yn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → (∀ᶠ (n : ℕ) in Filter.atTop, Xn n =ᵐ[μ] Yn n) → Causalean.Stat.Tendsto_dist_vec Yn Q μ hYn","l":219,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.map_continuous","k":"theorem","m":"Causalean.Stat.Limit.ConvergenceVec","a":"Stat/Limit","t":1,"d":"Pushforward of Tendsto_dist_vec under a continuous map. Suppose Xn is an E-valued sequence, measurable at every sample size, converging in distribution to a probability measure Q on E, and g : E → F is continuous. Then the composed sequence g ∘ Xn converges in distribution to the pushforward measure Q.map g.","s":"∀ {Ω : Type u_1} {E : Type u_2} {F : Type u_3} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : PseudoMetricSpace E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : PseudoMetricSpace F] [inst_6 : MeasurableSpace F] [inst_7 : BorelSpace F] {Xn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_8 : MeasureTheory.IsProbabilityMeasure Q] {g : E → F} (hg : Continuous g) (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => g (Xn n ω)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (AEMeasurable.comp_aemeasurable (Measurable.aemeasurable (Continuous.measurable hg)) (hXn n))⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map g Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable hg))⟩)","l":237,"f":"Causalean/Stat/Limit/ConvergenceVec.lean"},{"n":"Causalean.Stat.Tendsto_dist_vec.of_charFun_tendsto","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":1,"d":"Cramér–Wold / Lévy continuity wrapper. For an E-valued sequence Xn, if each Xn n is a.e. measurable and the characteristic functions of Xn n converge pointwise, at every point t, to the characteristic function of a probability measure Q, then the sequence Xn converges in distribution to Q.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → E} {Q : MeasureTheory.Measure E} [inst_7 : MeasureTheory.IsProbabilityMeasure Q] (hXn : ∀ (n : ℕ), AEMeasurable (Xn n) μ), (∀ (t : E), Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (Xn n) μ) t) Filter.atTop (nhds (MeasureTheory.charFun Q t))) → Causalean.Stat.Tendsto_dist_vec Xn Q μ hXn","l":219,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.Tendsto_inProb","k":"def","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":1,"d":"Given a sequence of real-valued random variables, a real-valued limiting random variable, and a measure on a measurable sample space, convergence in probability means that, for every positive tolerance, the measure of outcomes whose absolute error exceeds that tolerance tends to zero as the sample-size index tends to infinity.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → (ℕ → Ω → ℝ) → (Ω → ℝ) → MeasureTheory.Measure Ω → Prop","l":50,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.Tendsto_inProb.add_zero_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Sum of in-probability limits. If Xn →ₚ 0 and Yn →ₚ 0 then Xn + Yn →ₚ 0. Union bound: {ε ≤ |Xn+Yn|} ⊆ {ε/2 ≤ |Xn|} ∪ {ε/2 ≤ |Yn|}.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb Yn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Xn n ω + Yn n ω) (fun x => 0) μ","l":41,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":1,"d":"Continuous mapping for convergence in probability at a point. If a real-valued sequence Yn converges in probability to a point c under μ, and a function g is continuous at c, then the composed sequence g ∘ Yn converges in probability to g c. Generalizes Tendsto_inProb.inv (the case g = fun x => 1/x at a nonzero c).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {c : ℝ} {μ : MeasureTheory.Measure Ω} {g : ℝ → ℝ}, ContinuousAt g c → Causalean.Stat.Tendsto_inProb Yn (fun x => c) μ → Causalean.Stat.Tendsto_inProb (fun n ω => g (Yn n ω)) (fun x => g c) μ","l":40,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.const_mul_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Scalar multiple preserves in-probability convergence to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ} (c : ℝ), Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ → Causalean.Stat.Tendsto_inProb (fun n ω => c * Xn n ω) (fun x => 0) μ","l":351,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.inv","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Reciprocal continuity for convergence in probability at a nonzero constant: if Yn →_p Y₀ with Y₀ ≠ 0, then 1 / Yn →_p 1 / Y₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {Y₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => Y₀) μ → Y₀ ≠ 0 → Causalean.Stat.Tendsto_inProb (fun n ω => 1 / Yn n ω) (fun x => 1 / Y₀) μ","l":65,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.isBigOp_one","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"In-probability tightness. A sequence converging in probability to a constant is bounded in probability: Xₙ →ₚ c ⟹ Xₙ = O_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn : ℕ → Ω → ℝ} {c : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Xn (fun x => c) μ → Causalean.Stat.IsBigOp Xn (fun x => 1) μ","l":157,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.isLittleOp_one","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Convergence in probability to zero implies o_p(1): if Yn →_p 0, then Yn is IsLittleOp of the constant-one rate.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => 0) μ → Causalean.Stat.IsLittleOp Yn (fun x => 1) μ","l":139,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.matrix_comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Matrix continuous mapping in probability. If every entry of the random matrix Mₙ converges in probability to the corresponding entry of M₀, and g is continuous at M₀, then g(Mₙ) →ₚ g(M₀).","s":"∀ {Ω : Type u_1} {K : Type u_2} [inst : MeasurableSpace Ω] [Fintype K] {μ : MeasureTheory.Measure Ω} {Mn : ℕ → Ω → Matrix K K ℝ} {M₀ : Matrix K K ℝ} {g : Matrix K K ℝ → ℝ}, ContinuousAt g M₀ → (∀ (i j : K), Causalean.Stat.Tendsto_inProb (fun n ω => Mn n ω i j) (fun x => M₀ i j) μ) → Causalean.Stat.Tendsto_inProb (fun n ω => g (Mn n ω)) (fun x => g M₀) μ","l":249,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.of_isLittleOp_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Linearity","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Little-o in probability implies convergence in probability. An IsLittleOp _ 1 sequence converges to 0 in probability. Both unwind to the same μ{· < |·|} → 0 statement up to a harmless </≤ slack (handled with ε/2).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Causalean.Stat.Tendsto_inProb Xn (fun x => 0) μ","l":69,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Linearity.lean"},{"n":"Causalean.Stat.Tendsto_inProb.pi_comp_continuousAt","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Finite-Pi continuous mapping in probability. If every coordinate of a finite-dimensional random vector converges in probability to the corresponding constant coordinate, and g is continuous at the limiting vector, then g(Yₙ) →ₚ g(c).","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [Fintype ι] {μ : MeasureTheory.Measure Ω} {Yn : ℕ → Ω → ι → ℝ} {c : ι → ℝ} {g : (ι → ℝ) → ℝ}, ContinuousAt g c → (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => Yn n ω i) (fun x => c i) μ) → Causalean.Stat.Tendsto_inProb (fun n ω => g (Yn n ω)) (fun x => g c) μ","l":188,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sqrt","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":2,"d":"Square root preserves convergence in probability. If Vn →ₚ v₀ under μ, then √Vn →ₚ √v₀. Continuous mapping with the (everywhere continuous) Real.sqrt, via Tendsto_inProb.comp_continuousAt.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {Vn : ℕ → Ω → ℝ} {v₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Vn (fun x => v₀) μ → Causalean.Stat.Tendsto_inProb (fun n ω => √(Vn n ω)) (fun x => √v₀) μ","l":171,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sub","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Difference of convergent-in-probability sequences. If Xn →_p a and Yn →_p b (both to constant limits), then Xn − Yn →_p a − b. Standard ε/2-union-bound argument; the analogue of TendstoInMeasure.sub, which Mathlib does not currently provide for the constant-limit case.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Xn Yn : ℕ → Ω → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Xn (fun x => a) μ → Causalean.Stat.Tendsto_inProb Yn (fun x => b) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Xn n ω - Yn n ω) (fun x => a - b) μ","l":99,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb.sub_const","k":"theorem","m":"Causalean.Stat.Limit.ContinuousMapping","a":"Stat/Limit","t":2,"d":"Center a convergence-in-probability statement around a constant limit: if Yn →_p Y₀, then Yn - Y₀ →_p 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {Yn : ℕ → Ω → ℝ} {Y₀ : ℝ} {μ : MeasureTheory.Measure Ω}, Causalean.Stat.Tendsto_inProb Yn (fun x => Y₀) μ → Causalean.Stat.Tendsto_inProb (fun n ω => Yn n ω - Y₀) (fun x => 0) μ","l":88,"f":"Causalean/Stat/Limit/ContinuousMapping.lean"},{"n":"Causalean.Stat.Tendsto_inProb_iff","k":"theorem","m":"Causalean.Stat.Limit.Convergence","a":"Stat/Limit","t":2,"d":"Convergence in probability of a sequence of real random variables to a limit is exactly convergence in measure of that sequence along the natural numbers.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Xn : ℕ → Ω → ℝ) (X : Ω → ℝ) (μ : MeasureTheory.Measure Ω), Causalean.Stat.Tendsto_inProb Xn X μ ↔ MeasureTheory.TendstoInMeasure μ Xn Filter.atTop X","l":60,"f":"Causalean/Stat/Limit/Convergence.lean"},{"n":"Causalean.Stat.UniformlyBounded","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"For an arbitrary domain and a real-valued function on that domain, the uniform boundedness property holds exactly when there exists a nonnegative real constant that bounds the function's absolute value at every input.","s":"{A : Type u_1} → (A → ℝ) → Prop","l":50,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.ZEstimatorRegularity","k":"structure","m":"Causalean.Stat.MEstimation.ZEstimator","a":"Stat/MEstimation","t":1,"d":"Regularity conditions for the Z-estimator central limit theorem. Bundles, for a score function and target parameter θ₀ under a sampling law, the population identification condition that the score vanishes in mean at the truth, a Jacobian of the population score at θ₀ together with a witnessed inverse, finite variance of the score at the truth, measurability of the score at every parameter value, continuity of the population score map at θ₀, local integrability of the score on a neighborhood of θ₀, and an integrable L² envelope bounding local score differences.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {E : Type u_3} → [inst_1 : NormedAddCommGroup E] → [InnerProductSpace ℝ E] → [MeasurableSpace E] → (E → X → E) → E → MeasureTheory.Measure X → Type u_3","l":45,"f":"Causalean/Stat/MEstimation/ZEstimator.lean"},{"n":"Causalean.Stat.abs_cdfIF_le_one","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF is bounded by 1 in absolute value.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y z : ℝ), |Causalean.Stat.cdfIF P y z| ≤ 1","l":135,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.abs_empiricalMass_le_one","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":2,"d":"The absolute empirical mass of any point is at most one, including for the empty sample.","s":"∀ {𝒳 : Type u_1} {N : ℕ} (sample : Fin N → 𝒳) (x : 𝒳), |Causalean.Stat.empiricalMass sample x| ≤ 1","l":32,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.abs_exp_mul_one_sub_sub_one_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.GaussianBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"For a nonnegative real argument, the error in the exponential Euler factor exp(x) * (1 - x) is at most exp(x) * x² / 2.","s":"∀ (x : ℝ), 0 ≤ x → |Real.exp x * (1 - x) - 1| ≤ Real.exp x * x ^ 2 / 2","l":22,"f":"Causalean/Stat/CLT/MartingaleArray/GaussianBounds.lean"},{"n":"Causalean.Stat.abs_exp_sub_one_le_exp_abs_mul_abs","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.GaussianBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"For any real argument, the increment of the real exponential is bounded by exp(|x|) * |x|.","s":"∀ (x : ℝ), |Real.exp x - 1| ≤ Real.exp |x| * |x|","l":58,"f":"Causalean/Stat/CLT/MartingaleArray/GaussianBounds.lean"},{"n":"Causalean.Stat.abs_exp_varianceCorrection_sub_one_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.GaussianBounds","a":"Stat/CLT/MartingaleArray","t":1,"d":"If the exponential coefficient is nonnegative and a variance value lies between zero and a nonnegative budget, then its terminal Gaussian correction is Lipschitz in its distance from one, with a constant depending only on the budget.","s":"∀ (c V K : ℝ), 0 ≤ c → 0 ≤ V → V ≤ K → |Real.exp (c * (V - 1)) - 1| ≤ Real.exp (c * max K 1) * c * |V - 1|","l":99,"f":"Causalean/Stat/CLT/MartingaleArray/GaussianBounds.lean"},{"n":"Causalean.Stat.abs_integral_le_eLpNorm_two","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Constant-case Cauchy–Schwarz. If ν is a probability measure (or, more generally, satisfies ν univ ≤ 1) and f ∈ L²(ν), then |∫ f dν| ≤ ‖f‖₂.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {ν : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure ν] {f : X → ℝ}, MeasureTheory.MemLp f 2 ν → |∫ (x : X), f x ∂ν| ≤ (MeasureTheory.eLpNorm f 2 ν).toReal","l":306,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.abs_kernelMean_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If a proposed bound is nonnegative and bounds the estimator in absolute value at every output, then the kernel mean obeys the same absolute bound at every input when each kernel output is a probability distribution.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ} {M : ℝ}, 0 ≤ M → (∀ (y : Y), |T y| ≤ M) → ∀ (x : X), |Causalean.Stat.kernelMean K T x| ≤ M","l":70,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.abs_measureReal_sub_le_one","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Every term of the supremum defining tvDist is bounded by 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (A : Set Ω), |μ.real A - ν.real A| ≤ 1","l":55,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.abs_measureReal_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"For a measurable set A, the gap |μ.real A − ν.real A| is at most tvDist μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → |μ.real A - ν.real A| ≤ Causalean.Stat.tvDist μ ν","l":75,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.abs_setIntegral_le_half_integral_abs_of_integral_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":2,"d":"For an integrable function f whose integral vanishes, the integral over any measurable set is bounded in absolute value by half the L¹ norm of f.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ν : MeasureTheory.Measure Ω} {f : Ω → ℝ}, MeasureTheory.Integrable f ν → ∫ (x : Ω), f x ∂ν = 0 → ∀ {A : Set Ω}, MeasurableSet A → |∫ (x : Ω) in A, f x ∂ν| ≤ 1 / 2 * ∫ (x : Ω), |f x| ∂ν","l":50,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.abs_signedTail_le_one","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The signed tail indicator is bounded by 1 in absolute value.","s":"∀ (a s : ℝ), |Causalean.Stat.signedTail a s| ≤ 1","l":92,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.acceptanceRegion","k":"def","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"For a sample space, a pseudo-metric parameter space, an estimator from the sample space to the parameter space, a parameter value, and a real radius, the acceptance region is the set of sample points at which the estimator lies at distance strictly less than that radius from the parameter value.","s":"{Ω : Type u_1} → {Θ : Type u_2} → [PseudoMetricSpace Θ] → (Ω → Θ) → Θ → ℝ → Set Ω","l":61,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_compl","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"The acceptance region is the complement of the error region.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] (est : Ω → Θ) (θ : Θ) (s : ℝ), (Causalean.Stat.acceptanceRegion est θ s)ᶜ = {ω | s ≤ dist (est ω) θ}","l":78,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_disjoint","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Disjointness of acceptance regions. If two target values are 2s-separated, their acceptance regions are disjoint: a point within s of both would force the targets within 2s of each other.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] {est : Ω → Θ} {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → Disjoint (Causalean.Stat.acceptanceRegion est θ₀ s) (Causalean.Stat.acceptanceRegion est θ₁ s)","l":84,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.acceptanceRegion_pairwiseDisjoint","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Pairwise disjointness of the acceptance regions of a 2s-separated family.","s":"∀ {Ω : Type u_1} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] {ι : Type u_3} {est : Ω → Θ} {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → Pairwise (Function.onFun Disjoint fun i => Causalean.Stat.acceptanceRegion est (θ i) s)","l":101,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.adjoint_inv_self","k":"theorem","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":2,"d":"The adjoint (self-adjointness) of a right inverse of a self-adjoint operator: if adjoint M = M and M ∘L N = id, then adjoint N = N. (adjoint N is then a left inverse of M, and in finite dimension a one-sided inverse of an operator that already has a two-sided one is unique.) Shared with OverID.lean.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {M N : E →L[ℝ] E}, ContinuousLinearMap.adjoint M = M → M ∘SL N = ContinuousLinearMap.id ℝ E → ContinuousLinearMap.adjoint N = N","l":82,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.aemeasurable_quantile_unifOI","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The quantile function is a.e.-measurable with respect to the uniform measure on (0,1); it is monotone there, and a monotone function is measurable.","s":"∀ (μ : MeasureTheory.Measure ℝ), AEMeasurable (Causalean.Stat.quantile μ) Causalean.Stat.unifOI","l":58,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.affineInversionSet","k":"def","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":1,"d":"Given a real parameter region, real-valued intercept, slope, and radius parameters, the affine-inversion acceptance set consists exactly of those parameter values in the region whose affine discrepancy $|A-\\theta B|$ is at most $r$.","s":"Set ℝ → ℝ → ℝ → ℝ → Set ℝ","l":24,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affineInversionSet_restrictedVolume_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Affine inversion has restricted volume at most the parameter-region volume and at most twice the radius divided by the nonzero slope.","s":"∀ (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B r : ℝ), B ≠ 0 → 0 ≤ r → Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region A B r) ≤ min (MeasureTheory.volume region).toReal (2 * r / |B|)","l":30,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affineInversionSet_restrictedVolume_le_region","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Affine inversion is always bounded by the volume of its parameter region.","s":"∀ (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B r : ℝ), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region A B r) ≤ (MeasureTheory.volume region).toReal","l":66,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.affinePullbackEstimator","k":"def","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Given a source observation space, a target observation space, a deterministic observation rule from the source to the target, real affine slope and offset parameters, and a real-valued target-space estimator, the affine pullback estimator maps each source observation to the target estimator evaluated at its observed target value, minus the offset and divided by the slope.","s":"{X : Type uX} → {Y : Type uY} → (X → Y) → ℝ → ℝ → (Y → ℝ) → X → ℝ","l":185,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.affine_sqLoss_pullback_identity","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, then the squared error of a target estimator after deterministic observation equals the squared error of its affine pullback multiplied by the squared slope, point by point.","s":"∀ {X : Type uX} {Y : Type uY} {phi : X → Y} {a b theta : ℝ} {targetEst : Y → ℝ}, a ≠ 0 → ∀ (z : X), a ^ 2 * (Causalean.Stat.affinePullbackEstimator phi a b targetEst z - theta) ^ 2 = (targetEst (phi z) - (a * theta + b)) ^ 2","l":205,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.armGroupEvent","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a group-label map, a Boolean arm-assignment map, an arm label, and a group label, the arm--group event is the set of sample-space outcomes whose group and arm labels equal the requested labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → Bool → kappa → Set Omega","l":29,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armGroupResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a real-valued outcome, a supplied real center for every arm and group, an arm label, a group label, and a sample-space outcome, the arm--group residual is that outcome minus the center supplied for its requested arm and group.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":43,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armResidualMean","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, a real-valued outcome, arm--group centers, a sample, an arm label, and a group label, the totalized arm--group residual mean is its residual sum divided by its count when that count is positive, and zero otherwise.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → Bool → kappa → ℝ","l":114,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.armResidualSum","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, a real-valued outcome, arm--group centers, a sample, an arm label, and a group label, the arm--group residual sum adds the supported residuals of all sample coordinates.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → Bool → kappa → ℝ","l":104,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.assouad_average","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Assouad's lemma (average form). Assume each coordinate's decoded-bit event is measurable. If every hypercube vertex's law is within total variation β of each of its d neighbouring vertices' laws, then the average Hamming risk over the cube is at least (d / 2)(1 − β). Choosing the dimension d large and the per-coordinate divergence β small forces a large number of mis-decoded coordinates.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ {β : ℝ}, (∀ (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ β) → ↑d / 2 * (1 - β) ≤ (∑ τ, Causalean.Stat.hammingRisk P est τ) / ↑(Fintype.card (Fin d → Bool))","l":144,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.assouad_exists","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Assouad's lemma (existence form). Under the same hypotheses — each coordinate's decoded-bit event is measurable and every vertex's law is within total variation β of each of its d neighbours — some vertex τ forces Hamming risk at least (d / 2)(1 − β): no cube estimator can decode every vertex's coordinates reliably when neighbouring laws are statistically close.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ {β : ℝ}, (∀ (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ β) → ∃ τ, ↑d / 2 * (1 - β) ≤ Causalean.Stat.hammingRisk P est τ","l":183,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.attachKernel","k":"def","m":"Causalean.Stat.Minimax.ChiSquaredKernel","a":"Stat/Minimax","t":1,"d":"Given a measurable design space, a measurable mark space, a measure on the design space, and a probability kernel from design points to mark laws, the attached marked-observation law draws a design point from the measure and then a mark from the corresponding kernel, retaining both coordinates.","s":"{D : Type u_1} → {M : Type u_2} → [inst : MeasurableSpace D] → [inst_1 : MeasurableSpace M] → MeasureTheory.Measure D → ProbabilityTheory.Kernel D M → MeasureTheory.Measure (D × M)","l":22,"f":"Causalean/Stat/Minimax/ChiSquaredKernel.lean"},{"n":"Causalean.Stat.attachKernel.instIsProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.ChiSquaredKernel","a":"Stat/Minimax","t":2,"d":"A marked-observation law formed from a probability base and a probability kernel is itself a probability law.","s":"∀ {D : Type u_1} {M : Type u_2} [inst : MeasurableSpace D] [inst_1 : MeasurableSpace M] (m : MeasureTheory.Measure D) (kappa : ProbabilityTheory.Kernel D M) [MeasureTheory.IsProbabilityMeasure m] [ProbabilityTheory.IsMarkovKernel kappa], MeasureTheory.IsProbabilityMeasure (Causalean.Stat.attachKernel m kappa)","l":30,"f":"Causalean/Stat/Minimax/ChiSquaredKernel.lean"},{"n":"Causalean.Stat.auxPoissonCountKernel","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"Given a Poisson mean, the auxiliary-count Markov kernel sends each fixed array to that same array paired with an independent Poisson count.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → NNReal → ProbabilityTheory.Kernel (Fin n → X) ((Fin n → X) × ℕ)","l":446,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.auxPoissonCountKernel.isMarkovKernel","k":"instance","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"The auxiliary-count kernel is a Markov kernel.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {n : ℕ} (lambda : NNReal), ProbabilityTheory.IsMarkovKernel (Causalean.Stat.auxPoissonCountKernel lambda)","l":454,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.auxPoissonCountKernel_comp_pi","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For an observation probability law, a Poisson mean, and a fixed sample size, mixing the auxiliary-count kernel over the fixed iid sample law gives exactly the product of that iid law and the Poisson count law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ), (MeasureTheory.Measure.pi fun x => P).bind ⇑(Causalean.Stat.auxPoissonCountKernel lambda) = (MeasureTheory.Measure.pi fun x => P).prod (ProbabilityTheory.poissonMeasure lambda)","l":460,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.bddAbove_tvDist_range","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The family defining tvDist is bounded above (by 1).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], BddAbove (Set.range fun A => |μ.real ↑A - ν.real ↑A|)","l":68,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.bddBelow_quantileSet","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"For 0 < τ, the super-level set is bounded below: since cdf μ → 0 at -∞, any point where the cdf already drops below τ is a lower bound.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, 0 < τ → BddBelow (Causalean.Stat.quantileSet μ τ)","l":62,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.bddBelow_range_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"For a nonnegative risk the worst-case risks are bounded below by zero, which is the side condition the infimum over estimators needs.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk))","l":103,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.bootstrap_studentized_tendsto","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap studentized CLT. Let θ̂ₙ be asymptotically linear at θ₀ with influence function ψ along the i.i.d. sample S, where ψ is measurable, integrable, and square-integrable; suppose further that the influence function is nondegenerate, $\\int \\psi^2\\,dP > 0$, the rescaled estimator is a.e. measurable at every sample size, and the bootstrap studentized statistic is a.e. measurable at every sample size. Then the bootstrap-studentized statistic $\\sqrt n(\\hat\\theta_n-\\theta_0)/\\hat\\sigma_n$ converges in distribution to the standard normal law, where $\\hat\\sigma_n$ is the bootstrap standard error.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → 0 < ∫ (x : X), ψ x ^ 2 ∂P → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ) → ∀ (hStud_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ) (Causalean.Stat.gaussianMeasure 0 1) μ hStud_meas","l":107,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.bootstrap_wald_coverage","k":"theorem","m":"Causalean.Stat.Bootstrap.CI","a":"Stat/Bootstrap","t":1,"d":"Bootstrap Wald asymptotic coverage. Under the asymptotic-linearity hypothesis, measurability, integrability, and square-integrability of ψ, influence-function nondegeneracy, and measurability of the rescaled estimator and of the studentized statistic at every sample size — the hypotheses of bootstrap_studentized_tendsto — fix a positive critical value z and a coverage-probability sequence coverProb that asymptotically tracks the studentized interval's true coverage event; then coverProb converges to the standard normal probability of the interval [-z, z], so the bootstrap studentized interval has asymptotic $N(0,1)$-coverage.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] {θn : ℕ → Ω → ℝ} {θ₀ : ℝ} {ψ : X → ℝ} {S : Causalean.Stat.IIDSample Ω X μ P}, (Causalean.Stat.IsAsymLinear θn θ₀ ψ S fun m => Finset.range m) → Measurable ψ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω)) μ → MeasureTheory.Integrable (fun ω => ψ (S.Z 0 ω) ^ 2) μ → 0 < ∫ (x : X), ψ x ^ 2 ∂P → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator θn θ₀ (fun m => Finset.range m) n) μ) → (∀ (n : ℕ), AEMeasurable (Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n) μ) → ∀ {z : ℝ}, 0 < z → ∀ (coverProb : ℕ → ℝ), Filter.Tendsto (fun n => coverProb n - (μ {ω | Causalean.Stat.IIDSample.bootstrapStudentized θn θ₀ S ψ n ω ∈ Set.Icc (-z) z}).toReal) Filter.atTop (nhds 0) → Filter.Tendsto coverProb Filter.atTop (nhds ((Causalean.Stat.gaussianMeasure 0 1) (Set.Icc (-z) z)).toReal)","l":147,"f":"Causalean/Stat/Bootstrap/CI.lean"},{"n":"Causalean.Stat.bretagnolle_huber_affinity","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":1,"d":"Bretagnolle–Huber inequality. For probability measures μ, ν such that μ is absolutely continuous with respect to ν and their Kullback–Leibler divergence is finite, the two-point testing affinity 1 − tvDist μ ν is at least ½·exp(-KL(μ‖ν)):","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → InformationTheory.klDiv μ ν ≠ ⊤ → 1 / 2 * Real.exp (-(InformationTheory.klDiv μ ν).toReal) ≤ 1 - Causalean.Stat.tvDist μ ν","l":363,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.budgetMultiplier","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.StoppedBudget","a":"Stat/CLT/MartingaleArray","t":1,"d":"Given two sequences of budget charges, two budgets, and a time index, the budget multiplier at that time is one exactly when both cumulative charges through k remain within budget, and zero otherwise.","s":"(ℕ → ℝ) → (ℕ → ℝ) → ℝ → ℝ → ℕ → ℝ","l":22,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedBudget.lean"},{"n":"Causalean.Stat.budgetWeightedSums_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.StoppedBudget","a":"Stat/CLT/MartingaleArray","t":1,"d":"If both charge sequences are nonnegative before the row length and both budgets are nonnegative, then the total charges selected by the through-time budget multiplier do not exceed their respective budgets. The result permits the charge that first crosses a budget to be discarded and uses no probabilistic assumptions.","s":"∀ (x y : ℕ → ℝ) (K delta : ℝ) (r : ℕ), (∀ k < r, 0 ≤ x k) → (∀ k < r, 0 ≤ y k) → 0 ≤ K → 0 ≤ delta → ∑ k ∈ Finset.range r, Causalean.Stat.budgetMultiplier x y K delta k * x k ≤ K ∧ ∑ k ∈ Finset.range r, Causalean.Stat.budgetMultiplier x y K delta k * y k ≤ delta","l":29,"f":"Causalean/Stat/CLT/MartingaleArray/StoppedBudget.lean"},{"n":"Causalean.Stat.cappedStatistic","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"Given a statistic on finite samples, an overflow value, a fixed array, and an auxiliary count, the randomized capped statistic evaluates T on the requested prefix off overflow and equals the specified value on overflow.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ) → ℝ → (Fin n → X) → ℕ → ℝ","l":422,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.card_mul_exp_neg_log_eq_rpow","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"The exponential KL-overlap floor has the expected power-law scaling after multiplication by the number of coordinates.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ (κ : ℝ), ↑M * (1 / 2 * Real.exp (-(κ * Real.log ↑M))) = ↑M ^ (1 - κ) / 2","l":334,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.card_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For coordinate-set sizes r and s and matching size h, the number of partial matchings is the product of the two subset counts and the number of permutations of h objects.","s":"∀ (r s h : ℕ), (Causalean.Stat.partialMatchingsOfSize r s h).card = r.choose h * s.choose h * h.factorial","l":164,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.categoryIndicator","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given an observation space, a category-label space whose labels can be compared for equality, a category-label map, a category, and an observation, the category indicator is one when that observation has the designated category and zero otherwise.","s":"{X : Type u_2} → {Iota : Type u_3} → [DecidableEq Iota] → (X → Iota) → Iota → X → ℝ","l":41,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.categoryMass","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given a measurable observation space, a category-label space, a measure on the observation space, a category-label map, and a category, the category mass is the measure of observations assigned to that category, represented as a real number.","s":"{X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → Iota) → Iota → ℝ","l":34,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.cdfIF","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"For a measure on the real line and a real threshold, the pointwise influence function of the cumulative distribution function assigns to an observation the lower-ray indicator at that threshold minus the measure-assigned cumulative value at that threshold.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ → ℝ","l":113,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_mean_zero","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF has mean zero under P.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfIF P y z ∂P = 0","l":128,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_sq_integrable","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfIF is square-integrable (it is bounded).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), MeasureTheory.Integrable (fun z => Causalean.Stat.cdfIF P y z ^ 2) P","l":149,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfIF_variance","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Variance of the empirical-cdf influence function. At a fixed threshold y, the population second moment of the cdf influence function equals F(y)(1 − F(y)), the Bernoulli variance of the indicator event {Z ≤ y}.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfIF P y z ^ 2 ∂P = ↑(ProbabilityTheory.cdf P) y * (1 - ↑(ProbabilityTheory.cdf P) y)","l":159,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat","k":"def","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"For a real threshold, the lower-ray indicator statistic assigns one to each real-valued observation no greater than that threshold and zero to every other observation.","s":"ℝ → ℝ → ℝ","l":55,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_eq","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator statistic at level y is the indicator function of the lower ray (-∞, y], taking the value one on that ray and zero off it.","s":"∀ (y : ℝ), Causalean.Stat.cdfStat y = (Set.Iic y).indicator fun x => 1","l":60,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_le_one","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is bounded above by one.","s":"∀ (y z : ℝ), Causalean.Stat.cdfStat y z ≤ 1","l":77,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_nonneg","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is nonnegative.","s":"∀ (y z : ℝ), 0 ≤ Causalean.Stat.cdfStat y z","l":72,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdfStat_sq","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The indicator is idempotent: (1{z ≤ y})² = 1{z ≤ y}.","s":"∀ (y z : ℝ), Causalean.Stat.cdfStat y z ^ 2 = Causalean.Stat.cdfStat y z","l":82,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.cdf_increment_sqrt_tendsto","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Taylor increment. √n (F(q₀+a/√n) − F(q₀)) → f₀·a from HasDerivAt F f₀ q₀. Used by both the L3 oscillation mesh term and the L4 root-n rate (in Rate.lean).","s":"∀ {P : MeasureTheory.Measure ℝ} {τ q₀ f₀ : ℝ}, Causalean.Stat.SampleQuantileReg P τ q₀ f₀ → ∀ (a : ℝ), Filter.Tendsto (fun n => √↑n * (↑(ProbabilityTheory.cdf P) (q₀ + a / √↑n) - ↑(ProbabilityTheory.cdf P) q₀)) Filter.atTop (nhds (f₀ * a))","l":506,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.cellMoment","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"Given a cell-label space and an observation space, a real-valued cell-mass function, a map assigning each observation to a cell, a nonnegative integer source-sample size, a source sample, a real-valued observation statistic, and a cell label, the cell moment is the inverse source-sample size and inverse cell mass times the sum of the statistic over source observations projected to that cell.","s":"{𝒳 : Type u_1} → {Ω : Type u_2} → (𝒳 → ℝ) → (Ω → 𝒳) → {n : ℕ} → (Fin n → Ω) → (Ω → ℝ) → 𝒳 → ℝ","l":39,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.centeredCrossMoment","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"For a measure on a measurable sample space and two real-valued random variables on that sample space, the centered cross moment is the integral of their product minus the product of their separate integrals.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ℝ) → (Ω → ℝ) → ℝ","l":544,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.centeredCrossMoment_normalizedOrderedProductStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"Under a probability population law, for an i.i.d. sample, statistic orders and a sample size, if the first order does not exceed the sample size and the second order does not exceed the sample size, then for two coordinate-function families, when their product kernels are measurable, their product kernels are integrable, every merged kernel is measurable, and every merged kernel is integrable, their centered cross moment equals an explicit disjoint normalization correction plus the merged moments from every positive-size partial matching.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), Measurable (Causalean.Stat.orderedProductKernel f) → Measurable (Causalean.Stat.orderedProductKernel g) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel f) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel g) (MeasureTheory.Measure.pi fun x => P) → (∀ (M : Causalean.Stat.PartialMatching r s), Measurable (Causalean.Stat.mergedProductKernel f g M)) → (∀ (M : Causalean.Stat.PartialMatching r s), MeasureTheory.Integrable (Causalean.Stat.mergedProductKernel f g M) (MeasureTheory.Measure.pi fun x => P)) → Causalean.Stat.centeredCrossMoment μ (Causalean.Stat.normalizedOrderedProductStatistic S f n) (Causalean.Stat.normalizedOrderedProductStatistic S g n) = (Causalean.Stat.matchingNormalization n (Causalean.Stat.PartialMatching.empty r s) - 1) * Causalean.Stat.orderedProductMean P f * Causalean.Stat.orderedProductMean P g + ∑ h ∈ Finset.range (min r s + 1) with 0 < h, ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.mergedProductMoment P f g M","l":596,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.charFun_isGaussian_centered","k":"theorem","m":"Causalean.Stat.CLT.GaussianCharFunBridge","a":"Stat/CLT","t":1,"d":"Centered-Gaussian characteristic function. For a Gaussian measure Q on E with mean zero, its characteristic function at any point t equals $\\exp(-\\tfrac12\\,\\mathrm{covarianceBilin}\\ Q\\ t\\ t)$.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : MeasurableSpace E] [inst_3 : BorelSpace E] [SecondCountableTopology E] [CompleteSpace E] (Q : MeasureTheory.Measure E) [ProbabilityTheory.IsGaussian Q], ∫ (x : E), x ∂Q = 0 → ∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(((ProbabilityTheory.covarianceBilin Q) t) t) / 2)","l":50,"f":"Causalean/Stat/CLT/GaussianCharFunBridge.lean"},{"n":"Causalean.Stat.charFun_isGaussian_of_cov_eq","k":"theorem","m":"Causalean.Stat.CLT.GaussianCharFunBridge","a":"Stat/CLT","t":1,"d":"Bridge to the abstract CLT target hQ. For a centered Gaussian measure Q on E with mean zero whose covariance bilinear form at every t equals the influence-function second moment $\\int \\langle t,\\psi(x)\\rangle^2\\,dP$, its characteristic function at t equals $\\exp(-\\tfrac12\\int \\langle t,\\psi(x)\\rangle^2\\,dP)$ — the exact target shape consumed as the hypothesis hQ by IIDSample.clt_normalizedSum_vec_of_charFun and friends.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : MeasurableSpace E] [inst_3 : BorelSpace E] [SecondCountableTopology E] [CompleteSpace E] {X : Type u_2} [inst_6 : MeasurableSpace X] {P : MeasureTheory.Measure X} {ψ : X → E} (Q : MeasureTheory.Measure E) [ProbabilityTheory.IsGaussian Q], ∫ (x : E), x ∂Q = 0 → (∀ (t : E), ((ProbabilityTheory.covarianceBilin Q) t) t = ∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) → ∀ (t : E), MeasureTheory.charFun Q t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)","l":64,"f":"Causalean/Stat/CLT/GaussianCharFunBridge.lean"},{"n":"Causalean.Stat.chiSqDist","k":"def","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":1,"d":"For a nonnegative integer number of degrees of freedom, the chi-squared distribution is the probability law on the real line of the squared Euclidean norm of a standard Gaussian vector with that many coordinates.","s":"ℕ → MeasureTheory.Measure ℝ","l":48,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv","k":"def","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"For two measures on the same measurable space, the finite real chi-squared divergence of the first relative to the second is the integral, under the second measure, of the square of one less the Radon--Nikodym density of the first measure with respect to the second, with infinite density values represented by zero in the real-valued integrand.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → ℝ","l":54,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_bernoulliLaw_centerHalf","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredTwoPoint","a":"Stat/Minimax","t":2,"d":"For a perturbation whose magnitude is strictly below one half, the χ²-divergence of the Bernoulli law with success probability one half plus that perturbation, relative to the centered Bernoulli law, is four times the squared perturbation.","s":"∀ {u : ℝ}, |u| < 1 / 2 → Causalean.Stat.chiSqDiv (Causalean.Mathlib.Probability.bernoulliLaw (1 / 2 + u)) (Causalean.Mathlib.Probability.bernoulliLaw (1 / 2)) = 4 * u ^ 2","l":21,"f":"Causalean/Stat/Minimax/ChiSquaredTwoPoint.lean"},{"n":"Causalean.Stat.chiSqDiv_eq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Expansion of the χ²-divergence: χ²(μ‖ν) = ∫ (dμ/dν)² ∂ν − 1 when μ ≪ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.chiSqDiv μ ν = ∫ (x : Ω), (μ.rnDeriv ν x).toReal ^ 2 ∂ν - 1","l":118,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_map_measurableEquiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"χ²-divergence is invariant under a measurable equivalence. Pushing both measures forward through e : Ω ≃ᵐ Ω' leaves the χ²-divergence unchanged.","s":"∀ {Ω : Type u_4} {Ω' : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] (e : Ω ≃ᵐ Ω') (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], Causalean.Stat.chiSqDiv (MeasureTheory.Measure.map (⇑e) μ) (MeasureTheory.Measure.map (⇑e) ν) = Causalean.Stat.chiSqDiv μ ν","l":247,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The χ²-divergence is nonnegative: its integrand is a square.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω}, 0 ≤ Causalean.Stat.chiSqDiv μ ν","l":62,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_prod","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Binary tensorization of the χ²-divergence. For probability measures with μ₁ ≪ ν₁ and μ₂ ≪ ν₂ and integrable squared density deviations, the χ²-divergence tensorizes multiplicatively: 1 + χ²(μ₁⊗μ₂ ‖ ν₁⊗ν₂) = (1 + χ²(μ₁‖ν₁))·(1 + χ²(μ₂‖ν₂)).","s":"∀ {α : Type u_2} {mα : MeasurableSpace α} {β : Type u_3} {mβ : MeasurableSpace β} (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ₁] [MeasureTheory.IsProbabilityMeasure ν₁] [MeasureTheory.IsProbabilityMeasure μ₂] [MeasureTheory.IsProbabilityMeasure ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → MeasureTheory.Integrable (fun x => ((μ₁.rnDeriv ν₁ x).toReal - 1) ^ 2) ν₁ → MeasureTheory.Integrable (fun y => ((μ₂.rnDeriv ν₂ y).toReal - 1) ^ 2) ν₂ → 1 + Causalean.Stat.chiSqDiv (μ₁.prod μ₂) (ν₁.prod ν₂) = (1 + Causalean.Stat.chiSqDiv μ₁ ν₁) * (1 + Causalean.Stat.chiSqDiv μ₂ ν₂)","l":184,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_prod_ancillary","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Ancillary product factor leaves the χ²-divergence unchanged. Tensoring both measures with a *common* probability measure ρ (an ancillary coordinate, whose law is the same under μ and ν) does not change the χ²-divergence: χ²(μ⊗ρ ‖ ν⊗ρ) = χ²(μ‖ν). This is the formal content of \"an ancillary observation carries no information\": its contribution is 1 + χ²(ρ‖ρ) = 1. It is the bridge from the finite-cell lower bound to the continuous-covariate one (the within-cell position is ancillary).","s":"∀ {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [MeasureTheory.IsProbabilityMeasure ρ], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.chiSqDiv (μ.prod ρ) (ν.prod ρ) = Causalean.Stat.chiSqDiv μ ν","l":444,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_self","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The χ²-divergence of a (sigma-finite) measure against itself is zero.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (ρ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite ρ], Causalean.Stat.chiSqDiv ρ ρ = 0","l":261,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.chiSqDiv_twoPointMean_centerHalf","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredTwoPoint","a":"Stat/Minimax","t":1,"d":"For a mean perturbation whose magnitude is strictly below one half, the χ²-divergence of the symmetric two-point law with that mean, relative to the centered symmetric two-point law, is four times the squared perturbation.","s":"∀ {u : ℝ}, |u| < 1 / 2 → Causalean.Stat.chiSqDiv (Causalean.Mathlib.Probability.twoPointMean (1 / 2) u) (Causalean.Mathlib.Probability.twoPointMean (1 / 2) 0) = 4 * u ^ 2","l":89,"f":"Causalean/Stat/Minimax/ChiSquaredTwoPoint.lean"},{"n":"Causalean.Stat.classCoverage_liminf","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"Pointwise coverage rows with a vanishing uniform error imply asymptotic uniform coverage over any eventually inhabited sequence of model classes.","s":"∀ {Model : Type u_1} (cls : ℕ → Model → Prop) (coverage : ℕ → Model → ℝ) (alpha : ℝ) (delta : ℕ → ℝ), Filter.Tendsto delta Filter.atTop (nhds 0) → (∀ᶠ (n : ℕ) in Filter.atTop, ∃ P, cls n P) → (∀ (n : ℕ) (P : Model), cls n P → 0 ≤ coverage n P ∧ coverage n P ≤ 1) → (∀ (n : ℕ) (P : Model), cls n P → 1 - alpha - delta n ≤ coverage n P) → 1 - alpha ≤ Filter.liminf (fun n => ⨅ P, coverage n ↑P) Filter.atTop","l":172,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.classFrontierRisk","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"For a sequence of model classes, a real-valued strength and a real-valued expected-length criterion for each sample size and model, and a real strength threshold, the class frontier risk is the limit superior, across sample sizes, of the supremum expected length over the models in that class whose strength is at least the threshold.","s":"{Model : Type u_1} → (ℕ → Model → Prop) → (ℕ → Model → ℝ) → (ℕ → Model → ℝ) → ℝ → ℝ","l":226,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.classFrontierRisk_le","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"A pointwise capped inverse-square-root expected-length bound passes through both the class supremum and asymptotic limsup at the threshold value.","s":"∀ {Model : Type u_1} (cls : ℕ → Model → Prop) (strength expectedLength : ℕ → Model → ℝ) (C0 t0 : ℝ), 0 ≤ C0 → 0 < t0 → (∀ (n : ℕ) (P : Model), cls n P → 0 ≤ expectedLength n P) → (∀ (n : ℕ) (P : Model), cls n P → expectedLength n P ≤ C0 * min 1 (strength n P ^ (-1 / 2))) → Causalean.Stat.classFrontierRisk cls strength expectedLength t0 ≤ C0 * min 1 (t0 ^ (-1 / 2))","l":238,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.collisionKernel","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"Given a finite or infinite cell-label space, a real-valued cell-mass function, and two cell labels, the collision kernel equals the reciprocal mass of the first label when the labels coincide and equals zero otherwise.","s":"{𝒳 : Type u_1} → (𝒳 → ℝ) → 𝒳 → 𝒳 → ℝ","l":22,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.collisionScale","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"Given a cell-label space, a real-valued cell-mass function, a nonnegative integer target-sample size, and a target sample of that size, the collision scale is the average collision-kernel value over all ordered pairs of distinct target indices, with the displayed reciprocal convention also applying at sample sizes zero and one.","s":"{𝒳 : Type u_1} → (𝒳 → ℝ) → {N : ℕ} → (Fin N → 𝒳) → ℝ","l":30,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.comonotoneCoupling","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For two measures on the real line, the comonotone quantile coupling is the image of the uniform measure on the open unit interval under the pair of their quantile functions evaluated at the same uniform draw. Both coordinates are therefore driven by that common draw.","s":"MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → MeasureTheory.Measure (ℝ × ℝ)","l":43,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.compressedCoordinateLaw","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For a compression from a measurable observation space to a measurable summary space and a measure on the observation space, the compressed-coordinate law is the image measure of the summary obtained by applying the compression to an observation governed by that measure.","s":"{Z : Type u_1} → {S : Type u_2} → [inst : MeasurableSpace Z] → [inst_1 : MeasurableSpace S] → (Z → S) → MeasureTheory.Measure Z → MeasureTheory.Measure S","l":23,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.compressedCoordinateLaw_klDiv_le","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Measurable finite-coordinate compression cannot increase KL divergence.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : MeasurableSpace S] (compress : Z → S), Measurable compress → ∀ (μ ν : MeasureTheory.Measure Z) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν], InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw compress μ) (Causalean.Stat.compressedCoordinateLaw compress ν) ≤ InformationTheory.klDiv μ ν","l":32,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.compressionCoupling","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"Given two standard Borel measurable spaces, an observation space and a compressed-state space, two probability measures on the observation space, and a measurable compression map from observations to compressed states, the compression coupling is the joint law obtained by maximally coupling the two compressed laws and then, conditional on each coupled compressed state, drawing each observation from its corresponding regular conditional distribution.","s":"{Z : Type u_1} → {S : Type u_2} → [inst : MeasurableSpace Z] → [StandardBorelSpace Z] → [inst_2 : MeasurableSpace S] → [StandardBorelSpace S] → (Q0 Q1 : MeasureTheory.Measure Z) → [MeasureTheory.IsProbabilityMeasure Q0] → [MeasureTheory.IsProbabilityMeasure Q1] → (compress : Z → S) → Measurable compress → MeasureTheory.Measure (Z × Z)","l":401,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling.instIsProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress)","l":550,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_equal_compression_mass_ge","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"In the lifted coupling, the compressed observations agree with probability at least one minus the total variation of their compressed laws.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), ENNReal.ofReal (1 - Causalean.Stat.tvDist (MeasureTheory.Measure.map compress Q0) (MeasureTheory.Measure.map compress Q1)) ≤ (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) {p | compress p.1 = compress p.2}","l":565,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_compress_pair","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Compressing both coordinates of the lifted coupling recovers the maximal coupling of the compressed laws.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress) [inst_6 : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map compress Q0)] [inst_7 : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map compress Q1)], MeasureTheory.Measure.map (Prod.map compress compress) (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Causalean.Stat.maximalCoupling (MeasureTheory.Measure.map compress Q0) (MeasureTheory.Measure.map compress Q1)","l":496,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_fst","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The first marginal of the lifted compression coupling is the first raw law.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), MeasureTheory.Measure.map Prod.fst (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Q0","l":444,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.compressionCoupling_map_snd","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The second marginal of the lifted compression coupling is the second raw law.","s":"∀ {Z : Type u_1} {S : Type u_2} [inst : MeasurableSpace Z] [inst_1 : StandardBorelSpace Z] [inst_2 : MeasurableSpace S] [inst_3 : StandardBorelSpace S] (Q0 Q1 : MeasureTheory.Measure Z) [inst_4 : MeasureTheory.IsProbabilityMeasure Q0] [inst_5 : MeasureTheory.IsProbabilityMeasure Q1] (compress : Z → S) (hcompress : Measurable compress), MeasureTheory.Measure.map Prod.snd (Causalean.Stat.compressionCoupling Q0 Q1 compress hcompress) = Q1","l":470,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.condExp_cexp_ae_eq_quadratic_add_remainder","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTaylor","a":"Stat/CLT/MartingaleArray","t":2,"d":"For a conditioning sigma-algebra contained in the ambient sigma-algebra, a square-integrable real random variable with conditional mean zero, the conditional characteristic-function increment is exactly its constant term, conditional quadratic term, and conditional Taylor remainder.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ], m ≤ mΩ → ∀ (X : Ω → ℝ), MeasureTheory.MemLp X 2 μ → μ[X | m] =ᵐ[μ] 0 → ∀ (t : ℝ), μ[fun ω => Complex.exp (Complex.I * ↑(t * X ω)) | m] =ᵐ[μ] fun ω => 1 - ↑(t ^ 2 / 2) * ↑(μ[fun ω => X ω ^ 2 | m] ω) + μ[fun ω => Causalean.Stat.expQuadraticRemainder (t * X ω) | m] ω","l":42,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTaylor.lean"},{"n":"Causalean.Stat.condExp_ofReal_ae_eq_zero","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTaylor","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the conditioning sigma-algebra is contained in the ambient sigma-algebra, a real random variable is integrable, and has conditional mean zero, then its complex embedding also has conditional mean zero.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ], m ≤ mΩ → ∀ (X : Ω → ℝ), MeasureTheory.Integrable X μ → μ[X | m] =ᵐ[μ] 0 → μ[fun ω => ↑(X ω) | m] =ᵐ[μ] 0","l":25,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTaylor.lean"},{"n":"Causalean.Stat.conditionalFourthMoment_tendstoInProbability_of_fourthMomentSum","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Lyapunov","a":"Stat/CLT/MartingaleArray","t":2,"d":"If every increment has a finite fourth moment and the deterministic sum of unconditional fourth moments tends to zero, then the conditional fourth-moment row sums converge in probability to zero.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Filter.Tendsto A.fourthMomentSum Filter.atTop (nhds 0) → Causalean.Stat.TendstoInProbability μ A.conditionalFourthMoment 0","l":190,"f":"Causalean/Stat/CLT/MartingaleArray/Lyapunov.lean"},{"n":"Causalean.Stat.conditionalLindeberg_of_conditionalFourthMoment","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Lyapunov","a":"Stat/CLT/MartingaleArray","t":2,"d":"If every increment has a finite fourth moment and the conditional fourth-moment row sums converge in probability to zero, then the conditional Lindeberg condition holds at every positive threshold.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Causalean.Stat.TendstoInProbability μ A.conditionalFourthMoment 0 → ∀ (ε : ℝ), 0 < ε → Causalean.Stat.TendstoInProbability μ (A.conditionalLindeberg ε) 0","l":47,"f":"Causalean/Stat/CLT/MartingaleArray/Lyapunov.lean"},{"n":"Causalean.Stat.conditionalLindeberg_of_fourthMomentSum","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Lyapunov","a":"Stat/CLT/MartingaleArray","t":1,"d":"If every increment has a finite fourth moment and the deterministic sum of unconditional fourth moments tends to zero, then the conditional Lindeberg condition holds at every positive threshold.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Filter.Tendsto A.fourthMomentSum Filter.atTop (nhds 0) → ∀ (ε : ℝ), 0 < ε → Causalean.Stat.TendstoInProbability μ (A.conditionalLindeberg ε) 0","l":269,"f":"Causalean/Stat/CLT/MartingaleArray/Lyapunov.lean"},{"n":"Causalean.Stat.consistent_lt_norm_of_le_dist","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":2,"d":"Format bridge. On a normed group dist x θ₀ = ‖x − θ₀‖, so the consistency statement produced by mEstimator_consistent_of_glivenkoCantelli (phrased with ε ≤ dist (θn n) θ₀) implies the strictly-larger-radius form ε < ‖θn n − θ₀‖ consumed by the CLT layer. {ε < ‖·‖} ⊆ {ε ≤ dist}, so the measures are squeezed to 0.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_1 : NormedAddCommGroup E] (θn : ℕ → Ω → E) (θ₀ : E), (∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (θn n ω) θ₀}) Filter.atTop (nhds 0)) → ∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)","l":67,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.continuous_covarianceModelLoss","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given a compact covariance model that is nonempty, the population loss is continuous in the target second-moment matrix.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)}, IsCompact M → M.Nonempty → Continuous (Causalean.Stat.covarianceModelLoss M)","l":376,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.continuous_gaussianCovarianceDiscrepancy","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"The Gaussian covariance discrepancy varies continuously with its positive-definite candidate and target matrix.","s":"∀ {V : Type u_1} [inst : Fintype V] [inst_1 : DecidableEq V], Continuous fun p => Causalean.Stat.gaussianCovarianceDiscrepancy (↑p.1) p.2","l":41,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.continuous_maxDirDeriv","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"The directional derivative of the maximum functional is continuous.","s":"∀ (a b : ℝ), Continuous (Causalean.Stat.maxDirDeriv a b)","l":175,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.continuous_minDirDeriv","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"The directional derivative of the minimum functional is continuous.","s":"∀ (a b : ℝ), Continuous (Causalean.Stat.minDirDeriv a b)","l":184,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.continuous_normalizedCovarianceDiscrepancy","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"The normalized Gaussian covariance discrepancy varies continuously with both positive-definite covariances.","s":"∀ {V : Type u_1} [inst : Fintype V] [inst_1 : DecidableEq V], Continuous fun p => Causalean.Stat.normalizedCovarianceDiscrepancy p.1 p.2","l":69,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.coordinateOverlap","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For two probability laws at each coordinate, indexed by its binary state, a compression at each coordinate, and one coordinate, the common-part overlap at that coordinate is one minus the total-variation distance between the two compressed laws of that coordinate.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → [inst : (j : Fin M) → MeasurableSpace (Z j)] → [(j : Fin M) → MeasurableSpace (S j)] → ((j : Fin M) → Bool → MeasureTheory.Measure (Z j)) → ((j : Fin M) → Z j → S j) → Fin M → ℝ","l":67,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinateOverlap_product_le_of_nonnegative_kl","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Under a nonnegative logarithmic KL budget, the product of coordinatewise total-variation factors has the finite-M exponential bound.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ {Z : Fin M → Type u_1} {S : Fin M → Type u_2} [inst : (j : Fin M) → MeasurableSpace (Z j)] [inst_1 : (j : Fin M) → MeasurableSpace (S j)] (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) [∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] (compress : (j : Fin M) → Z j → S j), (∀ (j : Fin M), Measurable (compress j)) → ∀ {κ : ℝ}, 0 ≤ κ → (∀ (j : Fin M), InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j false)) (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j true)) ≤ ENNReal.ofReal (κ * Real.log ↑M)) → ∏ j, (1 - Causalean.Stat.coordinateOverlap Q compress j) ≤ Real.exp (-↑M ^ (1 - κ) / 2)","l":349,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewiseSuccessProbability","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For a nonnegative number of coordinates, measurable raw-observation and summary spaces at every coordinate, a measurable ancillary space, two probability laws for each coordinate, indexed by its binary state, a probability law for a common ancillary variable, one compression for each coordinate, and a decoder for each coordinate that uses its compressed observation, the full raw observation vector, and the ancillary variable, the coordinatewise success probability is the average, over all binary state vectors, of the probability that every decoder recovers its corresponding state under the associated independent product experiment.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → [inst : (j : Fin M) → MeasurableSpace (Z j)] → [(j : Fin M) → MeasurableSpace (S j)] → [inst_2 : MeasurableSpace A] → (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) → (R : MeasureTheory.Measure A) → [∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] → [MeasureTheory.IsProbabilityMeasure R] → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → ENNReal","l":43,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewise_overlap_direct_product","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"Coordinatewise-overlap direct-product bound. In a conditionally independent hypercube experiment with at least one coordinate, per-coordinate candidate laws Q j, indexed by a bit, a per-coordinate compression map compress that is measurable at every coordinate, and per-coordinate decoders decoder built from the compressed local summary, the other coordinates' raw data, and shared randomness that are jointly measurable and depend on the raw sample at coordinate j only through its compressed summary, not directly on the raw value at j, then decentralized coordinate decoders cannot on average be correct more often than the common-part product bound built from the per-coordinate total-variation overlaps; and if every compressed adjacent KL divergence is at most κ log M, the displayed finite-M simultaneous-error certificate follows.","s":"∀ {M : ℕ}, 1 ≤ M → ∀ {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} [inst : (j : Fin M) → MeasurableSpace (Z j)] [∀ (j : Fin M), StandardBorelSpace (Z j)] [inst_2 : (j : Fin M) → MeasurableSpace (S j)] [∀ (j : Fin M), StandardBorelSpace (S j)] [inst_4 : MeasurableSpace A] [StandardBorelSpace A] (Q : (j : Fin M) → Bool → MeasureTheory.Measure (Z j)) (R : MeasureTheory.Measure A) [inst_6 : ∀ (j : Fin M) (b : Bool), MeasureTheory.IsProbabilityMeasure (Q j b)] [inst_7 : MeasureTheory.IsProbabilityMeasure R] (compress : (j : Fin M) → Z j → S j), (∀ (j : Fin M), Measurable (compress j)) → ∀ (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M), Measurable fun p => decoder j p.1 p.2.1 p.2.2) → (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → Causalean.Stat.coordinatewiseSuccessProbability Q R compress decoder ≤ ENNReal.ofReal (1 / 2 * (1 + ∏ j, (1 - Causalean.Stat.coordinateOverlap Q compress j))) ∧ ∀ κ < 1, (∀ (j : Fin M), InformationTheory.klDiv (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j false)) (Causalean.Stat.compressedCoordinateLaw (compress j) (Q j true)) ≤ ENNReal.ofReal (κ * Real.log ↑M)) → ENNReal.ofReal (1 / 2 * (1 - Real.exp (-↑M ^ (1 - κ) / 2))) ≤ 1 - Causalean.Stat.coordinatewiseSuccessProbability Q R compress decoder","l":415,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coordinatewise_overlap_direct_product_asymptotic","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Along any sequence M_n → ∞ and for fixed κ < 1, the finite direct-product certificate tends to one half.","s":"∀ (Mseq : ℕ → ℕ), ∀ κ < 1, Filter.Tendsto Mseq Filter.atTop Filter.atTop → Filter.Tendsto (fun n => 1 / 2 * (1 - Real.exp (-↑(Mseq n) ^ (1 - κ) / 2))) Filter.atTop (nhds (1 / 2))","l":749,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.countermonotoneCoupling","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For two measures on the real line, the countermonotone quantile coupling is the image of the uniform measure on the open unit interval under the first measure's quantile function at a uniform draw and the second measure's quantile function at one minus that draw.","s":"MeasureTheory.Measure ℝ → MeasureTheory.Measure ℝ → MeasureTheory.Measure (ℝ × ℝ)","l":53,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.countermonotone_le_jointCdf","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Corollary: the countermonotone joint cdf is dominated pointwise by any other coupling's joint cdf.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.countermonotoneCoupling μ ν) x y ≤ Causalean.Stat.jointCdf π x y","l":218,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.countermonotone_le_product_expectation","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Lower optimality. For any coupling π of μ and ν, where both marginals have finite second moment, the expectation of the coordinate product under π is at least its expectation under the countermonotone coupling of μ and ν:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.countermonotoneCoupling μ ν ≤ ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π","l":69,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.coupledDecoderGood","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For a nonnegative number of coordinates, one compression for each coordinate, one decoder for each coordinate, a binary hypercube vertex, a pair of raw observations at every coordinate, and an ancillary-variable value, the coupled decoder-good condition holds exactly when every decoder, applied to its compressed selected observation together with the full selected raw vector and the ancillary value, returns its corresponding vertex bit.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → A → Prop","l":90,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledDecoderGood_count_le_half","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"Once one coupled coordinate has equal compressions, at most half of the hypercube vertices can be simultaneously decoded correctly.","s":"∀ {M : ℕ} {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} (compress : (j : Fin M) → Z j → S j) (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → ∀ (z : (j : Fin M) → Z j × Z j) (a : A) (j : Fin M), compress j (z j).1 = compress j (z j).2 → ∑ omega, Causalean.Stat.coupledGoodIndicator compress decoder omega z a ≤ 2 ^ M / 2","l":154,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledDecoderGood_flip_exclusive","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"If the two compressed versions agree at coordinate j, simultaneous correctness is impossible at both endpoints of the corresponding cube edge.","s":"∀ {M : ℕ} {Z : Fin M → Type u_1} {S : Fin M → Type u_2} {A : Type u_3} (compress : (j : Fin M) → Z j → S j) (decoder : (j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool), (∀ (j : Fin M) (s : S j) (z z' : (k : Fin M) → Z k) (a : A), (∀ (k : Fin M), k ≠ j → z k = z' k) → decoder j s z a = decoder j s z' a) → ∀ (omega : Fin M → Bool) (z : (j : Fin M) → Z j × Z j) (a : A) (j : Fin M), compress j (z j).1 = compress j (z j).2 → ¬(Causalean.Stat.coupledDecoderGood compress decoder omega z a ∧ Causalean.Stat.coupledDecoderGood compress decoder (Causalean.Stat.flipBit j omega) z a)","l":123,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupledGoodIndicator","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For a nonnegative number of coordinates, one compression for each coordinate, one decoder for each coordinate, a binary hypercube vertex, a pair of raw observations at every coordinate, and an ancillary-variable value, the coupled decoder-good condition holds exactly when every decoder, applied to its compressed selected observation together with the full selected raw vector and the ancillary value, returns its corresponding vertex bit.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → {S : Fin M → Type u_2} → {A : Type u_3} → ((j : Fin M) → Z j → S j) → ((j : Fin M) → S j → ((k : Fin M) → Z k) → A → Bool) → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → A → ENNReal","l":107,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.coupling_fst_memLp","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The first coordinate is in L²(π) when μ = π.map Prod.fst has a second moment.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun p => p.1) 2 π","l":47,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integrable_mul","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"XY ∈ L¹(π) by Cauchy–Schwarz from the two L² marginals.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun p => p.1 * p.2) π","l":81,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integral_fst","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"E_π[X] = E_μ[id]: the first moment of a coupling is that of its first marginal.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → ∫ (p : ℝ × ℝ), p.1 ∂π = ∫ (x : ℝ), x ∂μ","l":63,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_integral_snd","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"E_π[Y] = E_ν[id]: the first moment of a coupling is that of its second marginal.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.2 ∂π = ∫ (y : ℝ), y ∂ν","l":72,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.coupling_snd_memLp","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The second coordinate is in L²(π) when ν = π.map Prod.snd has a second moment.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.MemLp (fun p => p.2) 2 π","l":55,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.covarianceModelLoss","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For a covariance model and target matrix, the population loss is the infimum Gaussian discrepancy over positive-definite model covariances.","s":"{V : Type u_2} → [inst : Fintype V] → [DecidableEq V] → Set (Causalean.Stat.PositiveCovariance V) → Matrix V V ℝ → ℝ","l":369,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.coverageInfOrOne","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"For a real-valued criterion indexed by an arbitrary collection, the worst-case criterion with the empty-collection convention is its infimum when the collection is nonempty and is one when it is empty.","s":"{ι : Sort u_1} → (ι → ℝ) → ℝ","l":291,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverageInfOrOne_of_isEmpty","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"On an empty model class, worst-case coverage with the empty-class convention is one, expressing that the coverage requirement is vacuously satisfied.","s":"∀ {ι : Sort u_1} [IsEmpty ι] (f : ι → ℝ), Causalean.Stat.coverageInfOrOne f = 1","l":313,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverageInfOrOne_of_nonempty","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"On a nonempty model class, worst-case coverage with the empty-class convention is the ordinary infimum of coverage across the class.","s":"∀ {ι : Sort u_1} [Nonempty ι] (f : ι → ℝ), Causalean.Stat.coverageInfOrOne f = ⨅ i, f i","l":306,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.coverage_tv_expectedRestrictedVolume_lower","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"Coverage-to-expected-restricted-volume bound. For a family of laws Q u indexed by u : ℝ, a random set C, a subset I of a parameter region region, and a reference point reference, suppose every Q u is a probability measure, C covers u with probability at least coverage, for every u in I, Q u is within total variation tv of the reference law Q reference, for every u in I, the graph {(ω, u) | u ∈ C ω} is measurable, region is measurable, region has finite Lebesgue volume, I is measurable, and I is contained in region. Then the expected restricted volume of C under the reference law Q reference is at least (volume I) · (coverage − tv).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : ℝ → MeasureTheory.Measure Ω) (C : Ω → Set ℝ) (region I : Set ℝ) (reference coverage tv : ℝ), (∀ (u : ℝ), MeasureTheory.IsProbabilityMeasure (Q u)) → (∀ u ∈ I, coverage ≤ ((Q u) {ω | u ∈ C ω}).toReal) → (∀ u ∈ I, Causalean.Stat.tvDist (Q u) (Q reference) ≤ tv) → MeasurableSet {p | p.2 ∈ C p.1} → MeasurableSet region → MeasureTheory.volume region ≠ ⊤ → MeasurableSet I → I ⊆ region → (MeasureTheory.volume I).toReal * (coverage - tv) ≤ ∫ (ω : Ω), Causalean.Stat.restrictedSetVolume region (C ω) ∂Q reference","l":97,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.crossAverage","k":"def","m":"Causalean.Stat.Sample.CollisionEstimator","a":"Stat/Sample","t":1,"d":"Given a cell-label space and an observation space, a real-valued cell-mass function, an observation-to-cell map, nonnegative integer source and target sample sizes, a source sample, a target sample of cell labels, and a real-valued observation statistic, the cross-sample average is the average, over target observations, of the corresponding source-sample cell moments.","s":"{𝒳 : Type u_1} → {Ω : Type u_2} → (𝒳 → ℝ) → (Ω → 𝒳) → {n N : ℕ} → (Fin n → Ω) → (Fin N → 𝒳) → (Ω → ℝ) → ℝ","l":52,"f":"Causalean/Stat/Sample/CollisionEstimator.lean"},{"n":"Causalean.Stat.cumulative_risk_engine","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Sequential-experiment cumulative-risk engine.","s":"∀ (J d L Iq : ℝ), 0 < J → d ≠ 0 → 0 ≤ L → 0 ≤ Iq → ∀ (b B : ℕ → ℝ), (∀ (n : ℕ), B n = ∑ t ∈ Finset.Icc 1 n, b t) → (∀ (t : ℕ), 1 ≤ t → d ^ 2 / 4 / (Iq + 5 * J / 4 * ↑t + L * √(↑t * B (t - 1))) ≤ b t) → ∃ T₀, ∀ (T : ℕ), T₀ ≤ T → d ^ 2 / (32 * J) * Real.log ↑T ≤ B T","l":144,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.decode_ne_flip_compl","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"The j-th decoding-error set at the flipped vertex is the complement of the one at τ: the estimator's j-th bit either matches τ j or its flip, never both.","s":"∀ {Ω : Type u_1} {d : ℕ} (est : Ω → Fin d → Bool) (j : Fin d) (τ : Fin d → Bool), {ω | est ω j ≠ Causalean.Stat.flipBit j τ j} = {ω | est ω j ≠ τ j}ᶜ","l":102,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.deltaMethod","k":"theorem","m":"Causalean.Stat.Inference.DeltaMethod","a":"Stat/Inference","t":1,"d":"Multivariate delta method. Let Tn be a sequence of E-valued estimators of a target t₀, and let g : E → F be Fréchet-differentiable at t₀ with derivative Dg. Given that the rescaled deviations √n • (Tn − t₀) are measurable at every sample size and that their images √n • (g(Tn) − g(t₀)) are measurable at every sample size, if the laws of √n • (Tn − t₀) converge weakly to a probability measure Q on E, then the laws of √n • (g(Tn) − g(t₀)) converge weakly to the pushforward of Q along the linear map Dg.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {E : Type u_2} {F : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : NormedAddCommGroup F] [inst_7 : NormedSpace ℝ F] [inst_8 : MeasurableSpace F] [inst_9 : BorelSpace F] (Tn : ℕ → Ω → E) (t₀ : E) (g : E → F) (Dg : E →L[ℝ] F) (Q : MeasureTheory.ProbabilityMeasure E) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (g (Tn n ω) - g t₀)) μ), HasFDerivAt g Dg t₀ → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (Tn n ω - t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hTn n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (g (Tn n ω) - g t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hgTn n)⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map ⇑Dg ↑Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable (ContinuousLinearMap.continuous Dg)))⟩)","l":263,"f":"Causalean/Stat/Inference/DeltaMethod.lean"},{"n":"Causalean.Stat.deltaMethod_max_tie","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":1,"d":"Directional delta method for max at a tie. Let ân, b̂n be two real-valued estimator sequences of a common value a. Suppose the joint rescaled deviation √n • ((ân, b̂n) − (a, a)) is measurable at every sample size and it converges in distribution to a probability measure Q on ℝ × ℝ. Then the rescaled deviation of the pointwise maximum, √n · (max(ân, b̂n) − a), converges in distribution to the pushforward of Q under the map (x, y) ↦ max(x, y).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (an bn : ℕ → Ω → ℝ) (a : ℝ) (Q : MeasureTheory.Measure (ℝ × ℝ)) [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hSn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n • ((an n ω, bn n ω) - (a, a))) Q μ hSn_meas → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (max (an n ω) (bn n ω) - a)) (MeasureTheory.Measure.map (fun z => max z.1 z.2) Q) μ (Causalean.Stat.maxStat_aemeasurable an bn a hSn_meas)","l":165,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.deltaMethod_min_tie","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":1,"d":"Directional delta method for min at a tie. Let ân, b̂n be two real-valued estimator sequences of a common value a. Suppose the joint rescaled deviation √n • ((ân, b̂n) − (a, a)) is measurable at every sample size and it converges in distribution to a probability measure Q on ℝ × ℝ. Then the rescaled deviation of the pointwise minimum, √n · (min(ân, b̂n) − a), converges in distribution to the pushforward of Q under the map (x, y) ↦ min(x, y).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (an bn : ℕ → Ω → ℝ) (a : ℝ) (Q : MeasureTheory.Measure (ℝ × ℝ)) [inst_2 : MeasureTheory.IsProbabilityMeasure Q] (hSn_meas : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ), Causalean.Stat.Tendsto_dist_vec (fun n ω => √↑n • ((an n ω, bn n ω) - (a, a))) Q μ hSn_meas → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (min (an n ω) (bn n ω) - a)) (MeasureTheory.Measure.map (fun z => min z.1 z.2) Q) μ (Causalean.Stat.minStat_aemeasurable an bn a hSn_meas)","l":202,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.deltaMethod_ratio","k":"theorem","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":1,"d":"Ratio / quotient delta method. Let t₀ = (a, b) with b nonzero and let Tn n ω = (N̂ₙ, D̂ₙ) be a bivariate estimator sequence of t₀. Given that the rescaled deviation √n • (Tn − t₀) is measurable at every sample size and that the rescaled ratio √n • (Tn 0 / Tn 1 − a/b) is measurable at every sample size, if the laws of the rescaled deviation converge weakly to a probability measure Q on EuclideanSpace ℝ (Fin 2), then the laws of the rescaled ratio √n • (Tn 0 / Tn 1 − a/b) converge weakly to the pushforward of Q along the ratio derivative ratioDeriv t₀.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (Tn : ℕ → Ω → EuclideanSpace ℝ (Fin 2)) (t₀ : EuclideanSpace ℝ (Fin 2)), t₀.ofLp 1 ≠ 0 → ∀ (Q : MeasureTheory.ProbabilityMeasure (EuclideanSpace ℝ (Fin 2))) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((Tn n ω).ofLp 0 / (Tn n ω).ofLp 1 - t₀.ofLp 0 / t₀.ofLp 1)) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • (Tn n ω - t₀)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hTn n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (fun ω => √↑n • ((Tn n ω).ofLp 0 / (Tn n ω).ofLp 1 - t₀.ofLp 0 / t₀.ofLp 1)) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hgTn n)⟩) Filter.atTop (nhds ⟨MeasureTheory.Measure.map ⇑(Causalean.Stat.ratioDeriv t₀) ↑Q, MeasureTheory.Measure.isProbabilityMeasure_map (Measurable.aemeasurable (Continuous.measurable (ContinuousLinearMap.continuous (Causalean.Stat.ratioDeriv t₀))))⟩)","l":103,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.deltaMethod_scalar","k":"theorem","m":"Causalean.Stat.Inference.DeltaMethod","a":"Stat/Inference","t":1,"d":"Scalar delta method. For a real-valued estimator sequence Tn of a target t₀, suppose the rescaled deviation √n(Tn − t₀) is measurable at every sample size and the rescaled image √n(g(Tn) − g(t₀)) is measurable at every sample size, where g is differentiable at t₀ with derivative g'. If the rescaled deviation converges in distribution to the project's Gaussian law with mean zero and variance σ², then the rescaled image converges in distribution to the same Gaussian wrapper with variance g'² · σ².","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] (Tn : ℕ → Ω → ℝ) (t₀ : ℝ) (g : ℝ → ℝ) (g' σsq : ℝ) (hTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (Tn n ω - t₀)) μ) (hgTn : ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (g (Tn n ω) - g t₀)) μ), HasDerivAt g g' t₀ → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (Tn n ω - t₀)) (Causalean.Stat.gaussianMeasure 0 σsq) μ hTn → Causalean.Stat.Tendsto_dist (fun n ω => √↑n * (g (Tn n ω) - g t₀)) (Causalean.Stat.gaussianMeasure 0 (g' ^ 2 * σsq)) μ hgTn","l":50,"f":"Causalean/Stat/Inference/DeltaMethod.lean"},{"n":"Causalean.Stat.densityAffinity","k":"def","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"Given a measurable sample space, a measure on that space, and two real-valued functions on it, the Hellinger--Bhattacharyya affinity of the functions relative to the measure is the integral of the square root of their pointwise product.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → (α → ℝ) → (α → ℝ) → ℝ","l":54,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.densityAffinity_pi","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"Affinity tensorizes. On a finite product of σ-finite measure spaces, the affinity of two densities that each factor coordinatewise is the product of the coordinate affinities. This is what makes an n-fold product construction tractable: a single coordinate defect computation is enough.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {E : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (E i)] (μ : (i : ι) → MeasureTheory.Measure (E i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (f g : (i : ι) → E i → ℝ), (∀ (i : ι) (u : E i), 0 ≤ f i u) → (∀ (i : ι) (u : E i), 0 ≤ g i u) → (Causalean.Stat.densityAffinity (MeasureTheory.Measure.pi μ) (fun x => ∏ i, f i (x i)) fun x => ∏ i, g i (x i)) = ∏ i, ∫ (u : E i), √(f i u * g i u) ∂μ i","l":327,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.deterministicBayesRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a measurable parameter space, a finite measurable state space, an action space, a prior measure on the parameter space, a map assigning each parameter a state, and a real-valued loss for each action and state, the deterministic Bayes risk is the infimum, over all actions, of the prior-integrated loss evaluated at the state assigned to each parameter.","s":"{Θ : Type u_1} → {S : Type u_2} → {A : Type u_3} → [inst : MeasurableSpace Θ] → MeasureTheory.Measure Θ → (Θ → S) → (A → S → ℝ) → ℝ","l":217,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.deterministicBayesRisk_eq_realBayesRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A measurable deterministic state map gives the same Bayes risk whether viewed directly or as its deterministic Markov kernel.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (f : Θ → S) (hf : Measurable f) (loss : A → S → ℝ), Causalean.Stat.deterministicBayesRisk π f loss = Causalean.Stat.realBayesRisk π (ProbabilityTheory.Kernel.deterministic f hf) loss","l":225,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.deterministicBayesRisk_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"With at least one available action, a measurable deterministic state map, and nonnegative statewise loss, the deterministic Bayes risk is at most the finite-state minimax value.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] [Nonempty A] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (f : Θ → S), Measurable f → ∀ (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → Causalean.Stat.deterministicBayesRisk π f loss ≤ Causalean.Stat.minimaxValue loss","l":233,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.discrepancyAttainedOn_of_compactSublevel","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given a covariance model with a nonempty compact discrepancy sublevel, the global Gaussian discrepancy minimum is attained.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)} {T : Matrix V V ℝ}, Causalean.Stat.HasCompactDiscrepancySublevel M T → Causalean.Stat.IsDiscrepancyAttainedOn M T","l":287,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.discrepancyAttainedOn_of_isCompact","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given a compact covariance model, a nonempty model, and a target matrix, the Gaussian discrepancy minimum is attained.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)}, IsCompact M → M.Nonempty → ∀ (T : Matrix V V ℝ), Causalean.Stat.IsDiscrepancyAttainedOn M T","l":273,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.disintegrate_sum","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Every real test function has the same joint expectation as the marginally weighted guarded posterior expectation.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (f : S → X → ℝ), ∑ s, ∑ x, Causalean.Stat.jointMass ν L s x * f s x = ∑ x, Causalean.Stat.observationMass ν L x * ∑ s, Causalean.Stat.posteriorWeight ν L x s * f s x","l":158,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.empProcVec_atEstimator_tendsto_zero","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Empirical process at the estimator vanishes. If the score family ψ is asymptotically equicontinuous at θ₀ along the i.i.d. sample S and θn is a sequence of estimators consistent for θ₀, then for any fixed tolerance ε > 0, the centered empirical process of the score gap ψ(θn,·) − ψ(θ₀,·), evaluated at the random estimator θn, namely Gₙ(ψ(θn,·) − ψ(θ₀,·)), converges to 0 in probability — unconditionally, with no restriction to a shrinking neighborhood of θ₀.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (ψ : E → X → E) (θ₀ : E) (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε < ‖S.empProcVec (fun z => ψ (θn n ω) z - ψ θ₀ z) n ω‖}) Filter.atTop (nhds 0)","l":96,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.empProcVec_chebyshev","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Chebyshev bound for the vector empirical process. For an i.i.d. sample S and a function f that is measurable and square-integrable under P, and for any tolerance ε > 0, the probability that the centered empirical process Gₙ(f) exceeds ε in norm at sample size n is at most $(\\int\\|f\\|^2\\,dP)/\\varepsilon^2$:","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] [inst_7 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → MeasureTheory.MemLp f 2 P → ∀ (n : ℕ) {ε : ℝ}, 0 < ε → μ {ω | ε < ‖S.empProcVec f n ω‖} ≤ ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P) / ENNReal.ofReal (ε ^ 2)","l":217,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.empProcVec_isLittleOp_of_L2","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Deterministic-curve equicontinuity (non-vacuousness witness). Suppose that for every parameter value θ, the score gap ψ(θ,·) − ψ(θ₀,·) is measurable and square-integrable under P. Along a deterministic parameter sequence θn whose score perturbation shrinks in L²(P): $\\int\\|\\psi(\\theta_n,\\cdot)-\\psi(\\theta_0,\\cdot)\\|^2\\,dP \\to 0$, the centered empirical-process gap Gₙ(ψ(θn,·) − ψ(θ₀,·)) is $o_p(1)$, with no chaining and no consistency hypothesis: it is a direct consequence of the uniform-in-n Chebyshev bound empProcVec_chebyshev.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (ψ : E → X → E) (θ₀ : E) (θn : ℕ → E) (S : Causalean.Stat.IIDSample Ω X μ P), (∀ (θ : E), Measurable fun x => ψ θ x - ψ θ₀ x) → (∀ (θ : E), MeasureTheory.MemLp (fun x => ψ θ x - ψ θ₀ x) 2 P) → Filter.Tendsto (fun n => ∫ (x : X), ‖ψ (θn n) x - ψ θ₀ x‖ ^ 2 ∂P) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => ‖S.empProcVec (fun z => ψ (θn n) z - ψ θ₀ z) n ω‖) (fun x => 1) μ","l":182,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.empProcVec_sq_lintegral_le","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.SecondMoment","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Second-moment bound for the vector empirical process (variance identity at the empirical scale). For an i.i.d. sample S and a function f that is measurable and square-integrable under P, the second moment of the centered empirical process Gₙ(f) at sample size n is bounded by the population second moment of f:","s":"∀ {Ω : Type u_1} {X : Type u_2} {E : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] [inst_7 : MeasurableSpace E] [BorelSpace E] (S : Causalean.Stat.IIDSample Ω X μ P) (f : X → E), Measurable f → MeasureTheory.MemLp f 2 P → ∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (‖S.empProcVec f n ω‖ ^ 2) ∂μ ≤ ENNReal.ofReal (∫ (x : X), ‖f x‖ ^ 2 ∂P)","l":105,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/SecondMoment.lean"},{"n":"Causalean.Stat.empiricalCDF_isAsymLinear","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Asymptotic linearity of the empirical cdf. For an i.i.d. real sample S at a fixed point y, the empirical cdf F̂ₙ(y) is asymptotically linear at F(y) with influence function cdfIF P y and an identically-zero remainder.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ), Causalean.Stat.IsAsymLinear (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) (Causalean.Stat.cdfIF P y) S fun m => Finset.range m","l":222,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalCDF_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Empirical-cdf consistency (WLLN). For an i.i.d. real sample S at a fixed point y, the empirical cdf F̂ₙ(y) converges to F(y) in probability.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ), Causalean.Stat.Tendsto_inProb (S.empiricalCDF y) (fun x => ↑(ProbabilityTheory.cdf P) y) μ","l":260,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalCDF_tendsto_normal","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":1,"d":"Empirical-cdf central limit theorem. For an i.i.d. real sample and a threshold y, provided the rescaled empirical-cdf estimator $\\sqrt n\\,(\\hat F_n(y)-F(y))$ is almost-everywhere measurable at every sample size, the rescaled deviation converges in distribution, as $n\\to \\infty$, to the centered Gaussian law with variance $F(y)(1-F(y))$, where $F$ is the population cdf and $\\hat F_n$ the empirical cdf.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ) (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (S.empiricalCDF y) (↑(ProbabilityTheory.cdf P) y) fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (↑(ProbabilityTheory.cdf P) y * (1 - ↑(ProbabilityTheory.cdf P) y))) μ hθn_meas","l":239,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.empiricalKishDispersion","k":"def","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Given an observation space, a real-valued weight function on that space, a nonnegative sample size, and a sample indexed by the integers from zero through one less than that size, the empirical Kish dispersion is the reciprocal of the sample size multiplied by the sum of the squared weights of the sampled observations.","s":"{Ω : Type u_1} → (Ω → ℝ) → (n : ℕ) → (Fin n → Ω) → ℝ","l":26,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_lower_tail_le","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Lower-tail bound for empirical Kish dispersion. Given a positive sample size $n$, a positive population Kish dispersion $\\kappa$, the empirical Kish dispersion is square-integrable under the sampling measure Q, its expectation under Q equals $\\kappa$, and its variance under Q is at most $4k^2\\kappa/n$ for a weight envelope $k$, then the probability that the empirical Kish dispersion falls below half its mean $\\kappa/2$ is at most $16k^2/(n\\kappa)$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (n : ℕ) (Q : MeasureTheory.Measure (Fin n → Ω)) [MeasureTheory.IsProbabilityMeasure Q] (g : Ω → ℝ) (k kappa : ℝ), 0 < n → 0 < kappa → MeasureTheory.MemLp (Causalean.Stat.empiricalKishDispersion g n) 2 Q → ∫ (sample : Fin n → Ω), Causalean.Stat.empiricalKishDispersion g n sample ∂Q = kappa → ProbabilityTheory.variance (Causalean.Stat.empiricalKishDispersion g n) Q ≤ 4 * k ^ 2 * kappa / ↑n → (Q {sample | Causalean.Stat.empiricalKishDispersion g n sample < kappa / 2}).toReal ≤ 16 * k ^ 2 / (↑n * kappa)","l":110,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_mean","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":1,"d":"Expected empirical Kish dispersion. Given a positive sample size $n$ and an integrable squared weight statistic $g^2$ under the population measure, the expectation of the empirical Kish dispersion — the sample average of the squared observation-level weights — under the $n$-fold product sampling measure equals the population second moment $\\int g^2\\,d\\mu$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ) (n : ℕ), 0 < n → MeasureTheory.Integrable (fun o => g o ^ 2) μ → (∫ (sample : Fin n → Ω), Causalean.Stat.empiricalKishDispersion g n sample ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (o : Ω), g o ^ 2 ∂μ","l":45,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalKishDispersion_variance_le","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":2,"d":"If the fourth power of a weight is bounded by four times a squared envelope times its second power, empirical Kish dispersion has variance at most four times the squared envelope and the population second moment, divided by sample size.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (g : Ω → ℝ) (n : ℕ) (k kappa : ℝ), 0 < n → MeasureTheory.MemLp (fun o => g o ^ 2) 2 μ → ∫ (o : Ω), g o ^ 2 ∂μ = kappa → (∀ᵐ (o : Ω) ∂μ, g o ^ 4 ≤ 4 * k ^ 2 * g o ^ 2) → ProbabilityTheory.variance (fun sample => Causalean.Stat.empiricalKishDispersion g n sample) (MeasureTheory.Measure.pi fun x => μ) ≤ 4 * k ^ 2 * kappa / ↑n","l":62,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.empiricalMass","k":"def","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"Given an observation space, a nonnegative sample size, a sample indexed by the integers from zero through one less than that size, and a point in the observation space, the empirical mass of that point is the reciprocal of the sample size multiplied by the number of sampled observations equal to that point.","s":"{𝒳 : Type u_1} → {N : ℕ} → (Fin N → 𝒳) → 𝒳 → ℝ","l":24,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.empiricalScoreDiff_isLittleOp_sqrt","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(b) Empirical-process score difference is o_p(√n).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (_reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsLittleOp (fun n ω => ‖(√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • ∫ (z : X), ψ (θn n ω) z - ψ θ₀ z ∂P‖) (fun x => 1) μ","l":348,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.emptyMatchingNormalization_sub_one_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, and common order bound R, if the first order is at most R, the second order is at most R, and the sample size is at least four times R squared, the empty partial matching's normalization differs from one by at most twice R squared divided by n.","s":"∀ {n r s R : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → |Causalean.Stat.matchingNormalization n (Causalean.Stat.PartialMatching.empty r s) - 1| ≤ 2 * ↑R ^ 2 / ↑n","l":280,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.ennreal_error_lower_bound_of_success_upper_bound","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A real-valued product bound below one converts an ENNReal simultaneous success upper bound into the complementary error lower bound.","s":"∀ {s : ENNReal} {p e : ℝ}, 0 ≤ p → e ≤ 1 → p ≤ e → s ≤ ENNReal.ofReal (1 / 2 * (1 + p)) → ENNReal.ofReal (1 / 2 * (1 - e)) ≤ 1 - s","l":391,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.equivFintype","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For two finite sets, the collection of bijections between them is itself a finite collection.","s":"{α : Type u_1} → {β : Type u_2} → [Fintype α] → [Fintype β] → Fintype (α ≃ β)","l":126,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.eucl","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"For a nonnegative integer $k$ specifying a finite collection of coordinates and a real-valued vector indexed by those coordinates, the Euclidean representation of that vector is the corresponding point of the $k$-dimensional Euclidean space.","s":"{k : ℕ} → (Fin k → ℝ) → EuclideanSpace ℝ (Fin k)","l":56,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.eucl_apply","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Coordinate access for eucl: (eucl v) j = v j.","s":"∀ {k : ℕ} (v : Fin k → ℝ) (j : Fin k), (Causalean.Stat.eucl v).ofLp j = v j","l":140,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.event_pullback_along_iidSample","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"Event transport along an IID sample (Fin n version). Given a measurable event E in the space of length-n outcome tuples whose product-measure probability under n independent copies of the population law is at least $1-\\delta$, the pullback of E along the joint observable built from the first n coordinates of the i.i.d. sample is a measurable event on the underlying sample space, with probability at least $1-\\delta$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ) {E : Set (Fin n → X)}, MeasurableSet E → ∀ {δ : ℝ}, (MeasureTheory.Measure.pi fun x => P) E ≥ 1 - ENNReal.ofReal δ → have Ψ := fun ω k => S.Z (↑k) ω; have E' := Ψ ⁻¹' E; MeasurableSet E' ∧ μ E' ≥ 1 - ENNReal.ofReal δ","l":91,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.exists_grid_bracket","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileBahadur.Oscillation","a":"Stat/Quantile/SampleQuantileBahadur","t":2,"d":"Uniform grid bracketing. For the uniform mesh u_k = −M + 2M·k/K on [−M, M] (K ≥ 1, M > 0), any x ∈ [−M, M] lies in some cell [u_j, u_{j+1}] with j < K. Pure real-arithmetic fact (floor of the scaled coordinate, clamped to the last cell).","s":"∀ {M : ℝ}, 0 < M → ∀ {K : ℕ}, 1 ≤ K → ∀ {x : ℝ}, -M ≤ x → x ≤ M → ∃ j < K, -M + 2 * M * ↑j / ↑K ≤ x ∧ x ≤ -M + 2 * M * (↑j + 1) / ↑K","l":470,"f":"Causalean/Stat/Quantile/SampleQuantileBahadur/Oscillation.lean"},{"n":"Causalean.Stat.exists_positive_gap_of_attained","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given an attained covariance-model discrepancy that omits the positive-definite truth, every model covariance has a common strictly positive normalized gap.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)} {T : Causalean.Stat.PositiveCovariance V}, Causalean.Stat.IsDiscrepancyAttainedOn M ↑T → T ∉ M → ∃ gap, 0 < gap ∧ ∀ K ∈ M, gap ≤ Causalean.Stat.normalizedCovarianceDiscrepancy K T","l":304,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.exists_positive_gap_of_isCompact","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given a compact covariance model, a nonempty model, and a truth outside the model, the model is separated from the truth by a strictly positive population gap.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)}, IsCompact M → M.Nonempty → ∀ {T : Causalean.Stat.PositiveCovariance V}, T ∉ M → ∃ gap, 0 < gap ∧ ∀ K ∈ M, gap ≤ Causalean.Stat.normalizedCovarianceDiscrepancy K T","l":321,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.exists_real_ge_mixture","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"Witness extraction. Some component carries at least the mixture's .real-mass: since the mixture is a weighted average (weights summing to 1), its mass on A is at most the maximal component mass, attained over the finite index.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] [Nonempty ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] (A : Set Ω), ∃ i, (Causalean.Stat.mixture w P).real A ≤ (P i).real A","l":87,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.exists_stable_positive_gap","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given a compact covariance model, a nonempty model, and a truth outside the model, the strict positive population gap persists for nearby positive-definite truths.","s":"∀ {V : Type u_2} [inst : Fintype V] [inst_1 : DecidableEq V] {M : Set (Causalean.Stat.PositiveCovariance V)}, IsCompact M → M.Nonempty → ∀ {T : Causalean.Stat.PositiveCovariance V}, T ∉ M → ∃ gap, 0 < gap ∧ ∃ U ∈ nhds T, ∀ T' ∈ U, ∀ K ∈ M, gap ≤ Causalean.Stat.normalizedCovarianceDiscrepancy K T'","l":332,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.expCellUpdatedMean","k":"def","m":"Causalean.Stat.MEstimation.FinitePoissonDerivative","a":"Stat/MEstimation","t":1,"d":"Given a finite index set with decidable equality, a real-valued vector of cell means, one cell index, a real scale, and a real path coordinate, the exponentially updated cell-mean vector agrees with the original vector at every cell other than the selected one and assigns the selected cell the value $B\\exp(x)$.","s":"{I : Type u_1} → [DecidableEq I] → (I → ℝ) → I → ℝ → ℝ → I → ℝ","l":24,"f":"Causalean/Stat/MEstimation/FinitePoissonDerivative.lean"},{"n":"Causalean.Stat.expQuadraticRemainder","k":"def","m":"Causalean.Stat.CLT.MartingaleArray.ExponentialBounds","a":"Stat/CLT/MartingaleArray","t":1,"d":"The quadratic Taylor remainder of the characteristic-function kernel at a real argument u is exp(iu) - 1 - iu + u²/2.","s":"ℝ → ℂ","l":20,"f":"Causalean/Stat/CLT/MartingaleArray/ExponentialBounds.lean"},{"n":"Causalean.Stat.exp_neg_half_klDiv_le_bhattacharyya","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Jensen / Bhattacharyya step. exp(-½·KL) ≤ ∫ √p ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → Real.exp (-(1 / 2) * (InformationTheory.klDiv μ ν).toReal) ≤ ∫ (x : Ω), √(μ.rnDeriv ν x).toReal ∂ν","l":304,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.expectedRestrictedVolume_affineInversion_frontier_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":1,"d":"Affine-inversion frontier bound over a finite-volume region. Fix a probability space (Ω, Q), real-valued functions A, B, K on it, and a region of the line with finite Lebesgue measure. Suppose a nonnegative radius scale L, a positive centering level mu and a positive sample size n; suppose K is pointwise nonnegative and Q-integrable with Q-mean at most Kbar, and that the Q-probability that B deviates from mu by more than mu/2 is at most q. Suppose further a positive scale kappa, a nonnegative inflation factor inflation and nonnegative slack Y, with Kbar controlled by inflation · kappa and the bad-event contribution (vol region) · q controlled by Y / t, where t is defined as n · mu² / kappa. Then the Q-expected restricted volume of the affine-inversion set built from A, B and the shrinking radius L · √(K/n) is at most max(vol region, 4·√inflation·L + Y) · min(1, t^(-1/2)).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Q] (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B K : Ω → ℝ) (n : ℕ) (L mu Kbar q kappa inflation Y t : ℝ), 0 ≤ L → 0 < mu → 0 < n → (∀ (w : Ω), 0 ≤ K w) → MeasureTheory.Integrable K Q → ∫ (w : Ω), K w ∂Q ≤ Kbar → (Q {w | mu / 2 < |B w - mu|}).toReal ≤ q → 0 < kappa → 0 ≤ inflation → 0 ≤ Y → Kbar ≤ inflation * kappa → (MeasureTheory.volume region).toReal * q ≤ Y / t → t = ↑n * mu ^ 2 / kappa → ∫ (w : Ω), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region (A w) (B w) (L * √(K w / ↑n))) ∂Q ≤ max (MeasureTheory.volume region).toReal (4 * √inflation * L + Y) * min 1 (t ^ (-1 / 2))","l":295,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.expectedRestrictedVolume_affineInversion_le","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"Expected restricted volume for affine inversion is controlled by the mean radius and by the probability that the random slope is less than half its positive target value.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Q] (region : Set ℝ), MeasureTheory.volume region ≠ ⊤ → ∀ (A B K : Ω → ℝ) (n : ℕ) (L mu Kbar q : ℝ), 0 ≤ L → 0 < mu → 0 < n → (∀ (w : Ω), 0 ≤ K w) → MeasureTheory.Integrable K Q → ∫ (w : Ω), K w ∂Q ≤ Kbar → (Q {w | mu / 2 < |B w - mu|}).toReal ≤ q → ∫ (w : Ω), Causalean.Stat.restrictedSetVolume region (Causalean.Stat.affineInversionSet region (A w) (B w) (L * √(K w / ↑n))) ∂Q ≤ 4 * L * √(Kbar / ↑n) / mu + (MeasureTheory.volume region).toReal * q","l":90,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.expected_restrictedSetVolume_eq_integral_inclusion","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"The expected restricted volume of a jointly measurable random set equals the integral, over the parameter region, of its pointwise inclusion probabilities.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (Q : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure Q] (C : Ω → Set ℝ) (region : Set ℝ), MeasurableSet {p | p.2 ∈ C p.1} → MeasurableSet region → MeasureTheory.volume region ≠ ⊤ → ∫ (ω : Ω), Causalean.Stat.restrictedSetVolume region (C ω) ∂Q = ∫ (u : ℝ) in region, (Q {ω | u ∈ C ω}).toReal","l":34,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.factorialDisjointCorrection_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, and common order bound R, if the first order is at most R, the second order is at most R, and the sample size is at least four times R squared, the absolute disjoint normalization correction is at most twice R squared divided by n.","s":"∀ {n r s R : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → |↑(n.descFactorial (r + s)) / (↑(n.descFactorial r) * ↑(n.descFactorial s)) - 1| ≤ 2 * ↑R ^ 2 / ↑n","l":249,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.factorialMatchingRatio_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":1,"d":"For sample size n, orders r and s, common order bound R, and overlap size h, if the first order is at most R, the second order is at most R, the sample size is at least four times R squared, and the overlap fits inside both orders, the mixed falling-factorial ratio is at most exp(1) divided by n to the overlap size.","s":"∀ {n r s R h : ℕ}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → h ≤ min r s → ↑(n.descFactorial (r + s - h)) / (↑(n.descFactorial r) * ↑(n.descFactorial s)) ≤ Real.exp 1 / ↑n ^ h","l":217,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.fano_average_error","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"Fano average-error lower bound. For a measurable estimator est and a family of parameter values that are pairwise 2s-separated, the average probability of error over the N = card ι hypotheses is at least 1 − (1 + ∑ᵢ tvDist (P i₀) (P i)) / N.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), 1 - (1 + ∑ i, Causalean.Stat.tvDist (P i₀) (P i)) / ↑(Fintype.card ι) ≤ (∑ i, (P i).real {ω | s ≤ dist (est ω) (θ i)}) / ↑(Fintype.card ι)","l":159,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.fano_exists_error","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":1,"d":"Fano existence-of-bad-hypothesis bound (uniform β). For a measurable estimator est and a family of parameter values that are pairwise 2s-separated, if every hypothesis's law is within total variation β of the reference P i₀, then some hypothesis has error probability at least 1 − 1/N − β. This is the directly usable minimax statement: choosing the number of hypotheses N large and the divergence β small forces error.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι) {β : ℝ}, (∀ (i : ι), Causalean.Stat.tvDist (P i₀) (P i) ≤ β) → ∃ i, 1 - 1 / ↑(Fintype.card ι) - β ≤ (P i).real {ω | s ≤ dist (est ω) (θ i)}","l":188,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.fiber_integral_pi","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fibre integral. For a coupling π of two probability measures, the π-integral of the product of the two signed tail indicators at thresholds s and t equals the joint survival function of π at (s, t), adjusted by cross terms built from the two marginal survival functions and the sign indicators of s and t.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 s * Causalean.Stat.signedTail p.2 t ∂π = Causalean.Stat.jointSurv π s t - Causalean.Stat.tailInd 0 t * Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s * Causalean.Stat.survSnd π t + Causalean.Stat.tailInd 0 s * Causalean.Stat.tailInd 0 t","l":135,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.fiber_sub_mean_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Pointwise cancellation: the fibre integral minus the product of the centred marginal survival functions is exactly the Fréchet gap H_π - F·G.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (q : ℝ × ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π - (Causalean.Stat.survFst π q.1 - Causalean.Stat.tailInd 0 q.1) * (Causalean.Stat.survSnd π q.2 - Causalean.Stat.tailInd 0 q.2) = Causalean.Stat.jointCdf π q.1 q.2 - ↑(ProbabilityTheory.cdf μ) q.1 * ↑(ProbabilityTheory.cdf ν) q.2","l":45,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.finProductKernel","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"For an input measurable space, an output measurable space, a nonnegative integer number of coordinates, and a Markov kernel from the input space to the output space, the finite product kernel maps an input vector to the unique point-mass kernel at zero coordinates, and otherwise to independent kernel outputs coordinate by coordinate.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → (n : ℕ) → ProbabilityTheory.Kernel X Y → ProbabilityTheory.Kernel (Fin n → X) (Fin n → Y)","l":258,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finProductKernel_apply","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Each fibre of the finite product kernel equals the independent product of its coordinate output laws, including the unique empty product fibre.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (x : Fin n → X), (Causalean.Stat.finProductKernel n K) x = MeasureTheory.Measure.pi fun i => K (x i)","l":272,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finProductKernel_comp_pi","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Passing an independent finite product law through the coordinatewise product kernel produces the product of the one-coordinate garbled law, including when there are no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K], (MeasureTheory.Measure.pi fun x => P).bind ⇑(Causalean.Stat.finProductKernel n K) = MeasureTheory.Measure.pi fun x => P.bind ⇑K","l":300,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.finiteCategoryPilotGood","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given a measurable sample space and observation space, a finite category-label space whose labels can be compared for equality, a sample-space measure and population observation measure, an independent and identically distributed sample, a category-label map, a finite pilot-coordinate block, and real threshold, lower-band, and upper-band values, the finite-category pilot good event consists exactly of outcomes for which each selected category has population mass at least the lower band and each unselected category has population mass at most the upper band.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → [Fintype Iota] → [DecidableEq Iota] → Causalean.Stat.IIDSample Omega X mu P → (X → Iota) → Finset ℕ → ℝ → ℝ → ℝ → Set Omega","l":338,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteCategoryPilot_bad_probability","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given an iid sample, a measurable category label, a finite pilot block, a positive selection threshold, a lower population-mass band, an upper population-mass band, an upper-tail exponential tilt, a lower-tail exponential tilt, nonnegativity of the upper-tail tilt, and nonpositivity of the lower-tail tilt, failure of the simultaneous category-mass sandwich has probability at most the number of categories times the sum of the two explicit Chernoff tails. The bound also covers empty blocks and empty category types.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) {t : ℝ}, 0 < t → ∀ (lowerBand upperBand sUpper sLower : ℝ), 0 ≤ sUpper → sLower ≤ 0 → mu.real (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)ᶜ ≤ ↑(Fintype.card Iota) * (Real.exp (-sUpper * t + ↑block.card * (lowerBand * (Real.exp sUpper - 1))) + Real.exp (-sLower * t + ↑block.card * (upperBand * (Real.exp sLower - 1))))","l":388,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteCategoryPilot_bad_probability_log_two","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a positive selection threshold, a lower population-mass band, and an upper population-mass band, the simultaneous category-mass sandwich failure probability obeys the explicit bound obtained from opposite logarithmic tilts of magnitude log two.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) {t : ℝ}, 0 < t → ∀ (lowerBand upperBand : ℝ), mu.real (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)ᶜ ≤ ↑(Fintype.card Iota) * (Real.exp (-Real.log 2 * t + ↑block.card * lowerBand) + Real.exp (Real.log 2 * t - ↑block.card * upperBand / 2))","l":487,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.finiteCovarianceModel_populationGap","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"Given a finite family of covariance models whose members are compact and nonempty, and a positive-definite true covariance contained in at least one model, the population losses have the positive gap required by finite penalized model selection.","s":"∀ {V : Type u_2} {I : Type u_3} [inst : Fintype V] [inst_1 : DecidableEq V] [Fintype I] [Nonempty I] (models : I → Set (Causalean.Stat.PositiveCovariance V)), (∀ (i : I), IsCompact (models i)) → (∀ (i : I), (models i).Nonempty) → ∀ (T : Causalean.Stat.PositiveCovariance V), (∃ i, T ∈ models i) → ∃ gap, 0 < gap ∧ ∀ i ∈ Causalean.Stat.populationMinimizers fun j => Causalean.Stat.covarianceModelLoss (models j) ↑T, ∀ j ∉ Causalean.Stat.populationMinimizers fun k => Causalean.Stat.covarianceModelLoss (models k) ↑T, Causalean.Stat.covarianceModelLoss (models i) ↑T + gap ≤ Causalean.Stat.covarianceModelLoss (models j) ↑T","l":456,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.finiteDesignBayesRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a finite state space, an action space, a finite randomization design on the state space, and a real-valued loss for each action and state, the finite-design Bayes risk is the infimum, over all actions, of their expected losses under that design.","s":"{S : Type u_2} → {A : Type u_3} → [inst : Fintype S] → Causalean.Experimentation.DesignBased.FiniteDesign S → (A → S → ℝ) → ℝ","l":276,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finiteDesignBayesRisk_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"With at least one available action and nonnegative statewise loss, a finite design's Bayes risk is at most the finite-state minimax value.","s":"∀ {S : Type u_2} {A : Type u_3} [inst : Fintype S] [Nonempty S] [inst_2 : MeasurableSpace S] [MeasurableSingletonClass S] [Nonempty A] (D : Causalean.Experimentation.DesignBased.FiniteDesign S) (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → Causalean.Stat.finiteDesignBayesRisk D loss ≤ Causalean.Stat.minimaxValue loss","l":285,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finiteDesignBayesRisk_statewiseSquaredLoss","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"The real-valued finite-design Bayes-risk infimum for observation-dependent squared loss is exactly the posterior residual risk.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ), Causalean.Stat.finiteDesignBayesRisk ν (Causalean.Stat.statewiseSquaredLoss L t) = Causalean.Stat.posteriorResidual ν L t","l":326,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.finiteDesign_expectedLoss_eq_sum_eq_integral","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A finite design's expected action loss equals both its finite weighted sum and its induced probability-measure integral.","s":"∀ {S : Type u_2} {A : Type u_3} [inst : Fintype S] [Nonempty S] [inst_2 : MeasurableSpace S] [MeasurableSingletonClass S] (D : Causalean.Experimentation.DesignBased.FiniteDesign S) (loss : A → S → ℝ) (a : A), D.E (loss a) = ∑ s, D.p s * loss a s ∧ D.E (loss a) = ∫ (s : S), loss a s ∂D.toMeasure","l":251,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finiteDesign_expectedLoss_le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"If the statewise loss is nonnegative, then a finite design's expected loss for each action is no greater than that action's finite worst-case risk.","s":"∀ {S : Type u_2} {A : Type u_3} [inst : Fintype S] [Nonempty S] [inst_2 : MeasurableSpace S] [MeasurableSingletonClass S] (D : Causalean.Experimentation.DesignBased.FiniteDesign S) (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → ∀ (a : A), D.E (loss a) ≤ Causalean.Stat.worstCaseRisk loss a","l":260,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finiteDesign_expectedLoss_statewiseSquaredLoss","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Finite-design expected statewise squared loss is the joint-law squared risk.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (T : X → ℝ), ν.E (Causalean.Stat.statewiseSquaredLoss L t T) = Causalean.Stat.squaredRisk ν L t T","l":319,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.finiteDesign_integrable","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A real-valued statistic on a finite design space is integrable under the design's induced probability measure.","s":"∀ {S : Type u_2} [inst : Fintype S] [Nonempty S] [inst_2 : MeasurableSpace S] [MeasurableSingletonClass S] (D : Causalean.Experimentation.DesignBased.FiniteDesign S) (g : S → ℝ), MeasureTheory.Integrable g D.toMeasure","l":244,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finiteInjectiveTuples","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For a finite coordinate family and a sample size, the collection of injective sample assignments consists of all assignments sending distinct coordinates to distinct positions among the first $n$ observations.","s":"(ι : Type u_3) → [Fintype ι] → (n : ℕ) → Finset (ι → Fin n)","l":425,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.finiteInjectiveTuples_card","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For a finite coordinate family and sample size n, the number of injective sample assignments is the falling factorial of n with length equal to the number of coordinates.","s":"∀ (ι : Type u_3) [inst : Fintype ι] (n : ℕ), (Causalean.Stat.finiteInjectiveTuples ι n).card = n.descFactorial (Fintype.card ι)","l":431,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.finiteMaxError","k":"def","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":1,"d":"For an indexed real vector and a deterministic target vector, the largest absolute coordinate error is the maximum of their coordinatewise absolute differences over a finite nonempty index type.","s":"{ι : Type u_2} → [Fintype ι] → [Nonempty ι] → (ι → ℝ) → (ι → ℝ) → ℝ","l":28,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.finitePoissonObjective","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"Given a finite index set, cell weights, cell means, a real-linear map from the parameter space to cell predictors, and a parameter value, the finite Poisson objective is the sum over cells of each weight times its mean times its predictor minus its predictor's exponential.","s":"{E : Type u_1} → {I : Type u_2} → [inst : AddCommGroup E] → [inst_1 : Module ℝ E] → [Fintype I] → (I → ℝ) → (I → ℝ) → (E →ₗ[ℝ] I → ℝ) → E → ℝ","l":82,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_argmax_continuousAt_mean","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":1,"d":"Continuity of the finite Poisson maximizer in the means. If every cell weight is strictly positive and the base mean vector m₀ has every entry strictly positive, and the linear design map is injective, then the selected maximizer of the finite Poisson criterion is continuous, as a function of the mean vector, at m₀.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] [Nonempty I] (q : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (m₀ : I → ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m₀ i) → Function.Injective ⇑A → ContinuousAt (fun m => Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q m A)) m₀","l":238,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_argmax_tendsto","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":1,"d":"Stability of finite Poisson maximizers under convergence of weights and means. Given a sequence of finite Poisson criteria with weights qN N and means mN N, suppose the limiting weights are strictly positive, the limiting means are strictly positive, the linear design map is injective, the weights qN N converge cellwise to the limiting weights q, and the means mN N converge cellwise to the limiting means m. If argmax N maximizes the N-th criterion, eventually in N, and limitArgmax is the unique global maximizer of the limiting criterion, then the maximizer sequence argmax converges to limitArgmax.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [SecondCountableTopology E] [inst_4 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (argmax : ℕ → E) (limitArgmax : E), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ (y : E), Causalean.Stat.finitePoissonObjective (qN N) (mN N) A y ≤ Causalean.Stat.finitePoissonObjective (qN N) (mN N) A (argmax N)) → Causalean.Stat.IsUniqueGlobalMax (Causalean.Stat.finitePoissonObjective q m A) limitArgmax → Filter.Tendsto argmax Filter.atTop (nhds limitArgmax)","l":195,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_continuous","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"The finite Poisson criterion is continuous in its parameter whenever the linear design acts on a finite-dimensional normed space.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ), Continuous (Causalean.Stat.finitePoissonObjective q m A)","l":91,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_eventually_common_compact","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":2,"d":"Convergent positive coefficients and a fixed injective design put any eventual sequence of global maximizers in one common compact ball.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (argmax : ℕ → E), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ (y : E), Causalean.Stat.finitePoissonObjective (qN N) (mN N) A y ≤ Causalean.Stat.finitePoissonObjective (qN N) (mN N) A (argmax N)) → ∃ K, IsCompact K ∧ ∀ᶠ (N : ℕ) in Filter.atTop, argmax N ∈ K","l":65,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finitePoissonObjective_exists_unique_max","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"Existence and uniqueness of the finite Poisson pseudo-true parameter. If every cell weight is strictly positive, every cell mean is strictly positive, and the linear design map is injective, then the finite Poisson criterion attains its supremum over the parameter space at a unique point.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] [Nonempty I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → ∃! x, ∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x","l":101,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_expCell_argmax_snd_hasDerivAt","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonDerivative","a":"Stat/MEstimation","t":1,"d":"Derivative of the finite Poisson maximizer under one exponentially perturbed cell mean. Suppose every cell weight is strictly positive, every base cell mean is strictly positive, the perturbation scale B is strictly positive, and the linear design map is injective; consider replacing cell j's mean by the exponential path x ↦ B · exp x. If betaDot is the unique value solving the score equation linearized at the maximizer for base point x₀, for every perturbation direction, then the second (scalar) coordinate of the maximizer, as a function of x, has derivative betaDot at x₀.","s":"∀ {U : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup U] [inst_1 : NormedSpace ℝ U] [FiniteDimensional ℝ U] [inst_3 : Fintype I] [inst_4 : DecidableEq I] [Nonempty I] (q m : I → ℝ) (A : U × ℝ →ₗ[ℝ] I → ℝ) (j : I) (B x₀ betaDot : ℝ), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → 0 < B → Function.Injective ⇑A → (∀ (v : U × ℝ), (∀ (d : U × ℝ), ∑ i, q i * A d i * ((if i = j then B * Real.exp x₀ else 0) - Real.exp (A (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q (Causalean.Stat.expCellUpdatedMean m j B x₀) A)) i) * A v i) = 0) → v.2 = betaDot) → HasDerivAt (fun x => (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q (Causalean.Stat.expCellUpdatedMean m j B x) A)).2) betaDot x₀","l":32,"f":"Causalean/Stat/MEstimation/FinitePoissonDerivative.lean"},{"n":"Causalean.Stat.finitePoissonObjective_isMax_of_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"For a nonnegative weighted Poisson objective, vanishing of every directional score is sufficient for global maximality.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (x : E), (∀ (i : I), 0 ≤ q i) → (∀ (d : E), ∑ i, q i * A d i * (m i - Real.exp (A x i)) = 0) → ∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x","l":303,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"Every directional score vanishes at a global maximizer.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : Fintype I] (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (x d : E), (∀ (y : E), Causalean.Stat.finitePoissonObjective q m A y ≤ Causalean.Stat.finitePoissonObjective q m A x) → ∑ i, q i * A d i * (m i - Real.exp (A x i)) = 0","l":266,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.finitePoissonObjective_snd_sign_of_nuisance_score","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonSign","a":"Stat/MEstimation","t":1,"d":"Sign of the selected scalar coefficient from the nuisance score. Suppose every cell weight is strictly positive, every cell mean is strictly positive, the linear design map is injective, and the conditional nuisance fit u₀, with the scalar coordinate held at zero, clears every nuisance-direction score. Then the selected scalar coefficient of the finite Poisson maximizer has exactly the sign of the remaining scalar score: it is negative, zero, or positive exactly when the scalar score is.","s":"∀ {U : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup U] [inst_1 : NormedSpace ℝ U] [FiniteDimensional ℝ U] [inst_3 : Fintype I] [Nonempty I] (q m : I → ℝ) (A : U × ℝ →ₗ[ℝ] I → ℝ) (u₀ : U), (∀ (i : I), 0 < q i) → (∀ (i : I), 0 < m i) → Function.Injective ⇑A → (∀ (u : U), ∑ i, q i * A (u, 0) i * (m i - Real.exp (A (u₀, 0) i)) = 0) → have beta := (Causalean.Stat.maximizerOrZero (Causalean.Stat.finitePoissonObjective q m A)).2; have scalarScore := ∑ i, q i * A (0, 1) i * (m i - Real.exp (A (u₀, 0) i)); (beta < 0 ↔ scalarScore < 0) ∧ (beta = 0 ↔ scalarScore = 0) ∧ (0 < beta ↔ 0 < scalarScore)","l":21,"f":"Causalean/Stat/MEstimation/FinitePoissonSign.lean"},{"n":"Causalean.Stat.finitePoissonObjective_tendstoUniformlyOn","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoissonConsistency","a":"Stat/MEstimation","t":2,"d":"Pointwise convergence of the finitely many weights and means gives uniform convergence of the finite Poisson objective on every compact parameter set.","s":"∀ {E : Type u_1} {I : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [inst_3 : Fintype I] (qN mN : ℕ → I → ℝ) (q m : I → ℝ) (A : E →ₗ[ℝ] I → ℝ) (K : Set E), IsCompact K → (∀ (i : I), Filter.Tendsto (fun N => qN N i) Filter.atTop (nhds (q i))) → (∀ (i : I), Filter.Tendsto (fun N => mN N i) Filter.atTop (nhds (m i))) → TendstoUniformlyOn (fun N => Causalean.Stat.finitePoissonObjective (qN N) (mN N) A) (Causalean.Stat.finitePoissonObjective q m A) Filter.atTop K","l":26,"f":"Causalean/Stat/MEstimation/FinitePoissonConsistency.lean"},{"n":"Causalean.Stat.finite_integrable","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A real-valued statistic on a finite state space is integrable under every finite measure.","s":"∀ {S : Type u_2} [Fintype S] [Nonempty S] [inst : MeasurableSpace S] [MeasurableSingletonClass S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsFiniteMeasure μ] (f : S → ℝ), MeasureTheory.Integrable f μ","l":37,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finite_measure_halfline_tails_small","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":1,"d":"Finite-measure half-line tail control. For a finite measure Q on the real line and a positive tolerance ε, there is a positive cutoff R such that both the lower half-line tail $Q((-\\infty,-R])$ and the upper half-line tail $Q([R,\\infty))$ are at most ε.","s":"∀ (Q : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasure Q] {ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ Q (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ Q (Set.Ici R) ≤ ENNReal.ofReal ε","l":38,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.finite_one_add_chiSqDiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":1,"d":"Finite χ²-divergence formula. On a finite sample space, for probability measures μ, ν with μ absolutely continuous with respect to ν, the (shifted) χ²-divergence is the explicit sum of squared point-mass ratios: 1 + χ²(μ‖ν) = ∑ₓ (μ{x})² / (ν{x}). Combines chiSqDiv_eq, the finite integral formula integral_fintype, and the discrete RN bridge above.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : Fintype Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → 1 + Causalean.Stat.chiSqDiv μ ν = ∑ x, μ.real {x} ^ 2 / ν.real {x}","l":51,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.finite_penalized_argmin_consistent","k":"theorem","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":1,"d":"Let empirical losses have population losses under a probability measure, with deterministic penalties, a deterministic tie rank, and a selected model. If each empirical loss converges in probability, population minimizers are separated by a positive gap, each penalty is sublinear, the selector is always the ranked minimizer of the penalized empirical criterion, and each success event is measurable, then the probability of selecting a population minimizer converges to one.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [Fintype ι] [Nonempty ι] (empirical : ℕ → Ω → ι → ℝ) (population : ι → ℝ) (penalty : ℕ → ι → ℝ) (rank : ι → ℕ) (selector : ℕ → Ω → ι) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => empirical n ω i) (fun x => population i) μ) → (∃ gap, 0 < gap ∧ ∀ i ∈ Causalean.Stat.populationMinimizers population, ∀ j ∉ Causalean.Stat.populationMinimizers population, population i + gap ≤ population j) → (∀ (i : ι), Filter.Tendsto (fun n => penalty n i / ↑n) Filter.atTop (nhds 0)) → (∀ (n : ℕ) (ω : Ω), Causalean.Stat.IsRankedMinimizer rank (fun i => ↑n * empirical n ω i + penalty n i) (selector n ω)) → (∀ (n : ℕ), MeasurableSet {ω | selector n ω ∈ Causalean.Stat.populationMinimizers population}) → Filter.Tendsto (fun n => μ {ω | selector n ω ∈ Causalean.Stat.populationMinimizers population}) Filter.atTop (nhds 1)","l":178,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.finite_penalized_argmin_failure_tendsto_zero","k":"theorem","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":2,"d":"Let empirical losses have population losses under a probability measure, with deterministic penalties, a deterministic tie rank, and a selected model. If each empirical loss converges in probability, population minimizers are separated by a positive gap, each penalty is sublinear, and the selector is always the ranked minimizer of the penalized empirical criterion, then the probability of selecting outside the population-minimizer class converges to zero.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [Fintype ι] [Nonempty ι] (empirical : ℕ → Ω → ι → ℝ) (population : ι → ℝ) (penalty : ℕ → ι → ℝ) (rank : ι → ℕ) (selector : ℕ → Ω → ι) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ], (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => empirical n ω i) (fun x => population i) μ) → (∃ gap, 0 < gap ∧ ∀ i ∈ Causalean.Stat.populationMinimizers population, ∀ j ∉ Causalean.Stat.populationMinimizers population, population i + gap ≤ population j) → (∀ (i : ι), Filter.Tendsto (fun n => penalty n i / ↑n) Filter.atTop (nhds 0)) → (∀ (n : ℕ) (ω : Ω), Causalean.Stat.IsRankedMinimizer rank (fun i => ↑n * empirical n ω i + penalty n i) (selector n ω)) → Filter.Tendsto (fun n => μ {ω | selector n ω ∉ Causalean.Stat.populationMinimizers population}) Filter.atTop (nhds 0)","l":99,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.finite_range_bddAbove","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A real-valued function on a finite state space has a bounded range.","s":"∀ {S : Type u_2} [Fintype S] [Nonempty S] [inst : MeasurableSpace S] [MeasurableSingletonClass S] (f : S → ℝ), BddAbove (Set.range f)","l":29,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finite_range_bddBelow","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A real-valued function on a finite state space has a range bounded below.","s":"∀ {S : Type u_2} [Fintype S] [Nonempty S] [inst : MeasurableSpace S] [MeasurableSingletonClass S] (f : S → ℝ), BddBelow (Set.range f)","l":33,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.finsetFintype","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For a finite set, the collection of all its finite subsets is itself a finite collection.","s":"{α : Type u_1} → [Fintype α] → Fintype (Finset α)","l":138,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.flipBit","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Given a hypercube dimension, a coordinate of that hypercube, and a Boolean hypercube vertex, the coordinate-flipped vertex agrees with the given vertex at every coordinate except the specified one, where it takes the opposite Boolean value.","s":"{d : ℕ} → Fin d → (Fin d → Bool) → Fin d → Bool","l":63,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipBit_involutive","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Flipping the same coordinate twice is the identity.","s":"∀ {d : ℕ} (j : Fin d), Function.Involutive (Causalean.Stat.flipBit j)","l":71,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipBit_self","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Flipping a coordinate changes that coordinate to the opposite Boolean value.","s":"∀ {d : ℕ} (j : Fin d) (τ : Fin d → Bool), Causalean.Stat.flipBit j τ j = !τ j","l":67,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipPerm","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Given a hypercube dimension and a coordinate of that hypercube, the coordinate-flip permutation is the bijection of Boolean hypercube vertices that flips precisely the specified coordinate.","s":"{d : ℕ} → Fin d → Equiv.Perm (Fin d → Bool)","l":79,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.flipPerm_apply","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"The coordinate-flip permutation acts by flipping that coordinate.","s":"∀ {d : ℕ} (j : Fin d) (τ : Fin d → Bool), (Causalean.Stat.flipPerm j) τ = Causalean.Stat.flipBit j τ","l":83,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.foldB_centered_sum_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Centered fold-B empirical sum is o_p(1) from L²-rate o_p(1). For an i.i.d. sample and a one-shot split into a nuisance fold and an estimation fold, given a family of random functions f n ω : X → ℝ that is jointly measurable in the training data and the outcome, for every n, measurable with respect to the fold-A σ-algebra jointly with the outcome, for every n, square-integrable under the population measure, for every n and ω, and whose $L^2(P)$ norm is $o_p(1)$ under the sampling measure, the centered fold-B empirical sum of f — the estimation-fold sample average of f n ω minus its population mean, rescaled by $\\sqrt{|B(n)|}$ — is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ), Measurable (Function.uncurry (f n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => (√↑(split.foldB n).card)⁻¹ * ∑ i ∈ split.foldB n, (f n ω (S.Z i ω) - ∫ (x : X), f n ω x ∂P)) (fun x => 1) μ","l":629,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_deterministic_affine_transport","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Suppose the affine slope is nonzero, the observation rule is measurable, and each target-experiment law is the pushforward of its corresponding source law. If every measurable source estimator has squared risk at least a fixed level for some parameter, then every measurable target estimator has squared risk at least that level multiplied by the squared affine slope for some parameter, where the target parameter is transformed by the same affine map.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) (theta : Iota → ℝ) (phi : X → Y) (a b L : ℝ), a ≠ 0 → Measurable phi → (∀ (j : Iota), Q j = MeasureTheory.Measure.map phi (P j)) → (∀ (sourceEst : X → ℝ), Measurable sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (P j) sourceEst (theta j)) → ∀ (targetEst : Y → ℝ), Measurable targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (Q j) targetEst (a * theta j + b)","l":243,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_deterministic_affine_transport_pi","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Suppose every source law is a probability law, the affine slope is nonzero, the observation rule is measurable, and each target marginal law is the pushforward of its corresponding source marginal. If every measurable estimator based on the finite source product experiment has squared risk at least a fixed level for some parameter, then every measurable estimator based on the corresponding target product experiment has squared risk at least that level multiplied by the squared affine slope for some parameter, including when the sample has no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (theta : Iota → ℝ) (phi : X → Y) (a b L : ℝ), a ≠ 0 → Measurable phi → (∀ (j : Iota), Q j = MeasureTheory.Measure.map phi (P j)) → (∀ (sourceEst : (Fin n → X) → ℝ), Measurable sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P j) sourceEst (theta j)) → ∀ (targetEst : (Fin n → Y) → ℝ), Measurable targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => Q j) targetEst (a * theta j + b)","l":270,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_kernel_affine_transport","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Suppose the affine slope is nonzero, each target law is obtained by passing its source law through the common Markov kernel, and every measurable uniformly bounded source estimator incurs squared risk at least a fixed level for some parameter index. Then every measurable uniformly bounded target estimator incurs at least the source level multiplied by the squared slope for some parameter index, with the target parameter transformed by the same affine map.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (theta : Iota → ℝ) (a b L : ℝ), a ≠ 0 → (∀ (j : Iota), Q j = (P j).bind ⇑K) → (∀ (sourceEst : X → ℝ), Measurable sourceEst → Causalean.Stat.UniformlyBounded sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (P j) sourceEst (theta j)) → ∀ (targetEst : Y → ℝ), Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (Q j) targetEst (a * theta j + b)","l":226,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.forall_estimator_exists_sqRisk_ge_of_kernel_affine_transport_pi","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"Suppose the affine slope is nonzero, each one-coordinate target law is obtained by applying the common Markov kernel to its source law, and every measurable uniformly bounded estimator on the finite source product incurs squared risk at least a fixed level for some parameter index. Then every measurable uniformly bounded estimator on the target product incurs at least the source level multiplied by the squared slope for some parameter index, including when the sample has no coordinates.","s":"∀ {X : Type uX} {Y : Type uY} {Iota : Type uI} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : Iota → MeasureTheory.Measure X) (Q : Iota → MeasureTheory.Measure Y) [∀ (j : Iota), MeasureTheory.IsProbabilityMeasure (P j)] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] (theta : Iota → ℝ) (a b L : ℝ), a ≠ 0 → (∀ (j : Iota), Q j = (P j).bind ⇑K) → (∀ (sourceEst : (Fin n → X) → ℝ), Measurable sourceEst → Causalean.Stat.UniformlyBounded sourceEst → ∃ j, L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P j) sourceEst (theta j)) → ∀ (targetEst : (Fin n → Y) → ℝ), Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → ∃ j, a ^ 2 * L ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => Q j) targetEst (a * theta j + b)","l":380,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.frechet_hoeffding_lower","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fréchet–Hoeffding lower bound. For a coupling π of the probability measures μ and ν and reals x and y, the joint cdf of π at (x, y) is bounded below by the maximum of 0 and the sum of the marginal cdfs of μ at x and ν at y, minus 1.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (x y : ℝ), max (↑(ProbabilityTheory.cdf μ) x + ↑(ProbabilityTheory.cdf ν) y - 1) 0 ≤ Causalean.Stat.jointCdf π x y","l":101,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.frechet_hoeffding_upper","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Fréchet–Hoeffding upper bound. For a coupling π of the probability measures μ and ν and reals x and y, the joint cdf of π at (x, y) is bounded above by the smaller of the marginal cdfs of μ at x and ν at y.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (x y : ℝ), Causalean.Stat.jointCdf π x y ≤ min (↑(ProbabilityTheory.cdf μ) x) (↑(ProbabilityTheory.cdf ν) y)","l":43,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.gaussianCovarianceDiscrepancy","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For a candidate covariance matrix and target second-moment matrix, the Gaussian covariance discrepancy is log determinant plus inverse-weighted trace.","s":"{V : Type u_1} → [Fintype V] → [DecidableEq V] → Matrix V V ℝ → Matrix V V ℝ → ℝ","l":28,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.gaussianCovarianceDiscrepancy_le_truth_iff","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"For positive-definite candidate and target covariances, the candidate has no larger Gaussian discrepancy than the truth exactly when it is the truth.","s":"∀ {V : Type u_1} [inst : Fintype V] [inst_1 : DecidableEq V] (K T : Causalean.Stat.PositiveCovariance V), Causalean.Stat.gaussianCovarianceDiscrepancy ↑K ↑T ≤ Causalean.Stat.gaussianCovarianceDiscrepancy ↑T ↑T ↔ K = T","l":239,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.gaussianLimit","k":"def","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":1,"d":"Given a measurable observation space, a measure on that space, a finite-dimensional real inner-product outcome space equipped with its Borel σ-algebra, a measurable vector-valued function of an observation, and a finite integral of the squared norm of that function under the measure, the Gaussian limit law is the centered Gaussian measure on the outcome space whose covariance operator is the function's second-moment operator under the given measure.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → {E : Type u_3} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_4 : MeasurableSpace E] → [BorelSpace E] → {ψ : X → E} → Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → MeasureTheory.Measure E","l":60,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_charFun","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The characteristic function of the limiting Gaussian is exactly the abstract target exp(−½ ∫⟪t,ψ⟫² dP) of MultivariateCLT.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (t : E), MeasureTheory.charFun (Causalean.Stat.gaussianLimit hψ hvar) t = Complex.exp (-↑(∫ (x : X), inner ℝ t (ψ x) ^ 2 ∂P) / 2)","l":105,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_covarianceBilin","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The covariance form of the limiting Gaussian recovers the asymptotic-variance integral ∫⟪s,ψ⟫⟪t,ψ⟫ dP.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (s t : E), ((ProbabilityTheory.covarianceBilin (Causalean.Stat.gaussianLimit hψ hvar)) s) t = ∫ (x : X), inner ℝ s (ψ x) * inner ℝ t (ψ x) ∂P","l":82,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_mean","k":"theorem","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"The limiting Gaussian is centered.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), ∫ (x : E), x ∂Causalean.Stat.gaussianLimit hψ hvar = 0","l":74,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.gaussianLimit_waldForm_map","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"χ² identification of the Wald quadratic form. When the linear map induced by the asymptotic-variance operator Σ is injective, i.e. Σ is non-degenerate, the multivariate-CLT Gaussian limit, pushed through the Wald quadratic form S ↦ ⟪S, Σ⁻¹ S⟫, is exactly the χ²_d distribution, with d the dimension of the ambient space.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)), MeasureTheory.Measure.map (fun S => inner ℝ S ((Causalean.Stat.secondMomentInv hψ hvar hinj) S)) (Causalean.Stat.gaussianLimit hψ hvar) = Causalean.Stat.chiSqDist (Module.finrank ℝ E)","l":151,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.gaussianMeasure","k":"def","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":1,"d":"For a real-valued mean and a real-valued variance input, the Gaussian probability measure on the real line has the specified mean and variance $max(v,0)$; thus a negative variance input is replaced by zero and yields a point mass at the mean.","s":"ℝ → ℝ → MeasureTheory.Measure ℝ","l":50,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.gaussianMeasure_zero_one_frontier_Icc","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"The standard normal gives zero mass to the boundary of a symmetric closed interval.","s":"∀ {z : ℝ}, 0 < z → (Causalean.Stat.gaussianMeasure 0 1) (frontier (Set.Icc (-z) z)) = 0","l":82,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.gaussianMeasure_zero_one_singleton","k":"theorem","m":"Causalean.Stat.Inference.Studentize","a":"Stat/Inference","t":2,"d":"The standard normal has no atom at any real point.","s":"∀ (x : ℝ), (Causalean.Stat.gaussianMeasure 0 1) {x} = 0","l":74,"f":"Causalean/Stat/Inference/Studentize.lean"},{"n":"Causalean.Stat.gaussian_tail_small","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":2,"d":"Compatibility alias for probability-measure half-line tail control.","s":"∀ (Q : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure Q] {ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ Q (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ Q (Set.Ici R) ≤ ENNReal.ofReal ε","l":94,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.gaussian_tail_small_gaussian","k":"theorem","m":"Causalean.Stat.CLT.GaussianTail","a":"Stat/CLT","t":2,"d":"Both symmetric half-line tails of a Gaussian distribution can be made smaller than any positive tolerance by choosing a sufficiently large positive cutoff.","s":"∀ {m v ε : ℝ}, 0 < ε → ∃ R, 0 < R ∧ (Causalean.Stat.gaussianMeasure m v) (Set.Iic (-R)) ≤ ENNReal.ofReal ε ∧ (Causalean.Stat.gaussianMeasure m v) (Set.Ici R) ≤ ENNReal.ofReal ε","l":102,"f":"Causalean/Stat/CLT/GaussianTail.lean"},{"n":"Causalean.Stat.glivenkoCantelli_of_fintype","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.GlivenkoCantelli","a":"Stat/EmpiricalProcess","t":1,"d":"A finite class of integrable functions is Glivenko–Cantelli. Consider a finite family of real-valued functions f i on the sample space, observed along an i.i.d. sample S drawn from a probability distribution P. If every f i is measurable and every f i is integrable with respect to P, then the worst-case gap between the empirical mean and the population mean of f i, taken over all indices i, converges to zero in probability as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [Finite ι] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (f : ι → X → ℝ), (∀ (i : ι), Measurable (f i)) → (∀ (i : ι), MeasureTheory.Integrable (f i) P) → Causalean.Stat.GlivenkoCantelli S f","l":102,"f":"Causalean/Stat/EmpiricalProcess/GlivenkoCantelli.lean"},{"n":"Causalean.Stat.glivenkoCantelli_of_hasL1Bracketing","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.GlivenkoCantelli","a":"Stat/EmpiricalProcess","t":1,"d":"A class with finite L¹(P) brackets of arbitrarily small width is Glivenko-Cantelli. Consider a family of real-valued functions f i on the sample space, observed along an i.i.d. sample S drawn from a probability distribution P. If every f i is measurable and for every target width the family can be covered by finitely many upper/lower bracket pairs, each integrable and each sandwiching its assigned member almost everywhere with L¹(P)-gap between the bracket endpoints at most that width, then the worst-case gap between the empirical mean and the population mean of f i, taken over all indices i, converges to zero in probability as the sample size grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} {ι : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (f : ι → X → ℝ), (∀ (i : ι), Measurable (f i)) → Causalean.Stat.HasL1Bracketing f P → Causalean.Stat.GlivenkoCantelli S f","l":139,"f":"Causalean/Stat/EmpiricalProcess/GlivenkoCantelli.lean"},{"n":"Causalean.Stat.gmmBread","k":"def","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"Given a finite-dimensional real inner-product parameter space, a finite-dimensional real inner-product moment space, a linear Jacobian from the parameter space to the moment space, and a linear weighting operator on the moment space, the GMM bread operator is the parameter-space operator $G^{\\mathsf T}WG$.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → E →L[ℝ] E","l":71,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmmHatMatrix","k":"def","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"Given a finite-dimensional real inner-product parameter space, a finite-dimensional real inner-product moment space, a linear Jacobian from the parameter space to the moment space, and a parameter-space linear operator, the whitened GMM hat operator is $GAG^{\\mathsf T}$, where $A$ is the supplied operator.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (E →L[ℝ] E) → F →L[ℝ] F","l":62,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmIF","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"On finite-dimensional real inner-product parameter and moment spaces and a measurable data space, given a continuous linear map from the parameter space to the moment space, a continuous linear weighting operator, a continuous linear operator on the parameter space, a moment function, and a parameter value, the GMM influence function sends an observation $x$ to $-B G^{\\mathsf T}Wg(\\theta_0,x)$, where $B$ is the supplied parameter-space operator.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (E →L[ℝ] E) → (E → X → F) → E → X → E","l":73,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.gmmResidualMaker","k":"def","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"Given a finite-dimensional real inner-product parameter space, a finite-dimensional real inner-product moment space, a linear Jacobian from the parameter space to the moment space, and a parameter-space linear operator, the whitened GMM residual-maker operator is the identity on the moment space minus the corresponding hat operator.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (E →L[ℝ] E) → F →L[ℝ] F","l":68,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_finrank_range","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The rank of the residual maker's range is m − k (finrank F − finrank E), the Hansen–Sargan degrees of freedom.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, effInv ∘SL ContinuousLinearMap.adjoint G ∘SL G = ContinuousLinearMap.id ℝ E → (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → Module.finrank ℝ ↥(↑(Causalean.Stat.gmmResidualMaker G effInv)).range = Module.finrank ℝ F - Module.finrank ℝ E","l":135,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_idempotent","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The residual maker M = I − H is idempotent: M ∘L M = M.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmResidualMaker G effInv ∘SL Causalean.Stat.gmmResidualMaker G effInv = Causalean.Stat.gmmResidualMaker G effInv","l":124,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmResidualMaker_isSelfAdjoint","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":2,"d":"The residual maker M = I − H is self-adjoint.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → IsSelfAdjoint (Causalean.Stat.gmmResidualMaker G effInv)","l":114,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.gmmSandwich","k":"def","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"Given a finite-dimensional real inner-product parameter space, a finite-dimensional real inner-product moment space, a linear Jacobian from the parameter space to the moment space, a linear weighting operator, a moment covariance operator, and a parameter-space linear operator, the GMM sandwich variance operator is $AG^{\\mathsf T}W\\Sigma WGA$, where $A$ is the supplied operator.","s":"{E : Type u_1} → {F : Type u_2} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (F →L[ℝ] F) → (E →L[ℝ] E) → E →L[ℝ] E","l":77,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmmScore","k":"def","m":"Causalean.Stat.GMM.Setup","a":"Stat/GMM","t":1,"d":"On finite-dimensional real inner-product parameter and moment spaces and a measurable data space, given a continuous linear map from the parameter space to the moment space, a continuous linear weighting operator on the moment space, and a moment function of a parameter and an observation, the combined GMM score maps each parameter--observation pair to the adjoint-map image of its weighted moment, $G^{\\mathsf T}Wg(\\theta,x)$.","s":"{E : Type u_1} → {F : Type u_2} → {X : Type u_3} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace ℝ F] → [FiniteDimensional ℝ F] → (E →L[ℝ] F) → (F →L[ℝ] F) → (E → X → F) → E → X → E","l":59,"f":"Causalean/Stat/GMM/Setup.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":1,"d":"GMM asymptotic linearity. Consider a GMM problem prob (moment function, weight matrix, and target parameter θ₀) together with Z-estimator regularity conditions reg for its combined score, evaluated along an i.i.d. sample S. If the Jacobian inverse of the combined score's population expectation at θ₀, reg.J₀_inv, coincides with the GMM problem's \"bread\" inverse prob.breadInv, the estimator sequence θn is consistent for θ₀, the combined score is stochastically equicontinuous at θ₀ along θn, θn converges to θ₀ at the parametric rate, i.e. ‖θn n − θ₀‖ is Oₚ(n^(-1/2)), and θn eventually and almost surely solves the sample moment equation ∑ᵢ score(θn n, Zᵢ) = 0, then θn admits the usual GMM asymptotic-linear expansion around θ₀.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":67,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_donsker","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity with the equicontinuity hypothesis discharged. The GMM analogue of zEstimator_clt_of_donsker: gmm_asymptotically_linear with the opaque modulus hStochEquicont replaced by the class-level Donsker property AsymptoticEquicont prob.score prob.θ₀ P μ S, reconstructed via stochEquicontAt_of_asymptoticEquicont from hAEC and consistency.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.AsymptoticEquicont prob.score prob.θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":176,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_extremum","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity from extremum primitives. The GMM analogue of zEstimator_clt_of_extremum: gmm_asymptotically_linear with the consistency hypothesis discharged from a Glivenko–Cantelli GMM criterion m with a well-separated population maximum at θ₀ of which θn is a sample maximiser. The classical instance is m θ = −ḡ_n(θ)ᵀ W ḡ_n(θ) (the GMM objective), whose score is prob.score.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m prob.θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ prob.θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m prob.θ₀ x ∂P) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":146,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_asymptotically_linear_of_extremum_donsker","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic linearity from primitive conditions: both opaque hypotheses discharged. The GMM analogue of zEstimator_clt_of_extremum_donsker: neither hConsistent nor hStochEquicont is assumed. Consistency is derived from the Glivenko–Cantelli GMM criterion m with well-separated maximum, then fed to both the linearisation and the StochEquicontAt reduction applied to hAEC.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m prob.θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ prob.θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m prob.θ₀ x ∂P) → Causalean.Stat.AsymptoticEquicont prob.score prob.θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn prob.θ₀ prob.influence S fun n => Finset.range n","l":199,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.gmm_efficiency","k":"theorem","m":"Causalean.Stat.GMM.VarianceAlgebra","a":"Stat/GMM","t":1,"d":"GMM optimal-weighting theorem (Hansen 1982, Theorem 3.2). For a Jacobian G, a self-adjoint weighting operator W, and a positive-semidefinite covariance operator Cov admitting a two-sided inverse CovInv, suppose further that the \"bread\" GᵀWG has a two-sided inverse breadInv and that the efficient bread Gᵀ CovInv G has a two-sided inverse effInv. Then the sandwich asymptotic variance of the GMM estimator with weighting W dominates the efficient (optimally-weighted) asymptotic variance in the Löwner order, i.e. their difference is a positive-semidefinite operator.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] (G : E →L[ℝ] F) (W Cov CovInv : F →L[ℝ] F), ContinuousLinearMap.adjoint W = W → Cov.IsPositive → CovInv ∘SL Cov = ContinuousLinearMap.id ℝ F → Cov ∘SL CovInv = ContinuousLinearMap.id ℝ F → ∀ (breadInv : E →L[ℝ] E), breadInv ∘SL Causalean.Stat.gmmBread G W = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmBread G W ∘SL breadInv = ContinuousLinearMap.id ℝ E → ∀ (effInv : E →L[ℝ] E), effInv ∘SL Causalean.Stat.gmmBread G CovInv = ContinuousLinearMap.id ℝ E → Causalean.Stat.gmmBread G CovInv ∘SL effInv = ContinuousLinearMap.id ℝ E → (Causalean.Stat.gmmSandwich G W Cov breadInv - effInv).IsPositive","l":105,"f":"Causalean/Stat/GMM/VarianceAlgebra.lean"},{"n":"Causalean.Stat.gmm_tendsto_normal_vec","k":"theorem","m":"Causalean.Stat.GMM.AsymptoticNormality","a":"Stat/GMM","t":2,"d":"GMM asymptotic normality from a supplied vector CLT contact. Combining gmm_asymptotically_linear with IsAsymLinearVec.tendsto_normal_vec, the rescaled GMM estimator converges to the caller-supplied vector CLT target Q. The theorem is parametric in the limiting law: the concrete Gaussian or sandwich identification is supplied through hCLT and the target law Q.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} {F : Type u_4} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] [inst_9 : FiniteDimensional ℝ F] [inst_10 : MeasurableSpace F] [inst_11 : MeasureTheory.IsProbabilityMeasure μ] (prob : Causalean.Stat.GMMProblem P) (reg : Causalean.Stat.ZEstimatorRegularity prob.score prob.θ₀ P), reg.J₀_inv = prob.breadInv → ∀ (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - prob.θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt prob.score prob.θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - prob.θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, prob.score (θn n ω) (S.Z i ω) = 0) → ∀ (Q : MeasureTheory.ProbabilityMeasure E) (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn prob.θ₀ (fun m => Finset.range m) n) μ) (hSum_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinearVec.normalizedSum S prob.influence (fun m => Finset.range m) n) μ), Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.normalizedSum S prob.influence (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hSum_meas n)⟩) Filter.atTop (nhds Q) → Filter.Tendsto (fun n => ⟨MeasureTheory.Measure.map (Causalean.Stat.IsAsymLinearVec.rescaledEstimator θn prob.θ₀ (fun m => Finset.range m) n) μ, MeasureTheory.Measure.isProbabilityMeasure_map (hθn_meas n)⟩) Filter.atTop (nhds Q)","l":108,"f":"Causalean/Stat/GMM/AsymptoticNormality.lean"},{"n":"Causalean.Stat.groupArmCount","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, a sample, an arm label, and a group label, the arm--group count is the number of sample coordinates whose two labels equal those requested.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → Bool → kappa → ℕ","l":69,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.groupCount","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, a sample, and a group label, the group count is the sum of the sample counts in that group for the false and true arm labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → kappa → ℕ","l":77,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.groupEvent","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a group-label map, and a group label, the group event is the set of sample-space outcomes having that group label, irrespective of arm assignment.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → kappa → Set Omega","l":37,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.half_integral_abs_rnDeriv_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"For two probability measures dominated by a common finite measure, half the L¹ distance between their Radon--Nikodym densities is bounded by total variation. This is the reverse Scheffé inequality needed to construct the common submeasure in the maximal-coupling argument.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν ξ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [MeasureTheory.IsFiniteMeasure ξ], μ.AbsolutelyContinuous ξ → ν.AbsolutelyContinuous ξ → 1 / 2 * ∫ (x : Ω), |(μ.rnDeriv ξ x).toReal - (ν.rnDeriv ξ x).toReal| ∂ξ ≤ Causalean.Stat.tvDist μ ν","l":196,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.half_one_sub_tvDist_le_max_error","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound (max form). For a measurable estimator est, under 2s-separation of θ₀, θ₁, the worst-case error probability is at least ½ (1 − tvDist P₀ P₁).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - Causalean.Stat.tvDist P₀ P₁) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":73,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.hammingRisk","k":"def","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":1,"d":"Given a measurable sample space, a hypercube dimension, a measure for every Boolean hypercube vertex, an estimator returning a Boolean vertex from each sample outcome, and a true vertex, the Hamming risk is the sum, over coordinates, of the measure of the outcomes at which the estimator differs from that true vertex under its associated measure.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → {d : ℕ} → ((Fin d → Bool) → MeasureTheory.Measure Ω) → (Ω → Fin d → Bool) → (Fin d → Bool) → ℝ","l":87,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.hammingRisk_pair_ge","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Per-coordinate pair bound. For each vertex τ, the coordinate-j error mass at τ plus the one at its j-flip is at least 1 − tvDist (P τ) (P (flip j τ)). This is the two-point testing bound applied to the j-th decoded bit.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d) (τ : Fin d → Bool), 1 - Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ)) ≤ (P τ).real {ω | est ω j ≠ τ j} + (P (Causalean.Stat.flipBit j τ)).real {ω | est ω j ≠ Causalean.Stat.flipBit j τ j}","l":111,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.harmonic_sum_ge_log_sub_log","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Harmonic-tail lower bound: for natural numbers with 1 ≤ m, ∑_{t=m}^{n} 1/t ≥ log(n+1) - log m. Proof by telescoping the pointwise bound 1/t ≥ log(t+1) - log t.","s":"∀ (m n : ℕ), 1 ≤ m → Real.log (↑n + 1) - Real.log ↑m ≤ ∑ t ∈ Finset.Icc m n, 1 / ↑t","l":60,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.hasFDerivAt_ratio","k":"theorem","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":2,"d":"Fréchet derivative of the ratio map. On EuclideanSpace ℝ (Fin 2), the map v ↦ v 0 / v 1 is Fréchet-differentiable at any t₀ whose second coordinate is nonzero, with derivative ratioDeriv t₀.","s":"∀ {t₀ : EuclideanSpace ℝ (Fin 2)}, t₀.ofLp 1 ≠ 0 → HasFDerivAt (fun v => v.ofLp 0 / v.ofLp 1) (Causalean.Stat.ratioDeriv t₀) t₀","l":66,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.hasHadamardDirDerivAt_max","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"max is Hadamard directionally differentiable. At (a, b) the derivative is maxDirDeriv a b (linear off the diagonal, sublinear at a tie).","s":"∀ (a b : ℝ), Causalean.Stat.HasHadamardDirDerivAt (fun z => max z.1 z.2) (Causalean.Stat.maxDirDeriv a b) (a, b)","l":193,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.hasHadamardDirDerivAt_min","k":"theorem","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":2,"d":"min is Hadamard directionally differentiable. At (a, b) the derivative is minDirDeriv a b.","s":"∀ (a b : ℝ), Causalean.Stat.HasHadamardDirDerivAt (fun z => min z.1 z.2) (Causalean.Stat.minDirDeriv a b) (a, b)","l":266,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.hasIIDSample_iff_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The existence of an i.i.d. sample with a prescribed common marginal is equivalent to that marginal being a probability measure.","s":"∀ {X : Type u} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X), Causalean.Stat.HasIIDSample P ↔ MeasureTheory.IsProbabilityMeasure P","l":250,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.hasIIDSample_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"An i.i.d. sample with a given law always exists: every probability measure is the common marginal of some independent, identically distributed sample, realised on the infinite product space by the coordinate projections.","s":"∀ {X : Type u} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P], Causalean.Stat.HasIIDSample P","l":243,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.hellingerSqDensity","k":"def","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"Given a measurable sample space, a measure on that space, and two real-valued functions on it, the unhalved squared Hellinger discrepancy relative to the measure is the integral of the squared difference between their pointwise square roots.","s":"{α : Type u_1} → [inst : MeasurableSpace α] → MeasureTheory.Measure α → (α → ℝ) → (α → ℝ) → ℝ","l":65,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.hellingerSqDensity_eq_two_mul_one_sub_affinity","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"For two integrable nonnegative densities that each integrate to one, the unhalved squared Hellinger distance is exactly twice the affinity defect, i.e. twice one minus the affinity. This is the algebraic identity that lets an affinity computation be read as a Hellinger distance and back.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (f g : α → ℝ), MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → 0 ≤ f → 0 ≤ g → ∫ (x : α), f x ∂μ = 1 → ∫ (x : α), g x ∂μ = 1 → Causalean.Stat.hellingerSqDensity μ f g = 2 * (1 - Causalean.Stat.densityAffinity μ f g)","l":75,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.hoeffding_cov_identity","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Hoeffding's covariance identity. For a coupling π of μ and ν, where both μ and ν have finite second moment (are L²), the covariance of the coordinates under π equals the double integral, over the plane, of the Fréchet gap between the joint and product cumulative distribution functions, H_π - F·G:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π - (∫ (x : ℝ), x ∂μ) * ∫ (y : ℝ), y ∂ν = ∫ (x : ℝ) (y : ℝ), Causalean.Stat.jointCdf π x y - ↑(ProbabilityTheory.cdf μ) x * ↑(ProbabilityTheory.cdf ν) y","l":102,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.hoeffding_cov_identity_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Hoeffding's covariance identity, product form. For a coupling π of two L² probability measures μ, ν,","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π - (∫ (x : ℝ), x ∂μ) * ∫ (y : ℝ), y ∂ν = ∫ (q : ℝ × ℝ), Causalean.Stat.jointCdf π q.1 q.2 - ↑(ProbabilityTheory.cdf μ) q.1 * ↑(ProbabilityTheory.cdf ν) q.2 ∂MeasureTheory.volume.prod MeasureTheory.volume","l":80,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.hoeffding_decomp","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"Hoeffding decomposition (pointwise identity). For a kernel h, population law P, and points x and y, the kernel value decomposes as h(x, y) = θ + h₁(x) + h₁(y) + g(x, y), the population mean plus the two first-order Hoeffding projections plus the degenerate second-order residual.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] (h : X → X → ℝ) (P : MeasureTheory.Measure X) (x y : X), h x y = Causalean.Stat.uMean h P + Causalean.Stat.uProj h P x + Causalean.Stat.uProj h P y + Causalean.Stat.uDegen h P x y","l":80,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.hoeffding_decomp_order","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For an order-m kernel h, population law P, and an m-tuple of points z, the kernel value decomposes as the population mean plus the sum of the m coordinatewise first Hoeffding projections plus the degenerate higher-order residual kernel evaluated at z.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {m : ℕ} [inst_1 : NeZero m] (h : (Fin m → X) → ℝ) (P : MeasureTheory.Measure X) (z : Fin m → X), h z = Causalean.Stat.uMeanOrder h P + ∑ j, Causalean.Stat.uProjOrderAt j h P (z j) + Causalean.Stat.uDegenOrder h P z","l":120,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.iInf_squaredRisk_eq_posteriorResidual","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The infimum over all real estimators of finite squared risk is exactly the posterior residual Bayes risk.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ), ⨅ T, Causalean.Stat.squaredRisk ν L t T = Causalean.Stat.posteriorResidual ν L t","l":285,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.iidSample_finN_measurable","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The joint observable Ψ ω k = S.Z k ω is measurable Ω → (Fin n → X).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ), Measurable fun ω k => S.Z (↑k) ω","l":85,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iidSample_finN_pushforward","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"For an i.i.d. sample S and a fixed horizon n, the joint map of the first n sample points pushes μ forward to the product measure on Fin n → X.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (n : ℕ), MeasureTheory.Measure.map (fun ω k => S.Z (↑k) ω) μ = MeasureTheory.Measure.pi fun x => P","l":67,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iidSample_infinitePi","k":"def","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":1,"d":"Given a measurable observation space and a probability measure on it, the independent, identically distributed sample on the infinite product space is formed by coordinate projections, and every coordinate has the given probability measure as its marginal law.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (P : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure P] → Causalean.Stat.IIDSample (ℕ → X) X (MeasureTheory.Measure.infinitePi fun x => P) P","l":52,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.iid_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"n-sample structure-agnostic two-point bound. Given two single-observation laws P₀, P₁ and a real functional τ such that the functional values τ P₀, τ P₁ are 2s-separated, then for any measurable estimator est built from n i.i.d. samples (data law Measure.pi (fun _ ↦ Pⱼ)), the worst-case miss probability, using τ P₀ and τ P₁ as the two parameters, is at least ½(1 − tvDist) between the two n-fold product laws.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (P₀ P₁ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] (τ : MeasureTheory.Measure S → ℝ) (n : ℕ) {s : ℝ}, 2 * s ≤ |τ P₀ - τ P₁| → ∀ {est : (Fin n → S) → ℝ}, Measurable est → (1 - Causalean.Stat.tvDist (MeasureTheory.Measure.pi fun x => P₀) (MeasureTheory.Measure.pi fun x => P₁)) / 2 ≤ max ((MeasureTheory.Measure.pi fun x => P₀).real {x | s ≤ |est x - τ P₀|}) ((MeasureTheory.Measure.pi fun x => P₁).real {x | s ≤ |est x - τ P₁|})","l":82,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.inducedFiniteDesign","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a measurable parameter space, a nonempty finite measurable state space with measurable singletons, a probability prior on the parameter space, and a Markov kernel from parameters to states, the induced finite design is the design whose probability assigned to each state is the prior average of the kernel's probability of that state, these probabilities are nonnegative, and their sum over all states is one.","s":"{Θ : Type u_1} → {S : Type u_2} → [inst : MeasurableSpace Θ] → [inst_1 : Fintype S] → [Nonempty S] → [inst_3 : MeasurableSpace S] → [MeasurableSingletonClass S] → (π : MeasureTheory.Measure Θ) → [MeasureTheory.IsProbabilityMeasure π] → (K : ProbabilityTheory.Kernel Θ S) → [ProbabilityTheory.IsMarkovKernel K] → Causalean.Experimentation.DesignBased.FiniteDesign S","l":298,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.inducedFiniteDesign_expectedLoss_eq_mixedKernelLoss","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"Expected loss under the induced finite design equals the continuous-prior, kernel-averaged loss integral.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [inst_5 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_6 : ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ) (a : A), (Causalean.Stat.inducedFiniteDesign π K).E (loss a) = Causalean.Stat.mixedKernelLoss π K loss a","l":323,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.inducedFiniteDesign_p","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"The induced finite design's mass at each state is the prior integral of the kernel's singleton probability.","s":"∀ {Θ : Type u_1} {S : Type u_2} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [inst_5 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_6 : ProbabilityTheory.IsMarkovKernel K] (s : S), (Causalean.Stat.inducedFiniteDesign π K).p s = ∫ (θ : Θ), (K θ).real {s} ∂π","l":316,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.injectiveSubtypeEquivEmbedding","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For a domain size and a codomain size, the equivalence between injective maps and embeddings identifies every injective map from an $m$-element index set to an $n$-element index set with the corresponding embedding, and conversely.","s":"(m n : ℕ) → { t // Function.Injective t } ≃ (Fin m ↪ Fin n)","l":58,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For a tuple length and a number of available sample positions, the ordered injective-tuple count is the real-valued number of ordered selections of that length with distinct entries among the first specified number of observations.","s":"ℕ → ℕ → ℝ","l":54,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount_eq_descFactorial","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The real-valued ordered injective tuple count is the falling factorial.","s":"∀ (m n : ℕ), Causalean.Stat.injectiveTupleCount m n = ↑(n.descFactorial m)","l":78,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTupleCount_ne_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"If m ≤ n, there is at least one ordered injective m-tuple in Fin n.","s":"∀ {m n : ℕ}, m ≤ n → Causalean.Stat.injectiveTupleCount m n ≠ 0","l":111,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.injectiveTupleCount_two_eq_mul_sub_one","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"For a sample of size at least two, the number of ordered injective pairs is the sample size times one less than the sample size.","s":"∀ {n : ℕ}, 2 ≤ n → Causalean.Stat.injectiveTupleCount 2 n = ↑n * (↑n - 1)","l":71,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.injectiveTupleCount_two_eq_offDiag_card","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"The number of ordered injective pairs from a sample of size n equals the number of off-diagonal ordered pairs drawn from that sample.","s":"∀ (n : ℕ), Causalean.Stat.injectiveTupleCount 2 n = ↑(Finset.range n).offDiag.card","l":50,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.injectiveTuples","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"For a tuple length and a number of available sample positions, the collection of ordered injective tuples consists of all ordered selections of the specified length whose entries are distinct positions among the first specified number of observations.","s":"(m n : ℕ) → Finset (Fin m → Fin n)","l":49,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTuples_card_eq_descFactorial","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The ordered injective tuple count is the falling factorial n (n-1) ....","s":"∀ (m n : ℕ), (Causalean.Stat.injectiveTuples m n).card = n.descFactorial m","l":66,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.injectiveTuples_fiber_card_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"Fibres of a fixed coordinate map on ordered injective tuples have the same cardinality. The bijection composes tuples with the codomain transposition swapping the two fibre values.","s":"∀ {m n : ℕ} (j : Fin m) (y y' : Fin n), {t ∈ Causalean.Stat.injectiveTuples m n | t j = y}.card = {t ∈ Causalean.Stat.injectiveTuples m n | t j = y'}.card","l":126,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.inner_normalizedSum","k":"theorem","m":"Causalean.Stat.CLT.MultivariateCLT","a":"Stat/CLT","t":2,"d":"The inner product of t with the vector normalised sum is the scalar normalised sum of the projected influence function x ↦ ⟪t, ψ x⟫.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (S : Causalean.Stat.IIDSample Ω X μ P) (ψ : X → E) (t : E) (I : ℕ → Finset ℕ) (n : ℕ) (ω : Ω), inner ℝ t (Causalean.Stat.IsAsymLinearVec.normalizedSum S ψ I n ω) = Causalean.Stat.IsAsymLinear.normalizedSum S (fun x => inner ℝ t (ψ x)) I n ω","l":91,"f":"Causalean/Stat/CLT/MultivariateCLT.lean"},{"n":"Causalean.Stat.insertCoord","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an order $m$, a distinguished coordinate, a value in the observation space, and values for every remaining coordinate, the completed coordinate tuple assigns the given value to the distinguished coordinate and the supplied remaining values to all other coordinates.","s":"{X : Type u_2} → {m : ℕ} → (j : Fin m) → X → ({ k // k ≠ j } → X) → Fin m → X","l":95,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.instIsGaussianGaussianLimit","k":"instance","m":"Causalean.Stat.CLT.GaussianLimit","a":"Stat/CLT","t":2,"d":"Given a measurable observation space, a measure on that space, a finite-dimensional real inner-product outcome space equipped with its Borel σ-algebra, a measurable vector-valued function of an observation, and a finite integral of the squared norm of that function under the measure, the Gaussian limit law is Gaussian.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P), ProbabilityTheory.IsGaussian (Causalean.Stat.gaussianLimit hψ hvar)","l":66,"f":"Causalean/Stat/CLT/GaussianLimit.lean"},{"n":"Causalean.Stat.instIsMarkovKernelFinProductKernel","k":"instance","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For every nonnegative integer number of coordinates and every Markov kernel between measurable input and output spaces, the independent product kernel over that many coordinates is a Markov kernel, including when there are zero coordinates.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K], ProbabilityTheory.IsMarkovKernel (Causalean.Stat.finProductKernel n K)","l":291,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasureGaussianMeasure","k":"instance","m":"Causalean.Stat.CLT.AsymptoticLinearity","a":"Stat/CLT","t":2,"d":"For every real-valued mean and real-valued variance input, the Gaussian measure on the real line is a probability measure.","s":"∀ (m v : ℝ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.gaussianMeasure m v)","l":56,"f":"Causalean/Stat/CLT/AsymptoticLinearity.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasureRealChiSqDist","k":"instance","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"For every nonnegative integer number of degrees of freedom, the chi-squared distribution with that number of degrees of freedom is a probability law.","s":"∀ (d : ℕ), MeasureTheory.IsProbabilityMeasure (Causalean.Stat.chiSqDist d)","l":55,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_map_max","k":"instance","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"For any probability law on ordered pairs of real numbers, the law of the larger coordinate is a probability law.","s":"∀ (Q : MeasureTheory.Measure (ℝ × ℝ)) [MeasureTheory.IsProbabilityMeasure Q], MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map (fun z => max z.1 z.2) Q)","l":90,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_map_min","k":"instance","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"For any probability law on ordered pairs of real numbers, the law of the smaller coordinate is a probability law.","s":"∀ (Q : MeasureTheory.Measure (ℝ × ℝ)) [MeasureTheory.IsProbabilityMeasure Q], MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map (fun z => min z.1 z.2) Q)","l":97,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.instIsProbabilityMeasure_unifOI","k":"instance","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The uniform probability measure on the open unit interval is a probability law.","s":"MeasureTheory.IsProbabilityMeasure Causalean.Stat.unifOI","l":43,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.integrableOn_finiteSelector_sq_of_eligible","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, the designated pilot event is measurable, every branch selected there is eligible, and eligible branches have integrable squared error under the tail law, the selected squared error is integrable on that pilot event.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {mu : MeasureTheory.Measure Omega} {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ} {Eligible : Iota → Prop}, Measurable pilot → Measurable tail → Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → (∀ a ∈ good, Eligible (select a)) → (∀ (i : Iota), Eligible i → MeasureTheory.Integrable (fun b => err i b ^ 2) (MeasureTheory.Measure.map tail mu)) → MeasureTheory.IntegrableOn (fun omega => err (select (pilot omega)) (tail omega) ^ 2) (pilot ⁻¹' good) mu","l":201,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.integrable_bigPhi","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The key integrability. Let π be a coupling of μ and ν, where both marginals have finite second moment. Then the product tail kernel Φ q p = signedTail p.1 q.1 * signedTail p.2 q.2 is integrable for the product of Lebesgue measure on the plane with π, i.e. on (volume ⊗ volume) ⊗ π.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.1 z.1.1 * Causalean.Stat.signedTail z.2.2 z.1.2) ((MeasureTheory.volume.prod MeasureTheory.volume).prod π)","l":60,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"cdfStat y is integrable (bounded by 1 on a finite measure).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), MeasureTheory.Integrable (Causalean.Stat.cdfStat y) P","l":87,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.integrable_fiber","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The fibre q ↦ ∫ p, Φ q p ∂π is integrable on ℝ × ℝ, by Integrable.integral_prod_left applied to integrable_bigPhi.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun q => ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π) (MeasureTheory.volume.prod MeasureTheory.volume)","l":177,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_frechet_gap","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Hoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The Fréchet gap H_π - F·G is integrable on ℝ × ℝ for an L² coupling; this is the integrability side-condition consumed by hoeffding_cov_identity and by the monotone comparison of double integrals in Optimality.lean.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun p => Causalean.Stat.jointCdf π p.1 p.2 - ↑(ProbabilityTheory.cdf μ) p.1 * ↑(ProbabilityTheory.cdf ν) p.2) (MeasureTheory.volume.prod MeasureTheory.volume)","l":60,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Hoeffding.lean"},{"n":"Causalean.Stat.integrable_inner_smul","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":2,"d":"The integrand x ↦ ⟪t, ψ x⟫ • ψ x is integrable when ‖ψ‖² is, by the Cauchy–Schwarz bound ‖⟪t,ψ⟫ • ψ‖ ≤ ‖t‖ ‖ψ‖².","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] [MeasurableSMul₂ ℝ E] {ψ : X → E}, Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → ∀ (t : E), MeasureTheory.Integrable (fun x => inner ℝ t (ψ x) • ψ x) P","l":56,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.integrable_kernelAverageLoss","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"Under a probability prior, the kernel-averaged loss of a finite-state action is integrable.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ) (a : A), MeasureTheory.Integrable (Causalean.Stat.kernelAverageLoss K loss a) π","l":64,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.integrable_kernel_singletonReal","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A finite-state Markov kernel's probability of a fixed state is integrable under every probability prior.","s":"∀ {Θ : Type u_1} {S : Type u_2} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (s : S), MeasureTheory.Integrable (fun θ => (K θ).real {s}) π","l":91,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.integrable_loss_comp","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A finite-state loss composed with a measurable deterministic state map is integrable under every probability prior.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (f : Θ → S), Measurable f → ∀ (loss : A → S → ℝ) (a : A), MeasureTheory.Integrable (fun θ => loss a (f θ)) π","l":185,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.integrable_normalizedFiniteKernelStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"If a finite-arity kernel is measurable and integrable under the product of the population law, then the normalized statistic that averages it over all injective index tuples of the sample is integrable.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ι : Type u_3} [inst_2 : Fintype ι] {k : (ι → X) → ℝ} {n : ℕ}, Measurable k → MeasureTheory.Integrable k (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (Causalean.Stat.normalizedFiniteKernelStatistic S k n) μ","l":462,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.integrable_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is Lebesgue integrable: it is a bounded function supported on a bounded interval.","s":"∀ (a : ℝ), MeasureTheory.Integrable (Causalean.Stat.signedTail a) MeasureTheory.volume","l":97,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integrable_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The tensor product (s, t) ↦ signedTail x s * signedTail y t is integrable on ℝ × ℝ for Lebesgue×Lebesgue.","s":"∀ (x y : ℝ), MeasureTheory.Integrable (fun q => Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2) (MeasureTheory.volume.prod MeasureTheory.volume)","l":189,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integrable_sqrt_rnDeriv","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"√p is ν-integrable (dominated by (p+1)/2).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.Integrable (fun x => √(μ.rnDeriv ν x).toReal) ν","l":106,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.integrable_survFst_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The centred marginal survival function s ↦ SX s - 𝟙{s<0} is Lebesgue integrable (it is E[signedTail X ·], integrable by Integrable.integral_prod_left on integrable_tail_fst_prod).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.Integrable (fun s => Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s) MeasureTheory.volume","l":158,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_survSnd_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The centred marginal survival function t ↦ SY t - 𝟙{t<0} is Lebesgue integrable.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun t => Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t) MeasureTheory.volume","l":168,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_tail_fst_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The single-coordinate tail kernel is integrable on volume ⊗ π. Domination: ∫ s, |signedTail p.1 s| ds = |p.1|, which is π-integrable since X ∈ L²(π) and π is finite.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.1 z.1) (MeasureTheory.volume.prod π)","l":25,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integrable_tail_snd_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The single-coordinate tail kernel is integrable on volume ⊗ π (second coordinate).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → MeasureTheory.Integrable (fun z => Causalean.Stat.signedTail z.2.2 z.1) (MeasureTheory.volume.prod π)","l":43,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integral_abs_mul_le_eLpNorm_mul_eLpNorm","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"Cauchy–Schwarz for products of L²-functions. If f is square-integrable against the measure ν and g is square-integrable against ν, then the integral of |f · g| against ν is at most the product of the L² norms ‖f‖₂ · ‖g‖₂. Stated for the absolute value of the product because that is the form consumed by the AIPW remainder bound.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {ν : MeasureTheory.Measure X} {f g : X → ℝ}, MeasureTheory.MemLp f 2 ν → MeasureTheory.MemLp g 2 ν → ∫ (x : X), |f x * g x| ∂ν ≤ (MeasureTheory.eLpNorm f 2 ν).toReal * (MeasureTheory.eLpNorm g 2 ν).toReal","l":321,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_abs_mul_op_of_eLpNorm_op","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":1,"d":"L² product rate ⇒ integrated absolute-product rate. For a random family f_n and a fixed function g, suppose each f_n(ω) is square-integrable against ν, for every n and ω, g is square-integrable against ν, the rates s_n are positive, and the L² norm ‖f_n(ω)‖₂ is o_p(s_n). Then the integrated absolute product ∫ |f_n(ω) x · g x| dν is o_p(s_n · ‖g‖₂).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {ν : MeasureTheory.Measure X} {fn : ℕ → Ω → X → ℝ} {g : X → ℝ}, (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (fn n ω) 2 ν) → MeasureTheory.MemLp g 2 ν → ∀ {sn : ℕ → ℝ}, (∀ (n : ℕ), 0 < sn n) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fn n ω) 2 ν).toReal) sn μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (x : X), |fn n ω x * g x| ∂ν) (fun n => sn n * (MeasureTheory.eLpNorm g 2 ν).toReal) μ","l":385,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_abs_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The L¹ norm of the signed tail indicator is |a|: the integrand is ±1 on an interval of length |a|.","s":"∀ (a : ℝ), ∫ (s : ℝ), |Causalean.Stat.signedTail a s| = |a|","l":167,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_categoryIndicator","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given a measurable category label and a fixed category, the mean of its zero-one indicator under a probability law equals the category's population mass.","s":"∀ {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Iota] {P : MeasureTheory.Measure X} [inst_2 : DecidableEq Iota] [MeasurableSingletonClass Iota] [MeasureTheory.IsProbabilityMeasure P] {label : X → Iota}, Measurable label → ∀ (k : Iota), ∫ (x : X), Causalean.Stat.categoryIndicator label k x ∂P = Causalean.Stat.categoryMass P label k","l":57,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.integral_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The integral of the indicator statistic is the cdf value F(y).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] (y : ℝ), ∫ (z : ℝ), Causalean.Stat.cdfStat y z ∂P = ↑(ProbabilityTheory.cdf P) y","l":95,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.integral_comp_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Under a common probability law, if the observation rule is measurable and the real-valued statistic of the recoded sample is measurable, then its expectation after coordinatewise recoding equals its expectation under the product of the recoded marginal law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (mu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] {phi : X → Y}, Measurable phi → ∀ (g : (Fin n → Y) → ℝ), Measurable g → (∫ (z : Fin n → X), g fun i => phi (z i) ∂MeasureTheory.Measure.pi fun x => mu) = ∫ (y : Fin n → Y), g y ∂MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map phi mu","l":302,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_comp_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Averaging a function of the sub-index coordinates over the whole i.i.d. product sample is the same as averaging it over an i.i.d. product sample indexed by the sub-index alone.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} {E : Type u_5} [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_5 : DecidablePred p] (g : (Subtype p → X) → E), (∫ (s : ι → X), g fun i => s ↑i ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (z : Subtype p → X), g z ∂MeasureTheory.Measure.pi fun x => μ","l":163,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_comp_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Averaging a function of the coordinates in a finite index set over the whole i.i.d. product sample is the same as averaging it over an i.i.d. product sample indexed by that finite set.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} {E : Type u_5} [inst_1 : MeasurableSpace X] [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι) (g : (↥S → X) → E), (∫ (s : ι → X), g fun i => s ↑i ∂MeasureTheory.Measure.pi fun x => μ) = ∫ (z : ↥S → X), g z ∂MeasureTheory.Measure.pi fun x => μ","l":215,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.integral_designWeight_residual_cross_coordinates_eq_zero","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, each residual is centered within its arm/group cell, and the two sample coordinates differ, any finite-design weight times their two supported residuals has product-law integral zero.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ∂mu = 0) → ∀ (W : (Fin n → kappa × Bool) → ℝ) (i j : Fin n), i ≠ j → ∀ (a b : Bool) (k l : kappa), (∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * Causalean.Stat.supportedArmGroupResidual group arm Y center a k (z i) * Causalean.Stat.supportedArmGroupResidual group arm Y center b l (z j) ∂MeasureTheory.Measure.pi fun x => mu) = 0","l":225,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_designWeight_residual_sq_le_indicator","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, each cell's residual second moment obeys the stated envelope, and the design weight is nonnegative, the weighted residual square at one sample coordinate is bounded in expectation by the corresponding weighted cell indicator envelope.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (V : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ^ 2 ∂mu ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal * V ^ 2) → ∀ (W : (Fin n → kappa × Bool) → ℝ), (∀ (d : Fin n → kappa × Bool), 0 ≤ W d) → ∀ (i : Fin n) (a : Bool) (k : kappa), (∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * Causalean.Stat.supportedArmGroupResidual group arm Y center a k (z i) ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ V ^ 2 * ∫ (z : Fin n → Omega), W (Causalean.Stat.sampleDesign group arm z) * (Causalean.Stat.armGroupEvent group arm a k).indicator (fun x => 1) (z i) ∂MeasureTheory.Measure.pi fun x => mu","l":317,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_empiricalMass","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"Expected empirical mass. Given a positive sample size $m$ and a measurable singleton {a}, the expectation, under the $m$-fold product sampling measure, of the empirical mass of the point a — the fraction of sample observations equal to a — equals the population probability of {a}.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ}, 0 < m → ∀ (a : 𝒳), MeasurableSet {a} → (∫ (sample : Fin m → 𝒳), Causalean.Stat.empiricalMass sample a ∂MeasureTheory.Measure.pi fun x => μ) = μ.real {a}","l":60,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.integral_empiricalMass_sq","k":"theorem","m":"Causalean.Stat.Sample.EmpiricalMass","a":"Stat/Sample","t":1,"d":"Second moment of the empirical mass. Given a positive sample size $m$ and a measurable singleton {a}, the second moment, under the $m$-fold product sampling measure, of the empirical mass of the point a equals the squared population probability of {a} plus the usual binomial sampling correction $(1/m)(\\mu(\\{a\\})-\\mu(\\{a\\})^2)$.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] {m : ℕ}, 0 < m → ∀ (a : 𝒳), MeasurableSet {a} → (∫ (sample : Fin m → 𝒳), Causalean.Stat.empiricalMass sample a ^ 2 ∂MeasureTheory.Measure.pi fun x => μ) = μ.real {a} ^ 2 + (↑m)⁻¹ * (μ.real {a} - μ.real {a} ^ 2)","l":96,"f":"Causalean/Stat/Sample/EmpiricalMass.lean"},{"n":"Causalean.Stat.integral_kernelMean_eq_integral_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then its expectation after taking the kernel mean under a source probability law equals its expectation under the garbled law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∫ (x : X), Causalean.Stat.kernelMean K T x ∂P = ∫ (y : Y), T y ∂P.bind ⇑K","l":124,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.integral_le_sSup_range_of_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Bayes risk never exceeds worst-case risk. Averaging an integrable risk function over the parameter space against any prior *probability* distribution gives at most the supremum of that risk over the parameter space (assuming the risk is bounded above). This is the step that lets a minimax lower bound be certified by exhibiting a single prior and bounding its average risk from below.","s":"∀ {Θ : Type u_2} [inst : MeasurableSpace Θ] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (risk : Θ → ℝ), MeasureTheory.Integrable risk π → BddAbove (Set.range risk) → ∫ (θ : Θ), risk θ ∂π ≤ sSup (Set.range risk)","l":137,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.integral_loss_comp_le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"If the state map is measurable and the loss is nonnegative in every state, then the prior average of the composed loss is no greater than its finite worst-case risk.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (f : Θ → S), Measurable f → ∀ (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → ∀ (a : A), ∫ (θ : Θ), loss a (f θ) ∂π ≤ Causalean.Stat.worstCaseRisk loss a","l":207,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.integral_min_le_one_sub_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Affinity lower bound. ∫ min(p,1) ∂ν ≤ 1 - tvDist μ ν. Derived from Scheffé's ≤ inequality together with ∫ min(p,1) ∂ν = 1 - ½∫|p-1|∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∫ (x : Ω), min (μ.rnDeriv ν x).toReal 1 ∂ν ≤ 1 - Causalean.Stat.tvDist μ ν","l":122,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.integral_mul_condExp_eq_integral_mul_of_ae_bound","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTaylor","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the conditioning sigma-algebra is contained in the ambient sigma-algebra, an integrable complex function is multiplied by a predictable complex weight whose norm has an almost-sure finite bound, then integrating after conditional expectation gives the original weighted integral.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ], m ≤ mΩ → ∀ (f g : Ω → ℂ), MeasureTheory.Integrable f μ → MeasureTheory.StronglyMeasurable g → ∀ (B : ℝ), (∀ᵐ (ω : Ω) ∂μ, ‖g ω‖ ≤ B) → ∫ (ω : Ω), g ω * μ[f | m] ω ∂μ = ∫ (ω : Ω), g ω * f ω ∂μ","l":155,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTaylor.lean"},{"n":"Causalean.Stat.integral_norm_expQuadraticRemainder_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTaylor","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a square-integrable real random variable, a positive truncation level, and a frequency small at that level, the integrated quadratic exponential remainder is bounded by a cubic small-jump term plus the truncated second moment of the large jumps.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (X : Ω → ℝ), MeasureTheory.MemLp X 2 μ → ∀ (t η : ℝ), 0 < η → |t| * η ≤ 1 → ∫ (ω : Ω), ‖Causalean.Stat.expQuadraticRemainder (t * X ω)‖ ∂μ ≤ |t| ^ 3 * η * ∫ (ω : Ω), X ω ^ 2 ∂μ + (2 / η ^ 2 + |t| / η + t ^ 2 / 2) * ∫ (ω : Ω), if η < |X ω| then X ω ^ 2 else 0 ∂μ","l":215,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTaylor.lean"},{"n":"Causalean.Stat.integral_norm_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The L¹ norm of the product tail representation is |x| * |y|. This is the domination bound that makes the Fubini swap against an L² coupling valid: its π-integral is E|XY| < ∞ by Cauchy–Schwarz.","s":"∀ (x y : ℝ), ∫ (q : ℝ × ℝ), ‖Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2‖ ∂MeasureTheory.volume.prod MeasureTheory.volume = |x| * |y|","l":205,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_normalizedFiniteKernelStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Variance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, a finite coordinate family, a kernel, and sample size n, if the number of coordinates does not exceed the sample size, the kernel is measurable, and the kernel is integrable under the product law, the expected normalized statistic equals the kernel's product-law mean.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {ι : Type u_3} [inst_2 : Fintype ι] {k : (ι → X) → ℝ} {n : ℕ}, Fintype.card ι ≤ n → Measurable k → MeasureTheory.Integrable k (MeasureTheory.Measure.pi fun x => P) → ∫ (ω : Ω), Causalean.Stat.normalizedFiniteKernelStatistic S k n ω ∂μ = ∫ (z : ι → X), k z ∂MeasureTheory.Measure.pi fun x => P","l":489,"f":"Causalean/Stat/UStatistic/OrderM/Variance.lean"},{"n":"Causalean.Stat.integral_normalizedOrderedProductStatistic_mul","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, statistic orders and a sample size, if the first order and the second order do not exceed the sample size, then for two coordinate-function families, when every merged kernel is measurable and integrable under its product law, the expected product of their normalized statistics is the partial-matching sum of merged product-law moments.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), (∀ (M : Causalean.Stat.PartialMatching r s), Measurable (Causalean.Stat.mergedProductKernel f g M)) → (∀ (M : Causalean.Stat.PartialMatching r s), MeasureTheory.Integrable (Causalean.Stat.mergedProductKernel f g M) (MeasureTheory.Measure.pi fun x => P)) → ∫ (ω : Ω), Causalean.Stat.normalizedOrderedProductStatistic S f n ω * Causalean.Stat.normalizedOrderedProductStatistic S g n ω ∂μ = ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.mergedProductMoment P f g M","l":494,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.integral_occupancyDesignVarianceFactor_le_reciprocal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If group and arm labels are measurable, the overlap margin is positive and at most one half, and both arms receive at least that share in every positive-mass group, the expected design variance factor is at most sixteen divided by the squared margin times one minus the margin, multiplied by expected reciprocal usable occupancy. This includes zero-mass groups, empty samples, empty group types, and every zero-count boundary.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (epsilon : ℝ), 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : kappa), 0 < (mu (Causalean.Stat.groupEvent group k)).toReal → ∀ (a : Bool), epsilon * (mu (Causalean.Stat.groupEvent group k)).toReal ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal) → (∫ (z : Fin n → Omega), Causalean.Stat.occupancyDesignVarianceFactor group arm z ∂MeasureTheory.Measure.pi fun x => mu) ≤ 16 / (epsilon ^ 2 * (1 - epsilon)) * ∫ (z : Fin n → Omega), Causalean.Stat.inverseUsableGroupTotal group arm z ∂MeasureTheory.Measure.pi fun x => mu","l":443,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.integral_occupancyWeightedResidual_sq_le_reciprocal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Variance","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If group labels, arm labels, and outcomes are measurable, every supported residual has a finite second moment, residuals are centered within every arm/group cell, their cellwise second moments obey the stated envelope, the overlap margin is positive and at most one half, and both arms receive at least that share in every positive-mass group, the squared occupancy-weighted residual has product-law expectation at most sixteen divided by the squared margin times one minus the margin, multiplied by the squared envelope and expected reciprocal usable occupancy.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (V epsilon : ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ∂mu = 0) → (∀ (a : Bool) (k : kappa), ∫ (omega : Omega) in Causalean.Stat.armGroupEvent group arm a k, Causalean.Stat.armGroupResidual Y center a k omega ^ 2 ∂mu ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal * V ^ 2) → 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : kappa), 0 < (mu (Causalean.Stat.groupEvent group k)).toReal → ∀ (a : Bool), epsilon * (mu (Causalean.Stat.groupEvent group k)).toReal ≤ (mu (Causalean.Stat.armGroupEvent group arm a k)).toReal) → (∫ (z : Fin n → Omega), Causalean.Stat.occupancyWeightedResidual mu group arm Y center z ^ 2 ∂MeasureTheory.Measure.pi fun x => mu) ≤ 16 / (epsilon ^ 2 * (1 - epsilon)) * V ^ 2 * ∫ (z : Fin n → Omega), if 0 < Causalean.Stat.usableGroupTotal group arm z then (↑(Causalean.Stat.usableGroupTotal group arm z))⁻¹ else 0 ∂MeasureTheory.Measure.pi fun x => mu","l":264,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Variance.lean"},{"n":"Causalean.Stat.integral_op_of_eLpNorm_op","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"L² rate ⇒ integrated bias rate (probability measure case). If eLpNorm (f n ω) 2 ν = o_p(rₙ) and ν is a probability measure, then (∫ x, f n ω x ∂ν) = o_p(rₙ).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X : Type u_2} [inst_1 : MeasurableSpace X] {ν : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure ν] {fn : ℕ → Ω → X → ℝ}, (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (fn n ω) 2 ν) → ∀ {rn : ℕ → ℝ}, (∀ (n : ℕ), 0 < rn n) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (fn n ω) 2 ν).toReal) rn μ → Causalean.Stat.IsLittleOp (fun n ω => ∫ (x : X), fn n ω x ∂ν) rn μ","l":359,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.integral_pi_eval_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"Integrating a function of one coordinate under a finite product law recovers the one-dimensional integral.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {ι : Type u_3} [inst_2 : Fintype ι] {E : Type u_4} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace ℝ E] (i : ι) {f : X → E}, MeasureTheory.Integrable f P → (∫ (z : ι → X), f (z i) ∂MeasureTheory.Measure.pi fun x => P) = ∫ (x : X), f x ∂P","l":182,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.integral_prod_eq_integral_fiber","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"First swap. For a coupling π of μ and ν, where both marginals have finite second moment, the expectation E_π[XY] equals the Lebesgue double integral, over the plane, of the fibre integrals ∫ p, Φ q p ∂π, i.e. ∫ p, p.1 * p.2 ∂π = ∫ q, (∫ p, Φ q p ∂π) dq.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π = ∫ (q : ℝ × ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 q.1 * Causalean.Stat.signedTail p.2 q.2 ∂π ∂MeasureTheory.volume.prod MeasureTheory.volume","l":94,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.integral_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Signed tail representation. ∫ s, (𝟙{s < a} - 𝟙{s < 0}) ds = a.","s":"∀ (a : ℝ), ∫ (s : ℝ), Causalean.Stat.signedTail a s = a","l":112,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_signedTail_fst","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The π-integral of the signed tail indicator of the first coordinate is the marginal survival function minus the constant 𝟙{s<0}: ∫ p, (𝟙{s < p.1} - 𝟙{s < 0}) ∂π = SX s - 𝟙{s < 0}. Uses ∫ p, 𝟙{s < p.1} ∂π = π.real (Prod.fst ⁻¹' Ioi s) and π univ = 1.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.1 s ∂π = Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s","l":90,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.integral_signedTail_prod","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Product tail representation. For any two reals x and y, their product x * y equals the two-dimensional Lebesgue integral of the product of their signed tail indicators, ∫∫ (𝟙{s<x} - 𝟙{s<0})(𝟙{t<y} - 𝟙{t<0}) ds dt.","s":"∀ (x y : ℝ), ∫ (q : ℝ × ℝ), Causalean.Stat.signedTail x q.1 * Causalean.Stat.signedTail y q.2 ∂MeasureTheory.volume.prod MeasureTheory.volume = x * y","l":197,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.integral_signedTail_snd","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubini","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The π-integral of the signed tail indicator of the second coordinate.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (t : ℝ), ∫ (p : ℝ × ℝ), Causalean.Stat.signedTail p.2 t ∂π = Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t","l":114,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubini.lean"},{"n":"Causalean.Stat.integratedLowerBound_le_inducedFiniteDesignBayesRisk","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"If an integrated continuous-prior risk agrees estimator by estimator with the induced finite squared risk and has a common lower bound, that constant lower-bounds the real-valued finite-design Bayes-risk infimum.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [inst_8 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_9 : ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (integratedRisk : (X → ℝ) → ℝ) (B : ℝ), (∀ (T : X → ℝ), integratedRisk T = Causalean.Stat.squaredRisk (Causalean.Stat.inducedFiniteDesign π K) L t T) → (∀ (T : X → ℝ), B ≤ integratedRisk T) → B ≤ Causalean.Stat.finiteDesignBayesRisk (Causalean.Stat.inducedFiniteDesign π K) (Causalean.Stat.statewiseSquaredLoss L t)","l":366,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.integratedLowerBound_le_inducedPosteriorResidual","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"If an integrated continuous-prior risk agrees estimator by estimator with the induced finite squared risk and has a common lower bound, that bound is at most the posterior residual risk of the induced finite experiment.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [inst_8 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_9 : ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (integratedRisk : (X → ℝ) → ℝ) (B : ℝ), (∀ (T : X → ℝ), integratedRisk T = Causalean.Stat.squaredRisk (Causalean.Stat.inducedFiniteDesign π K) L t T) → (∀ (T : X → ℝ), B ≤ integratedRisk T) → B ≤ Causalean.Stat.posteriorResidual (Causalean.Stat.inducedFiniteDesign π K) L t","l":383,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.interiorIndicator","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a total count and a success count, the binomial interior indicator equals one when both the success and failure counts are positive, and equals zero when either count is zero.","s":"ℕ → ℕ → ℝ","l":99,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.inverseSqrtCap_anti","k":"theorem","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":2,"d":"The capped inverse-square-root rate is antitone on positive strengths.","s":"∀ {t0 t : ℝ}, 0 < t0 → t0 ≤ t → min 1 (t ^ (-1 / 2)) ≤ min 1 (t0 ^ (-1 / 2))","l":220,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.inverseStrength_to_frontier","k":"theorem","m":"Causalean.Stat.Inference.AffineInversion","a":"Stat/Inference","t":2,"d":"An inverse-root plus inverse-strength bound, capped by a nonnegative region volume, is bounded by the compact inverse-square-root frontier form.","s":"∀ {cap A B t : ℝ}, 0 ≤ B → 0 < t → min cap (A / √t + B / t) ≤ max cap (A + B) * min 1 (t ^ (-1 / 2))","l":240,"f":"Causalean/Stat/Inference/AffineInversion.lean"},{"n":"Causalean.Stat.inverseTwoCounts","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a total count and a success count, the two-arm inverse-count contribution is the sum of the reciprocal success and failure counts when both counts are positive, and is zero when either count is zero.","s":"ℕ → ℕ → ℝ","l":93,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.inverseUsableGroupTotal","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a finite group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, and a sample, the inverse usable-group total is the reciprocal usable-group total when that total is positive and is zero otherwise.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℝ","l":147,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.isBigOp_centered_crossFit_sum","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.CrossFitRate","a":"Stat/EmpiricalProcess","t":1,"d":"Cross-fit empirical-increment rate. Given observations W i, each of which is measurable, grouped into evaluation folds s n, each nonempty, and training σ-algebras m_A n, each contained in the ambient σ-algebra on the sample space such that the training σ-algebra m_A n is independent of the observations indexed by the fold s n and those fold observations are, conditionally, i.i.d. draws from P: for a score g n that viewed jointly in the sample point and its argument is measurable with respect to the training σ-algebra m_A n (the cross-fitting case of a fixed integrand evaluated at a nuisance estimated on the other folds) and is square-integrable under P at every sample point, and for any deterministic, nonnegative sequence Vn that dominates the average, over the training draw, of the squared L²(P)-norm of g n, the centered and rescaled evaluation-fold average of g n is stochastically bounded at the rate $\\sqrt{V_n}$.","s":"∀ {Ω : Type u_2} {X : Type u_3} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure P] (W : ℕ → Ω → X), (∀ (i : ℕ), Measurable (W i)) → ∀ (s : ℕ → Finset ℕ), (∀ (n : ℕ), 0 < (s n).card) → ∀ (m_A : ℕ → MeasurableSpace Ω), (∀ (n : ℕ), m_A n ≤ mΩ) → (∀ (n : ℕ), ProbabilityTheory.Indep (m_A n) (MeasurableSpace.comap (fun ω i => W (↑i) ω) inferInstance) μ) → (∀ (n : ℕ), MeasureTheory.Measure.map (fun ω i => W (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P) → ∀ (g : ℕ → Ω → X → ℝ), (∀ (n : ℕ), Measurable (Function.uncurry (g n))) → (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (g n ω) 2 P) → ∀ {Vn : ℕ → ℝ}, (∀ (n : ℕ), 0 ≤ Vn n) → (∀ (n : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal ((MeasureTheory.eLpNorm (g n ω) 2 P).toReal ^ 2) ∂μ ≤ ENNReal.ofReal (Vn n)) → Causalean.Stat.IsBigOp (fun n ω => (√↑(s n).card)⁻¹ * ∑ i ∈ s n, (g n ω (W i ω) - ∫ (x : X), g n ω x ∂P)) (fun n => √(Vn n)) μ","l":450,"f":"Causalean/Stat/EmpiricalProcess/CrossFitRate.lean"},{"n":"Causalean.Stat.isCoupling_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For probability measures μ and ν on the reals, the comonotone coupling of μ and ν is indeed a coupling of the pair, i.e. its two marginals recover μ and ν.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.IsCoupling (Causalean.Stat.comonotoneCoupling μ ν) μ ν","l":92,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.isCoupling_countermonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For probability measures μ and ν on the reals, the countermonotone coupling of μ and ν is indeed a coupling of the pair, i.e. its two marginals recover μ and ν.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.IsCoupling (Causalean.Stat.countermonotoneCoupling μ ν) μ ν","l":125,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.isLittleOp_abs","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The absolute value of an o_p(1) sequence is o_p(1) (the threshold events coincide).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {R : ℕ → Ω → ℝ}, Causalean.Stat.IsLittleOp R (fun x => 1) μ → Causalean.Stat.IsLittleOp (fun n ω => |R n ω|) (fun x => 1) μ","l":206,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_finset_sum_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"A finite sum of o_p(1) sequences is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} (s : Finset ι) (g : ι → ℕ → Ω → ℝ), (∀ i ∈ s, Causalean.Stat.IsLittleOp (g i) (fun x => 1) μ) → Causalean.Stat.IsLittleOp (fun n ω => ∑ i ∈ s, g i n ω) (fun x => 1) μ","l":183,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_of_abs_le_const_mul_one","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Domination: if |Xn| ≤ C·|Yn| with Yn o_p(1) and C > 0, then Xn is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {X Y : ℕ → Ω → ℝ} {C : ℝ}, 0 < C → Causalean.Stat.IsLittleOp Y (fun x => 1) μ → (∀ (n : ℕ) (ω : Ω), |X n ω| ≤ C * |Y n ω|) → Causalean.Stat.IsLittleOp X (fun x => 1) μ","l":197,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.isLittleOp_of_eventuallyEq","k":"theorem","m":"Causalean.Stat.UStatistic.Hajek","a":"Stat/UStatistic","t":1,"d":"Transfer of o_p along eventual equality. For sequences of random variables f, g : ℕ → Ω → ℝ and a rate r : ℕ → ℝ, if g is o_p(r) under μ and f n = g n for all sufficiently large n, then f is also o_p(r) under μ.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : ℕ → Ω → ℝ} {r : ℕ → ℝ}, Causalean.Stat.IsLittleOp g r μ → (∀ᶠ (n : ℕ) in Filter.atTop, f n = g n) → Causalean.Stat.IsLittleOp f r μ","l":42,"f":"Causalean/Stat/UStatistic/Hajek.lean"},{"n":"Causalean.Stat.isLittleOp_one_of_measure_ne_tendsto_zero","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"If Yₙ and Xₙ agree except on an event of vanishing probability, then Yₙ − Xₙ = o_p(1). This is the device that handles the *off-diagonal* case of the directional delta method (where the lattice functional locally selects one coordinate, so the rescaled image equals that coordinate's marginal except when the estimated ordering is wrong — an event whose probability tends to 0).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {Xn Yn : ℕ → Ω → ℝ}, Filter.Tendsto (fun n => μ {ω | Yn n ω ≠ Xn n ω}) Filter.atTop (nhds 0) → Causalean.Stat.IsLittleOp (fun n ω => Yn n ω - Xn n ω) (fun x => 1) μ","l":66,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.isLittleOp_zero","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"o_p-triviality of the zero sequence at the constant rate rₙ = 1.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp (fun x x_1 => 0) (fun x => 1) μ","l":185,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.isLittleOp_zero_one'","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The zero sequence is o_p(1).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω}, Causalean.Stat.IsLittleOp (fun x x_1 => 0) (fun x => 1) μ","l":168,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.jStatistic_chiSq","k":"theorem","m":"Causalean.Stat.GMM.OverID","a":"Stat/GMM","t":1,"d":"Headline: the whitened GMM J-statistic limit law is χ²_{m−k}. Under a standard Gaussian random vector on the moment space, the distribution of the whitened GMM J-statistic — the squared norm of the residual-maker projection of that Gaussian vector — is the chi-squared law with finrank F − finrank E = m − k degrees of freedom, the Hansen–Sargan over-identification test statistic.","s":"∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : FiniteDimensional ℝ F] [inst_6 : MeasurableSpace F] [BorelSpace F] {G : E →L[ℝ] F} {effInv : E →L[ℝ] E}, effInv ∘SL ContinuousLinearMap.adjoint G ∘SL G = ContinuousLinearMap.id ℝ E → (ContinuousLinearMap.adjoint G ∘SL G) ∘SL effInv = ContinuousLinearMap.id ℝ E → MeasureTheory.Measure.map (fun w => ‖(Causalean.Stat.gmmResidualMaker G effInv) w‖ ^ 2) (Causalean.Mathlib.stdGaussian F) = Causalean.Stat.chiSqDist (Module.finrank ℝ F - Module.finrank ℝ E)","l":205,"f":"Causalean/Stat/GMM/OverID.lean"},{"n":"Causalean.Stat.jointCdf","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffding","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a measure on pairs of real-valued quantities and real thresholds, the joint cumulative distribution function is the real-valued mass that the measure assigns to the lower-left quadrant consisting of pairs whose first coordinate is at most the first threshold and whose second coordinate is at most the second threshold.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ → ℝ","l":35,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffding.lean"},{"n":"Causalean.Stat.jointCdf_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Comonotone attainment. The joint cdf of the comonotone coupling of two probability measures μ and ν, evaluated at a point (x, y), equals the Fréchet–Hoeffding upper bound min (cdf μ x) (cdf ν y).","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.comonotoneCoupling μ ν) x y = min (↑(ProbabilityTheory.cdf μ) x) (↑(ProbabilityTheory.cdf ν) y)","l":26,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointCdf_countermonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Countermonotone attainment. The joint cdf of the countermonotone coupling of two probability measures μ and ν, evaluated at a point (x, y), equals the Fréchet–Hoeffding lower bound max (cdf μ x + cdf ν y - 1) 0.","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf (Causalean.Stat.countermonotoneCoupling μ ν) x y = max (↑(ProbabilityTheory.cdf μ) x + ↑(ProbabilityTheory.cdf ν) y - 1) 0","l":98,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointCdf_le_comonotone","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.FrechetHoeffdingAttainment","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Corollary: for any coupling π of (μ, ν), its joint cdf is dominated pointwise by the comonotone joint cdf. This is the pointwise inequality that the covariance identity turns into optimality of E[XY].","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (x y : ℝ), Causalean.Stat.jointCdf π x y ≤ Causalean.Stat.jointCdf (Causalean.Stat.comonotoneCoupling μ ν) x y","l":210,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/FrechetHoeffdingAttainment.lean"},{"n":"Causalean.Stat.jointMass","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, a latent state, and an observation, the joint mass of that state and observation is the prior mass of the state times the conditional singleton mass of the observation.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [inst_1 : MeasurableSpace S] → [inst_2 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → S → X → ℝ","l":52,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.jointMass_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Every latent-state--observation joint mass is nonnegative.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (s : S) (x : X), 0 ≤ Causalean.Stat.jointMass ν L s x","l":60,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.jointMass_sum","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The joint masses of a normalized finite prior and Markov observation kernel sum to one.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) [ProbabilityTheory.IsMarkovKernel L], ∑ s, ∑ x, Causalean.Stat.jointMass ν L s x = 1","l":65,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.jointSurv","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a measure on pairs of real-valued quantities and real thresholds, the joint survival function is the real-valued mass of pairs for which the first coordinate exceeds the first threshold and the second coordinate exceeds the second threshold.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ → ℝ","l":46,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.jointSurv_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Inclusion–exclusion. The joint survival function of a coupling is S s t = 1 - F s - G t + H_π s t. Proof: Ioi s ×ˢ Ioi t is the complement of (Prod.fst ⁻¹' Iic s) ∪ (Prod.snd ⁻¹' Iic t), and the mass of that union is F s + G t - H_π s t by measureReal_union_add_inter, the intersection being the lower-left quadrant Iic s ×ˢ Iic t.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), Causalean.Stat.jointSurv π s t = 1 - ↑(ProbabilityTheory.cdf μ) s - ↑(ProbabilityTheory.cdf ν) t + Causalean.Stat.jointCdf π s t","l":118,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.kernelAffinePullback","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"For an input measurable space, an output measurable space, and a Markov kernel from the input space to the output space, a real slope, a real offset, and a real-valued estimator on the output space, the affine kernel pullback assigns to each input the kernel mean of the estimator, minus the offset, divided by the slope.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → ProbabilityTheory.Kernel X Y → ℝ → ℝ → (Y → ℝ) → X → ℝ","l":90,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.kernelAverageLoss","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a measurable parameter space, a finite measurable state space, an action space, a probability kernel from parameters to states, a real-valued loss for each action and state, an action, and a parameter value, the kernel-average loss is the expected loss of that action under the state distribution selected by the kernel at that parameter value.","s":"{Θ : Type u_1} → {S : Type u_2} → {A : Type u_3} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace S] → ProbabilityTheory.Kernel Θ S → (A → S → ℝ) → A → Θ → ℝ","l":50,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.kernelAverageLoss_eq_sum","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"Kernel averaging on a finite state space equals the sum of statewise losses weighted by the kernel's singleton probabilities.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ) (a : A) (θ : Θ), Causalean.Stat.kernelAverageLoss K loss a θ = ∑ s, (K θ).real {s} * loss a s","l":78,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.kernelMass","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a measurable latent-state space, a measurable observation space, a Markov observation kernel from latent states to observations, a latent state, and an observation, the real-valued singleton observation mass is the kernel's mass at that observation conditional on that latent state.","s":"{S : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace S] → [inst_1 : MeasurableSpace X] → ProbabilityTheory.Kernel S X → S → X → ℝ","l":33,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.kernelMass_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Every singleton observation mass emitted by a Markov kernel is nonnegative.","s":"∀ {S : Type u_1} {X : Type u_2} [Fintype S] [Fintype X] [inst : MeasurableSpace S] [inst_1 : MeasurableSpace X] [MeasurableSingletonClass X] (L : ProbabilityTheory.Kernel S X) (s : S) (x : X), 0 ≤ Causalean.Stat.kernelMass L s x","l":40,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.kernelMass_sum","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The singleton observation masses emitted in each latent state sum to one.","s":"∀ {S : Type u_1} {X : Type u_2} [Fintype S] [inst : Fintype X] [inst_1 : MeasurableSpace S] [inst_2 : MeasurableSpace X] [MeasurableSingletonClass X] (L : ProbabilityTheory.Kernel S X) [ProbabilityTheory.IsMarkovKernel L] (s : S), ∑ x, Causalean.Stat.kernelMass L s x = 1","l":45,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.kernelMean","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"For an input measurable space, an output measurable space, and a Markov kernel from the input space to the output space and a real-valued function on the output space, the kernel mean assigns to each input the expectation of that function under the output distribution selected by the kernel.","s":"{X : Type uX} → {Y : Type uY} → [inst : MeasurableSpace X] → [inst_1 : MeasurableSpace Y] → ProbabilityTheory.Kernel X Y → (Y → ℝ) → X → ℝ","l":56,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.klForm_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"KL-form Le Cam two-point lower bound (unconditional). For probability measures P₀ and P₁ and two points θ₀, θ₁ in a pseudometric parameter space, if P₀ is absolutely continuous with respect to P₁ and their Kullback–Leibler divergence is finite, while est is a measurable estimator of the parameter and θ₀ and θ₁ are separated by at least 2s, then the worse of the two error probabilities — that est misses θ₀ by at least s under P₀, or misses θ₁ by at least s under P₁ — is at least (1 - √(klDiv(P₀,P₁)/2))/2.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ], P₀.AbsolutelyContinuous P₁ → InformationTheory.klDiv P₀ P₁ ≠ ⊤ → ∀ {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - √((InformationTheory.klDiv P₀ P₁).toReal / 2)) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":491,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klForm_two_point_lower_bound_of_pinsker","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"KL-form Le Cam two-point lower bound. Given a Pinsker-type bound h controlling the total variation distance between P₀ and P₁ by their Kullback–Leibler divergence, if est is a measurable estimator of the parameter and the parameter values θ₀ and θ₁ are separated by at least 2s, then the worse of the two error probabilities — that est misses θ₀ by at least s under P₀, or misses θ₁ by at least s under P₁ — is at least (1 - √(klDiv(P₀,P₁)/2))/2.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ], Causalean.Stat.PinskerBound P₀ P₁ → ∀ {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → (1 - √((InformationTheory.klDiv P₀ P₁).toReal / 2)) / 2 ≤ max (P₀.real {ω | s ≤ dist (est ω) θ₀}) (P₁.real {ω | s ≤ dist (est ω) θ₁})","l":465,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klFun_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":2,"d":"Pinsker scalar inequality. For x ≥ 0, (3/2) * (x - 1)^2 / (x + 2) ≤ klFun x. This is the elementary bound that powers Pinsker's inequality through Cauchy–Schwarz.","s":"∀ {x : ℝ}, 0 ≤ x → 3 / 2 * (x - 1) ^ 2 / (x + 2) ≤ InformationTheory.klFun x","l":229,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.klFun_mul_lower_aux","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":2,"d":"Auxiliary: pinskerPhi x ≥ 0 for x ≥ 0, i.e. (x + 2) * klFun x - (3/2) * (x - 1)^2 ≥ 0.","s":"∀ {x : ℝ}, 0 ≤ x → 0 ≤ (x + 2) * InformationTheory.klFun x - 3 / 2 * (x - 1) ^ 2","l":209,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.le_cam_two_point_chisq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Le Cam two-point testing floor under a χ²-budget (uniform-constant form). Two statements share this structure: for every χ²-budget C ≥ 0 there is a single floor constant c > 0, fixed before the laws are chosen, such that every pair of probability laws with Q-integrable squared density deviation and χ²-divergence at most C has combined testing error P(Aᶜ) + Q(A) at least c on every measurable test region A; and for every per-observation χ²-budget c0 ≥ 0 there is a single floor c' > 0 such that whenever the single-observation χ²-divergence is at most c0/m, the m-fold i.i.d. product experiment retains that same constant testing floor c', by the product χ²-identity 1 + χ²(P^{⊗m}‖Q^{⊗m}) = (1 + χ²)^m ≤ exp(c0).","s":"(∀ (Cchi : ℝ), 0 ≤ Cchi → ∃ c, 0 < c ∧ ∀ {Ω : Type} [inst : MeasurableSpace Ω] (Pp Pm : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Pp] [MeasureTheory.IsProbabilityMeasure Pm], Pp.AbsolutelyContinuous Pm → MeasureTheory.Integrable (fun x => ((Pp.rnDeriv Pm x).toReal - 1) ^ 2) Pm → Causalean.Stat.chiSqDiv Pp Pm ≤ Cchi → ∀ (A : Set Ω), MeasurableSet A → Pp.real Aᶜ + Pm.real A ≥ c) ∧ ∀ (c0 : ℝ), 0 ≤ c0 → ∃ c', 0 < c' ∧ ∀ {Ω : Type} [inst : MeasurableSpace Ω] (Pp Pm : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure Pp] [MeasureTheory.IsProbabilityMeasure Pm], Pp.AbsolutelyContinuous Pm → MeasureTheory.Integrable (fun x => ((Pp.rnDeriv Pm x).toReal - 1) ^ 2) Pm → ∀ (m : ℕ), 0 < m → Causalean.Stat.chiSqDiv Pp Pm ≤ c0 / ↑m → ∀ (A : Set (Fin m → Ω)), MeasurableSet A → (MeasureTheory.Measure.pi fun x => Pp).real Aᶜ + (MeasureTheory.Measure.pi fun x => Pm).real A ≥ c'","l":619,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.le_cdf_of_quantile_le","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Galois connection (one direction). If the quantile lies at or below x, then the cdf has already reached level τ at x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, τ < 1 → Causalean.Stat.quantile μ τ ≤ x → τ ≤ ↑(ProbabilityTheory.cdf μ) x","l":97,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.le_cdf_quantile","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Key membership lemma. For interior τ ∈ (0,1), the quantile lands in the super-level set: τ ≤ cdf μ (quantile μ τ). This is where right-continuity of the cdf is used.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, τ < 1 → τ ≤ ↑(ProbabilityTheory.cdf μ) (Causalean.Stat.quantile μ τ)","l":77,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A bound uniform over admissible estimators is a lower bound. If every admissible estimator has worst-case risk at least c, the minimax value is at least c. This is the converse half of a minimax rate; it needs at least one admissible estimator to exist, since otherwise the minimax value is zero by convention.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty E] {risk : E → Θ → ℝ} {c : ℝ}, (∀ (e : E), c ≤ Causalean.Stat.worstCaseRisk risk e) → c ≤ Causalean.Stat.minimaxValue risk","l":144,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.le_minimaxValue_of_two_point","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"Two-point reduction to the minimax value. Fix two models θ₀ and θ₁ from the parameter class. If every estimator's risk is bounded above across the parameter class and for each estimator, the larger of its risks at the two fixed models is at least c, then the minimax value of the problem is at least c.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty E] {risk : E → Θ → ℝ} {c : ℝ} (θ₀ θ₁ : Θ), (∀ (e : E), BddAbove (Set.range (risk e))) → (∀ (e : E), c ≤ max (risk e θ₀) (risk e θ₁)) → c ≤ Causalean.Stat.minimaxValue risk","l":153,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The risk at any single model in the class is at most the estimator's worst-case risk, provided that estimator's risk is bounded over the class.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ} {e : E}, BddAbove (Set.range (risk e)) → ∀ (θ : Θ), risk e θ ≤ Causalean.Stat.worstCaseRisk risk e","l":115,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.lintegral_ofReal_tendsto_zero_of_bdd_isLittleOp","k":"theorem","m":"Causalean.Stat.Orthogonality.ConditionalOp","a":"Stat/Orthogonality","t":2,"d":"Bounded convergence in probability ⇒ L¹ convergence. If Xn ω ∈ [0, M] with M ≥ 0 deterministic, each Xn n is measurable, and Xn = o_p(1) under μ (with μ a probability measure), then ∫⁻ ω, ENNReal.ofReal (Xn n ω) ∂μ → 0 in ℝ≥0∞.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Xn : ℕ → Ω → ℝ} {M : ℝ}, 0 ≤ M → (∀ (n : ℕ), Measurable (Xn n)) → (∀ (n : ℕ) (ω : Ω), 0 ≤ Xn n ω) → (∀ (n : ℕ) (ω : Ω), Xn n ω ≤ M) → Causalean.Stat.IsLittleOp Xn (fun x => 1) μ → Filter.Tendsto (fun n => ∫⁻ (ω : Ω), ENNReal.ofReal (Xn n ω) ∂μ) Filter.atTop (nhds 0)","l":188,"f":"Causalean/Stat/Orthogonality/ConditionalOp.lean"},{"n":"Causalean.Stat.localStochasticExpansion","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":1,"d":"Local stochastic expansion for Z-estimators. For a criterion function ψ with population Fréchet-Jacobian reg.J₀ at θ₀, an i.i.d. sample S, and an estimator sequence θn, suppose θn is consistent for θ₀, the centered empirical process of ψ is stochastically equicontinuous at θ₀ along θn, and θn approaches θ₀ at the parametric rate, ‖θn − θ₀‖ = O_p(1/√n). Then the empirical average score difference (1/√n) Σᵢ (ψ(θn,Zᵢ) − ψ(θ₀,Zᵢ)) agrees with its linearization √n · reg.J₀(θn − θ₀) up to an o_p(1) remainder in norm.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → Causalean.Stat.IsLittleOp (fun n ω => ‖(√↑n)⁻¹ • ∑ i ∈ Finset.range n, (ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)) - √↑n • reg.J₀ (θn n ω - θ₀)‖) (fun x => 1) μ","l":495,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.log_add_one_sub_log_le_inv","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Elementary per-term bound log(x+1) - log x ≤ 1/x for x > 0, i.e. log(1 + 1/x) ≤ 1/x. This is the pointwise ingredient of the harmonic-tail lower bound.","s":"∀ {x : ℝ}, 0 < x → Real.log (x + 1) - Real.log x ≤ 1 / x","l":45,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.log_lin_log_le_half_log","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":2,"d":"Sub-logarithmic growth of log (c·log T + 1): for any c > 0 there is a threshold N beyond which log (c · log T + 1) ≤ (1/2) · log T. This is the log log T = o(log T) book-keeping that lets the harmonic tail dominate the t_* = Θ(log T) cutoff. (Route: c·log T + 1 ≤ (log T)²/16 ≤ exp((log T)/2) for large T, then take log.)","s":"∀ {c : ℝ}, 0 < c → ∃ N, ∀ (T : ℕ), N ≤ T → Real.log (c * Real.log ↑T + 1) ≤ 1 / 2 * Real.log ↑T","l":127,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.loss_integrable_kernel","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"The loss from one action is integrable under every probability distribution emitted by a Markov kernel into a finite state space.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ) (a : A) (θ : Θ), MeasureTheory.Integrable (loss a) (K θ)","l":44,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mEstimator_consistent_of_bracketing","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.MEstimatorConsistency","a":"Stat/EmpiricalProcess","t":1,"d":"Bracketing corollary (the econometrician's headline). Let m be a criterion function of a parameter ranging over a pseudo-metric space Θ, observed along an i.i.d. sample S drawn from P. If each m(θ,·) is measurable, the criterion class admits, for every target width, a finite collection of integrable upper/lower bracket functions sandwiching the class members almost everywhere with L¹(P)-gap at most that width, the estimator sequence thetaHat attains a sample-objective value at every sample size and outcome that is at least as large as the sample objective at θ₀, and the population objective has a well-separated maximum at θ₀, meaning that for every ε>0 there is a gap η>0 such that the objective at any θ at distance at least ε from θ₀ falls short of the objective at θ₀ by at least η, then thetaHat is consistent for θ₀: for every ε>0 the probability that thetaHat n lies at distance at least ε from θ₀ tends to zero as the sample size n grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {Θ : Type u_3} [inst_2 : PseudoMetricSpace Θ] [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (m : Θ → X → ℝ) (θ₀ : Θ) (thetaHat : ℕ → Ω → Θ), (∀ (θ : Θ), Measurable (m θ)) → Causalean.Stat.HasL1Bracketing m P → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (thetaHat n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : Θ), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (thetaHat n ω) θ₀}) Filter.atTop (nhds 0)","l":94,"f":"Causalean/Stat/EmpiricalProcess/MEstimatorConsistency.lean"},{"n":"Causalean.Stat.mEstimator_consistent_of_glivenkoCantelli","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.MEstimatorConsistency","a":"Stat/EmpiricalProcess","t":1,"d":"Consistency of extremum estimators (Newey–McFadden 1994, Thm 2.1). Let m be a criterion function of a parameter ranging over a pseudo-metric space Θ, with population objective M(θ) equal to the expectation of m(θ,·) under P and sample objective the empirical mean of m(θ,·) along an i.i.d. sample S. If the criterion class {m(θ,·) : θ ∈ Θ} obeys the Glivenko–Cantelli uniform law, so the worst-case gap between the sample and population objectives vanishes in probability, the estimator sequence thetaHat attains a sample-objective value at every sample size and outcome that is at least as large as the sample objective at θ₀, and the population objective has a well-separated maximum at θ₀, meaning that for every ε>0 there is a gap η>0 such that the objective at any θ at distance at least ε from θ₀ falls short of the objective at θ₀ by at least η, then thetaHat is consistent for θ₀: for every ε>0 the probability that thetaHat n lies at distance at least ε from θ₀ tends to zero as the sample size n grows.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {Θ : Type u_3} [inst_2 : PseudoMetricSpace Θ] (S : Causalean.Stat.IIDSample Ω X μ P) (m : Θ → X → ℝ) (θ₀ : Θ) (thetaHat : ℕ → Ω → Θ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (thetaHat n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : Θ), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ dist (thetaHat n ω) θ₀}) Filter.atTop (nhds 0)","l":49,"f":"Causalean/Stat/EmpiricalProcess/MEstimatorConsistency.lean"},{"n":"Causalean.Stat.map_one_sub_unifOI","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Coupling","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The reflection u ↦ 1 - u preserves the uniform measure on (0,1): unifOI.map (fun u => 1 - u) = unifOI.","s":"MeasureTheory.Measure.map (fun u => 1 - u) Causalean.Stat.unifOI = Causalean.Stat.unifOI","l":63,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Coupling.lean"},{"n":"Causalean.Stat.map_pi_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Under a common probability law, applying a measurable observation rule coordinate by coordinate turns the finite product law into the finite product of the recoded marginal law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (mu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] {phi : X → Y}, Measurable phi → MeasureTheory.Measure.map (fun z i => phi (z i)) (MeasureTheory.Measure.pi fun x => mu) = MeasureTheory.Measure.pi fun x => MeasureTheory.Measure.map phi mu","l":290,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The law of the sub-index coordinates of an i.i.d. product sample is the product law over that sub-index; this is the pushforward packaging of the measure-preserving statement.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.Measure.map (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) = MeasureTheory.Measure.pi fun x => μ","l":155,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"The law of the coordinates in a finite index set of an i.i.d. product sample is the product law over that finite set; this is the pushforward packaging of the measure-preserving statement.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι), MeasureTheory.Measure.map (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) = MeasureTheory.Measure.pi fun x => μ","l":207,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.map_weighted_sum_gaussian","k":"theorem","m":"Causalean.Stat.Bootstrap.GaussianMultiplier","a":"Stat/Bootstrap","t":2,"d":"Core multiplier lemma. Let ξ : Fin n → Ω → ℝ be a mutually independent family of measurable random variables, each with law gaussianReal 0 1 (standard Gaussian). Then for any fixed weights a : Fin n → ℝ, the weighted sum ω ↦ ∑ i, a i * ξ i ω has law","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : Fin n → Ω → ℝ), ProbabilityTheory.iIndepFun ξ μ → (∀ (i : Fin n), Measurable (ξ i)) → (∀ (i : Fin n), MeasureTheory.Measure.map (ξ i) μ = ProbabilityTheory.gaussianReal 0 1) → ∀ (a : Fin n → ℝ), MeasureTheory.Measure.map (fun ω => ∑ i, a i * ξ i ω) μ = ProbabilityTheory.gaussianReal 0 ⟨∑ i, a i ^ 2, Finset.sum_nonneg fun i x => Even.pow_nonneg (even_two_mul 1) (a i)⟩","l":53,"f":"Causalean/Stat/Bootstrap/GaussianMultiplier.lean"},{"n":"Causalean.Stat.martingaleArrayCLT","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Main","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a square-integrable real martingale-difference triangular array, if the sum of conditional variances in each row converges in probability to one and the conditional Lindeberg sum converges in probability to zero at every positive threshold, then the row sums converge in distribution to the standard normal law. Row lengths may vary with the row index, and no independence between increments is assumed.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), Causalean.Stat.TendstoInProbability μ A.predictableQuadraticVariation 1 → (∀ (ε : ℝ), 0 < ε → Causalean.Stat.TendstoInProbability μ (A.conditionalLindeberg ε) 0) → Causalean.Stat.TendstoInDistribution μ A.rowSum (ProbabilityTheory.gaussianReal 0 1) (Causalean.Stat.MartingaleDifferenceArray.rowSum_aemeasurable A)","l":25,"f":"Causalean/Stat/CLT/MartingaleArray/Main.lean"},{"n":"Causalean.Stat.martingaleArrayCLT_of_conditionalFourthMoment","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Main","a":"Stat/CLT/MartingaleArray","t":2,"d":"For a square-integrable martingale-difference array whose predictable quadratic variations converge in probability to one, if all increments have finite fourth moments and the conditional fourth-moment row sums vanish in probability, then the row sums converge in distribution to the standard normal law.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), Causalean.Stat.TendstoInProbability μ A.predictableQuadraticVariation 1 → (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Causalean.Stat.TendstoInProbability μ A.conditionalFourthMoment 0 → Causalean.Stat.TendstoInDistribution μ A.rowSum (ProbabilityTheory.gaussianReal 0 1) (Causalean.Stat.MartingaleDifferenceArray.rowSum_aemeasurable A)","l":43,"f":"Causalean/Stat/CLT/MartingaleArray/Main.lean"},{"n":"Causalean.Stat.martingaleArrayCLT_of_deterministicVariance_fourthMomentSum","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Main","a":"Stat/CLT/MartingaleArray","t":2,"d":"For a square-integrable martingale-difference array, suppose its predictable quadratic variation equals a deterministic row variance, those variances converge to one, all increments have finite fourth moments, and the deterministic fourth-moment row sums vanish. Then the row sums converge in distribution to the standard normal law.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (v : ℕ → ℝ), (∀ (n : ℕ), A.predictableQuadraticVariation n =ᵐ[μ n] fun x => v n) → Filter.Tendsto v Filter.atTop (nhds 1) → (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Filter.Tendsto A.fourthMomentSum Filter.atTop (nhds 0) → Causalean.Stat.TendstoInDistribution μ A.rowSum (ProbabilityTheory.gaussianReal 0 1) (Causalean.Stat.MartingaleDifferenceArray.rowSum_aemeasurable A)","l":74,"f":"Causalean/Stat/CLT/MartingaleArray/Main.lean"},{"n":"Causalean.Stat.martingaleArrayCLT_of_fourthMomentSum","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Main","a":"Stat/CLT/MartingaleArray","t":2,"d":"For a square-integrable martingale-difference array whose predictable quadratic variations converge in probability to one, if all increments have finite fourth moments and the deterministic sum of their unconditional fourth moments vanishes, then the row sums converge in distribution to the standard normal law.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [inst : ∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), Causalean.Stat.TendstoInProbability μ A.predictableQuadraticVariation 1 → (∀ (n k : ℕ), k < A.rowLength n → MeasureTheory.MemLp (A.increment n k) 4 (μ n)) → Filter.Tendsto A.fourthMomentSum Filter.atTop (nhds 0) → Causalean.Stat.TendstoInDistribution μ A.rowSum (ProbabilityTheory.gaussianReal 0 1) (Causalean.Stat.MartingaleDifferenceArray.rowSum_aemeasurable A)","l":59,"f":"Causalean/Stat/CLT/MartingaleArray/Main.lean"},{"n":"Causalean.Stat.martingaleArrayCharFun_tendsto","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.CharacteristicFunction","a":"Stat/CLT/MartingaleArray","t":1,"d":"For a square-integrable martingale-difference triangular array, if its predictable quadratic variations converge in probability to one and its conditional Lindeberg sums converge in probability to zero at every positive threshold, then the characteristic function of each row sum converges pointwise to the characteristic function of the standard normal law.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ), Causalean.Stat.TendstoInProbability μ A.predictableQuadraticVariation 1 → (∀ (ε : ℝ), 0 < ε → Causalean.Stat.TendstoInProbability μ (A.conditionalLindeberg ε) 0) → ∀ (t : ℝ), Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map (A.rowSum n) (μ n)) t) Filter.atTop (nhds (MeasureTheory.charFun (ProbabilityTheory.gaussianReal 0 1) t))","l":123,"f":"Causalean/Stat/CLT/MartingaleArray/CharacteristicFunction.lean"},{"n":"Causalean.Stat.matchingNormalization","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"For a nonnegative sample size, two nonnegative factor orders, and a partial matching between their indices, the matching normalization is the falling factorial of the sample size at the number of distinct observations induced by the matching, divided by the product of the two marginal falling factorials.","s":"ℕ → {r s : ℕ} → Causalean.Stat.PartialMatching r s → ℝ","l":49,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.matchingNormalization_le","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderBounds","a":"Stat/UStatistic/OrderM","t":2,"d":"For sample size n, orders r and s, common order bound R, and overlap size h, a partial matching has normalization at most exp(1) divided by n to the overlap size when the first order is at most R, the second order is at most R, the sample size is at least four times R squared, and the matching has size h as claimed.","s":"∀ {n r s R h : ℕ} {M : Causalean.Stat.PartialMatching r s}, r ≤ R → s ≤ R → 4 * R ^ 2 ≤ n → M ∈ Causalean.Stat.partialMatchingsOfSize r s h → Causalean.Stat.matchingNormalization n M ≤ Real.exp 1 / ↑n ^ h","l":234,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderBounds.lean"},{"n":"Causalean.Stat.matchingNormalization_of_mem","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Given sample size n, statistic orders r and s, and overlap size h, a partial matching belonging to the size-h family, its normalization has the falling factorial of length r + s - h in the numerator.","s":"∀ {n r s h : ℕ} {M : Causalean.Stat.PartialMatching r s}, M ∈ Causalean.Stat.partialMatchingsOfSize r s h → Causalean.Stat.matchingNormalization n M = ↑(n.descFactorial (r + s - h)) / (↑(n.descFactorial r) * ↑(n.descFactorial s))","l":422,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.maxDirDeriv","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Given a first real coordinate and a second real coordinate, the directional derivative of the maximum functional at their pair sends a direction $(u,v)$ to $u$ when $b<a$, to $v$ when $a<b$, and to $\\max(u,v)$ when $a=b$.","s":"ℝ → ℝ → ℝ × ℝ → ℝ","l":161,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.maxStat_aemeasurable","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The image statistic √n (max âₙ b̂ₙ − a) is AEMeasurable for every n, *derived* from the joint rescaled deviation's measurability via sqrt_mul_max_sub — so deltaMethod_max_tie need not assume it.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (an bn : ℕ → Ω → ℝ) (a : ℝ), (∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ) → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (max (an n ω) (bn n ω) - a)) μ","l":141,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.maximalCoupling","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"Given a measurable sample space and two probability measures on it, the maximal coupling is the measure on pairs whose first branch is the diagonal coupling when their total-variation distance is zero, and whose second branch otherwise combines their common part on the diagonal with the normalized product of their residual measures.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → (mu nu : MeasureTheory.Measure X) → [MeasureTheory.IsProbabilityMeasure mu] → [MeasureTheory.IsProbabilityMeasure nu] → MeasureTheory.Measure (X × X)","l":218,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling.instIsProbabilityMeasure","k":"instance","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure mu] [inst_2 : MeasureTheory.IsProbabilityMeasure nu], MeasureTheory.IsProbabilityMeasure (Causalean.Stat.maximalCoupling mu nu)","l":350,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_eq_mass_ge","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"For two probability measures mu and nu on a standard Borel space X, the two coordinates of their maximal coupling agree with probability at least one minus their total variation distance.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] [MeasurableEq X] (mu nu : MeasureTheory.Measure X) [inst_2 : MeasureTheory.IsProbabilityMeasure mu] [inst_3 : MeasureTheory.IsProbabilityMeasure nu], ENNReal.ofReal (1 - Causalean.Stat.tvDist mu nu) ≤ (Causalean.Stat.maximalCoupling mu nu) {p | p.1 = p.2}","l":362,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_map_fst","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The first marginal of the maximal coupling is the first law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure mu] [inst_2 : MeasureTheory.IsProbabilityMeasure nu], MeasureTheory.Measure.map Prod.fst (Causalean.Stat.maximalCoupling mu nu) = mu","l":283,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximalCoupling_map_snd","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The second marginal of the maximal coupling is the second law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure mu] [inst_2 : MeasureTheory.IsProbabilityMeasure nu], MeasureTheory.Measure.map Prod.snd (Causalean.Stat.maximalCoupling mu nu) = nu","l":316,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.maximizerOrZero","k":"def","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":1,"d":"Given a domain equipped with a distinguished zero and a real-valued criterion on that domain, the selected maximizer or zero is a global maximizer when one exists and is the distinguished zero otherwise.","s":"{A : Type u_1} → [Zero A] → (A → ℝ) → A","l":33,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.mean_fst_tail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Tail formula for the mean. E[X] = ∫ s, (SX s - 𝟙{s<0}) ds, by the same swap on volume ⊗ π using integral_signedTail pointwise.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun x => x) 2 μ → ∫ (x : ℝ), x ∂μ = ∫ (s : ℝ), Causalean.Stat.survFst π s - Causalean.Stat.tailInd 0 s","l":121,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.mean_snd_tail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.HoeffdingFubiniIntegrability","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"Tail formula for the mean (second coordinate).","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (y : ℝ), y ∂ν = ∫ (t : ℝ), Causalean.Stat.survSnd π t - Causalean.Stat.tailInd 0 t","l":140,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/HoeffdingFubiniIntegrability.lean"},{"n":"Causalean.Stat.measurableEqOfStandardBorel","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"For a standard Borel measurable space, the measurability of its equality relation is defined.","s":"∀ (X : Type u_1) [inst : MeasurableSpace X] [StandardBorelSpace X], MeasurableEq X","l":21,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.measurableSet_acceptanceRegion","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"A measurable estimator has a measurable acceptance region around any target.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ (θ : Θ) (s : ℝ), MeasurableSet (Causalean.Stat.acceptanceRegion est θ s)","l":69,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.measurableSet_armGroupEvent","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels have measurable joint arm/group fibers.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), MeasurableSet (Causalean.Stat.armGroupEvent group arm a k)","l":180,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurableSet_decode_ne","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Coordinate-decoding error sets are measurable, being complements of the measurable level sets {ω | est ω j = b}.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j ≠ b}","l":95,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.measurableSet_error","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":2,"d":"The error region {ω | s ≤ dist (est ω) θ} of an estimator is measurable.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ (θ : Θ) (s : ℝ), MeasurableSet {ω | s ≤ dist (est ω) θ}","l":40,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.measurableSet_finiteCategoryPilotGood","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a selection threshold, a lower mass band, and an upper mass band, the simultaneous finite-category pilot good event is measurable.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : Fintype Iota] [inst_4 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (t lowerBand upperBand : ℝ), MeasurableSet (Causalean.Stat.finiteCategoryPilotGood S label block t lowerBand upperBand)","l":354,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurableSet_groupEvent","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"A measurable group label has measurable group fibers.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa), Measurable group → ∀ (k : kappa), MeasurableSet (Causalean.Stat.groupEvent group k)","l":174,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurableSet_selectorCell","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"When the selection rule is measurable, the designated pilot event is measurable, and a branch is fixed, its selector cell is measurable.","s":"∀ {Alpha : Type u_2} {Iota : Type u_4} [inst : MeasurableSpace Alpha] [inst_1 : MeasurableSpace Iota] [MeasurableSingletonClass Iota] {select : Alpha → Iota}, Measurable select → ∀ {good : Set Alpha}, MeasurableSet good → ∀ (i : Iota), MeasurableSet (Causalean.Stat.selectorCell select good i)","l":45,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurableSet_usableGroup","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the event that a fixed empirical group has both arms represented measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), MeasurableSet {z | Causalean.Stat.usableGroup group arm z k}","l":244,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_affinePullbackEstimator","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the observation rule is measurable and the target estimator is measurable, then undoing an affine change after pulling the estimator back through the observation rule is measurable.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {phi : X → Y} {a b : ℝ} {targetEst : Y → ℝ}, Measurable phi → Measurable targetEst → Measurable (Causalean.Stat.affinePullbackEstimator phi a b targetEst)","l":194,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.measurable_armGroupEvent_indicator","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Restricting a measurable function to a joint arm/group cell — keeping its values on the cell and setting it to zero outside — leaves it measurable, whenever the group and arm labels are measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {beta : Type u_3} [inst_5 : MeasurableSpace beta] [inst_6 : Zero beta] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa) {f : Omega → beta}, Measurable f → Measurable ((Causalean.Stat.armGroupEvent group arm a k).indicator f)","l":190,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_armGroupResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"A measurable outcome makes each arm/group-centered residual measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable Y → ∀ (a : Bool) (k : kappa), Measurable (Causalean.Stat.armGroupResidual Y center a k)","l":283,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_armResidualMean","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each zero-safe arm/group residual mean measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.armResidualMean group arm Y center z a k","l":322,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_armResidualSum","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each arm/group residual sum measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.armResidualSum group arm Y center z a k","l":305,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_cappedStatistic","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the finite-sample statistic is measurable, then the capped statistic is jointly measurable in the fixed array and auxiliary count.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {n : ℕ} {T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ}, Measurable T → ∀ (zOver : ℝ), Measurable fun z => Causalean.Stat.cappedStatistic T zOver z.1 z.2","l":430,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.measurable_categoryIndicator","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given a measurable category label and a fixed category, the corresponding zero-one category indicator is measurable.","s":"∀ {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Iota] [inst_2 : DecidableEq Iota] [MeasurableSingletonClass Iota] {label : X → Iota}, Measurable label → ∀ (k : Iota), Measurable (Causalean.Stat.categoryIndicator label k)","l":47,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurable_cdfIF","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The empirical-cdf influence function is measurable.","s":"∀ {P : MeasureTheory.Measure ℝ} (y : ℝ), Measurable (Causalean.Stat.cdfIF P y)","l":123,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.measurable_cdfStat","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"The lower-ray indicator is measurable.","s":"∀ (y : ℝ), Measurable (Causalean.Stat.cdfStat y)","l":67,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.measurable_finCoordinatewise","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Applying a measurable observation rule separately to every position of a finite sample produces a measurable recoded sample.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) {phi : X → Y}, Measurable phi → Measurable fun z i => phi (z i)","l":283,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurable_finiteSelector_apply","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"When finite branch selection is measurable and each branch-specific error is measurable, evaluating the selected error on a pilot-tail pair is measurable.","s":"∀ {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Alpha] [inst_1 : MeasurableSpace Beta] [inst_2 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {select : Alpha → Iota}, Measurable select → ∀ {err : Iota → Beta → ℝ}, (∀ (i : Iota), Measurable (err i)) → Measurable fun z => err (select z.1) z.2","l":54,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurable_finiteSelector_sq","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":2,"d":"If the pilot coordinate is measurable, the tail coordinate is measurable, the finite selection rule is measurable, and every branch error is measurable, the selected squared error is measurable.","s":"∀ {Omega : Type u_1} {Alpha : Type u_2} {Beta : Type u_3} {Iota : Type u_4} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace Alpha] [inst_2 : MeasurableSpace Beta] [inst_3 : MeasurableSpace Iota] [Fintype Iota] [MeasurableSingletonClass Iota] {pilot : Omega → Alpha} {tail : Omega → Beta} {select : Alpha → Iota} {err : Iota → Beta → ℝ}, Measurable pilot → Measurable tail → Measurable select → (∀ (i : Iota), Measurable (err i)) → Measurable fun omega => err (select (pilot omega)) (tail omega) ^ 2","l":75,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.measurable_groupArmCount","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make each fixed arm/group count measurable on a finite product sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (a : Bool) (k : kappa), Measurable fun z => Causalean.Stat.groupArmCount group arm z a k","l":218,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_groupCount","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make each fixed group count measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa), Measurable fun z => Causalean.Stat.groupCount group arm z k","l":231,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_inverseUsableGroupTotal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the zero-safe reciprocal usable occupancy measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.inverseUsableGroupTotal group arm)","l":382,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_kernelAffinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Measurability of a target estimator ensures that its affine kernel pullback is measurable on the source experiment.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) {a b : ℝ} {targetEst : Y → ℝ}, Measurable targetEst → Measurable (Causalean.Stat.kernelAffinePullback K a b targetEst)","l":99,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.measurable_kernelMean","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Measurability of a real-valued estimator ensures that its expectation under each kernel output distribution varies measurably with the kernel input.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) {T : Y → ℝ}, Measurable T → Measurable (Causalean.Stat.kernelMean K T)","l":63,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.measurable_kernel_singletonReal","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A finite-state Markov kernel's probability of a fixed state is measurable in its source parameter.","s":"∀ {Θ : Type u_1} {S : Type u_2} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (s : S), Measurable fun θ => (K θ).real {s}","l":85,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.measurable_occupancyDesignVarianceFactor","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the zero-safe occupancy design variance factor measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.occupancyDesignVarianceFactor group arm)","l":400,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_occupancyWeightedResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Measurable group labels, arm labels, and outcomes make the zero-safe occupancy-weighted residual measurable on the finite product sample space.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → Measurable (Causalean.Stat.occupancyWeightedResidual mu group arm Y center)","l":344,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_pilotCategoryCount","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite coordinate block, and a fixed category, the pilot count of that category is measurable.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota), Measurable (Causalean.Stat.pilotCategoryCount S label block k)","l":81,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.measurable_quantileIF","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The sample-quantile influence function is measurable.","s":"∀ (τ q₀ f₀ : ℝ), Measurable (Causalean.Stat.quantileIF τ q₀ f₀)","l":78,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.measurable_quantileIFVec","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The joint quantile influence function is measurable.","s":"∀ {k : ℕ} (τ q f : Fin k → ℝ), Measurable (Causalean.Stat.quantileIFVec τ q f)","l":77,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.measurable_sampleDesign","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the coordinatewise finite design measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable (Causalean.Stat.sampleDesign group arm)","l":205,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_signedTail","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is measurable in the tail variable s.","s":"∀ (a : ℝ), Measurable (Causalean.Stat.signedTail a)","l":68,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.measurable_signedTail_uncurry","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail is jointly measurable in (a, s), since {z | z.2 < z.1} is an open (hence measurable) subset of ℝ × ℝ.","s":"Measurable fun z => Causalean.Stat.signedTail z.1 z.2","l":75,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.measurable_supportedArmGroupResidual","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group labels, arm labels, and outcomes make each residual restricted to its own arm/group cell measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → ∀ (a : Bool) (k : kappa), Measurable (Causalean.Stat.supportedArmGroupResidual group arm Y center a k)","l":293,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_usableGroupTotal","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"Measurable group and arm labels make the total occupancy in empirically usable groups measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → Measurable fun z => Causalean.Stat.usableGroupTotal group arm z","l":270,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurable_usableGroup_ite","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"A quantity defined by two branches, selected according to whether a fixed empirical group has both arms represented, is measurable whenever both branches are and the group and arm labels are measurable.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} {beta : Type u_3} [inst_5 : MeasurableSpace beta] (group : Omega → kappa) (arm : Omega → Bool), Measurable group → Measurable arm → ∀ (k : kappa) [inst_6 : DecidablePred fun z => Causalean.Stat.usableGroup group arm z k] {f g : (Fin n → Omega) → beta}, Measurable f → Measurable g → Measurable fun z => if Causalean.Stat.usableGroup group arm z k then f z else g z","l":255,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a decidable sub-index of an i.i.d. product sample leaves the i.i.d. product sample over that sub-index.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => μ)","l":147,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict_dep","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a decidable sub-index of a product of probability measures leaves the product measure over that sub-index: the retained coordinates carry exactly their own product law, with no trace of the discarded ones.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : ι → Type u_4} [inst_1 : (i : ι) → MeasurableSpace (X i)] (μ : (i : ι) → MeasureTheory.Measure (X i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (p : ι → Prop) [inst_3 : DecidablePred p], MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.pi fun i => μ ↑i)","l":130,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measurePreserving_pi_restrict_finset","k":"theorem","m":"Causalean.Stat.Sample.PiTransport","a":"Stat/Sample","t":2,"d":"Dropping the coordinates outside a finite set of indices of an i.i.d. product sample leaves the i.i.d. product sample indexed by that finite set.","s":"∀ {ι : Type u_3} [inst : Fintype ι] {X : Type u_4} [inst_1 : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (S : Finset ι), MeasureTheory.MeasurePreserving (fun s i => s ↑i) (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => μ)","l":198,"f":"Causalean/Stat/Sample/PiTransport.lean"},{"n":"Causalean.Stat.measureReal_sub_eq_setIntegral_rnDeriv_sub_one","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":2,"d":"With density p = dμ/dν and μ ≪ ν, the signed gap μ.real A − ν.real A equals ∫ x in A, (p x − 1) ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∀ {A : Set Ω}, MeasurableSet A → μ.real A - ν.real A = ∫ (x : Ω) in A, (μ.rnDeriv ν x).toReal - 1 ∂ν","l":94,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.measureReal_sub_le_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The signed gap ν.real A − μ.real A is at most tvDist μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → ν.real A - μ.real A ≤ Causalean.Stat.tvDist μ ν","l":80,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.measure_eq_of_tvDist_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Probability measures at total-variation distance zero are equal.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu], Causalean.Stat.tvDist mu nu = 0 → mu = nu","l":202,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.mem_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"A partial matching belongs to the family with h pairs exactly when its matching size is h.","s":"∀ {r s h : ℕ} (M : Causalean.Stat.PartialMatching r s), M ∈ Causalean.Stat.partialMatchingsOfSize r s h ↔ M.size = h","l":158,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.mergedProductKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"For two nonnegative orders, a family of real-valued left factors indexed by the first order, a family of real-valued right factors indexed by the second order, and a partial matching between their indices, the merged product kernel assigns to each collection of observations indexed by the merged coordinates the product of all left and right factors, evaluating matched factors at their common observation and unmatched factors at separate observations.","s":"{X : Type u_2} → {r s : ℕ} → (Fin r → X → ℝ) → (Fin s → X → ℝ) → (M : Causalean.Stat.PartialMatching r s) → (M.MergedIndex → X) → ℝ","l":26,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.mergedProductMoment","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"For two nonnegative orders, a measure on a measurable observation space, families of real-valued left and right factors, and a partial matching between their indices, the merged product moment is the integral of the associated merged product kernel under independent draws from that measure, one draw for each merged coordinate.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {r s : ℕ} → MeasureTheory.Measure X → (Fin r → X → ℝ) → (Fin s → X → ℝ) → Causalean.Stat.PartialMatching r s → ℝ","l":40,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.mergedProductMoment_empty","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"Under a probability law, for two coordinate-function families of orders r and s, if the first product kernel is measurable, the second product kernel is measurable, the first product kernel is integrable, and the second product kernel is integrable, the empty matching's merged moment factors into the two separate product-law means.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {r s : ℕ} (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), Measurable (Causalean.Stat.orderedProductKernel f) → Measurable (Causalean.Stat.orderedProductKernel g) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel f) (MeasureTheory.Measure.pi fun x => P) → MeasureTheory.Integrable (Causalean.Stat.orderedProductKernel g) (MeasureTheory.Measure.pi fun x => P) → Causalean.Stat.mergedProductMoment P f g (Causalean.Stat.PartialMatching.empty r s) = Causalean.Stat.orderedProductMean P f * Causalean.Stat.orderedProductMean P g","l":550,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.minDirDeriv","k":"def","m":"Causalean.Stat.Inference.HadamardDeriv","a":"Stat/Inference","t":1,"d":"Given a first real coordinate and a second real coordinate, the directional derivative of the minimum functional at their pair sends a direction $(u,v)$ to $u$ when $a<b$, to $v$ when $b<a$, and to $\\min(u,v)$ when $a=b$.","s":"ℝ → ℝ → ℝ × ℝ → ℝ","l":169,"f":"Causalean/Stat/Inference/HadamardDeriv.lean"},{"n":"Causalean.Stat.minStat_aemeasurable","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The image statistic √n (min âₙ b̂ₙ − a) is AEMeasurable for every n, derived likewise via sqrt_mul_min_sub.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (an bn : ℕ → Ω → ℝ) (a : ℝ), (∀ (n : ℕ), AEMeasurable (fun ω => √↑n • ((an n ω, bn n ω) - (a, a))) μ) → ∀ (n : ℕ), AEMeasurable (fun ω => √↑n * (min (an n ω) (bn n ω) - a)) μ","l":153,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.minimaxValue","k":"def","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"Given an estimator class, a model class, and a real-valued risk for each estimator-model pair, the minimax value is the infimum, over all estimators in the class, of their worst-case risks over the model class.","s":"{E : Sort u_1} → {Θ : Sort u_3} → (E → Θ → ℝ) → ℝ","l":54,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_eq_sInf_range","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value is the infimum of the set of worst-case risks the admissible estimators attain. Rewriting handle for arguments phrased on that set.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} (risk : E → Θ → ℝ), Causalean.Stat.minimaxValue risk = sInf (Set.range (Causalean.Stat.worstCaseRisk risk))","l":254,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Comparison of two minimax problems. If every admissible estimator of the second problem is matched by an estimator of the first whose worst-case risk is no larger, the first minimax value is at most the second. Both restricting the model class and enlarging the estimator family are instances; the boundedness hypothesis is on the first problem's worst-case risks, and the second problem must have at least one estimator.","s":"∀ {E : Sort u_1} {E' : Sort u_2} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E'] {risk : E → Θ → ℝ} {risk' : E' → Θ' → ℝ}, BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → (∀ (e' : E'), ∃ e, Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e') → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":196,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Any admissible estimator is an upper bound. Exhibiting one estimator and bounding its worst-case risk bounds the minimax value, which is how achievability half of a minimax rate is certified. The hypothesis rules out a worst-case risk that decreases without bound across the estimator family.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → ∀ (e : E), Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.worstCaseRisk risk e","l":128,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_le_worstCaseRisk_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"For a nonnegative risk, any admissible estimator is an upper bound on the minimax value with no further side condition, since zero already bounds the worst-case risks below.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → ∀ (e : E), Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.worstCaseRisk risk e","l":137,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_mono_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Restricting the model class cannot raise the minimax value. If each model of the smaller class sits, through the map, inside the larger class with no larger risk, then the minimax value over the smaller class is at most the minimax value over the larger one — the same estimators face a weaker adversary. This is the standard step that transfers a published converse proved on a convenient subclass to the full class.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E] [Nonempty Θ] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} (φ : Θ → Θ'), BddBelow (Set.range (Causalean.Stat.worstCaseRisk risk)) → (∀ (e : E), BddAbove (Set.range (risk' e))) → (∀ (e : E) (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":209,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_mono_class_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"For a map φ embedding the parameter class of the first problem into that of the second, if the first problem's risk is nonnegative, the second problem's risk is nonnegative, each estimator's risk is bounded above across the second parameter class, and the first risk at any model is dominated by the second risk at that model's image under φ, then the minimax value of the first problem is at most that of the second.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty E] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} (φ : Θ → Θ'), (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → (∀ (e : E) (θ' : Θ'), 0 ≤ risk' e θ') → (∀ (e : E), BddAbove (Set.range (risk' e))) → (∀ (e : E) (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.minimaxValue risk ≤ Causalean.Stat.minimaxValue risk'","l":223,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value of a nonnegative-risk problem is nonnegative, with no boundedness or nonemptiness caveat. Squared-error, absolute-error and regret losses all qualify.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ}, (∀ (e : E) (θ : Θ), 0 ≤ risk e θ) → 0 ≤ Causalean.Stat.minimaxValue risk","l":97,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_of_isEmpty_estimators","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"When no estimator is admissible the minimax value is zero, so no positive lower bound on the minimax value can hold vacuously.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [IsEmpty E] (risk : E → Θ → ℝ), Causalean.Stat.minimaxValue risk = 0","l":73,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_subtype_eq_sInf_setOf","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value over an admissible-estimator condition and a model-class condition, written with set comprehensions instead of subtypes: the infimum of the set of worst-case risks attained by admissible estimators, each of which is the supremum of the set of risks attained on the class. Runs that spell their minimax risk this way rewrite with this lemma and then use the rest of this file.","s":"∀ {A : Type u_5} {B : Type u_6} (Adm : A → Prop) (Cls : B → Prop) (risk : A → B → ℝ), (Causalean.Stat.minimaxValue fun e θ => risk ↑e ↑θ) = sInf {r | ∃ e, Adm e ∧ r = sSup {q | ∃ θ, Cls θ ∧ q = risk e θ}}","l":295,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.minimaxValue_subtype_eq_sInf_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The minimax value over the estimators satisfying an admissibility condition, written as the infimum of the set of worst-case risks the admissible estimators attain. This is the bridge for developments that state admissibility by a predicate rather than by a subtype while leaving the inner supremum as it stands.","s":"∀ {Θ : Sort u_3} {A : Type u_5} {Adm : A → Prop} (risk : A → Θ → ℝ), (Causalean.Stat.minimaxValue fun e => risk ↑e) = sInf {r | ∃ e, Adm e ∧ r = Causalean.Stat.worstCaseRisk risk e}","l":262,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.mixedIntegratedLowerBound_le_inducedFiniteDesignBayesRisk","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"If an integrated risk agrees estimator by estimator with the generic continuous mixture  and has a common lower bound, that bound transfers to the corresponding real-valued induced finite-design Bayes risk.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [inst_8 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_9 : ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (integratedRisk : (X → ℝ) → ℝ) (B : ℝ), (∀ (T : X → ℝ), integratedRisk T = Causalean.Stat.mixedKernelLoss π K (Causalean.Stat.statewiseSquaredLoss L t) T) → (∀ (T : X → ℝ), B ≤ integratedRisk T) → B ≤ Causalean.Stat.finiteDesignBayesRisk (Causalean.Stat.inducedFiniteDesign π K) (Causalean.Stat.statewiseSquaredLoss L t)","l":397,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.mixedIntegratedLowerBound_le_realBayesRisk","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"If an integrated risk agrees estimator by estimator with the generic continuous mixture  and has a common lower bound, that bound transfers to the real-valued continuous-mixture Bayes risk.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (integratedRisk : (X → ℝ) → ℝ) (B : ℝ), (∀ (T : X → ℝ), integratedRisk T = Causalean.Stat.mixedKernelLoss π K (Causalean.Stat.statewiseSquaredLoss L t) T) → (∀ (T : X → ℝ), B ≤ integratedRisk T) → B ≤ Causalean.Stat.realBayesRisk π K (Causalean.Stat.statewiseSquaredLoss L t)","l":417,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.mixedKernelLoss","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a measurable parameter space, a finite measurable state space, an action space, a prior measure on the parameter space, a probability kernel from parameters to states, a real-valued loss for each action and state, and an action, the mixed kernel loss is that action's kernel-average loss integrated with respect to the prior.","s":"{Θ : Type u_1} → {S : Type u_2} → {A : Type u_3} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace S] → MeasureTheory.Measure Θ → ProbabilityTheory.Kernel Θ S → (A → S → ℝ) → A → ℝ","l":102,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mixedKernelLoss_deterministic","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A measurable deterministic state map has the same mixed kernel loss as the ordinary prior integral of its composed loss.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (f : Θ → S) (hf : Measurable f) (loss : A → S → ℝ) (a : A), Causalean.Stat.mixedKernelLoss π (ProbabilityTheory.Kernel.deterministic f hf) loss a = ∫ (θ : Θ), loss a (f θ) ∂π","l":195,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mixedKernelLoss_eq_sum_integral","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A continuous-prior mixture through a finite-state kernel equals the sum of statewise losses weighted by the prior-averaged singleton probabilities.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ) (a : A), Causalean.Stat.mixedKernelLoss π K loss a = ∑ s, (∫ (θ : Θ), (K θ).real {s} ∂π) * loss a s","l":109,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mixedKernelLoss_le_worstCaseRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"If loss is nonnegative in every state, then the continuous-prior average of each action's finite-state kernel loss is no greater than that action's finite worst-case risk.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → ∀ (a : A), ∫ (θ : Θ), ∫ (s : S), loss a s ∂K θ ∂π ≤ Causalean.Stat.worstCaseRisk loss a","l":129,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mixedKernelLoss_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"If statewise loss is nonnegative, then its prior-and-kernel mixture is nonnegative.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → ∀ (a : A), 0 ≤ Causalean.Stat.mixedKernelLoss π K loss a","l":121,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.mixedKernelLoss_statewiseSquaredLoss_eq_inducedSquaredRisk","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The continuous-prior mixture of an estimator's finite-observation squared loss is the real-valued squared risk under the induced finite design.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [inst_8 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_9 : ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (T : X → ℝ), Causalean.Stat.mixedKernelLoss π K (Causalean.Stat.statewiseSquaredLoss L t) T = Causalean.Stat.squaredRisk (Causalean.Stat.inducedFiniteDesign π K) L t T","l":343,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.mixture","k":"def","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":1,"d":"Given a sample space equipped with a σ-algebra, a finite index set, nonnegative extended-real weights indexed by that set, and a measure for each index, the finite mixture measure is the sum of the component measures, each scaled by its corresponding weight.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → {ι : Type u_2} → [Fintype ι] → (ι → ENNReal) → (ι → MeasureTheory.Measure Ω) → MeasureTheory.Measure Ω","l":35,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixtureReal_le","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":1,"d":"Domination. If the mixture weights w sum to 1, each component measure P i is a probability measure, and every component assigns .real-mass at most B to the set A, then the mixture measure also assigns .real-mass at most B to A.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] (A : Set Ω) (B : ℝ), (∀ (i : ι), (P i).real A ≤ B) → (Causalean.Stat.mixture w P).real A ≤ B","l":57,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixture_apply","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"Evaluation: the mixture's mass on a set is the weighted sum of the parts' masses.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal) (P : ι → MeasureTheory.Measure Ω) (A : Set Ω), (Causalean.Stat.mixture w P) A = ∑ i, w i * (P i) A","l":42,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.mixture_isProbabilityMeasure","k":"theorem","m":"Causalean.Stat.Minimax.Mixture","a":"Stat/Minimax","t":2,"d":"If the weights sum to 1 and each part is a probability measure, the mixture is one.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {ι : Type u_2} [inst : Fintype ι] (w : ι → ENNReal), ∑ i, w i = 1 → ∀ (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)], MeasureTheory.IsProbabilityMeasure (Causalean.Stat.mixture w P)","l":48,"f":"Causalean/Stat/Minimax/Mixture.lean"},{"n":"Causalean.Stat.monotoneOn_quantile","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The quantile function is monotone on the open unit interval (0,1) (immediate from quantile_mono, whose hypotheses 0 < τ, τ' < 1 hold throughout the interior).","s":"∀ (μ : MeasureTheory.Measure ℝ), MonotoneOn (Causalean.Stat.quantile μ) (Set.Ioo 0 1)","l":50,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.mse_integrable_of_estimator_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"Squared-loss integrability for a truncated estimator. If a measurable estimator T takes values in the bounded interval [-M, M] (with M ≥ 0), then under any finite measure Q its squared loss (T − θ)² against an arbitrary target θ is integrable, because it is bounded by the constant (M + |θ|)². This is the routine integrability bookkeeping needed before the worst-case squared risk of a truncated estimator can be compared in a two-point lower bound.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (Q : MeasureTheory.Measure S) [MeasureTheory.IsFiniteMeasure Q] (T : S → ℝ), Measurable T → ∀ {M theta : ℝ}, 0 ≤ M → (∀ (s : S), T s ∈ Set.Icc (-M) M) → MeasureTheory.Integrable (fun s => (T s - theta) ^ 2) Q","l":116,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.mul_exp_neg_half_log","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"For y ≥ 0, y * exp(-(1/2)·log y) = √y.","s":"∀ {y : ℝ}, 0 ≤ y → y * Real.exp (-(1 / 2) * Real.log y) = √y","l":77,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.multiplierBootstrap_law","k":"theorem","m":"Causalean.Stat.Bootstrap.GaussianMultiplier","a":"Stat/Bootstrap","t":1,"d":"Exact law of the √n-scaled multiplier-bootstrap mean. Fix a sample size n, data x : Fin n → ℝ, and a multiplier family ξ. If ξ is independent across coordinates, each ξ i is measurable, and each ξ i has the standard Gaussian law, then the scaled multiplier-bootstrap statistic $n^{-1/2}\\sum_i (x_i-\\bar x)\\,\\xi_i$ has exactly the centered Gaussian law with variance equal to the sample variance $n^{-1}\\sum_i(x_i-\\bar x)^2$.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {n : ℕ} (ξ : Fin n → Ω → ℝ), ProbabilityTheory.iIndepFun ξ μ → (∀ (i : Fin n), Measurable (ξ i)) → (∀ (i : Fin n), MeasureTheory.Measure.map (ξ i) μ = ProbabilityTheory.gaussianReal 0 1) → ∀ (x : Fin n → ℝ), MeasureTheory.Measure.map (fun ω => (√↑n)⁻¹ * ∑ i, (x i - (↑n)⁻¹ * ∑ j, x j) * ξ i ω) μ = ProbabilityTheory.gaussianReal 0 ⟨(↑n)⁻¹ * ∑ i, (x i - (↑n)⁻¹ * ∑ j, x j) ^ 2, mul_nonneg (inv_nonneg_of_nonneg (Nat.cast_nonneg' n)) (Finset.sum_nonneg fun i x_1 => Even.pow_nonneg (even_two_mul 1) (x i - (↑n)⁻¹ * ∑ j, x j))⟩","l":128,"f":"Causalean/Stat/Bootstrap/GaussianMultiplier.lean"},{"n":"Causalean.Stat.noAtoms_chiSqDist","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"χ²_d has no atoms for d ≥ 1.","s":"∀ {d : ℕ}, 1 ≤ d → MeasureTheory.NullSingletonClass (Causalean.Stat.chiSqDist d)","l":167,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.noAtoms_pi_normSq","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"The law of the sum of n ≥ 1 squared independent standard normals has no atoms. The squared first coordinate already has an atomless law (its preimage under squaring is finite, and gaussianReal is atomless), and adding the independent remainder preserves atomlessness via Fubini.","s":"∀ {n : ℕ}, 1 ≤ n → MeasureTheory.NullSingletonClass (MeasureTheory.Measure.map (fun w => ∑ i, w i ^ 2) (MeasureTheory.Measure.pi fun x => ProbabilityTheory.gaussianReal 0 1))","l":110,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.nonempty_quantileSet","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"For τ < 1, the super-level set is nonempty: since cdf μ → 1 at +∞, some point has cdf above τ.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ : ℝ}, τ < 1 → (Causalean.Stat.quantileSet μ τ).Nonempty","l":71,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.norm_eucl_le_sum_abs","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"The Euclidean norm of a packed vector is bounded by the ℓ¹ norm of its coordinates: ‖eucl v‖ ≤ ∑ j, |v j|.","s":"∀ {k : ℕ} (v : Fin k → ℝ), ‖Causalean.Stat.eucl v‖ ≤ ∑ j, |v j|","l":151,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.norm_expQuadraticRemainder_le_cube","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ExponentialBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the real argument has absolute value at most one, then the norm of the quadratic Taylor remainder of exp(iu) is at most |u|³.","s":"∀ (u : ℝ), |u| ≤ 1 → ‖Causalean.Stat.expQuadraticRemainder u‖ ≤ |u| ^ 3","l":25,"f":"Causalean/Stat/CLT/MartingaleArray/ExponentialBounds.lean"},{"n":"Causalean.Stat.norm_expQuadraticRemainder_le_global","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ExponentialBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"For every real argument, the norm of the quadratic Taylor remainder of exp(iu) is bounded by 2 + |u| + u²/2.","s":"∀ (u : ℝ), ‖Causalean.Stat.expQuadraticRemainder u‖ ≤ 2 + |u| + u ^ 2 / 2","l":63,"f":"Causalean/Stat/CLT/MartingaleArray/ExponentialBounds.lean"},{"n":"Causalean.Stat.norm_expQuadraticRemainder_le_truncated","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ExponentialBounds","a":"Stat/CLT/MartingaleArray","t":1,"d":"Given a frequency, an increment, and a positive truncation level small enough that |t| · η ≤ 1 as assumed, the quadratic exponential remainder is bounded by a small cubic contribution plus a quadratic contribution supported on increments larger than η.","s":"∀ (t x η : ℝ), 0 < η → |t| * η ≤ 1 → ‖Causalean.Stat.expQuadraticRemainder (t * x)‖ ≤ |t| ^ 3 * η * x ^ 2 + (2 / η ^ 2 + |t| / η + t ^ 2 / 2) * if η < |x| then x ^ 2 else 0","l":95,"f":"Causalean/Stat/CLT/MartingaleArray/ExponentialBounds.lean"},{"n":"Causalean.Stat.norm_exp_mul_quadraticFactor_sub_one_le","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.GaussianBounds","a":"Stat/CLT/MartingaleArray","t":2,"d":"For nonnegative coefficients and a complex Taylor error, one compensated Gaussian step differs from one by at most the exponential weight times the Taylor error plus a quadratic Euler error.","s":"∀ (c v : ℝ) (r : ℂ), 0 ≤ c → 0 ≤ v → ‖↑(Real.exp (c * v)) * (1 - ↑(c * v) + r) - 1‖ ≤ Real.exp (c * v) * (‖r‖ + (c * v) ^ 2 / 2)","l":70,"f":"Causalean/Stat/CLT/MartingaleArray/GaussianBounds.lean"},{"n":"Causalean.Stat.norm_integral_mul_condExp_le_of_ae_bound","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ConditionalTaylor","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the conditioning sigma-algebra is contained in the ambient sigma-algebra, an integrable complex error is multiplied by a predictable complex weight whose norm is bounded by a nonnegative constant, then the norm of the integral of the weight times the conditional error is at most that constant times the integral norm of the original error.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ], m ≤ mΩ → ∀ (f g : Ω → ℂ), MeasureTheory.Integrable f μ → MeasureTheory.StronglyMeasurable g → ∀ (B : ℝ), 0 ≤ B → (∀ᵐ (ω : Ω) ∂μ, ‖g ω‖ ≤ B) → ‖∫ (ω : Ω), g ω * μ[f | m] ω ∂μ‖ ≤ B * ∫ (ω : Ω), ‖f ω‖ ∂μ","l":179,"f":"Causalean/Stat/CLT/MartingaleArray/ConditionalTaylor.lean"},{"n":"Causalean.Stat.norm_sq_eucl","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":2,"d":"Euclidean norm of a packed vector, squared, is the coordinatewise sum of squares.","s":"∀ {k : ℕ} (v : Fin k → ℝ), ‖Causalean.Stat.eucl v‖ ^ 2 = ∑ j, v j ^ 2","l":143,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.normalizedCovarianceDiscrepancy","k":"def","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":1,"d":"For positive-definite candidate and target covariances, the normalized Gaussian covariance discrepancy subtracts the criterion's value at the target.","s":"{V : Type u_1} → [inst : Fintype V] → [DecidableEq V] → Causalean.Stat.PositiveCovariance V → Causalean.Stat.PositiveCovariance V → ℝ","l":34,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.normalizedCovarianceDiscrepancy_eq_zero_iff","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"For positive-definite candidate and target covariances, the normalized Gaussian covariance discrepancy is zero exactly when the two covariances agree.","s":"∀ {V : Type u_1} [inst : Fintype V] [inst_1 : DecidableEq V] (K T : Causalean.Stat.PositiveCovariance V), Causalean.Stat.normalizedCovarianceDiscrepancy K T = 0 ↔ K = T","l":229,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.normalizedCovarianceDiscrepancy_nonneg","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"For positive-definite candidate and target covariances, the normalized Gaussian covariance discrepancy is nonnegative.","s":"∀ {V : Type u_1} [inst : Fintype V] [inst_1 : DecidableEq V] (K T : Causalean.Stat.PositiveCovariance V), 0 ≤ Causalean.Stat.normalizedCovarianceDiscrepancy K T","l":218,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.normalizedFiniteKernelStatistic","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, a finite coordinate family, a real-valued kernel indexed by that family, and a sample size, the normalized finite-kernel statistic maps each sample outcome to the average kernel value over every injective assignment of the coordinate family to the first $n$ sample positions. This is an average when the family has at most $n$ members; otherwise there are no such assignments, the normalizing count is zero, and the value is zero by the inverse-of-zero convention.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {ι : Type u_3} → [Fintype ι] → ((ι → X) → ℝ) → ℕ → Ω → ℝ","l":449,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedFiniteKernelStatistic_fin_eq_uStatisticOrder","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, an order-r kernel, and sample size n, the normalized finite-kernel statistic agrees with the existing fixed-order U-statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r : ℕ} (k : (Fin r → X) → ℝ) (n : ℕ), Causalean.Stat.normalizedFiniteKernelStatistic S k n = Causalean.Stat.uStatisticOrder S k n","l":465,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, an order, one real-valued function of an observation for each coordinate, and a sample size, the normalized ordered-product statistic maps each sample outcome to the average, over all injective ordered $r$-tuples from its first $n$ observations, of the product of the corresponding coordinate-specific function values. For $r > n$ there are no such tuples and the value is zero by convention.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {r : ℕ} → (Fin r → X → ℝ) → ℕ → Ω → ℝ","l":459,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic_eq_uStatisticOrder","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, a family of order-r coordinate functions, and sample size n, the normalized ordered-product statistic is the existing fixed-order U-statistic applied to their product kernel.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r : ℕ} (f : Fin r → X → ℝ) (n : ℕ), Causalean.Stat.normalizedOrderedProductStatistic S f n = Causalean.Stat.uStatisticOrder S (Causalean.Stat.orderedProductKernel f) n","l":483,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.normalizedOrderedProductStatistic_mul_expansion","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":2,"d":"For an i.i.d. sample, statistic orders and a sample size, if the first order does not exceed the sample size and the second order does not exceed the sample size, then the pointwise product of the statistics built from two coordinate-function families equals the sum of normalized merged-kernel statistics over all partial matchings, grouped by overlap size.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {r s n : ℕ}, r ≤ n → s ≤ n → ∀ (f : Fin r → X → ℝ) (g : Fin s → X → ℝ), (fun ω => Causalean.Stat.normalizedOrderedProductStatistic S f n ω * Causalean.Stat.normalizedOrderedProductStatistic S g n ω) = fun ω => ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, Causalean.Stat.matchingNormalization n M * Causalean.Stat.normalizedFiniteKernelStatistic S (Causalean.Stat.mergedProductKernel f g M) n ω","l":433,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.observationMass","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, and an observation, the observation marginal mass is the sum, over all latent states, of their joint masses with that observation.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [inst_1 : MeasurableSpace S] → [inst_2 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → X → ℝ","l":73,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.observationMass_mul_posteriorMean","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Observation mass times the guarded posterior mean equals the corresponding joint weighted target numerator, including on null fibers.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (x : X), Causalean.Stat.observationMass ν L x * Causalean.Stat.posteriorMean ν L t x = ∑ s, Causalean.Stat.jointMass ν L s x * t s x","l":178,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.observationMass_mul_posteriorWeight","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Multiplying an observation marginal by its guarded posterior weight recovers the joint atom, including on null fibers.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (x : X) (s : S), Causalean.Stat.observationMass ν L x * Causalean.Stat.posteriorWeight ν L x s = Causalean.Stat.jointMass ν L s x","l":140,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.observationMass_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Every observation marginal mass is nonnegative.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (x : X), 0 ≤ Causalean.Stat.observationMass ν L x","l":80,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.observationMass_sum","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The observation marginal masses sum to one.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) [ProbabilityTheory.IsMarkovKernel L], ∑ x, Causalean.Stat.observationMass ν L x = 1","l":85,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.occupancyDesignVarianceFactor","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a finite group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, and a sample, the occupancy design-variance factor is the squared reciprocal usable-group total times the sum, over usable groups, of squared group counts times the two reciprocal arm counts, and is zero when no group is usable.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℝ","l":157,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.occupancyWeightedResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a measurable sample space, a finite group-label space whose labels can be compared for equality, a nonnegative integer sample size, a measure on the sample space, a group-label map, a Boolean arm-assignment map, a real-valued outcome, arm--group centers, and a sample, the occupancy-weighted residual statistic is the usable-group-occupancy-weighted average of treated-minus-control residual means, totalized to zero when no group is usable.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [inst : MeasurableSpace Omega] → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → MeasureTheory.Measure Omega → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → (Fin n → Omega) → ℝ","l":127,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.occupancyWeightedResidual_memLp_two","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Variance","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"If group labels, arm labels, and outcomes are measurable and every arm/group-supported residual has a finite second moment, the zero-safe occupancy-weighted residual has a finite second moment under every finite independent product sample.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [inst : MeasurableSpace Omega] [inst_1 : Fintype kappa] [inst_2 : DecidableEq kappa] [inst_3 : MeasurableSpace kappa] [MeasurableSingletonClass kappa] {n : ℕ} (mu : MeasureTheory.Measure Omega) [MeasureTheory.IsProbabilityMeasure mu] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ), Measurable group → Measurable arm → Measurable Y → (∀ (a : Bool) (k : kappa), MeasureTheory.MemLp (Causalean.Stat.supportedArmGroupResidual group arm Y center a k) 2 mu) → MeasureTheory.MemLp (Causalean.Stat.occupancyWeightedResidual mu group arm Y center) 2 (MeasureTheory.Measure.pi fun x => mu)","l":156,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Variance.lean"},{"n":"Causalean.Stat.oneShot_iid","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":2,"d":"OneShot evaluation-fold i.i.d. product law. Public because the orthogonal-learning modulus chain consumes it as the joint-law bridge between fold B and Measure.pi (see Estimation/OrthogonalLearning/LocalEmpProcess/Rademacher.lean).","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) (n : ℕ), MeasureTheory.Measure.map (fun ω i => S.Z (↑i) ω) μ = MeasureTheory.Measure.pi fun x => P","l":319,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.one_add_chiSqDiv_attachKernel","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredKernel","a":"Stat/Minimax","t":1,"d":"For a probability design law, two measurable probability marking kernels, pointwise absolute continuity of the first mark law with respect to the second, and integrability of their jointly measurable squared density deviation, one plus the χ²-divergence of the resulting marked laws equals the design-average of one plus the conditional χ²-divergences.","s":"∀ {D : Type u_1} {M : Type u_2} [inst : MeasurableSpace D] [inst_1 : MeasurableSpace M] [inst_2 : MeasurableSpace.CountableOrCountablyGenerated D M] (m : MeasureTheory.Measure D) [MeasureTheory.IsProbabilityMeasure m] (kappa eta : ProbabilityTheory.Kernel D M) [ProbabilityTheory.IsMarkovKernel kappa] [ProbabilityTheory.IsMarkovKernel eta], (∀ (x : D), (kappa x).AbsolutelyContinuous (eta x)) → MeasureTheory.Integrable (fun p => ((kappa.rnDeriv eta p.1 p.2).toReal - 1) ^ 2) (Causalean.Stat.attachKernel m eta) → 1 + Causalean.Stat.chiSqDiv (Causalean.Stat.attachKernel m kappa) (Causalean.Stat.attachKernel m eta) = ∫ (x : D), 1 + Causalean.Stat.chiSqDiv (kappa x) (eta x) ∂m","l":40,"f":"Causalean/Stat/Minimax/ChiSquaredKernel.lean"},{"n":"Causalean.Stat.one_add_chiSqDiv_pi_iid","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"n-fold i.i.d. tensorization of the χ²-divergence on a finite sample space. For probability measures μ, ν on a finite space Ω with μ absolutely continuous with respect to ν, the χ²-divergence of the n-fold i.i.d. product laws satisfies 1 + χ²(μ^⊗n ‖ ν^⊗n) = (1 + χ²(μ‖ν))^n.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] [Fintype Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), 1 + Causalean.Stat.chiSqDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) = (1 + Causalean.Stat.chiSqDiv μ ν) ^ n","l":294,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.one_add_chiSqDiv_pi_iid_general","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"n-fold i.i.d. tensorization of the χ²-divergence on a GENERAL measurable space. For probability measures μ, ν on any measurable space S such that μ is absolutely continuous with respect to ν and the single-sample squared density deviation (dμ/dν − 1)² is ν-integrable, the χ²-divergence of the n-fold i.i.d. product tensorizes multiplicatively: 1 + χ²(μ^⊗n ‖ ν^⊗n) = (1 + χ²(μ‖ν))^n.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (μ ν : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → ∀ (n : ℕ), 1 + Causalean.Stat.chiSqDiv (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν) = (1 + Causalean.Stat.chiSqDiv μ ν) ^ n","l":411,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.one_le_secondMoment_of_mean_one","k":"theorem","m":"Causalean.Stat.Sample.EffectiveSampleSize","a":"Stat/Sample","t":2,"d":"A mean-one square-integrable weight has second moment at least one, so its Kish design effect cannot improve on an equally weighted sample.","s":"∀ {𝒳 : Type u_1} [inst : MeasurableSpace 𝒳] (μ : MeasureTheory.Measure 𝒳) [MeasureTheory.IsProbabilityMeasure μ] (w : 𝒳 → ℝ), MeasureTheory.MemLp w 2 μ → ∫ (x : 𝒳), w x ∂μ = 1 → 1 ≤ ∫ (x : 𝒳), w x ^ 2 ∂μ","l":33,"f":"Causalean/Stat/Sample/EffectiveSampleSize.lean"},{"n":"Causalean.Stat.one_sub_prod_le_sum","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":2,"d":"For finitely many numbers in the unit interval, the amount by which their product falls short of one is at most the total shortfall of the individual factors. Applied to coordinate affinities this is the union-bound step that turns a tensorized affinity into a sum of coordinate defects.","s":"∀ {ι : Type u_1} [inst : Fintype ι] (a : ι → ℝ), (∀ (i : ι), 0 ≤ a i) → (∀ (i : ι), a i ≤ 1) → 1 - ∏ i, a i ≤ ∑ i, (1 - a i)","l":298,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.one_sub_tvDist_le_error_sum","k":"theorem","m":"Causalean.Stat.Minimax.LeCam","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound (summed form). For a measurable estimator est, if the parameter values θ₀, θ₁ are 2s-separated, then the two error probabilities sum to at least 1 − tvDist P₀ P₁.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {est : Ω → Θ}, Measurable est → ∀ {θ₀ θ₁ : Θ} {s : ℝ}, 2 * s ≤ dist θ₀ θ₁ → 1 - Causalean.Stat.tvDist P₀ P₁ ≤ P₀.real {ω | s ≤ dist (est ω) θ₀} + P₁.real {ω | s ≤ dist (est ω) θ₁}","l":47,"f":"Causalean/Stat/Minimax/LeCam.lean"},{"n":"Causalean.Stat.one_sub_tvDist_le_test","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":1,"d":"Le Cam testing bound. For probability measures μ and ν on the same space, any measurable rejection region A yields a total testing error — the probability of A under μ plus the probability of the complement of A under ν — that is at least 1 − tvDist(μ,ν).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {A : Set Ω}, MeasurableSet A → 1 - Causalean.Stat.tvDist μ ν ≤ μ.real A + ν.real Aᶜ","l":102,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.orderDegenerateNegligible_of_residual","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.RemainderNegligible","a":"Stat/UStatistic/OrderM","t":1,"d":"The higher-order remainder of a fixed-order U-statistic is negligible. For an i.i.d. sample S and order-m kernel h, write g for the higher-order Hájek residual of h. If g is measurable and square-integrable under the m-fold product law, and if for every coordinate integrating h over the remaining m − 1 coordinates yields an integrable function of that coordinate with the same population mean uMeanOrder h P in every coordinate and h remains integrable in the remaining coordinates for every fixed value of that coordinate, then the √n-rescaled higher-order residual U-statistic √n · Gₙ converges to zero in probability, i.e. it is o_p(1). This discharges the OrderDegenerateNegligible hypothesis consumed by the order-m CLT uStatisticOrder_clt (Causalean.Stat.UStatistic.OrderM.CLT).","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uDegenOrder h P) → MeasureTheory.Integrable (fun z => Causalean.Stat.uDegenOrder h P z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → Causalean.Stat.OrderDegenerateNegligible S h","l":130,"f":"Causalean/Stat/UStatistic/OrderM/RemainderNegligible.lean"},{"n":"Causalean.Stat.orderedProductKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an observation space, an order, and one real-valued function of an observation for each coordinate, the ordered-product kernel maps an ordered $r$-tuple to the product of its coordinate-specific function values.","s":"{X : Type u_2} → {r : ℕ} → (Fin r → X → ℝ) → (Fin r → X) → ℝ","l":455,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.orderedProductMean","k":"def","m":"Causalean.Stat.UStatistic.OrderM.MixedOrderCovariance","a":"Stat/UStatistic/OrderM","t":1,"d":"For a nonnegative order, a measure on a measurable observation space, and a family of real-valued coordinate factors, the ordered-product mean is the integral of their coordinatewise product under independent draws from that measure, one draw for every coordinate.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {r : ℕ} → MeasureTheory.Measure X → (Fin r → X → ℝ) → ℝ","l":536,"f":"Causalean/Stat/UStatistic/OrderM/MixedOrderCovariance.lean"},{"n":"Causalean.Stat.overlap_ge_exp_neg_klBudget","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A nonnegative finite KL budget yields the corresponding Bretagnolle--Huber lower bound on testing overlap.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {B : ℝ}, 0 ≤ B → InformationTheory.klDiv μ ν ≤ ENNReal.ofReal B → 1 / 2 * Real.exp (-B) ≤ 1 - Causalean.Stat.tvDist μ ν","l":261,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.pairKernel","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an observation space and a real-valued binary kernel, the corresponding two-coordinate kernel maps an ordered pair to the binary kernel evaluated at its first and second entries.","s":"{X : Type u_2} → (X → X → ℝ) → (Fin 2 → X) → ℝ","l":320,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.partialMatchingEquivSigma","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For two nonnegative coordinate-set sizes, the partial-matching representation equivalence bijects partial matchings with a selected subset of each coordinate set and a bijection between the two selected subsets.","s":"(r s : ℕ) → Causalean.Stat.PartialMatching r s ≃ (left : Finset (Fin r)) × (right : Finset (Fin s)) × (↑↑left ≃ ↑↑right)","l":114,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.partialMatchingFintype","k":"instance","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"For every pair of nonnegative integer coordinate-set sizes, the collection of partial matchings between coordinate sets of those sizes is finite.","s":"(r s : ℕ) → Fintype (Causalean.Stat.PartialMatching r s)","l":145,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.partialMatchingsOfSize","k":"def","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":1,"d":"For two nonnegative coordinate-set sizes and a prescribed number of pairs, the fixed-size partial-matching family is the finite set of all partial matchings having exactly that prescribed number of paired coordinates.","s":"(r s : ℕ) → ℕ → Finset (Causalean.Stat.PartialMatching r s)","l":151,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.pi_iid_absolutelyContinuous","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"The n-fold product of μ is absolutely continuous w.r.t. that of ν whenever μ ≪ ν (for sigma-finite factors). Proved by induction via the piFinSuccAbove equivalence and the binary AbsolutelyContinuous.prod.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν], μ.AbsolutelyContinuous ν → ∀ (n : ℕ), (MeasureTheory.Measure.pi fun x => μ).AbsolutelyContinuous (MeasureTheory.Measure.pi fun x => ν)","l":269,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.pi_iid_integrable_sq_dev","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Integrability of the squared density deviation (dμ/dν − 1)² propagates from one sample to the n-fold i.i.d. product (d(μ^⊗n)/d(ν^⊗n) − 1)². On a general (possibly continuous) measurable space this is the side-condition that makes chiSqDiv_prod applicable inside the tensorization induction — it is free on a finite space (Integrable.of_finite) but must be derived here, by L²(ν)-tensorization of the single-sample density dμ/dν.","s":"∀ {S : Type u_2} [inst : MeasurableSpace S] (μ ν : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → ∀ (n : ℕ), MeasureTheory.Integrable (fun x => (((MeasureTheory.Measure.pi fun x => μ).rnDeriv (MeasureTheory.Measure.pi fun x => ν) x).toReal - 1) ^ 2) (MeasureTheory.Measure.pi fun x => ν)","l":324,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.pi_real_singleton","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":2,"d":"Real-valued product point mass. Under the product measure Measure.pi μ of a family of probability measures, the mass of a single point factorizes as the product of the marginal point masses: (Measure.pi μ).real {ω} = ∏ᵢ (μ i).real {ωᵢ}. Rewrites the singleton as Set.univ.pi (fun i => {ωᵢ}), applies Measure.pi_pi, then distributes toReal over the finite product.","s":"∀ {ι : Type u_1} [inst : Fintype ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] [∀ (i : ι), MeasurableSingletonClass (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (ω : (i : ι) → Ω i), (MeasureTheory.Measure.pi μ).real {ω} = ∏ i, (μ i).real {ω i}","l":81,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.pilotCategoryCount","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given a measurable sample space and observation space, a category-label space whose labels can be compared for equality, a sample-space measure and population observation measure, an independent and identically distributed sample, a category-label map, a finite set of sample coordinates, a category, and a sample-space outcome, the pilot category count is the number of selected coordinates whose observed label equals that category.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Omega X mu P → [DecidableEq Iota] → (X → Iota) → Finset ℕ → Iota → Omega → ℕ","l":70,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_cast_eq_sum","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a category label, a finite coordinate block, a category, and a sample outcome, the real-valued pilot count equals the sum of category indicators over the block.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] (S : Causalean.Stat.IIDSample Omega X mu P) (label : X → Iota) (block : Finset ℕ) (k : Iota) (omega : Omega), ↑(Causalean.Stat.pilotCategoryCount S label block k omega) = ∑ j ∈ block, Causalean.Stat.categoryIndicator label k (S.Z j omega)","l":116,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_lower_tail_of_tilt","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a fixed category, a lower bound on that category's population mass, an exponential tilt, and nonpositivity of the tilt, the probability that the category count is at most the specified level is bounded by the corresponding Chernoff exponent.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota) {p a : ℝ}, p ≤ Causalean.Stat.categoryMass P label k → ∀ s ≤ 0, mu.real {omega | ↑(Causalean.Stat.pilotCategoryCount S label block k omega) ≤ a} ≤ Real.exp (-s * a + ↑block.card * (p * (Real.exp s - 1)))","l":230,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotCategoryCount_upper_tail_of_tilt","k":"theorem","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":2,"d":"Given an iid sample, a measurable category label, a finite pilot block, a fixed category, an upper bound on that category's population mass, an exponential tilt, and nonnegativity of the tilt, the probability that the category count exceeds the specified level is bounded by the corresponding Chernoff exponent.","s":"∀ {Omega : Type u_1} {X : Type u_2} {Iota : Type u_3} [inst : MeasurableSpace Omega] [inst_1 : MeasurableSpace X] [inst_2 : MeasurableSpace Iota] {mu : MeasureTheory.Measure Omega} {P : MeasureTheory.Measure X} [inst_3 : DecidableEq Iota] [MeasurableSingletonClass Iota] (S : Causalean.Stat.IIDSample Omega X mu P) {label : X → Iota}, Measurable label → ∀ (block : Finset ℕ) (k : Iota) {p a : ℝ}, Causalean.Stat.categoryMass P label k ≤ p → ∀ (s : ℝ), 0 ≤ s → mu.real {omega | a < ↑(Causalean.Stat.pilotCategoryCount S label block k omega)} ≤ Real.exp (-s * a + ↑block.card * (p * (Real.exp s - 1)))","l":128,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pilotSelected","k":"def","m":"Causalean.Stat.SampleSplit.FiniteCategoryPilot","a":"Stat/SampleSplit","t":1,"d":"Given a measurable sample space and observation space, a finite category-label space whose labels can be compared for equality, a sample-space measure and population observation measure, an independent and identically distributed sample, a category-label map, a finite pilot-coordinate block, a real threshold, and a sample-space outcome, the selected pilot categories are exactly the categories whose pilot counts strictly exceed the threshold.","s":"{Omega : Type u_1} → {X : Type u_2} → {Iota : Type u_3} → [inst : MeasurableSpace Omega] → [inst_1 : MeasurableSpace X] → {mu : MeasureTheory.Measure Omega} → {P : MeasureTheory.Measure X} → [Fintype Iota] → [DecidableEq Iota] → Causalean.Stat.IIDSample Omega X mu P → (X → Iota) → Finset ℕ → ℝ → Omega → Finset Iota","l":327,"f":"Causalean/Stat/SampleSplit/FiniteCategoryPilot.lean"},{"n":"Causalean.Stat.pinskerBound_of_ac_of_ne_top","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"Pinsker's inequality (unconditional). For probability measures μ and ν on the same space, if μ is absolutely continuous with respect to ν and their Kullback–Leibler divergence is finite, then the total variation distance between μ and ν is at most the square root of half their Kullback–Leibler divergence: tvDist μ ν ≤ √(klDiv(μ,ν)/2).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → InformationTheory.klDiv μ ν ≠ ⊤ → Causalean.Stat.PinskerBound μ ν","l":288,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.pinskerBound_pi_iid","k":"theorem","m":"Causalean.Stat.Minimax.Pinsker","a":"Stat/Minimax","t":1,"d":"Pinsker's inequality for i.i.d. finite products (unconditional). For one-sample probability measures μ and ν, if μ is absolutely continuous with respect to ν and the log-likelihood ratio of μ against ν is integrable under μ, then the n-fold product measures μ^{⊗n} and ν^{⊗n} satisfy Pinsker's inequality: their total variation distance is at most the square root of half their Kullback–Leibler divergence.","s":"∀ {α : Type u_2} [inst : MeasurableSpace α] (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ → ∀ (n : ℕ), Causalean.Stat.PinskerBound (MeasureTheory.Measure.pi fun x => μ) (MeasureTheory.Measure.pi fun x => ν)","l":442,"f":"Causalean/Stat/Minimax/Pinsker.lean"},{"n":"Causalean.Stat.poissonCell_linear_coercive_bound","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"A positive-mean Poisson cell log likelihood is bounded above by a linearly coercive function. This is the elementary tail estimate used in the finite Poisson maximizer existence argument.","s":"∀ (m x : ℝ), 0 < m → m * x - Real.exp x ≤ (m + 1) ^ 2 / 2 - min m 1 * |x|","l":42,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.poissonCell_strictConcave_midpoint","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"Strict midpoint concavity of a positive-weight Poisson cell whenever the two linear predictors differ.","s":"∀ (q m x y : ℝ), 0 < q → x ≠ y → (q * (m * x - Real.exp x) + q * (m * y - Real.exp y)) / 2 < q * (m * ((x + y) / 2) - Real.exp ((x + y) / 2))","l":63,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.populationMinimizers","k":"def","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":1,"d":"For a population-loss vector, the population-minimizer class contains exactly the model indices attaining its smallest loss.","s":"{ι : Type u_2} → (ι → ℝ) → Set ι","l":78,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.populationMinimizers_covarianceModelLoss_iff","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given covariance models that are compact and nonempty, a positive-definite truth contained in some model, and a model index, the population minimizers are exactly the indices whose models contain the truth.","s":"∀ {V : Type u_2} {I : Type u_3} [inst : Fintype V] [inst_1 : DecidableEq V] [Fintype I] [Nonempty I] (models : I → Set (Causalean.Stat.PositiveCovariance V)), (∀ (i : I), IsCompact (models i)) → (∀ (i : I), (models i).Nonempty) → ∀ (T : Causalean.Stat.PositiveCovariance V), (∃ i, T ∈ models i) → ∀ (i : I), (i ∈ Causalean.Stat.populationMinimizers fun j => Causalean.Stat.covarianceModelLoss (models j) ↑T) ↔ T ∈ models i","l":429,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.populationScoreDiff_eq_jacobian_plus_remainder","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(c) Population Fréchet expansion of the score difference.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P), (fun θ => ∫ (z : X), ψ θ z - ψ θ₀ z ∂P - reg.J₀ (θ - θ₀)) =o[nhds θ₀] fun θ => ‖θ - θ₀‖","l":455,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.posteriorMean","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, an observation-dependent real target, and an observation, the guarded posterior mean is the sum over latent states of the guarded posterior weight times that target at the state and observation.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [inst_1 : MeasurableSpace S] → [inst_2 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → (S → X → ℝ) → X → ℝ","l":169,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorMean_minimizes","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The guarded posterior mean minimizes squared risk among all real estimators.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (T : X → ℝ), Causalean.Stat.squaredRisk ν L t (Causalean.Stat.posteriorMean ν L t) ≤ Causalean.Stat.squaredRisk ν L t T","l":277,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorResidual","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a finite measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, and an observation-dependent real target, the posterior residual risk is the sum over observations of their marginal mass times the posterior-weighted squared deviation of the target from its guarded posterior mean.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [Fintype X] → [inst_2 : MeasurableSpace S] → [inst_3 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → (S → X → ℝ) → ℝ","l":211,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, an observation, and a latent state, the guarded posterior weight is zero when the observation has zero marginal mass and is otherwise the joint mass of the state and observation divided by that marginal mass.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [inst_1 : MeasurableSpace S] → [inst_2 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → X → S → ℝ","l":93,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"Every guarded posterior weight is nonnegative.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (x : X) (s : S), 0 ≤ Causalean.Stat.posteriorWeight ν L x s","l":114,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight_of_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"On a null observation fiber, every guarded posterior weight is zero.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) {x : X}, Causalean.Stat.observationMass ν L x = 0 → ∀ (s : S), Causalean.Stat.posteriorWeight ν L x s = 0","l":108,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight_of_ne_zero","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"On a nonzero observation fiber, the guarded posterior weight is the usual Bayes ratio.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) {x : X}, Causalean.Stat.observationMass ν L x ≠ 0 → ∀ (s : S), Causalean.Stat.posteriorWeight ν L x s = Causalean.Stat.jointMass ν L s x / Causalean.Stat.observationMass ν L x","l":101,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight_sum_of_eq_zero","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"On a null observation fiber, the guarded posterior weights sum to zero.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) {x : X}, Causalean.Stat.observationMass ν L x = 0 → ∑ s, Causalean.Stat.posteriorWeight ν L x s = 0","l":134,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posteriorWeight_sum_of_pos","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"On a positive-mass observation fiber, the guarded posterior weights sum to one.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) {x : X}, 0 < Causalean.Stat.observationMass ν L x → ∑ s, Causalean.Stat.posteriorWeight ν L x s = 1","l":123,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.posterior_centered_sum_of_pos","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"On a positive-mass observation fiber, posterior target residuals have weighted mean zero.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) {x : X}, 0 < Causalean.Stat.observationMass ν L x → ∑ s, Causalean.Stat.posteriorWeight ν L x s * (t s x - Causalean.Stat.posteriorMean ν L t x) = 0","l":188,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.predictableQuadraticVariation_tendstoInProbability_of_ae_eq","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.Lyapunov","a":"Stat/CLT/MartingaleArray","t":2,"d":"If the predictable quadratic variation equals a deterministic scalar in every row and those scalars converge to one, then the predictable quadratic variation converges in probability to one.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (A : Causalean.Stat.MartingaleDifferenceArray Ω μ) (v : ℕ → ℝ), (∀ (n : ℕ), A.predictableQuadraticVariation n =ᵐ[μ n] fun x => v n) → Filter.Tendsto v Filter.atTop (nhds 1) → Causalean.Stat.TendstoInProbability μ A.predictableQuadraticVariation 1","l":25,"f":"Causalean/Stat/CLT/MartingaleArray/Lyapunov.lean"},{"n":"Causalean.Stat.prod_one_sub_le_exp_neg_card_mul","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A common coordinatewise overlap floor c bounds the complementary product by exp (-M * c).","s":"∀ {M : ℕ} (ρ : Fin M → ℝ) (c : ℝ), (∀ (j : Fin M), ρ j ≤ 1) → (∀ (j : Fin M), c ≤ ρ j) → ∏ j, (1 - ρ j) ≤ Real.exp (-↑M * c)","l":302,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.prod_one_sub_le_exp_neg_sum","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A product of complementary overlap probabilities is bounded by the exponential of minus their sum.","s":"∀ {M : ℕ} (ρ : Fin M → ℝ), (∀ (j : Fin M), ρ j ≤ 1) → ∏ j, (1 - ρ j) ≤ Real.exp (-∑ j, ρ j)","l":283,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.product_expectation_comonotoneCoupling","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Closed form of the optimum. For two probability measures μ and ν, each with finite second moment, the expectation of the coordinate product under the comonotone (quantile) coupling of μ and ν equals the integral, over (0,1), of the product of their quantile functions:","s":"∀ (μ ν : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.comonotoneCoupling μ ν = ∫ (u : ℝ) in Set.Ioo 0 1, Causalean.Stat.quantile μ u * Causalean.Stat.quantile ν u","l":99,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.product_expectation_le_comonotone","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Optimality","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Upper optimality. For any coupling π of μ and ν, where both marginals have finite second moment, the expectation of the coordinate product under π is at most its expectation under the comonotone (quantile) coupling of μ and ν:","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], MeasureTheory.MemLp (fun x => x) 2 μ → MeasureTheory.MemLp (fun y => y) 2 ν → ∫ (p : ℝ × ℝ), p.1 * p.2 ∂π ≤ ∫ (p : ℝ × ℝ), p.1 * p.2 ∂Causalean.Stat.comonotoneCoupling μ ν","l":36,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Optimality.lean"},{"n":"Causalean.Stat.quantile","k":"def","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"Given a measure on the real line and a real level, the lower quantile is the infimum of all real numbers at which the measure's cumulative distribution function is at least that level.","s":"MeasureTheory.Measure ℝ → ℝ → ℝ","l":50,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantileIF","k":"def","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"For a quantile index $\\tau$, a population quantile $q_0$, and a density value $f_0$, the sample-quantile influence function maps a real observation $z$ to $(\\tau-\\mathbf{1}\\{z\\le q_0\\})/f_0$.","s":"ℝ → ℝ → ℝ → ℝ → ℝ","l":69,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIFVec","k":"def","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"For a nonnegative integer $k$ specifying a finite collection of quantile coordinates and coordinatewise quantile levels, population quantiles, and density values, the joint quantile influence function maps each real-valued observation $z$ to the Euclidean vector whose $j$th coordinate is $(\\tau_j-\\mathbf{1}\\{z\\le q_j\\})/f_j$.","s":"{k : ℕ} → (Fin k → ℝ) → (Fin k → ℝ) → (Fin k → ℝ) → ℝ → EuclideanSpace ℝ (Fin k)","l":64,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.quantileIF_cross","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantileJoint","a":"Stat/Quantile","t":1,"d":"Covariance entry. Given qⱼ is the population τⱼ-quantile: $F(q_j)=\\tau_j$ and qₗ is the population τₗ-quantile: $F(q_l)=\\tau_l$, the cross-moment of the two quantile influence functions $\\psi_{\\tau_j}$ and $\\psi_{\\tau_l}$ under the population measure equals $(\\min(\\tau_j,\\tau_l)-\\tau_j\\tau_l)/(f_jf_l)$.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τj qj fj τl ql fl : ℝ}, ↑(ProbabilityTheory.cdf P) qj = τj → ↑(ProbabilityTheory.cdf P) ql = τl → ∫ (z : ℝ), Causalean.Stat.quantileIF τj qj fj z * Causalean.Stat.quantileIF τl ql fl z ∂P = (min τj τl - τj * τl) / (fj * fl)","l":86,"f":"Causalean/Stat/Quantile/SampleQuantileJoint.lean"},{"n":"Causalean.Stat.quantileIF_mean_zero","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The quantile influence function has mean zero under P, given that q₀ is the population τ-quantile (F(q₀) = τ).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, ↑(ProbabilityTheory.cdf P) q₀ = τ → ∫ (z : ℝ), Causalean.Stat.quantileIF τ q₀ f₀ z ∂P = 0","l":83,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIF_sq_integrable","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":2,"d":"The quantile influence function is square-integrable (it is bounded).","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, MeasureTheory.Integrable (fun z => Causalean.Stat.quantileIF τ q₀ f₀ z ^ 2) P","l":93,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileIF_variance","k":"theorem","m":"Causalean.Stat.Quantile.SampleQuantile","a":"Stat/Quantile","t":1,"d":"Variance of the quantile influence function. Provided q₀ is the population τ-quantile, i.e. the population cdf satisfies $F(q_0)=\\tau$, the second moment of the influence function $\\psi_\\tau$ under the population measure equals $\\tau(1-\\tau)/f_0^2$, the classical sample-quantile asymptotic variance.","s":"∀ {P : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure P] {τ q₀ f₀ : ℝ}, ↑(ProbabilityTheory.cdf P) q₀ = τ → ∫ (z : ℝ), Causalean.Stat.quantileIF τ q₀ f₀ z ^ 2 ∂P = τ * (1 - τ) / f₀ ^ 2","l":115,"f":"Causalean/Stat/Quantile/SampleQuantile.lean"},{"n":"Causalean.Stat.quantileSet","k":"def","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"Given a measure on the real line and a real level, the quantile super-level set is the set of all real numbers at which the measure's cumulative distribution function is at least that level.","s":"MeasureTheory.Measure ℝ → ℝ → Set ℝ","l":45,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantileSet_up_closed","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"The super-level set is up-closed (monotonicity of the cdf).","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x x' : ℝ}, x ∈ Causalean.Stat.quantileSet μ τ → x ≤ x' → x' ∈ Causalean.Stat.quantileSet μ τ","l":57,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_le_iff","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":1,"d":"Quantile / cdf Galois connection. For an interior probability level $\\tau\\in(0,1)$, the quantile of a real measure at level $\\tau$ is at most a point x exactly when $\\tau$ is at most the cdf of that measure at x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, 0 < τ → τ < 1 → (Causalean.Stat.quantile μ τ ≤ x ↔ τ ≤ ↑(ProbabilityTheory.cdf μ) x)","l":109,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_le_of_le_cdf","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"Galois connection (other direction). If the cdf reaches τ at x, then the quantile is at or below x.","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ x : ℝ}, 0 < τ → τ ≤ ↑(ProbabilityTheory.cdf μ) x → Causalean.Stat.quantile μ τ ≤ x","l":103,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.quantile_map_uniform","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Probability integral transform. For a Borel probability measure μ on ℝ, the pushforward of the uniform distribution on (0,1) under μ's quantile function equals μ itself.","s":"∀ (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsProbabilityMeasure μ], MeasureTheory.Measure.map (Causalean.Stat.quantile μ) Causalean.Stat.unifOI = μ","l":88,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.quantile_mono","k":"theorem","m":"Causalean.Stat.Quantile.Quantile","a":"Stat/Quantile","t":2,"d":"The quantile function is monotone in the probability level τ on (0,1).","s":"∀ {μ : MeasureTheory.Measure ℝ} {τ τ' : ℝ}, 0 < τ → τ' < 1 → τ ≤ τ' → Causalean.Stat.quantile μ τ ≤ Causalean.Stat.quantile μ τ'","l":116,"f":"Causalean/Stat/Quantile/Quantile.lean"},{"n":"Causalean.Stat.raoBlackwellStatistic","k":"def","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"Given a Poisson mean, a statistic, and an overflow value, the fixed-sample Rao--Blackwell statistic is the mean of the capped statistic under the auxiliary-count kernel.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {n : ℕ} → NNReal → (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ) → ℝ → (Fin n → X) → ℝ","l":490,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.raoBlackwellStatistic_eq_integral","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Given a Poisson mean, a finite-sample statistic, its measurability, an overflow value, and a fixed array, the Rao--Blackwell statistic equals the Poisson integral of the capped prefix statistic.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {n : ℕ} (lambda : NNReal) (T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ), Measurable T → ∀ (zOver : ℝ) (x : Fin n → X), Causalean.Stat.raoBlackwellStatistic lambda T zOver x = ∫ (m : ℕ), Causalean.Stat.cappedStatistic T zOver x m ∂ProbabilityTheory.poissonMeasure lambda","l":499,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.ratioDeriv","k":"def","m":"Causalean.Stat.Inference.RatioDeltaMethod","a":"Stat/Inference","t":1,"d":"For a two-dimensional real vector with coordinates $(a,b)$, the ratio derivative is the linear map sending an increment $(u,v)$ to $u/b-av/b^2$, namely the derivative of the ratio $a/b$ whenever $b$ is nonzero.","s":"EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) →L[ℝ] ℝ","l":50,"f":"Causalean/Stat/Inference/RatioDeltaMethod.lean"},{"n":"Causalean.Stat.realBayesRisk","k":"def","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"Given a measurable parameter space, a finite measurable state space, an action space, a prior measure on the parameter space, a probability kernel from parameters to states, and a real-valued loss for each action and state, the real Bayes risk is the infimum, over all actions, of their mixed kernel losses.","s":"{Θ : Type u_1} → {S : Type u_2} → {A : Type u_3} → [inst : MeasurableSpace Θ] → [inst_1 : MeasurableSpace S] → MeasureTheory.Measure Θ → ProbabilityTheory.Kernel Θ S → (A → S → ℝ) → ℝ","l":154,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.realBayesRisk_eq_inducedFiniteDesignBayesRisk","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"A continuous prior mixed through a finite-state Markov kernel has the same real Bayes risk as its induced finite design.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [inst_5 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_6 : ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ), Causalean.Stat.realBayesRisk π K loss = Causalean.Stat.finiteDesignBayesRisk (Causalean.Stat.inducedFiniteDesign π K) loss","l":331,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.realBayesRisk_le_minimaxValue","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":1,"d":"With at least one available action and nonnegative statewise loss, the real Bayes risk under a continuous prior and finite-state Markov kernel is at most the finite-state minimax value.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] [Nonempty A] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → Causalean.Stat.realBayesRisk π K loss ≤ Causalean.Stat.minimaxValue loss","l":170,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.realBayesRisk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"If statewise loss is nonnegative, then the real Bayes risk is nonnegative.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [ProbabilityTheory.IsMarkovKernel K] (loss : A → S → ℝ), (∀ (a : A) (s : S), 0 ≤ loss a s) → 0 ≤ Causalean.Stat.realBayesRisk π K loss","l":162,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.realBayesRisk_statewiseSquaredLoss_eq_posteriorResidual","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"The real Bayes risk of continuously mixed finite-state squared loss equals the posterior residual risk computed from the induced finite design.","s":"∀ {Θ : Type u_3} {S : Type u_4} {X : Type u_5} [inst : MeasurableSpace Θ] [inst_1 : Fintype S] [inst_2 : Nonempty S] [inst_3 : MeasurableSpace S] [inst_4 : MeasurableSingletonClass S] [inst_5 : Fintype X] [inst_6 : MeasurableSpace X] [MeasurableSingletonClass X] (π : MeasureTheory.Measure Θ) [inst_8 : MeasureTheory.IsProbabilityMeasure π] (K : ProbabilityTheory.Kernel Θ S) [inst_9 : ProbabilityTheory.IsMarkovKernel K] (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ), Causalean.Stat.realBayesRisk π K (Causalean.Stat.statewiseSquaredLoss L t) = Causalean.Stat.posteriorResidual (Causalean.Stat.inducedFiniteDesign π K) L t","l":354,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.real_two_point_lower_bound","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound for a real-valued parameter. If two candidate values θ₀, θ₁ : ℝ are 2s-separated, then for any measurable estimator est : Ω → ℝ, the worst-case probability of missing the truth by ≥ s is at least ½(1 − tvDist P₀ P₁). Specialization of half_one_sub_tvDist_le_max_error to Θ = ℝ with dist a b = |a − b|.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s : ℝ}, 2 * s ≤ |θ₀ - θ₁| → (1 - Causalean.Stat.tvDist P₀ P₁) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":56,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.rescaledEmpiricalCDF_eq_normalizedSum","k":"theorem","m":"Causalean.Stat.Quantile.EmpiricalCDF","a":"Stat/Quantile","t":2,"d":"Key identity. The rescaled empirical-cdf deviation equals the normalized influence-function sum: √n (F̂ₙ(y) − F(y)) = (1/√n) Σ cdfIF(Z_i).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure ℝ} (S : Causalean.Stat.IIDSample Ω ℝ μ P) (y : ℝ) (n : ℕ) (ω : Ω), √↑(Finset.range n).card * (S.empiricalCDF y n ω - ↑(ProbabilityTheory.cdf P) y) = (√↑(Finset.range n).card)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.cdfIF P y (S.Z i ω)","l":195,"f":"Causalean/Stat/Quantile/EmpiricalCDF.lean"},{"n":"Causalean.Stat.restrictedSetVolume","k":"def","m":"Causalean.Stat.Minimax.HonestConfidenceSet","a":"Stat/Minimax","t":1,"d":"For a parameter region on the real line and a set on the real line, the restricted set volume is the real-valued Lebesgue volume of their intersection, with infinite volume represented by zero.","s":"Set ℝ → Set ℝ → ℝ","l":27,"f":"Causalean/Stat/Minimax/HonestConfidenceSet.lean"},{"n":"Causalean.Stat.rnCommonPart","k":"def","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":1,"d":"Given a measurable sample space and three measures on it, consisting of two target measures and a reference measure, the common Radon--Nikodym submeasure is the reference measure weighted by the pointwise minimum of the two target measures' Radon--Nikodym densities relative to that reference measure.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → MeasureTheory.Measure X → MeasureTheory.Measure X → MeasureTheory.Measure X","l":109,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_le_left","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The RN common part is dominated by its first law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure mu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → Causalean.Stat.rnCommonPart mu nu xi ≤ mu","l":120,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_le_right","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The RN common part is dominated by its second law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure nu] [MeasureTheory.IsFiniteMeasure xi], nu.AbsolutelyContinuous xi → Causalean.Stat.rnCommonPart mu nu xi ≤ nu","l":132,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnCommonPart_mass_eq_one_sub_tvDist","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"The mass of the RN common part is exactly one minus total variation.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → nu.AbsolutelyContinuous xi → (Causalean.Stat.rnCommonPart mu nu xi) Set.univ = ENNReal.ofReal (1 - Causalean.Stat.tvDist mu nu)","l":144,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.rnDeriv_mul_measure_singleton","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquaredFinite","a":"Stat/Minimax","t":2,"d":"Discrete Radon–Nikodym bridge. On a measurable singleton {x}, the Radon–Nikodym density scaled by the base mass recovers the numerator mass: ν {x} · (dμ/dν)(x) = μ {x}, for μ ≪ ν. Proved by integrating the density over {x} (via setLIntegral_rnDeriv) and collapsing the singleton integral with restrict_singleton/lintegral_dirac'.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.SFinite ν] [μ.HaveLebesgueDecomposition ν], μ.AbsolutelyContinuous ν → ∀ (x : Ω), ν {x} * μ.rnDeriv ν x = μ {x}","l":35,"f":"Causalean/Stat/Minimax/ChiSquaredFinite.lean"},{"n":"Causalean.Stat.rnDeriv_prod_eq","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Product density factorization. For μ₁ ≪ ν₁ and μ₂ ≪ ν₂, the Radon–Nikodym density of the product is (a.e.) the product of the marginal densities.","s":"∀ {α : Type u_2} {mα : MeasurableSpace α} {β : Type u_3} {mβ : MeasurableSpace β} (μ₁ ν₁ : MeasureTheory.Measure α) (μ₂ ν₂ : MeasureTheory.Measure β) [MeasureTheory.SigmaFinite ν₁] [MeasureTheory.SigmaFinite ν₂] [MeasureTheory.SFinite ν₂] [μ₁.HaveLebesgueDecomposition ν₁] [μ₂.HaveLebesgueDecomposition ν₂], μ₁.AbsolutelyContinuous ν₁ → μ₂.AbsolutelyContinuous ν₂ → (μ₁.prod μ₂).rnDeriv (ν₁.prod ν₂) =ᵐ[ν₁.prod ν₂] fun z => μ₁.rnDeriv ν₁ z.1 * μ₂.rnDeriv ν₂ z.2","l":158,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.rnDeriv_setIntegral_le_sqrt_chi","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":2,"d":"Cauchy–Schwarz mass transfer under a χ²-budget. For probability measures P ≪ Q whose squared density deviation (dP/dQ − 1)² is Q-integrable, and whose χ²-divergence is at most C, the P-mass of any measurable set A is controlled by its Q-mass through P(A) ≤ √((C + 1)·Q(A)). This is the Cauchy–Schwarz step underlying the two-point testing floor: a set that is small under the reference measure Q cannot be large under P when the χ²-budget is finite.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q], P.AbsolutelyContinuous Q → MeasureTheory.Integrable (fun x => ((P.rnDeriv Q x).toReal - 1) ^ 2) Q → ∀ {C : ℝ}, Causalean.Stat.chiSqDiv P Q ≤ C → ∀ {A : Set Ω}, MeasurableSet A → P.real A ≤ √((C + 1) * Q.real A)","l":465,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.sampleDesign","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, and a sample of that size, the sample design maps every sample coordinate to its pair of group and arm labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → Fin n → kappa × Bool","l":61,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.score_diff_L2_isLittleOp_sqrt","k":"theorem","m":"Causalean.Stat.MEstimation.EmpiricalExpansion","a":"Stat/MEstimation","t":2,"d":"(a) Score difference is o_p(1) in mean square.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [inst_4 : MeasurableSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.IsLittleOp (fun n ω => (↑n)⁻¹ * ∑ i ∈ Finset.range n, ‖ψ (θn n ω) (S.Z i ω) - ψ θ₀ (S.Z i ω)‖ ^ 2) (fun x => 1) μ","l":85,"f":"Causalean/Stat/MEstimation/EmpiricalExpansion.lean"},{"n":"Causalean.Stat.secondMomentInv","k":"def","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":1,"d":"Let $P$ be a measure on a measurable sample space, and let a measurable function take values in a finite-dimensional real inner-product space equipped with its Borel σ-algebra. Given that function is measurable, its squared norm is integrable under $P$, and its second-moment operator is injective, the inverse second-moment operator is the continuous linear map obtained by applying twice the inverse of the positive square root of that operator.","s":"{X : Type u_1} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → {E : Type u_2} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [inst_3 : FiniteDimensional ℝ E] → [inst_4 : MeasurableSpace E] → [inst_5 : BorelSpace E] → {ψ : X → E} → (hψ : Measurable ψ) → (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) → Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar) → E →L[ℝ] E","l":87,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.secondMomentInv_secondMomentLM","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":2,"d":"Σ⁻¹ is a left inverse of Σ.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)) (x : E), (Causalean.Stat.secondMomentInv hψ hvar hinj) ((Causalean.Stat.secondMomentLM hψ hvar) x) = x","l":107,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.secondMomentLM","k":"def","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"Let $P$ be a measure on a measurable sample space, and let a function take values in a second-countable real inner-product space equipped with its Borel σ-algebra and measurable scalar multiplication. Given that function is measurable and its squared norm is integrable under $P$, the second-moment operator maps each vector $t$ to the integral of the vector-valued function that multiplies the function value by its inner product with $t$.","s":"{X : Type u_1} → {E : Type u_2} → [inst : MeasurableSpace X] → {P : MeasureTheory.Measure X} → [inst_1 : NormedAddCommGroup E] → [inst_2 : InnerProductSpace ℝ E] → [inst_3 : MeasurableSpace E] → [OpensMeasurableSpace E] → [SecondCountableTopology E] → [MeasurableSMul₂ ℝ E] → {ψ : X → E} → Measurable ψ → MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P → E →ₗ[ℝ] E","l":72,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_inner","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"For vectors t and s, the inner product of the second-moment operator applied to t with s equals the expectation under P of the product of the inner products ⟪t, ψ⟩ and ⟪s, ψ⟩.","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : SecondCountableTopology E] [inst_6 : MeasurableSMul₂ ℝ E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) [CompleteSpace E] (t s : E), inner ℝ ((Causalean.Stat.secondMomentLM hψ hvar) t) s = ∫ (x : X), inner ℝ t (ψ x) * inner ℝ s (ψ x) ∂P","l":94,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_isPositive","k":"theorem","m":"Causalean.Stat.CLT.SecondMomentOperator","a":"Stat/CLT","t":1,"d":"The second-moment operator is a positive linear operator, hence symmetric and self-adjoint.","s":"∀ {X : Type u_1} {E : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : OpensMeasurableSpace E] [inst_5 : SecondCountableTopology E] [inst_6 : MeasurableSMul₂ ℝ E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) [CompleteSpace E], (Causalean.Stat.secondMomentLM hψ hvar).IsPositive","l":105,"f":"Causalean/Stat/CLT/SecondMomentOperator.lean"},{"n":"Causalean.Stat.secondMomentLM_secondMomentInv","k":"theorem","m":"Causalean.Stat.Inference.ChiSquaredWald","a":"Stat/Inference","t":2,"d":"Σ⁻¹ is a right inverse of Σ.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] {ψ : X → E} (hψ : Measurable ψ) (hvar : MeasureTheory.Integrable (fun x => ‖ψ x‖ ^ 2) P) (hinj : Function.Injective ⇑(Causalean.Stat.secondMomentLM hψ hvar)) (x : E), (Causalean.Stat.secondMomentLM hψ hvar) ((Causalean.Stat.secondMomentInv hψ hvar hinj) x) = x","l":118,"f":"Causalean/Stat/Inference/ChiSquaredWald.lean"},{"n":"Causalean.Stat.selectCoupledRaw","k":"def","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":1,"d":"For a binary hypercube vertex and a pair of raw observations at every coordinate, the selected raw observation vector takes the first member of each pair when the corresponding vertex bit is false and the second member when it is true.","s":"{M : ℕ} → {Z : Fin M → Type u_1} → (Fin M → Bool) → ((j : Fin M) → Z j × Z j) → (j : Fin M) → Z j","l":80,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.selectorCell","k":"def","m":"Causalean.Stat.SampleSplit.FiniteSelector","a":"Stat/SampleSplit","t":1,"d":"For a selection rule from pilot samples to branch labels, a designated set of pilot samples, and a branch label, the selector cell for that branch is the subset of the designated set on which the selection rule chooses that label.","s":"{Alpha : Type u_2} → {Iota : Type u_4} → (Alpha → Iota) → Set Alpha → Iota → Set Alpha","l":35,"f":"Causalean/Stat/SampleSplit/FiniteSelector.lean"},{"n":"Causalean.Stat.sequential_cumulative_risk_regret","k":"theorem","m":"Causalean.Stat.Limit.SequentialCumulativeRisk","a":"Stat/Limit","t":1,"d":"Radius-monotone cumulative Neyman-regret lower bound (lem:local-neighborhood-cumulative-risk). Fix a per-round Fisher information scale J > 0, an oracle-sensitivity d ≠ 0, a nonnegative feedback constant L, and a nonnegative prior information Iq. Let b be a per-round mean-squared-error sequence with running totals B defined by B n = ∑_{t≤n} b t, satisfying the predictable van Trees / Bayes–Cramér–Rao recursion b t ≥ (d²/4) / (Iq + (5J/4)·t + L·√(t·B(t-1))) for every round t ≥ 1. Suppose further the Neyman-gap quadratic conversion 2S²·B T ≤ RB T for every T (the Bayes-average cumulative regret RB dominates twice the scaled cumulative Bayes MSE) and the worst-case cumulative regret Rsup dominates the Bayes average, RB T ≤ Rsup T for every T. Then there is a threshold T₀ beyond which the worst-case cumulative regret grows at least logarithmically: Rsup T ≥ (S² d² / (16J)) · log T for all T ≥ T₀, with universal constant 1/16.","s":"∀ (S J d L Iq : ℝ), 0 < J → d ≠ 0 → 0 ≤ L → 0 ≤ Iq → ∀ (b B RB Rsup : ℕ → ℝ), (∀ (n : ℕ), B n = ∑ t ∈ Finset.Icc 1 n, b t) → (∀ (t : ℕ), 1 ≤ t → d ^ 2 / 4 / (Iq + 5 * J / 4 * ↑t + L * √(↑t * B (t - 1))) ≤ b t) → (∀ (T : ℕ), 2 * S ^ 2 * B T ≤ RB T) → (∀ (T : ℕ), RB T ≤ Rsup T) → ∃ T₀, ∀ (T : ℕ), T₀ ≤ T → S ^ 2 * d ^ 2 / (16 * J) * Real.log ↑T ≤ Rsup T","l":387,"f":"Causalean/Stat/Limit/SequentialCumulativeRisk.lean"},{"n":"Causalean.Stat.signedTail","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a real threshold and a real argument, the signed tail indicator is the indicator that the argument is below the threshold minus the indicator that it is below zero.","s":"ℝ → ℝ → ℝ","l":44,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.signedTail_eq_indicator_sub","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"signedTail a is, as a function of s, the difference of the indicators of the two intervals Ico 0 a and Ico a 0 (at most one of which is nonempty). This is the normal form used to compute its integral and to see it is integrable.","s":"∀ (a : ℝ), Causalean.Stat.signedTail a = ((Set.Ico 0 a).indicator fun x => 1) - (Set.Ico a 0).indicator fun x => 1","l":57,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.sqLoss_kernelMean_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then the squared error of its kernel mean is no greater than the kernel average of its squared error at each source observation.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∀ (c : ℝ) (x : X), (Causalean.Stat.kernelMean K T x - c) ^ 2 ≤ ∫ (y : Y), (T y - c) ^ 2 ∂K x","l":138,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk","k":"def","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Given a measurable observation space, a measure on that space, a real-valued estimator, and a real target value, the squared risk is the expected value, under that measure, of the estimator's squared error from the target.","s":"{X : Type uX} → [inst : MeasurableSpace X] → MeasureTheory.Measure X → (X → ℝ) → ℝ → ℝ","l":178,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.sqRisk_cappedStatistic_restrict_nonoverflow_eq","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For an observation probability law, a Poisson mean, a fixed sample size, a measurable finite-sample statistic, an overflow value, and a target, the squared risk of the capped statistic on the nonoverflow restriction equals the squared risk of the original statistic under the correspondingly restricted finite-Poisson law.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) {T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ}, Measurable T → ∀ (zOver theta : ℝ), Causalean.Stat.sqRisk (((MeasureTheory.Measure.pi fun x => P).prod (ProbabilityTheory.poissonMeasure lambda)).restrict (Prod.snd ⁻¹' Set.Iic n)) (fun z => Causalean.Stat.cappedStatistic T zOver z.1 z.2) theta = Causalean.Stat.sqRisk ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Iic n)) T theta","l":523,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_cappedStatistic_restrict_overflow_eq","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For an observation probability law, a Poisson mean, a fixed sample size, a statistic, an overflow value, and a target, the squared risk of the capped statistic on the overflow restriction is its constant overflow loss times the Poisson overflow probability.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) (T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ) (zOver theta : ℝ), Causalean.Stat.sqRisk (((MeasureTheory.Measure.pi fun x => P).prod (ProbabilityTheory.poissonMeasure lambda)).restrict (Prod.snd ⁻¹' Set.Ioi n)) (fun z => Causalean.Stat.cappedStatistic T zOver z.1 z.2) theta = (zOver - theta) ^ 2 * (ProbabilityTheory.poissonMeasure lambda).real (Set.Ioi n)","l":557,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_finProductKernel_affinePullback_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, a finite-sample target estimator is measurable, and that estimator is uniformly bounded, then the source-product risk of its affine product-kernel pullback, multiplied by the squared slope, is at most its target-product risk.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (n : ℕ) (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b theta : ℝ}, a ≠ 0 → ∀ {targetEst : (Fin n → Y) → ℝ}, Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → a ^ 2 * Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P) (Causalean.Stat.kernelAffinePullback (Causalean.Stat.finProductKernel n K) a b targetEst) theta ≤ Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P.bind ⇑K) targetEst (a * theta + b)","l":355,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_finitePoisson_restrict_nonoverflow_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For an observation probability law, a Poisson mean, a fixed sample size, a measurable uniformly bounded statistic, and a target, restricting the finite-Poisson law to counts at most the fixed sample size cannot increase squared risk.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) {T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∀ (theta : ℝ), Causalean.Stat.sqRisk ((Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda).restrict (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample.count ⁻¹' Set.Iic n)) T theta ≤ Causalean.Stat.sqRisk (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda) T theta","l":591,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_kernelAffinePullback_le_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, the target estimator is measurable, and the target estimator is uniformly bounded, then the source risk of the affine kernel pullback, multiplied by the squared slope, is at most the target risk under the garbled law.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b theta : ℝ}, a ≠ 0 → ∀ {targetEst : Y → ℝ}, Measurable targetEst → Causalean.Stat.UniformlyBounded targetEst → a ^ 2 * Causalean.Stat.sqRisk P (Causalean.Stat.kernelAffinePullback K a b targetEst) theta ≤ Causalean.Stat.sqRisk (P.bind ⇑K) targetEst (a * theta + b)","l":199,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_kernelMean_le_comp","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the target estimator is measurable and uniformly bounded, then Rao--Blackwellizing it through a Markov kernel cannot increase squared risk: the source-law risk of the kernel mean is at most the target-law risk after garbling.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (P : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure P] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Measurable T → Causalean.Stat.UniformlyBounded T → ∀ (c : ℝ), Causalean.Stat.sqRisk P (Causalean.Stat.kernelMean K T) c ≤ Causalean.Stat.sqRisk (P.bind ⇑K) T c","l":166,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_map_affinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero, the deterministic observation rule is measurable, and the target estimator is measurable, then its squared risk under the pushed-forward law equals the pullback estimator's squared risk multiplied by the squared slope.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] {law : MeasureTheory.Measure X} {phi : X → Y} {a b theta : ℝ} {targetEst : Y → ℝ}, a ≠ 0 → Measurable phi → Measurable targetEst → a ^ 2 * Causalean.Stat.sqRisk law (Causalean.Stat.affinePullbackEstimator phi a b targetEst) theta = Causalean.Stat.sqRisk (MeasureTheory.Measure.map phi law) targetEst (a * theta + b)","l":217,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.sqRisk_raoBlackwellStatistic_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":1,"d":"For an observation probability law, a Poisson mean, and a fixed sample size, if the statistic is measurable, the interval is ordered, the statistic stays in the interval, the target is in the interval, and the overflow value is in the interval, then the fixed-iid squared risk of the Rao--Blackwell statistic is at most the uncapped finite-Poisson risk plus the squared interval diameter times the Poisson overflow probability.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) {T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ}, Measurable T → ∀ {a b theta zOver : ℝ}, a ≤ b → (∀ (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), T s ∈ Set.Icc a b) → theta ∈ Set.Icc a b → zOver ∈ Set.Icc a b → Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P) (Causalean.Stat.raoBlackwellStatistic lambda T zOver) theta ≤ Causalean.Stat.sqRisk (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda) T theta + (b - a) ^ 2 * (ProbabilityTheory.poissonMeasure lambda).real (Set.Ioi n)","l":629,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sqRisk_raoBlackwellStatistic_unitInterval_le","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"For an observation probability law, a Poisson mean, and a fixed sample size, under measurability of the statistic, unit-interval bounds on the statistic, target, and overflow value, the fixed-iid Rao--Blackwell risk is at most the uncapped finite-Poisson risk plus exactly the overflow probability.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (P : MeasureTheory.Measure X) [inst_1 : MeasureTheory.IsProbabilityMeasure P] (lambda : NNReal) (n : ℕ) {T : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X → ℝ}, Measurable T → ∀ {theta zOver : ℝ}, (∀ (s : Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.FiniteSample X), T s ∈ Set.Icc 0 1) → theta ∈ Set.Icc 0 1 → zOver ∈ Set.Icc 0 1 → Causalean.Stat.sqRisk (MeasureTheory.Measure.pi fun x => P) (Causalean.Stat.raoBlackwellStatistic lambda T zOver) theta ≤ Causalean.Stat.sqRisk (Causalean.Mathlib.Probability.FiniteMarkedPoissonPartition.finitePoissonSampleLaw P lambda) T theta + (ProbabilityTheory.poissonMeasure lambda).real (Set.Ioi n)","l":740,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.sq_bhattacharyya_le_two_mul_integral_min","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"Cauchy–Schwarz / Bhattacharyya step. (∫ √p ∂ν)² ≤ 2·∫ min(p,1) ∂ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → (∫ (x : Ω), √(μ.rnDeriv ν x).toReal ∂ν) ^ 2 ≤ 2 * ∫ (x : Ω), min (μ.rnDeriv ν x).toReal 1 ∂ν","l":160,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.sqrtFoldB_integral_isLittleOp_one","k":"theorem","m":"Causalean.Stat.SampleSplit.FoldBEmpiricalProcess","a":"Stat/SampleSplit","t":1,"d":"Bias term is o_p(1) under fixed-ratio split. Given a positive limiting fold-B sampling ratio $c$ with the fold-B fraction $|B(n)|/n$ converging to $c$, and a family of random functions f n ω that is square-integrable under the population measure, for every n, ω with $L^2(P)$ norm that is $o_p(n^{-1/2})$ under the sampling measure, the bias term $\\sqrt{|B(n)|}\\cdot\\int f(n,\\omega,\\cdot)\\,dP$ is $o_p(1)$.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] (S : Causalean.Stat.IIDSample Ω X μ P) (split : Causalean.Stat.OneShotSplit S) {c : ℝ}, 0 < c → Filter.Tendsto (fun n => ↑(split.foldB n).card / ↑n) Filter.atTop (nhds c) → ∀ (f : ℕ → Ω → X → ℝ), (∀ (n : ℕ) (ω : Ω), MeasureTheory.MemLp (f n ω) 2 P) → Causalean.Stat.IsLittleOp (fun n ω => (MeasureTheory.eLpNorm (f n ω) 2 P).toReal) (fun n => ↑n ^ (-(1 / 2))) μ → Causalean.Stat.IsLittleOp (fun n ω => √↑(split.foldB n).card * ∫ (x : X), f n ω x ∂P) (fun x => 1) μ","l":706,"f":"Causalean/Stat/SampleSplit/FoldBEmpiricalProcess.lean"},{"n":"Causalean.Stat.sqrt_le_half_add_one","k":"theorem","m":"Causalean.Stat.Minimax.BretagnolleHuber","a":"Stat/Minimax","t":2,"d":"√y ≤ (y + 1)/2 for y ≥ 0 (AM–GM); used to dominate √p by an integrable function.","s":"∀ {y : ℝ}, 0 ≤ y → √y ≤ (y + 1) / 2","l":91,"f":"Causalean/Stat/Minimax/BretagnolleHuber.lean"},{"n":"Causalean.Stat.sqrt_mul_max_sub","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The tie identity for max. √n (max âₙ b̂ₙ − a) = max (√n(âₙ−a)) (√n(b̂ₙ−a)) (no remainder).","s":"∀ {Ω : Type u_1} (an bn : ℕ → Ω → ℝ) (a : ℝ) (n : ℕ) (ω : Ω), √↑n * (max (an n ω) (bn n ω) - a) = (fun z => max z.1 z.2) (√↑n • ((an n ω, bn n ω) - (a, a)))","l":112,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.sqrt_mul_min_sub","k":"theorem","m":"Causalean.Stat.Inference.FunctionalDelta","a":"Stat/Inference","t":2,"d":"The tie identity for min. Companion to sqrt_mul_max_sub.","s":"∀ {Ω : Type u_1} (an bn : ℕ → Ω → ℝ) (a : ℝ) (n : ℕ) (ω : Ω), √↑n * (min (an n ω) (bn n ω) - a) = (fun z => min z.1 z.2) (√↑n • ((an n ω, bn n ω) - (a, a)))","l":127,"f":"Causalean/Stat/Inference/FunctionalDelta.lean"},{"n":"Causalean.Stat.sqrt_var_tendsto_inProb","k":"theorem","m":"Causalean.Stat.Inference.VarianceEstimation","a":"Stat/Inference","t":1,"d":"Standard-error consistency from variance consistency. Fix a positive scale σ₀. If a variance-estimator sequence varhat converges in probability to σ₀², then the standard-error estimator √varhat converges in probability to σ₀. This is exactly the σ̂ →ₚ σ₀ input required by the generic studentized CLT Tendsto_dist.div_tendsto_inProb_gaussian; callers feed fun N ω => Real.sqrt (varhat N ω) to it.","s":"∀ {Ω : Type u_3} [inst : MeasurableSpace Ω] {varhat : ℕ → Ω → ℝ} {σ₀ : ℝ} {μ : MeasureTheory.Measure Ω}, 0 < σ₀ → Causalean.Stat.Tendsto_inProb varhat (fun x => σ₀ ^ 2) μ → Causalean.Stat.Tendsto_inProb (fun n ω => √(varhat n ω)) (fun x => σ₀) μ","l":180,"f":"Causalean/Stat/Inference/VarianceEstimation.lean"},{"n":"Causalean.Stat.squaredRisk","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a finite measurable latent-state space, a finite measurable observation space, a finite probability design on the latent states, a Markov observation kernel from latent states to observations, an observation-dependent real target, and a real-valued estimator based on the observation, the squared risk is the sum over all latent states and observations of their joint mass times the estimator's squared error relative to the target.","s":"{S : Type u_1} → {X : Type u_2} → [inst : Fintype S] → [Fintype X] → [inst_2 : MeasurableSpace S] → [inst_3 : MeasurableSpace X] → Causalean.Experimentation.DesignBased.FiniteDesign S → ProbabilityTheory.Kernel S X → (S → X → ℝ) → (X → ℝ) → ℝ","l":201,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.squaredRisk_eq_posteriorResidual_add","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Every estimator's squared risk equals posterior residual risk plus its marginally weighted squared distance from the posterior mean.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ) (T : X → ℝ), Causalean.Stat.squaredRisk ν L t T = Causalean.Stat.posteriorResidual ν L t + ∑ x, Causalean.Stat.observationMass ν L x * (T x - Causalean.Stat.posteriorMean ν L t x) ^ 2","l":221,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.squaredRisk_posteriorMean","k":"theorem","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":2,"d":"The guarded posterior-mean estimator has squared risk exactly equal to posterior residual risk.","s":"∀ {S : Type u_1} {X : Type u_2} [inst : Fintype S] [inst_1 : Fintype X] [inst_2 : MeasurableSpace S] [inst_3 : MeasurableSpace X] [MeasurableSingletonClass X] (ν : Causalean.Experimentation.DesignBased.FiniteDesign S) (L : ProbabilityTheory.Kernel S X) (t : S → X → ℝ), Causalean.Stat.squaredRisk ν L t (Causalean.Stat.posteriorMean ν L t) = Causalean.Stat.posteriorResidual ν L t","l":269,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.statewiseSquaredLoss","k":"def","m":"Causalean.Stat.Minimax.FinitePosteriorBayesRisk","a":"Stat/Minimax","t":1,"d":"Given a measurable latent-state space, a finite measurable observation space, a Markov observation kernel from latent states to observations, an observation-dependent real target, a real-valued estimator based on the observation, and a latent state, the statewise squared loss is the sum over observations of their conditional singleton masses times the estimator's squared error relative to the target.","s":"{S : Type u_1} → {X : Type u_2} → [Fintype X] → [inst : MeasurableSpace S] → [inst_1 : MeasurableSpace X] → ProbabilityTheory.Kernel S X → (S → X → ℝ) → (X → ℝ) → S → ℝ","l":310,"f":"Causalean/Stat/Minimax/FinitePosteriorBayesRisk.lean"},{"n":"Causalean.Stat.stdGaussian_map_linearIsometryEquiv","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":2,"d":"stdGaussian is invariant under linear isometry equivalences. Pushing the standard Gaussian forward along an inner-product isometry yields the standard Gaussian on the target. Proved by matching characteristic functions: both sides are centered Gaussians whose covariance form is the inner product (preserved by the isometry).","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E] {F : Type u_2} [inst_5 : NormedAddCommGroup F] [inst_6 : InnerProductSpace ℝ F] [inst_7 : FiniteDimensional ℝ F] [inst_8 : MeasurableSpace F] [BorelSpace F] (ι : E ≃ₗᵢ[ℝ] F), MeasureTheory.Measure.map (⇑ι) (Causalean.Mathlib.stdGaussian E) = Causalean.Mathlib.stdGaussian F","l":64,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.stdGaussian_map_normSq","k":"theorem","m":"Causalean.Stat.CLT.ChiSquared","a":"Stat/CLT","t":1,"d":"The law of the squared norm under the standard Gaussian distribution on a finite-dimensional real inner product space equals the chi-squared distribution whose degrees of freedom is the space's dimension.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E], MeasureTheory.Measure.map (fun x => ‖x‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = Causalean.Stat.chiSqDist (Module.finrank ℝ E)","l":95,"f":"Causalean/Stat/CLT/ChiSquared.lean"},{"n":"Causalean.Stat.stdGaussian_map_normSq_orthogonalProjection","k":"theorem","m":"Causalean.Stat.CLT.ChiSquaredProjection","a":"Stat/CLT","t":1,"d":"For a continuous linear self-map P of a finite-dimensional real inner-product space that is self-adjoint and idempotent, i.e. P composed with itself equals P — so that P is an orthogonal projection — the law of $\\|Pz\\|^2$ under the standard Gaussian on the space equals the chi-squared distribution with degrees of freedom equal to the dimension of the range of P.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E] (P : E →L[ℝ] E), IsSelfAdjoint P → P ∘SL P = P → MeasureTheory.Measure.map (fun z => ‖P z‖ ^ 2) (Causalean.Mathlib.stdGaussian E) = Causalean.Stat.chiSqDist (Module.finrank ℝ ↥(↑P).range)","l":65,"f":"Causalean/Stat/CLT/ChiSquaredProjection.lean"},{"n":"Causalean.Stat.stochEquicontAt_of_asymptoticEquicont","k":"theorem","m":"Causalean.Stat.EmpiricalProcess.Equicontinuity.Modulus","a":"Stat/EmpiricalProcess/Equicontinuity","t":1,"d":"Reduction: StochEquicontAt from class-level equicontinuity + consistency. If the score family ψ is asymptotically equicontinuous at θ₀ along the i.i.d. sample S and θn is a sequence of estimators consistent for θ₀, then the pair (ψ, θ₀) satisfies the stochastic-equicontinuity-at-the-estimator condition StochEquicontAt along S and θn.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] (ψ : E → X → E) (θ₀ : E) (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn","l":152,"f":"Causalean/Stat/EmpiricalProcess/Equicontinuity/Modulus.lean"},{"n":"Causalean.Stat.stronglyMeasurable_kernelAverageLoss","k":"theorem","m":"Causalean.Stat.Minimax.FiniteKernelBayes","a":"Stat/Minimax","t":2,"d":"The kernel-averaged loss of a finite-state action is measurable as a function of the parameter.","s":"∀ {Θ : Type u_1} {S : Type u_2} {A : Type u_3} [inst : MeasurableSpace Θ] [Fintype S] [Nonempty S] [inst_3 : MeasurableSpace S] [MeasurableSingletonClass S] (K : ProbabilityTheory.Kernel Θ S) (loss : A → S → ℝ) (a : A), MeasureTheory.StronglyMeasurable (Causalean.Stat.kernelAverageLoss K loss a)","l":58,"f":"Causalean/Stat/Minimax/FiniteKernelBayes.lean"},{"n":"Causalean.Stat.sum_bernoulli_local_variance_le_share","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.BinomialDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If the overlap margin is positive and one group's success probability lies between the margin and one minus the margin, that group's expected inverse-arm variance contribution is controlled by its expected share of reciprocal usable occupancy.","s":"∀ {ι : Type u_1} [inst : Fintype ι] [inst_1 : DecidableEq ι] (r : ℕ) (p epsilon : ℝ), 0 < epsilon → epsilon ≤ p → p ≤ 1 - epsilon → ∑ b, (∏ i, if b i = true then p else 1 - p) * (↑(Fintype.card ι) ^ 2 * (↑(r + Fintype.card ι))⁻¹ ^ 2 * Causalean.Stat.inverseTwoCounts (Fintype.card ι) {i | b i = true}.card) ≤ 4 / (epsilon ^ 2 * (1 - epsilon)) * ∑ b, (∏ i, if b i = true then p else 1 - p) * (↑(Fintype.card ι) * (↑(r + Fintype.card ι))⁻¹ ^ 2 * Causalean.Stat.interiorIndicator (Fintype.card ι) {i | b i = true}.card)","l":103,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/BinomialDesign.lean"},{"n":"Causalean.Stat.sum_correct_le","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"Heart of Fano. The total probability of correct recovery, summed over the family, is at most 1 + ∑ᵢ tvDist (P i₀) (P i): comparing each Pᵢ(A i) to the reference P i₀(A i) costs one tvDist, and the reference masses sum to ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), ∑ i, (P i).real (Causalean.Stat.acceptanceRegion est (θ i) s) ≤ 1 + ∑ i, Causalean.Stat.tvDist (P i₀) (P i)","l":136,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.sum_decode_ge","k":"theorem","m":"Causalean.Stat.Minimax.Assouad","a":"Stat/Minimax","t":2,"d":"Per-coordinate lower bound. Summed over the cube, the coordinate-j error mass is at least half the cube-sum of 1 − tvDist (P τ) (P (flip j τ)), via pairing each vertex with its j-flip (an involution that preserves the sum).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {d : ℕ} (P : (Fin d → Bool) → MeasureTheory.Measure Ω) [∀ (τ : Fin d → Bool), MeasureTheory.IsProbabilityMeasure (P τ)] (est : Ω → Fin d → Bool), (∀ (j : Fin d) (b : Bool), MeasurableSet {ω | est ω j = b}) → ∀ (j : Fin d), ∑ τ, (1 - Causalean.Stat.tvDist (P τ) (P (Causalean.Stat.flipBit j τ))) ≤ 2 * ∑ τ, (P τ).real {ω | est ω j ≠ τ j}","l":123,"f":"Causalean/Stat/Minimax/Assouad.lean"},{"n":"Causalean.Stat.sum_injectiveTuples_apply_eq_range","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"For each coordinate of an ordered injective tuple, every sample index appears equally often.","s":"∀ {m n : ℕ}, m ≤ n → ∀ (j : Fin m) (f : ℕ → ℝ), ∑ t ∈ Causalean.Stat.injectiveTuples m n, f ↑(t j) = Causalean.Stat.injectiveTupleCount m n / ↑n * ∑ i ∈ Finset.range n, f i","l":160,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.sum_injectiveTuples_two_eq_offDiag","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"Ordered injective Fin 2 tuples are the same data as off-diagonal ordered pairs.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ) (n : ℕ) (ω : Ω), (∑ t ∈ Causalean.Stat.injectiveTuples 2 n, Causalean.Stat.pairKernel h fun j => S.Z (↑(t j)) ω) = ∑ p ∈ (Finset.range n).offDiag, h (S.Z p.1 ω) (S.Z p.2 ω)","l":324,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.sum_jointWeight_variance_le_inverse","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.FiniteDesign","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"If all joint group/arm masses are nonnegative, their two arm masses sum to the group mass, the overlap margin is positive and at most one half, and each positive-mass group gives both arms at least the overlap share, the joint-design average of the occupancy variance factor is bounded by a margin-dependent multiple of average reciprocal usable occupancy. Zero-mass groups contribute zero before any conditional arm probability is formed.","s":"∀ {n : ℕ} {κ : Type u_1} [inst : Fintype κ] [inst_1 : DecidableEq κ] (q : κ → Bool → ℝ) (g : κ → ℝ) (epsilon : ℝ), (∀ (k : κ) (a : Bool), 0 ≤ q k a) → (∀ (k : κ), q k false + q k true = g k) → 0 < epsilon → epsilon ≤ 1 / 2 → (∀ (k : κ), 0 < g k → ∀ (a : Bool), epsilon * g k ≤ q k a) → ∑ d, (∏ i, q (d i).1 (d i).2) * Causalean.Stat.occupancyDesignVarianceFactor Prod.fst Prod.snd d ≤ 4 / (epsilon ^ 2 * (1 - epsilon)) * ∑ d, (∏ i, q (d i).1 (d i).2) * Causalean.Stat.inverseUsableGroupTotal Prod.fst Prod.snd d","l":504,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/FiniteDesign.lean"},{"n":"Causalean.Stat.sum_partialMatchingsOfSize","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.PartialMatching","a":"Stat/UStatistic/OrderM","t":2,"d":"Given a quantity assigned to each partial matching, summing it over all matchings equals summing first by matching size through the smaller coordinate-set size.","s":"∀ {r s : ℕ} {α : Type u_1} [inst : AddCommMonoid α] (F : Causalean.Stat.PartialMatching r s → α), ∑ M, F M = ∑ h ∈ Finset.range (min r s + 1), ∑ M ∈ Causalean.Stat.partialMatchingsOfSize r s h, F M","l":216,"f":"Causalean/Stat/UStatistic/OrderM/PartialMatching.lean"},{"n":"Causalean.Stat.sum_refReal_acceptance_le","k":"theorem","m":"Causalean.Stat.Minimax.Fano","a":"Stat/Minimax","t":2,"d":"A reference hypothesis assigns total mass ≤ 1 across the disjoint acceptance regions, since their union has measure ≤ 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {Θ : Type u_2} [inst : PseudoMetricSpace Θ] [inst_1 : MeasurableSpace Θ] [OpensMeasurableSpace Θ] {ι : Type u_3} [inst_3 : Fintype ι] (P : ι → MeasureTheory.Measure Ω) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (P i)] {est : Ω → Θ}, Measurable est → ∀ {θ : ι → Θ} {s : ℝ}, (∀ (i k : ι), i ≠ k → 2 * s ≤ dist (θ i) (θ k)) → ∀ (i₀ : ι), ∑ i, (P i₀).real (Causalean.Stat.acceptanceRegion est (θ i) s) ≤ 1","l":109,"f":"Causalean/Stat/Minimax/Fano.lean"},{"n":"Causalean.Stat.supportedArmGroupResidual","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space, a group-label map, a Boolean arm-assignment map, a real-valued outcome, arm--group centers, an arm label, and a group label, the supported arm--group residual equals the corresponding residual on the requested arm--group event and zero outside that event.","s":"{Omega : Type u_1} → {kappa : Type u_2} → (Omega → kappa) → (Omega → Bool) → (Omega → ℝ) → (Bool → kappa → ℝ) → Bool → kappa → Omega → ℝ","l":51,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.supportedArmGroupResidual_mul_eq_zero_of_ne","k":"theorem","m":"Causalean.Stat.Sample.OccupancyWeightedMean.MomentBounds","a":"Stat/Sample/OccupancyWeightedMean","t":2,"d":"When two arm/group labels differ, their supported residuals have zero pointwise product because their supports are disjoint.","s":"∀ {Omega : Type u_1} {kappa : Type u_2} [MeasurableSpace Omega] [Fintype kappa] [DecidableEq kappa] [inst : MeasurableSpace kappa] [MeasurableSingletonClass kappa] (group : Omega → kappa) (arm : Omega → Bool) (Y : Omega → ℝ) (center : Bool → kappa → ℝ) (a b : Bool) (k l : kappa), (a, k) ≠ (b, l) → ∀ (omega : Omega), Causalean.Stat.supportedArmGroupResidual group arm Y center a k omega * Causalean.Stat.supportedArmGroupResidual group arm Y center b l omega = 0","l":28,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/MomentBounds.lean"},{"n":"Causalean.Stat.survFst","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a measure on pairs of real-valued quantities and a real threshold, the first marginal survival function is the real-valued mass of pairs whose first coordinate exceeds that threshold.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ","l":34,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survFst_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The first marginal survival function of a coupling is 1 - F, where F = cdf μ. Proof: Prod.fst ⁻¹' Ioi s is the complement of Prod.fst ⁻¹' Iic s, whose π-mass is μ (Iic s) = F s by h.map_fst.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s : ℝ), Causalean.Stat.survFst π s = 1 - ↑(ProbabilityTheory.cdf μ) s","l":53,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survSnd","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a measure on pairs of real-valued quantities and a real threshold, the second marginal survival function is the real-valued mass of pairs whose second coordinate exceeds that threshold.","s":"MeasureTheory.Measure (ℝ × ℝ) → ℝ → ℝ","l":40,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.survSnd_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The second marginal survival function of a coupling is 1 - G, where G = cdf ν.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (t : ℝ), Causalean.Stat.survSnd π t = 1 - ↑(ProbabilityTheory.cdf ν) t","l":86,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.surv_gap_eq","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.Survival","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"Survival gap = cdf gap. For a coupling π of two probability measures μ and ν, the gap between the joint and product-of-marginals survival functions of π at (s, t) equals the corresponding gap between the joint and product-of-marginals cumulative distribution functions.","s":"∀ {π : MeasureTheory.Measure (ℝ × ℝ)} {μ ν : MeasureTheory.Measure ℝ}, Causalean.Stat.IsCoupling π μ ν → ∀ (s t : ℝ), Causalean.Stat.jointSurv π s t - Causalean.Stat.survFst π s * Causalean.Stat.survSnd π t = Causalean.Stat.jointCdf π s t - ↑(ProbabilityTheory.cdf μ) s * ↑(ProbabilityTheory.cdf ν) t","l":178,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/Survival.lean"},{"n":"Causalean.Stat.tailInd","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"For a real threshold and a real argument, the tail indicator equals one when the argument is strictly below the threshold and zero otherwise.","s":"ℝ → ℝ → ℝ","l":36,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.tailInd_apply","k":"theorem","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.TailIntegral","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":2,"d":"The tail indicator equals one below the threshold and zero at or above it.","s":"∀ (a s : ℝ), Causalean.Stat.tailInd a s = if s < a then 1 else 0","l":53,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/TailIntegral.lean"},{"n":"Causalean.Stat.tendstoInProb_covarianceModelLoss","k":"theorem","m":"Causalean.Stat.GaussianCovariance","a":"Stat","t":2,"d":"Given covariance models that are compact and nonempty, an empirical covariance sequence, a target matrix, a sampling measure, and entrywise convergence in probability, the model losses converge coordinatewise in probability.","s":"∀ {V : Type u_2} {I : Type u_3} {Ω : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : Fintype V] [inst_2 : DecidableEq V] [Fintype I] (models : I → Set (Causalean.Stat.PositiveCovariance V)), (∀ (i : I), IsCompact (models i)) → (∀ (i : I), (models i).Nonempty) → ∀ (Sn : ℕ → Ω → Matrix V V ℝ) (T : Matrix V V ℝ) (P : MeasureTheory.Measure Ω), (∀ (a b : V), Causalean.Stat.Tendsto_inProb (fun n ω => Sn n ω a b) (fun x => T a b) P) → ∀ (i : I), Causalean.Stat.Tendsto_inProb (fun n ω => Causalean.Stat.covarianceModelLoss (models i) (Sn n ω)) (fun x => Causalean.Stat.covarianceModelLoss (models i) T) P","l":502,"f":"Causalean/Stat/GaussianCovariance.lean"},{"n":"Causalean.Stat.tendsto_argmax_of_eventually_mem_compact","k":"theorem","m":"Causalean.Stat.MEstimation.ArgmaxStability","a":"Stat/MEstimation","t":1,"d":"Deterministic argmax stability. Let criterion : ℕ → E → ℝ be a sequence of objective functions with pointwise limit limitCriterion, over a compact set K. Suppose the limiting maximizer limitArgmax lies in K, limitCriterion is continuous on K, limitArgmax maximizes limitCriterion over K, and it is the unique maximizer over K, up to equal criterion values. If criterion converges to limitCriterion uniformly on K, each argmax N eventually lies in K, and argmax N eventually maximizes criterion N over K, then the sequence argmax converges to limitArgmax.","s":"∀ {E : Type u_1} [inst : PseudoMetricSpace E] [SecondCountableTopology E] (criterion : ℕ → E → ℝ) (limitCriterion : E → ℝ) (argmax : ℕ → E) (limitArgmax : E) (K : Set E), IsCompact K → limitArgmax ∈ K → ContinuousOn limitCriterion K → (∀ y ∈ K, limitCriterion y ≤ limitCriterion limitArgmax) → TendstoUniformlyOn criterion limitCriterion Filter.atTop K → (∀ᶠ (N : ℕ) in Filter.atTop, argmax N ∈ K) → (∀ᶠ (N : ℕ) in Filter.atTop, ∀ y ∈ K, criterion N y ≤ criterion N (argmax N)) → (∀ y ∈ K, limitCriterion y = limitCriterion limitArgmax → y = limitArgmax) → Filter.Tendsto argmax Filter.atTop (nhds limitArgmax)","l":23,"f":"Causalean/Stat/MEstimation/ArgmaxStability.lean"},{"n":"Causalean.Stat.tendsto_inProb_finiteMaxError","k":"theorem","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":2,"d":"If empirical loss vectors have coordinatewise population targets under a sampling measure, and each coordinate converges in probability, then the largest absolute coordinate error converges in probability to zero.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [inst_2 : Nonempty ι] (empirical : ℕ → Ω → ι → ℝ) (population : ι → ℝ) (μ : MeasureTheory.Measure Ω), (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => empirical n ω i) (fun x => population i) μ) → Causalean.Stat.Tendsto_inProb (fun n ω => Causalean.Stat.finiteMaxError (empirical n ω) population) (fun x => 0) μ","l":35,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.tendsto_integral_abs_sub_of_tendstoInProbability_of_ae_bound","k":"theorem","m":"Causalean.Stat.CLT.MartingaleArray.ProbabilityBounds","a":"Stat/CLT/MartingaleArray","t":1,"d":"If measurable real row variables converge in probability to a constant, the bound is nonnegative, and their absolute deviations are uniformly bounded almost everywhere, then their expected absolute deviations tend to zero. The probability spaces may vary with the row.","s":"∀ {Ω : ℕ → Type u_1} {mΩ : (n : ℕ) → MeasurableSpace (Ω n)} {μ : (n : ℕ) → MeasureTheory.Measure (Ω n)} [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] (Y : (n : ℕ) → Ω n → ℝ) (c B : ℝ), (∀ (n : ℕ), AEMeasurable (Y n) (μ n)) → 0 ≤ B → (∀ (n : ℕ), ∀ᵐ (ω : Ω n) ∂μ n, |Y n ω - c| ≤ B) → Causalean.Stat.TendstoInProbability μ Y c → Filter.Tendsto (fun n => ∫ (ω : Ω n), |Y n ω - c| ∂μ n) Filter.atTop (nhds 0)","l":24,"f":"Causalean/Stat/CLT/MartingaleArray/ProbabilityBounds.lean"},{"n":"Causalean.Stat.tendsto_measure_finiteMaxError_ge_zero","k":"theorem","m":"Causalean.Stat.MEstimation.FiniteModelSelection","a":"Stat/MEstimation","t":2,"d":"If empirical loss vectors have coordinatewise population targets under a sampling measure, and each coordinate converges in probability, then, for a positive tolerance, the probability that the largest coordinate error exceeds that tolerance converges to zero.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : Fintype ι] [inst_2 : Nonempty ι] (empirical : ℕ → Ω → ι → ℝ) (population : ι → ℝ) (μ : MeasureTheory.Measure Ω), (∀ (i : ι), Causalean.Stat.Tendsto_inProb (fun n ω => empirical n ω i) (fun x => population i) μ) → ∀ {ε : ℝ}, 0 < ε → Filter.Tendsto (fun n => μ {ω | ε ≤ Causalean.Stat.finiteMaxError (empirical n ω) population}) Filter.atTop (nhds 0)","l":54,"f":"Causalean/Stat/MEstimation/FiniteModelSelection.lean"},{"n":"Causalean.Stat.testing_error_lower_of_chi","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Two-point testing-error floor from a finite χ²-budget. For probability measures P, Q such that P is absolutely continuous with respect to Q and the squared density deviation (dP/dQ − 1)² is Q-integrable, if C is nonnegative and the χ²-divergence chiSqDiv P Q is at most C, then for every measurable test region A, the combined testing error is at least 1/(4(C + 1)): P(Aᶜ) + Q(A) ≥ 1/(4(C + 1)). This is the positive two-point testing floor that powers χ²-budget minimax lower bounds — no test can separate P from Q better than this when their χ²-divergence is bounded.","s":"∀ {Ω : Type u_2} [inst : MeasurableSpace Ω] (P Q : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure Q], P.AbsolutelyContinuous Q → MeasureTheory.Integrable (fun x => ((P.rnDeriv Q x).toReal - 1) ^ 2) Q → ∀ {C : ℝ}, 0 ≤ C → Causalean.Stat.chiSqDiv P Q ≤ C → ∀ {A : Set Ω}, MeasurableSet A → P.real Aᶜ + Q.real A ≥ 1 / (4 * (C + 1))","l":561,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.tvDist","k":"def","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":1,"d":"For a sample space equipped with a σ-algebra and two measures on that space, the statistical total variation distance is the supremum, over all measurable events $A$, of the absolute difference between the two measures' real-valued masses of $A$.","s":"{Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → MeasureTheory.Measure Ω → MeasureTheory.Measure Ω → ℝ","l":47,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_eq_half_integral_abs_rnDeriv_sub","k":"theorem","m":"Causalean.Stat.Minimax.MaximalCoupling","a":"Stat/Minimax","t":2,"d":"Scheffé's identity with both probability laws dominated by an arbitrary finite reference measure.","s":"∀ {X : Type u_1} [inst : MeasurableSpace X] (mu nu xi : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure mu] [MeasureTheory.IsProbabilityMeasure nu] [MeasureTheory.IsFiniteMeasure xi], mu.AbsolutelyContinuous xi → nu.AbsolutelyContinuous xi → Causalean.Stat.tvDist mu nu = 1 / 2 * ∫ (x : X), |(mu.rnDeriv xi x).toReal - (nu.rnDeriv xi x).toReal| ∂xi","l":34,"f":"Causalean/Stat/Minimax/MaximalCoupling.lean"},{"n":"Causalean.Stat.tvDist_eq_zero_of_klBudget_nonpos","k":"theorem","m":"Causalean.Stat.Minimax.CoordinatewiseOverlap","a":"Stat/Minimax","t":2,"d":"A nonpositive real KL budget forces two probability measures to coincide, and hence forces their total variation distance to vanish.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {B : ℝ}, B ≤ 0 → InformationTheory.klDiv μ ν ≤ ENNReal.ofReal B → Causalean.Stat.tvDist μ ν = 0","l":317,"f":"Causalean/Stat/Minimax/CoordinatewiseOverlap.lean"},{"n":"Causalean.Stat.tvDist_integral_le_of_abs_le_ae","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"A measurable statistic bounded in absolute value by M almost surely under both laws has an expectation gap no larger than 2M times their total-variation distance.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (M : ℝ), 0 ≤ M → (∀ᵐ (x : Ω) ∂μ, |f x| ≤ M) → (∀ᵐ (x : Ω) ∂ν, |f x| ≤ M) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ 2 * M * Causalean.Stat.tvDist μ ν","l":221,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_integral_le_of_range_ae","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"A measurable statistic that lies almost surely in an interval of width c under both laws has an expectation gap no larger than c times their total-variation distance.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (a c : ℝ), 0 ≤ c → (∀ᵐ (x : Ω) ∂μ, f x ∈ Set.Icc a (a + c)) → (∀ᵐ (x : Ω) ∂ν, f x ∈ Set.Icc a (a + c)) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ Causalean.Stat.tvDist μ ν * c","l":194,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_integral_range","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"The expectation gap of a measurable statistic confined to an interval of width c is at most that width times the total-variation distance between the two probability laws.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (f : Ω → ℝ), Measurable f → ∀ (a c : ℝ), 0 ≤ c → (∀ (x : Ω), f x ∈ Set.Icc a (a + c)) → |∫ (x : Ω), f x ∂μ - ∫ (x : Ω), f x ∂ν| ≤ Causalean.Stat.tvDist μ ν * c","l":116,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_le_half_integral_abs_rnDeriv","k":"theorem","m":"Causalean.Stat.Minimax.Scheffe","a":"Stat/Minimax","t":1,"d":"Scheffé's inequality (≤ direction). For probability measures μ and ν on the same space with μ absolutely continuous with respect to ν, the total variation distance between μ and ν is at most half the L¹(ν)-distance of the Radon–Nikodym density dμ/dν to the constant 1.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → Causalean.Stat.tvDist μ ν ≤ 1 / 2 * ∫ (x : Ω), |(μ.rnDeriv ν x).toReal - 1| ∂ν","l":109,"f":"Causalean/Stat/Minimax/Scheffe.lean"},{"n":"Causalean.Stat.tvDist_le_half_sqrt_chiSqDiv","k":"theorem","m":"Causalean.Stat.Minimax.ChiSquared","a":"Stat/Minimax","t":1,"d":"Cauchy–Schwarz on Scheffé. For probability measures μ, ν with μ absolutely continuous with respect to ν and whose squared Radon–Nikodym density deviation (dμ/dν − 1)² is ν-integrable, the total variation distance tvDist μ ν is bounded by half the square root of the χ²-divergence chiSqDiv μ ν.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], μ.AbsolutelyContinuous ν → MeasureTheory.Integrable (fun x => ((μ.rnDeriv ν x).toReal - 1) ^ 2) ν → Causalean.Stat.tvDist μ ν ≤ 1 / 2 * √(Causalean.Stat.chiSqDiv μ ν)","l":66,"f":"Causalean/Stat/Minimax/ChiSquared.lean"},{"n":"Causalean.Stat.tvDist_le_one","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Total variation distance between probability measures is at most one.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], Causalean.Stat.tvDist μ ν ≤ 1","l":91,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_le_sqrt_two_mul_one_sub_affinity","k":"theorem","m":"Causalean.Stat.Minimax.HellingerAffinity","a":"Stat/Minimax","t":1,"d":"Cauchy–Schwarz on the Hellinger affinity. For a dominating measure μ and functions f, g such that f is μ-integrable, g is μ-integrable, f is pointwise nonnegative, g is pointwise nonnegative, f integrates to 1 against μ, and g integrates to 1 against μ — so that f dμ and g dμ are probability densities — the total variation distance between the two weighted laws is at most the square root of twice the affinity defect, √(2(1 − densityAffinity μ f g)). This is the Cauchy–Schwarz half of the standard total-variation–Hellinger comparison, and it is what converts an affinity computation into a Le Cam two-point bound.","s":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (f g : α → ℝ), MeasureTheory.Integrable f μ → MeasureTheory.Integrable g μ → 0 ≤ f → 0 ≤ g → ∫ (x : α), f x ∂μ = 1 → ∫ (x : α), g x ∂μ = 1 → Causalean.Stat.tvDist (μ.withDensity fun x => ENNReal.ofReal (f x)) (μ.withDensity fun x => ENNReal.ofReal (g x)) ≤ √(2 * (1 - Causalean.Stat.densityAffinity μ f g))","l":125,"f":"Causalean/Stat/Minimax/HellingerAffinity.lean"},{"n":"Causalean.Stat.tvDist_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"Total variation distance between probability measures is nonnegative.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ ν : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν], 0 ≤ Causalean.Stat.tvDist μ ν","l":86,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.tvDist_symm","k":"theorem","m":"Causalean.Stat.Minimax.TotalVariation","a":"Stat/Minimax","t":2,"d":"tvDist is symmetric.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} (μ ν : MeasureTheory.Measure Ω), Causalean.Stat.tvDist μ ν = Causalean.Stat.tvDist ν μ","l":95,"f":"Causalean/Stat/Minimax/TotalVariation.lean"},{"n":"Causalean.Stat.two_point_lower_bound_of_chiSqDiv_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"χ²-form two-point lower bound. For a measurable estimator est, if two candidate values θ₀, θ₁ are 2s-separated, P₀ is absolutely continuous with respect to P₁, the squared density deviation (dP₀/dP₁ − 1)² is P₁-integrable, and the χ²-divergence chiSqDiv P₀ P₁ is at most c, then the worst-case miss probability is at least (1 − ½√c)/2, via tvDist ≤ ½√χ². Since chiSqDiv tensorizes over i.i.d. samples (chiSqDiv_prod) and is computable for explicit families, this is the form used to certify minimax rates.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s : ℝ}, 2 * s ≤ |θ₀ - θ₁| → P₀.AbsolutelyContinuous P₁ → MeasureTheory.Integrable (fun x => ((P₀.rnDeriv P₁ x).toReal - 1) ^ 2) P₁ → ∀ {c : ℝ}, Causalean.Stat.chiSqDiv P₀ P₁ ≤ c → (1 - 1 / 2 * √c) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":98,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.two_point_lower_bound_of_tvDist_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxRisk","a":"Stat/Minimax","t":1,"d":"Le Cam two-point bound with an explicit total-variation bound. If two candidate values θ₀, θ₁ : ℝ are 2s-separated, est : Ω → ℝ is a measurable estimator, and the total variation distance tvDist P₀ P₁ is at most c, then the worst-case probability of missing the truth by ≥ s is at least (1 − c)/2. Variant of real_two_point_lower_bound with an explicit total-variation upper bound.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P₀ P₁ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P₀] [MeasureTheory.IsProbabilityMeasure P₁] {est : Ω → ℝ}, Measurable est → ∀ {θ₀ θ₁ s c : ℝ}, 2 * s ≤ |θ₀ - θ₁| → Causalean.Stat.tvDist P₀ P₁ ≤ c → (1 - c) / 2 ≤ max (P₀.real {ω | s ≤ |est ω - θ₀|}) (P₁.real {ω | s ≤ |est ω - θ₁|})","l":69,"f":"Causalean/Stat/Minimax/MinimaxRisk.lean"},{"n":"Causalean.Stat.uDegen","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"For an outcome space equipped with a σ-algebra, a real-valued two-argument kernel, and a measure $P$ on that space, the degenerate second-order kernel assigns to outcomes $x$ and $y$ the original kernel value minus its population mean and minus its first centered Hoeffding projections at $x$ and $y$.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → X → X → ℝ","l":72,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given a positive order $m$, an order-$m$ real-valued kernel, and a measure on the observation space, the higher-order residual kernel maps each $m$-tuple to the original kernel value minus its population mean and minus the sum of all coordinate-specific first Hoeffding projections at that tuple.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → (Fin m → X) → ℝ","l":115,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder_integral_tail_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The first-order Hoeffding residual has zero conditional mean in each coordinate after integrating over all other coordinates, provided the usual finite-product/Fubini identities and slice integrability assumptions hold.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} [inst_2 : NeZero m] {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → ∀ (j : Fin m) (x : X), (∫ (tail : { k // k ≠ j } → X), Causalean.Stat.uDegenOrder h P (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) = 0","l":198,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uDegenOrder_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 degenerate residual of pairKernel h is the order-2 degenerate kernel uDegen h P, evaluated at the two coordinates.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → Causalean.Stat.uDegenOrder (Causalean.Stat.pairKernel h) P = fun z => Causalean.Stat.uDegen h P (z 0) (z 1)","l":150,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uDegen_integral_right_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"Degeneracy of the second-order kernel. For every fixed x, ∫ g(x, ·) dP = 0. This is the defining property that makes the U-statistic's quadratic remainder negligible at the √n scale.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → ∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P → ∫ (y : X), Causalean.Stat.uDegen h P x y ∂P = 0","l":107,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"For a measurable observation space, an order $m$, a real-valued kernel of that order, and an observation distribution, the first-order influence function maps each observation to the sum of the kernel's first Hoeffding projections over all $m$ coordinate positions.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":41,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uInfluenceOrder_eq_card_mul_of_common_projection","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"If all coordinatewise first projections agree with a common projection, then the order-m influence function is m times that projection.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ} {φ : X → ℝ}, (∀ (j : Fin m) (x : X), Causalean.Stat.uProjOrderAt j h P x = φ x) → Causalean.Stat.uInfluenceOrder h P = fun x => ↑m * φ x","l":51,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uInfluenceOrder_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The summed first-order influence function is integrable if every coordinatewise first projection is integrable.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P) → MeasureTheory.Integrable (fun x => ∑ j, Causalean.Stat.uProjOrderAt j h P x) P","l":164,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The summed first-order influence function is centered if every coordinatewise first projection is centered.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {m : ℕ} {h : (Fin m → X) → ℝ}, (∀ (j : Fin m), MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P) → (∀ (j : Fin m), ∫ (x : X), Causalean.Stat.uProjOrderAt j h P x ∂P = 0) → ∫ (x : X), ∑ j, Causalean.Stat.uProjOrderAt j h P x ∂P = 0","l":173,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uInfluenceOrder_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 influence function of pairKernel h is 2 · h₁, matching the order-2 CLT influence function ψ = 2 · uProj h P.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → Causalean.Stat.uInfluenceOrder (Causalean.Stat.pairKernel h) P = fun x => 2 * Causalean.Stat.uProj h P x","l":138,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uMean","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"For an outcome space equipped with a σ-algebra, a real-valued two-argument kernel, and a measure $P$ on that space, the population mean of the kernel is its iterated integral with respect to $P$ in both arguments.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → ℝ","l":60,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uMeanOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given a measurable observation space and an order-$m$ real-valued kernel on it and a measure on that space, the population mean of the kernel is its integral under the $m$-fold product of that measure.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → ℝ","l":91,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uMeanOrder_pairKernel","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 population mean of pairKernel h is the order-2 mean of h. Requires product-integrability of h so that the joint integral over Fin 2 → X agrees (via Fubini) with the iterated integral defining uMean.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} {h : X → X → ℝ} [MeasureTheory.SigmaFinite P], MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → Causalean.Stat.uMeanOrder (Causalean.Stat.pairKernel h) P = Causalean.Stat.uMean h P","l":55,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uProj","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"For an outcome space equipped with a σ-algebra, a real-valued two-argument kernel, and a measure $P$ on that space, the first centered Hoeffding projection assigns to each outcome $x$ the integral of the kernel with first argument $x$ under $P$, minus the population mean of the kernel.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → (X → X → ℝ) → MeasureTheory.Measure X → X → ℝ","l":66,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uProjOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given a positive order $m$, an order-$m$ real-valued kernel, and a measure on the observation space, the first Hoeffding projection is the coordinate-specific first projection obtained by treating the first coordinate as distinguished.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":110,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an order-$m$ real-valued kernel, a distinguished coordinate, and a measure on the observation space, the coordinate-specific first Hoeffding projection maps a proposed value at that coordinate to the kernel integrated over the product measure for all other coordinates, minus the kernel's population mean.","s":"{X : Type u_2} → [inst : MeasurableSpace X] → {m : ℕ} → Fin m → ((Fin m → X) → ℝ) → MeasureTheory.Measure X → X → ℝ","l":100,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The coordinatewise first projection is integrable whenever the corresponding slice-averaged kernel is integrable.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure P] {m : ℕ} (j : Fin m) {h : (Fin m → X) → ℝ}, MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P → MeasureTheory.Integrable (Causalean.Stat.uProjOrderAt j h P) P","l":133,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The coordinatewise first projection integrates to zero once the slice-averaged representation of the population mean is available. The additional equality is the standard finite-product/Fubini identity for the chosen coordinate.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {m : ℕ} (j : Fin m) {h : (Fin m → X) → ℝ}, MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P → ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P → ∫ (x : X), Causalean.Stat.uProjOrderAt j h P x ∂P = 0","l":145,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uProjOrderAt_pairKernel_of_symm","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":2,"d":"Each coordinate first projection of pairKernel h equals the order-2 first projection of h, for a symmetric kernel. (For j = 0 no symmetry is needed; for j = 1 it is used to swap the integration coordinate.)","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, (∀ (x y : X), h x y = h y x) → MeasureTheory.Integrable (fun p => h p.1 p.2) (P.prod P) → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → ∀ (j : Fin 2), Causalean.Stat.uProjOrderAt j (Causalean.Stat.pairKernel h) P = Causalean.Stat.uProj h P","l":72,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.uProj_integrable","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"The first projection is integrable whenever x ↦ ∫ h(x, ·) dP is.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → MeasureTheory.Integrable (Causalean.Stat.uProj h P) P","l":99,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uProj_integral_eq_zero","k":"theorem","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":2,"d":"The first projection integrates to zero: ∫ h₁ dP = 0.","s":"∀ {X : Type u_2} [inst : MeasurableSpace X] {P : MeasureTheory.Measure X} [MeasureTheory.IsProbabilityMeasure P] {h : X → X → ℝ}, MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → ∫ (x : X), Causalean.Stat.uProj h P x ∂P = 0","l":90,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uRemainderOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"For an i.i.d. sample on a measurable sample space, with observations in a measurable observation space and their probability distribution, a positive integer order, a real-valued kernel of that order, and a sample size, the higher-order Hájek remainder statistic is the order-$m$ U-statistic of the kernel remaining after all of its first-order Hoeffding projections have been removed.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → [NeZero m] → ((Fin m → X) → ℝ) → ℕ → Ω → ℝ","l":60,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatistic","k":"def","m":"Causalean.Stat.UStatistic.Basic","a":"Stat/UStatistic","t":1,"d":"For a sample space and an outcome space, each equipped with a σ-algebra, an independent and identically distributed sample with probability law $\\mu$ and marginal law $P$, a real-valued two-argument kernel, and a nonnegative integer $n$, the order-two U-statistic is the random variable that, at each sample outcome, equals $(n(n-1))^{-1}$ times the sum of the kernel over all ordered distinct pairs among the first $n$ observations.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → (X → X → ℝ) → ℕ → Ω → ℝ","l":52,"f":"Causalean/Stat/UStatistic/Basic.lean"},{"n":"Causalean.Stat.uStatisticOrder","k":"def","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":1,"d":"Given an independent and identically distributed sample on a measurable sample space, with a specified sample-space measure and observation-space measure, an order-$m$ real-valued kernel, and a sample size, the fixed-order U-statistic maps each sample outcome to the average of the kernel over all ordered $m$-tuples of distinct observations among its first $n$ observations.","s":"{Ω : Type u_1} → {X : Type u_2} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace X] → {μ : MeasureTheory.Measure Ω} → {P : MeasureTheory.Measure X} → Causalean.Stat.IIDSample Ω X μ P → {m : ℕ} → ((Fin m → X) → ℝ) → ℕ → Ω → ℝ","l":85,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uStatisticOrder_clt","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.CLT","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic CLT. For an i.i.d. sample S and an order-m kernel h, write ψ for the summed coordinatewise first Hoeffding projection of h. If ψ is measurable, has population mean zero, and is square-integrable, if the higher-order Hájek remainder of the order-m U-statistic is negligible at the √n scale, and if the √n-rescaled U-statistic is almost-everywhere measurable at every sample size, then the √n-rescaled U-statistic converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_3 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uInfluenceOrder h P) → ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → Causalean.Stat.OrderDegenerateNegligible S h → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (fun r => Finset.range r) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) fun r => Finset.range r) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ^ 2 ∂P)) μ hθn_meas","l":52,"f":"Causalean/Stat/UStatistic/OrderM/CLT.lean"},{"n":"Causalean.Stat.uStatisticOrder_clt_of_regular","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.CLT","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic CLT (end-to-end). For an i.i.d. sample S and an order-m kernel h, write g for the higher-order Hájek residual of h and ψ for the summed coordinatewise first Hoeffding projection of h. If the residual g is measurable and square-integrable under the m-fold product law, if for every coordinate integrating h over the remaining m − 1 coordinates yields an integrable function of that coordinate with the same population mean uMeanOrder h P in every coordinate and h remains integrable in the remaining coordinates for every fixed value of that coordinate, and if ψ is measurable, mean zero, square-integrable, and the √n-rescaled U-statistic is almost-everywhere measurable at every sample size, then the √n-rescaled order-m U-statistic converges in distribution to the centered Gaussian law with variance ∫ψ²dP.","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} [inst_2 : MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_3 : NeZero m] (h : (Fin m → X) → ℝ), Measurable (Causalean.Stat.uDegenOrder h P) → MeasureTheory.Integrable (fun z => Causalean.Stat.uDegenOrder h P z ^ 2) (MeasureTheory.Measure.pi fun x => P) → (∀ (j : Fin m), MeasureTheory.Integrable (fun x => ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P) P) → (∀ (j : Fin m), ∫ (x : X), ∫ (tail : { k // k ≠ j } → X), h (Causalean.Stat.insertCoord j x tail) ∂MeasureTheory.Measure.pi fun x => P ∂P = Causalean.Stat.uMeanOrder h P) → (∀ (j : Fin m) (x : X), MeasureTheory.Integrable (fun tail => h (Causalean.Stat.insertCoord j x tail)) (MeasureTheory.Measure.pi fun x => P)) → Measurable (Causalean.Stat.uInfluenceOrder h P) → ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (fun r => Finset.range r) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) fun r => Finset.range r) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ^ 2 ∂P)) μ hθn_meas","l":80,"f":"Causalean/Stat/UStatistic/OrderM/CLT.lean"},{"n":"Causalean.Stat.uStatisticOrder_isAsymLinear","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Fixed-order U-statistic asymptotic linearity. For an i.i.d. sample S and an order-m kernel h, write ψ for the summed coordinatewise first Hoeffding projection of h. If ψ has population mean zero and is square-integrable, and if the higher-order Hájek remainder of the order-m U-statistic is negligible at the √n scale, then the order-m U-statistic is asymptotically linear toward its population mean uMeanOrder h P, with influence function ψ.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ), ∫ (x : X), Causalean.Stat.uInfluenceOrder h P x ∂P = 0 → MeasureTheory.Integrable (fun x => Causalean.Stat.uInfluenceOrder h P x ^ 2) P → Causalean.Stat.OrderDegenerateNegligible S h → Causalean.Stat.IsAsymLinear (Causalean.Stat.uStatisticOrder S h) (Causalean.Stat.uMeanOrder h P) (Causalean.Stat.uInfluenceOrder h P) S fun r => Finset.range r","l":298,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_remainder_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":2,"d":"The Hájek remainder in IsAsymLinear form is the rescaled higher-order residual U-statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ) {n : ℕ}, m ≤ n → (fun ω => √↑(Finset.range n).card * (Causalean.Stat.uStatisticOrder S h n ω - Causalean.Stat.uMeanOrder h P) - (√↑(Finset.range n).card)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.uInfluenceOrder h P (S.Z i ω)) = fun ω => √↑n * Causalean.Stat.uRemainderOrder S h n ω","l":266,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_sub_uMean_eq","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Hajek","a":"Stat/UStatistic/OrderM","t":1,"d":"Hájek decomposition for a fixed-order U-statistic. For an i.i.d. sample S, order-m kernel h, and sample outcome ω, if the sample size is at least m, then the order-m U-statistic centered at its population mean decomposes exactly as the average of the first-order influence function over the first n sample points plus the higher-order Hájek remainder statistic.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) {m : ℕ} [inst_2 : NeZero m] (h : (Fin m → X) → ℝ) {n : ℕ}, m ≤ n → ∀ (ω : Ω), Causalean.Stat.uStatisticOrder S h n ω - Causalean.Stat.uMeanOrder h P = (↑n)⁻¹ * ∑ i ∈ Finset.range n, Causalean.Stat.uInfluenceOrder h P (S.Z i ω) + Causalean.Stat.uRemainderOrder S h n ω","l":211,"f":"Causalean/Stat/UStatistic/OrderM/Hajek.lean"},{"n":"Causalean.Stat.uStatisticOrder_two_eq_uStatistic","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.Basic","a":"Stat/UStatistic/OrderM","t":2,"d":"The order-2 fixed-order statistic agrees with the existing ordered off-diagonal U-statistic for the corresponding pair kernel.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ) (n : ℕ), Causalean.Stat.uStatisticOrder S (Causalean.Stat.pairKernel h) n = Causalean.Stat.uStatistic S h n","l":362,"f":"Causalean/Stat/UStatistic/OrderM/Basic.lean"},{"n":"Causalean.Stat.uStatistic_clt_of_symmetric_via_orderM","k":"theorem","m":"Causalean.Stat.UStatistic.OrderM.OrderTwo","a":"Stat/UStatistic/OrderM","t":1,"d":"Order-2 U-statistic CLT, derived from the fixed-order-m CLT. For an i.i.d. sample S and a two-argument kernel h : X → X → ℝ that is symmetric and jointly measurable, suppose the order-2 Hájek residual of h is square-integrable under the product law P × P, the row integral x ↦ ∫h(x,y)dP(y) is integrable and each row y ↦ h(x,y) is itself integrable for every x, the first Hoeffding projection of h is square-integrable, and the √n-rescaled U-statistic is almost-everywhere measurable at every sample size. Then the √n-rescaled order-2 U-statistic converges in distribution to the centered Gaussian law with variance 4ζ₁, where ζ₁ is the variance of the first Hoeffding projection.","s":"∀ {Ω : Type u_3} {X : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {P : MeasureTheory.Measure X} (S : Causalean.Stat.IIDSample Ω X μ P) (h : X → X → ℝ), (∀ (x y : X), h x y = h y x) → (Measurable fun p => h p.1 p.2) → MeasureTheory.Integrable (fun p => Causalean.Stat.uDegen h P p.1 p.2 ^ 2) (P.prod P) → MeasureTheory.Integrable (fun x => ∫ (y : X), h x y ∂P) P → (∀ (x : X), MeasureTheory.Integrable (fun y => h x y) P) → MeasureTheory.Integrable (fun x => Causalean.Stat.uProj h P x ^ 2) P → ∀ (hθn_meas : ∀ (n : ℕ), AEMeasurable (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatistic S h) (Causalean.Stat.uMean h P) (fun m => Finset.range m) n) μ), Causalean.Stat.Tendsto_dist (Causalean.Stat.IsAsymLinear.rescaledEstimator (Causalean.Stat.uStatistic S h) (Causalean.Stat.uMean h P) fun m => Finset.range m) (Causalean.Stat.gaussianMeasure 0 (∫ (x : X), (fun x => 2 * Causalean.Stat.uProj h P x) x ^ 2 ∂P)) μ hθn_meas","l":173,"f":"Causalean/Stat/UStatistic/OrderM/OrderTwo.lean"},{"n":"Causalean.Stat.unifOI","k":"def","m":"Causalean.Stat.Coupling.ProductLossMonotoneCoupling.PIT","a":"Stat/Coupling/ProductLossMonotoneCoupling","t":1,"d":"The uniform probability measure on the open unit interval is Lebesgue measure restricted to the interval $(0,1)$.","s":"MeasureTheory.Measure ℝ","l":38,"f":"Causalean/Stat/Coupling/ProductLossMonotoneCoupling/PIT.lean"},{"n":"Causalean.Stat.uniformlyBounded_kernelAffinePullback","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"If the affine slope is nonzero and the target estimator is uniformly bounded, then the affine kernel pullback is uniformly bounded on the source experiment.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {a b : ℝ}, a ≠ 0 → ∀ {targetEst : Y → ℝ}, Causalean.Stat.UniformlyBounded targetEst → Causalean.Stat.UniformlyBounded (Causalean.Stat.kernelAffinePullback K a b targetEst)","l":107,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.uniformlyBounded_kernelMean","k":"theorem","m":"Causalean.Stat.Minimax.MarkovKernelTransport","a":"Stat/Minimax","t":2,"d":"Uniform boundedness of an estimator implies that averaging it against a Markov kernel is uniformly bounded by the same witness.","s":"∀ {X : Type uX} {Y : Type uY} [inst : MeasurableSpace X] [inst_1 : MeasurableSpace Y] (K : ProbabilityTheory.Kernel X Y) [ProbabilityTheory.IsMarkovKernel K] {T : Y → ℝ}, Causalean.Stat.UniformlyBounded T → Causalean.Stat.UniformlyBounded (Causalean.Stat.kernelMean K T)","l":82,"f":"Causalean/Stat/Minimax/MarkovKernelTransport.lean"},{"n":"Causalean.Stat.uniqueGlobalMax_maximizerOrZero","k":"theorem","m":"Causalean.Stat.MEstimation.FinitePoisson","a":"Stat/MEstimation","t":2,"d":"A criterion with a unique global maximum is maximized uniquely by its total maximizer selector.","s":"∀ {E : Type u_1} [inst : Zero E] (f : E → ℝ), (∃! x, ∀ (y : E), f y ≤ f x) → Causalean.Stat.IsUniqueGlobalMax f (Causalean.Stat.maximizerOrZero f)","l":335,"f":"Causalean/Stat/MEstimation/FinitePoisson.lean"},{"n":"Causalean.Stat.usableGroup","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, a sample, and a group label, the usable-group condition holds exactly when the count in the false arm is positive and the count in the true arm is positive.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → kappa → Prop","l":85,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.usableGroupTotal","k":"def","m":"Causalean.Stat.Sample.OccupancyWeightedMean.Basic","a":"Stat/Sample/OccupancyWeightedMean","t":1,"d":"Given a sample space, a finite group-label space whose labels can be compared for equality, a nonnegative integer sample size, a group-label map, a Boolean arm-assignment map, and a sample, the usable-group total is the sum of group counts over precisely those groups represented by both arm labels.","s":"{Omega : Type u_1} → {kappa : Type u_2} → [Fintype kappa] → [DecidableEq kappa] → {n : ℕ} → (Omega → kappa) → (Omega → Bool) → (Fin n → Omega) → ℕ","l":95,"f":"Causalean/Stat/Sample/OccupancyWeightedMean/Basic.lean"},{"n":"Causalean.Stat.variance_offDiag_kernel_le","k":"theorem","m":"Causalean.Stat.UStatistic.Variance","a":"Stat/UStatistic","t":2,"d":"A bounded kernel's off-diagonal average over an independent sample has variance at most thirty-two times the squared kernel bound divided by the sample size.","s":"∀ {X : Type u_3} [inst : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbabilityMeasure μ] (N : ℕ) (kernel : X → X → ℝ), Measurable (Function.uncurry kernel) → ∀ (M : ℝ), (∀ (x y : X), |kernel x y| ≤ M) → ProbabilityTheory.variance (fun target => (↑N * ↑(N - 1))⁻¹ * ∑ i, ∑ j, if i ≠ j then kernel (target i) (target j) else 0) (MeasureTheory.Measure.pi fun x => μ) ≤ 32 * M ^ 2 / ↑N","l":324,"f":"Causalean/Stat/UStatistic/Variance.lean"},{"n":"Causalean.Stat.worstCaseRisk","k":"def","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":1,"d":"Given an estimator class, a model class, a real-valued risk for each estimator-model pair, and an estimator, the worst-case risk is the supremum, over all models in the class, of that estimator's risk.","s":"{E : Sort u_1} → {Θ : Sort u_3} → (E → Θ → ℝ) → E → ℝ","l":45,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_eq_sSup_range","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The worst-case risk is the supremum of the set of risks the estimator attains across the class. Rewriting handle for arguments phrased on the set of attained risks.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = sSup (Set.range (risk e))","l":248,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_le","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A bound holding at every model in a nonempty class bounds the worst-case risk.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [Nonempty Θ] {risk : E → Θ → ℝ} {e : E} {c : ℝ}, (∀ (θ : Θ), risk e θ ≤ c) → Causalean.Stat.worstCaseRisk risk e ≤ c","l":122,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_mono_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Comparing two model classes at a fixed estimator: if every model of the first class is matched, through the map, by a model of the second class whose risk is at least as large, the worst-case risk over the first class is at most that over the second. The second class must bound the estimator's risk, and the first must be nonempty; see worstCaseRisk_mono_class_of_nonneg for the version that drops the nonemptiness.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} [Nonempty Θ] {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} {e : E} (φ : Θ → Θ'), BddAbove (Set.range (risk' e)) → (∀ (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e","l":172,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_mono_class_of_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Version of worstCaseRisk_mono_class for a nonnegative larger risk: no nonemptiness assumption on the smaller class is needed, because an empty class contributes worst-case risk zero, which the larger nonnegative worst-case risk already dominates.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {Θ' : Sort u_4} {risk : E → Θ → ℝ} {risk' : E → Θ' → ℝ} {e : E} (φ : Θ → Θ'), BddAbove (Set.range (risk' e)) → (∀ (θ' : Θ'), 0 ≤ risk' e θ') → (∀ (θ : Θ), risk e θ ≤ risk' e (φ θ)) → Causalean.Stat.worstCaseRisk risk e ≤ Causalean.Stat.worstCaseRisk risk' e","l":184,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_nonneg","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"A worst-case risk is nonnegative as soon as the risk is, with no boundedness or nonemptiness caveat: an empty class or an unbounded risk both give the value zero.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} {risk : E → Θ → ℝ} {e : E}, (∀ (θ : Θ), 0 ≤ risk e θ) → 0 ≤ Causalean.Stat.worstCaseRisk risk e","l":91,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_of_isEmpty_class","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"Over an empty model class the worst-case risk of every estimator is zero.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [IsEmpty Θ] (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = 0","l":67,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_of_unique","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"When the model class consists of a single law, the worst-case risk is just the risk at that law. This covers decision problems that minimise a functional of the estimator alone, with no adversarial choice of model.","s":"∀ {E : Sort u_1} {Θ : Sort u_3} [inst : Unique Θ] (risk : E → Θ → ℝ) (e : E), Causalean.Stat.worstCaseRisk risk e = risk e default","l":80,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.worstCaseRisk_subtype_eq_sSup_setOf","k":"theorem","m":"Causalean.Stat.Minimax.MinimaxValue","a":"Stat/Minimax","t":2,"d":"The worst-case risk over a class carved out by a membership condition, written as the supremum of the set of risks attained on the class. This is the bridge to the spelling that states the class by a predicate rather than by a subtype.","s":"∀ {A : Type u_5} {B : Type u_6} {Cls : B → Prop} (risk : A → B → ℝ) (e : A), Causalean.Stat.worstCaseRisk (fun a θ => risk a ↑θ) e = sSup {q | ∃ θ, Cls θ ∧ q = risk e θ}","l":279,"f":"Causalean/Stat/Minimax/MinimaxValue.lean"},{"n":"Causalean.Stat.zEstimator_clt","k":"theorem","m":"Causalean.Stat.MEstimation.ZEstimatorCLT","a":"Stat/MEstimation","t":1,"d":"Z-estimator asymptotic linearity. An estimator sequence that solves the empirical estimating equation eventually, almost surely, is consistent at the target parameter, converges to the target at the parametric $\\sqrt n$-rate, and whose score process is stochastically equicontinuous at the target parameter along the estimator sequence, admits the influence-function representation obtained by applying the negative inverse Jacobian to the target score.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":58,"f":"Causalean/Stat/MEstimation/ZEstimatorCLT.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_donsker","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT with the equicontinuity hypothesis discharged. Identical conclusion to zEstimator_clt, but the estimator-sequence-specific equicontinuity modulus is replaced by a class-level Donsker condition: if the estimator converges in probability to the target parameter, the score family is asymptotically equicontinuous at the target parameter — a property of the family alone, independent of the particular estimator sequence, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, then the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E), (∀ ε > 0, Filter.Tendsto (fun n => μ {ω | ε < ‖θn n ω - θ₀‖}) Filter.atTop (nhds 0)) → Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":126,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_extremum","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT from extremum primitives. Identical conclusion to zEstimator_clt, but with consistency derived rather than assumed: given an auxiliary criterion function m, suppose the criterion class indexed by the parameter obeys a uniform law of large numbers, i.e. is Glivenko–Cantelli, the estimator sample-maximises the empirical criterion at every sample size, and the population criterion has a well-separated maximum at the target parameter. Then, provided the score process is stochastically equicontinuous at the target parameter along the estimator sequence, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → Causalean.Stat.StochEquicontAt ψ θ₀ P μ S θn → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":88,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.Stat.zEstimator_clt_of_extremum_donsker","k":"theorem","m":"Causalean.Stat.MEstimation.ExtremumConsistency","a":"Stat/MEstimation","t":1,"d":"Z-estimator CLT from primitive conditions: both opaque hypotheses discharged. Combines zEstimator_clt_of_extremum and zEstimator_clt_of_donsker: if the criterion class indexed by the parameter obeys a uniform law of large numbers, the estimator sample-maximises the empirical criterion at every sample size, the population criterion has a well-separated maximum at the target parameter, the score family is asymptotically equicontinuous at the target parameter, the estimator converges to the target at the parametric $\\sqrt n$-rate, and the estimator solves the empirical estimating equation eventually, almost surely, then the estimator is asymptotically linear at the target parameter, with influence function minus the inverse Jacobian applied to the score at the target.","s":"∀ {Ω : Type u_1} {X : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace X] {μ : MeasureTheory.Measure Ω} {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [inst_5 : MeasurableSpace E] [BorelSpace E] (ψ : E → X → E) (θ₀ : E) (P : MeasureTheory.Measure X) (reg : Causalean.Stat.ZEstimatorRegularity ψ θ₀ P) [MeasureTheory.IsProbabilityMeasure μ] (S : Causalean.Stat.IIDSample Ω X μ P) (θn : ℕ → Ω → E) (m : E → X → ℝ), Causalean.Stat.GlivenkoCantelli S m → (∀ (n : ℕ) (ω : Ω), S.sampleMean (m θ₀) n ω ≤ S.sampleMean (m (θn n ω)) n ω) → (∀ (ε : ℝ), 0 < ε → ∃ η, 0 < η ∧ ∀ (θ : E), ε ≤ dist θ θ₀ → ∫ (x : X), m θ x ∂P + η ≤ ∫ (x : X), m θ₀ x ∂P) → Causalean.Stat.AsymptoticEquicont ψ θ₀ P μ S → Causalean.Stat.IsBigOp (fun n ω => ‖θn n ω - θ₀‖) (fun n => (√↑n)⁻¹) μ → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ᵐ (ω : Ω) ∂μ, ∑ i ∈ Finset.range n, ψ (θn n ω) (S.Z i ω) = 0) → Causalean.Stat.IsAsymLinearVec θn θ₀ (fun z => -reg.J₀_inv (ψ θ₀ z)) S fun n => Finset.range n","l":158,"f":"Causalean/Stat/MEstimation/ExtremumConsistency.lean"},{"n":"Causalean.SteinMethod.DepGraph","k":"structure","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"A dependency graph for a family of real-valued random variables X on a probability space with law μ: bundles an adjacency relation on the index set that is reflexive and symmetric, together with measurability of each variable in the family and the defining property that any two index sets joined by no edge carry independent variable tuples. The closed neighborhood of an index is the set of indices adjacent to it.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {ι : Type u_2} → (ι → Ω → ℝ) → MeasureTheory.Measure Ω → Type u_2","l":45,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_cov_locProd_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If every variable is bounded in absolute value by a nonnegative constant and each closed dependency neighborhood has at most m members, the absolute covariance of any two localized products is at most twice the square of m times that bound squared.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i j : ι), |ProbabilityTheory.covariance (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) (fun ω => X j ω * ∑ k ∈ D.nbhd j, X k ω) μ| ≤ 2 * (↑m * B ^ 2) ^ 2","l":211,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_locProd_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If each summand has absolute value at most B and each neighborhood has at most m members, then a summand times its neighborhood sum has absolute value at most m times B squared.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i : ι) (ω : Ω), |X i ω * ∑ k ∈ D.nbhd i, X k ω| ≤ ↑m * B ^ 2","l":117,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.abs_nbhdSum_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If each summand is bounded in absolute value by B and every closed neighborhood has at most m indices, then the absolute value of each neighborhood sum is at most m times B.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ} (i : ι), (D.nbhd i).card ≤ m → ∀ (ω : Ω), |∑ k ∈ D.nbhd i, X k ω| ≤ ↑m * B","l":104,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.cov_mul_nbhd_eq_zero","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Covariance vanishing for separated indices. If Nᵢ and Nⱼ have no edges between them, the localized products Xᵢ·Tᵢ and Xⱼ·Tⱼ are uncorrelated.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {i j : ι}, (∀ a ∈ D.nbhd i, ∀ b ∈ D.nbhd j, ¬D.G a b) → ∫ (x : Ω), (fun ω => (X i ω * ∑ k ∈ D.nbhd i, X k ω) * (X j ω * ∑ k ∈ D.nbhd j, X k ω)) x ∂μ = (∫ (x : Ω), (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) x ∂μ) * ∫ (x : Ω), (fun ω => X j ω * ∑ k ∈ D.nbhd j, X k ω) x ∂μ","l":161,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.indepFun_leaveOut","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Leave-out independence (the stein_cdf_clt hypothesis hindep).","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] [inst_2 : DecidableEq ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), ProbabilityTheory.IndepFun (X i) (fun ω => ∑ j ∈ Finset.univ \\ D.nbhd i, X j ω) μ","l":130,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.measurable_locProd","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The localized product gᵢ = Xᵢ · Tᵢ is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), Measurable fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω","l":96,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.measurable_nbhdSum","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The neighborhood sum is measurable.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), Measurable fun ω => ∑ k ∈ D.nbhd i, X k ω","l":89,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.memLp_locProd","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"If every variable is bounded in absolute value by a nonnegative constant and each closed dependency neighborhood has bounded size, each variable times its neighborhood sum has a finite second moment.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ∀ (i : ι), MeasureTheory.MemLp (fun ω => X i ω * ∑ k ∈ D.nbhd i, X k ω) 2 μ","l":200,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.mem_nbhd_iff","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Membership in the neighborhood is exactly adjacency.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {i j : ι}, j ∈ D.nbhd i ↔ D.G i j","l":79,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.nbhd","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For any finite index set and a dependency graph for a real-valued random-variable family, and for an index $i$, the closed dependency neighborhood of $i$ is the finite set of all indices adjacent to $i$ in that graph, including $i$ itself.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → {μ : MeasureTheory.Measure Ω} → {ι : Type u_2} → [Fintype ι] → {X : ι → Ω → ℝ} → Causalean.SteinMethod.DepGraph X μ → ι → Finset ι","l":70,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.self_mem_nbhd","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Each index is in its own neighborhood.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_2} [inst_1 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) (i : ι), i ∈ D.nbhd i","l":85,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.sum_E_nbhd_sq_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Negligibility bound (herr2). For any finite neighborhoods of size at most m, measurable summands bounded by B satisfy ∑ᵢ E[|Xᵢ|·Tᵢ²] ≤ m²·N·B³.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (N : ι → Finset ι), (∀ (i : ι), Measurable (X i)) → ∀ {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (N i).card ≤ m) → ∑ i, ∫ (ω : Ω), |X i ω| * (∑ k ∈ N i, X k ω) ^ 2 ∂μ ≤ ↑m ^ 2 * ↑(Fintype.card ι) * B ^ 3","l":342,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.DepGraph.var_nbhd_prod_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Pair-counting variance bound (herr1). With bounded summands |Xᵢ| ≤ B and degree ≤ m, the covariance double sum collapses to the ≤ N·m³ pairs at graph distance at most three, each bounded by 2(mB²)², giving Var(∑ᵢ Xᵢ·Tᵢ) ≤ 2·m⁵·N·B⁴.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] {X : ι → Ω → ℝ} (D : Causalean.SteinMethod.DepGraph X μ) {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → ∀ {m : ℕ}, (∀ (i : ι), (D.nbhd i).card ≤ m) → ProbabilityTheory.variance (fun ω => ∑ i, X i ω * ∑ k ∈ D.nbhd i, X k ω) μ ≤ 2 * ↑m ^ 5 * ↑(Fintype.card ι) * B ^ 4","l":260,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.abs_gExpect_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute value of the standard-normal expectation of a uniformly bounded function is at most the same bound. This is useful throughout Stein-method estimates.","s":"∀ {h : ℝ → ℝ} {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → |Causalean.SteinMethod.gExpect h| ≤ C","l":65,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.abs_integral_mul_le_sqrt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For square-integrable real-valued functions, the absolute value of their product integral is at most the product of the square roots of their two squared integrals.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} (f g : Ω → ℝ), MeasureTheory.MemLp f 2 μ → MeasureTheory.MemLp g 2 μ → |∫ (ω : Ω), f ω * g ω ∂μ| ≤ √(∫ (ω : Ω), f ω ^ 2 ∂μ) * √(∫ (ω : Ω), g ω ^ 2 ∂μ)","l":60,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.abs_le_exp_sq_div_four","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute value of any real number is at most the exponential of one quarter of its square.","s":"∀ (x : ℝ), |x| ≤ Real.exp (x ^ 2 / 4)","l":126,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"Bounded-degree dependency-graph CLT (raw, variance-floor form). Fix, for each n, a probability measure μ n, triangular-array summands X n i, and a dependency graph Dep n on the index set, with every dependency neighborhood of size at most a fixed bound Dmax. Suppose the summands are uniformly bounded in absolute value by a nonnegative constant M and mean zero, with the total variance v n satisfying, for a strictly positive constant c, a floor v n ≥ c · card (ι n) eventually in n; suppose also the index-set cardinality diverges to infinity. Then for every threshold s, the CDF of the standardized sum depSum (X n) / √(v n) under μ n at s converges, as n → ∞, to the standard-normal CDF at s. The index-size divergence together with the variance floor forces the Lyapunov ratio card (ι n) · (M / √(v n))³ → 0 that drives the underlying Stein bound.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":174,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt_eventually_bounded","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Eventual-bound variant of bounded_degree_dependency_clt: only the tail of a triangular array affects the limiting CDF, so the uniform summand bound need only hold after a finite prefix.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ᶠ (n : ℕ) in Filter.atTop, ∀ (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ᶠ (n : ℕ) in Filter.atTop, c * ↑(Fintype.card (ι n)) ≤ v n) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":230,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.bounded_degree_dependency_clt_of_variance_floor_all","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Dependency-graph CLT wrapper when the linear variance floor holds at every index. The public theorem below removes this all-index convenience by shifting to a tail.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (Dep : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (Dmax : ℕ), (∀ (n : ℕ) (i : ι n), ((Dep n).nbhd i).card ≤ Dmax) → ∀ (M : ℝ), 0 ≤ M → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ M) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → ∀ (v : ℕ → ℝ), (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = v n) → ∀ (c : ℝ), 0 < c → (∀ (n : ℕ), c * ↑(Fintype.card (ι n)) ≤ v n) → (∀ (n : ℕ), 0 < Fintype.card (ι n)) → Filter.Tendsto (fun n => Fintype.card (ι n)) Filter.atTop Filter.atTop → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (fun ω => Causalean.SteinMethod.depSum (X n) ω / √(v n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":59,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.cdf_tendsto_of_charFun_tendsto","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"CDF convergence from characteristic-function convergence. A sequence of real probability laws whose characteristic functions converge to those of an atomless target law has convergent CDF values at every threshold.","s":"∀ (lawn : ℕ → MeasureTheory.ProbabilityMeasure ℝ) (ν : MeasureTheory.ProbabilityMeasure ℝ) [MeasureTheory.NullSingletonClass ↑ν], (∀ (t : ℝ), Filter.Tendsto (fun n => MeasureTheory.charFun (↑(lawn n)) t) Filter.atTop (nhds (MeasureTheory.charFun (↑ν) t))) → ∀ (s : ℝ), Filter.Tendsto (fun n => (↑(lawn n)).real (Set.Iic s)) Filter.atTop (nhds ((↑ν).real (Set.Iic s)))","l":42,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.charFun_map_eq_cos_sin","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"A measurable real random variable under a finite measure has a characteristic function whose real and imaginary components are the corresponding cosine and sine integrals.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure ν] (W : Ω → ℝ), Measurable W → ∀ (t : ℝ), MeasureTheory.charFun (MeasureTheory.Measure.map W ν) t = ↑(∫ (ω : Ω), Real.cos (t * W ω) ∂ν) + ↑(∫ (ω : Ω), Real.sin (t * W ω) ∂ν) * Complex.I","l":77,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.depGraph_div_const","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.StandardizedDepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For every measurable sample space, finite index set, measure, and real-valued random-variable family, a dependency graph for that family, and any real constant $s$, the rescaled dependency graph is a dependency graph for the variables obtained by dividing each original variable by $s$. It retains the original adjacency relation, including its reflexivity and symmetry, and its measurability and independence properties.","s":"{Ω : Type u_1} → {ι : Type u_2} → [inst : MeasurableSpace Ω] → [Fintype ι] → {μ : MeasureTheory.Measure Ω} → {X : ι → Ω → ℝ} → Causalean.SteinMethod.DepGraph X μ → (s : ℝ) → Causalean.SteinMethod.DepGraph (fun i ω => X i ω / s) μ","l":29,"f":"Causalean/Mathlib/Probability/SteinMethod/StandardizedDepGraphCLT.lean"},{"n":"Causalean.SteinMethod.depSum","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For any sample space, any finite index set, and family of real-valued random variables indexed by that set, the aggregate random-variable function assigns to each sample point the sum of the family’s values at that point.","s":"{Ω : Type u_1} → {ι : Type u_2} → [Fintype ι] → (ι → Ω → ℝ) → Ω → ℝ","l":48,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.gExpect","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For every real-valued function, the standard-normal expectation of that function is its integral with respect to the normal distribution having mean zero and variance one.","s":"(ℝ → ℝ) → ℝ","l":40,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.nbhdSum","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For any sample space, any index set, a family of real-valued random variables, a finite neighborhood assigned to each index, and a selected index, the neighborhood-sum random-variable function assigns to each sample point the sum of the variables in that selected index’s neighborhood.","s":"{Ω : Type u_1} → {ι : Type u_2} → (ι → Ω → ℝ) → (ι → Finset ι) → ι → Ω → ℝ","l":53,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.phi","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For every real argument, the Gaussian weight is $e^{-x^2/2}$.","s":"ℝ → ℝ","l":34,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.phi_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The Gaussian weight φ(x) = e^{-x²/2} is continuous on the real line.","s":"Continuous Causalean.SteinMethod.phi","l":39,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.phi_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The Gaussian weight φ is integrable over the real line.","s":"MeasureTheory.Integrable Causalean.SteinMethod.phi MeasureTheory.volume","l":74,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinIntegrand","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For every real-valued test function and real argument, the Stein integrand value is the test function at that argument minus its standard-normal expectation, multiplied by $e^{-x^2/2}$.","s":"(ℝ → ℝ) → ℝ → ℝ","l":52,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinIntegrand_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The Stein integrand is continuous whenever the test function h is.","s":"∀ (h : ℝ → ℝ), Continuous h → Continuous (Causalean.SteinMethod.steinIntegrand h)","l":58,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol","k":"def","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For every real-valued test function and real evaluation point, the Stein-solution value is the exponential factor $e^{w^2/2}$ times the integral from $-\\infty$ to $w$ of the test function minus its standard-normal expectation, weighted by $e^{-x^2/2}$.","s":"(ℝ → ℝ) → ℝ → ℝ","l":45,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":1,"d":"For a real-valued test function h that is bounded in absolute value by a constant C, has derivative bounded in absolute value by a constant L, and is differentiable everywhere, the Stein equation's solution steinSol h, evaluated at any point w, is bounded in absolute value by the derivative bound L.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |Causalean.SteinMethod.steinSol h w| ≤ L","l":488,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Sup bound on the derivative of the Stein solution in terms of the h-derivative bound.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |deriv (Causalean.SteinMethod.steinSol h) w| ≤ 2 * L","l":776,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_continuous","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"A continuous differentiable test function with bounded values and bounded derivative has a Stein solution whose derivative is continuous.","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C L : ℝ}, 0 ≤ L → (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → Continuous (deriv (Causalean.SteinMethod.steinSol h))","l":86,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For a bounded differentiable test function, the derivative of its standard-normal Stein solution is differentiable at each point, with derivative equal to the solution value plus the point times its first derivative plus the derivative of the test function.","s":"∀ (h : ℝ → ℝ) {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → Differentiable ℝ h → ∀ (w : ℝ), HasDerivAt (deriv (Causalean.SteinMethod.steinSol h)) (Causalean.SteinMethod.steinSol h w + (w * deriv (Causalean.SteinMethod.steinSol h) w + deriv h w)) w","l":962,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_deriv_lipschitz","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"Lipschitz bound on the derivative of the Stein solution (equivalent to ‖f_h''‖ ≤ 2L; Chen–Goldstein–Shao Lemma 2.5). The form f_h' is 2L-Lipschitz is what the second-order Taylor step in the local-dependence Stein bound consumes.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (u v : ℝ), |deriv (Causalean.SteinMethod.steinSol h) u - deriv (Causalean.SteinMethod.steinSol h) v| ≤ 2 * L * |u - v|","l":1007,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_hasDerivAt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The Stein equation. For a continuous test function h that is bounded in absolute value by a constant C, the Stein solution steinSol h is differentiable at every point w, with derivative w·steinSol h w + (h w − E[h(Z)]) for a standard normal Z, i.e. it solves the Stein equation f'(w) − w·f(w) = h(w) − E[h(Z)].","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → ∀ (w : ℝ), HasDerivAt (Causalean.SteinMethod.steinSol h) (w * Causalean.SteinMethod.steinSol h w + (h w - Causalean.SteinMethod.gExpect h)) w","l":95,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_secondDeriv_abs_le","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The absolute second derivative of the standard-normal Stein solution at each point is at most twice the uniform bound on the derivative of the test function.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (w : ℝ), |Causalean.SteinMethod.steinSol h w + (w * deriv (Causalean.SteinMethod.steinSol h) w + deriv h w)| ≤ 2 * L","l":986,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.SteinMethod.steinSol_stein_eq","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Solution","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The Stein equation in subtractive form.","s":"∀ (h : ℝ → ℝ), Continuous h → ∀ {C : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → ∀ (w : ℝ), deriv (Causalean.SteinMethod.steinSol h) w - w * Causalean.SteinMethod.steinSol h w = h w - Causalean.SteinMethod.gExpect h","l":147,"f":"Causalean/Mathlib/Probability/SteinMethod/Solution.lean"},{"n":"Causalean.SteinMethod.steinSol_taylor_right","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The standard-normal Stein solution has a first-order Taylor error that grows no faster than the test function's derivative bound times the squared step size.","s":"∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → ∀ (a t : ℝ), |Causalean.SteinMethod.steinSol h (a + t) - Causalean.SteinMethod.steinSol h a - t * deriv (Causalean.SteinMethod.steinSol h) (a + t)| ≤ L * t ^ 2","l":97,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.stein_cdf_clt","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The bounded local-dependence CLT (CDF form). Fix, for each sample size n, a probability measure μ n, triangular-array summands X n i, and an index-dependent neighborhood set N n i for each i, all jointly measurable. Suppose the summands are uniformly bounded by a nonnegative constant sequence B n, mean zero, and each summand is independent of the sum of the summands outside its neighborhood; suppose the standardized sum has unit variance, and that the variance of the neighborhood-weighted cross term and the aggregate third-absolute-moment error term both tend to zero as n → ∞. Then for every threshold s, the CDF of the dependency sum under μ n at s converges, as n → ∞, to the standard-normal CDF at s.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] [inst_3 : (n : ℕ) → DecidableEq (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (N : (n : ℕ) → ι n → Finset (ι n)), (∀ (n : ℕ) (i : ι n), Measurable (X n i)) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ) (i : ι n), ProbabilityTheory.IndepFun (X n i) (fun ω => ∑ j ∈ Finset.univ \\ N n i, X n j ω) (μ n)) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => ∑ i, X n i ω * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω) (μ n)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, ∫ (ω : Ω n), |X n i ω| * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω ^ 2 ∂μ n) Filter.atTop (nhds 0) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (X n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":162,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.stein_cdf_clt_of_depGraph","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DepGraphCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The bounded dependency-graph CLT. Fix, for each n, a probability measure μ n, triangular-array summands X n i, and a dependency graph D n on the index set recording which pairs of summands may be dependent, with every dependency neighborhood of size at most m. Suppose the summands are uniformly bounded in absolute value by a constant sequence B n that tends to zero, with the product of the index-set cardinality and the cube of the bound also tending to zero, and suppose each summand is mean zero with the standardized sum having unit total variance. Then for every threshold s, the CDF of the dependency sum under μ n at s converges, as n → ∞, to the standard-normal CDF at s. The two Stein negligibility limits are derived internally.","s":"∀ {Ω : ℕ → Type u_3} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_4} [inst_2 : (n : ℕ) → Fintype (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (D : (n : ℕ) → Causalean.SteinMethod.DepGraph (X n) (μ n)) (m : ℕ), (∀ (n : ℕ) (i : ι n), ((D n).nbhd i).card ≤ m) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → Filter.Tendsto B Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ↑(Fintype.card (ι n)) * B n ^ 3) Filter.atTop (nhds 0) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → ∀ (s : ℝ), Filter.Tendsto (fun n => (MeasureTheory.Measure.map (Causalean.SteinMethod.depSum (X n)) (μ n)).real (Set.Iic s)) Filter.atTop (nhds ((ProbabilityTheory.gaussianReal 0 1).real (Set.Iic s)))","l":395,"f":"Causalean/Mathlib/Probability/SteinMethod/DepGraphCLT.lean"},{"n":"Causalean.SteinMethod.stein_expect_tendsto","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.CLT","a":"Mathlib/Probability/SteinMethod","t":2,"d":"For uniformly bounded, mean-zero locally dependent sums with unit variance whose Stein error terms vanish, expectations of a differentiable test function converge to its standard-normal expectation.","s":"∀ {Ω : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (Ω n)] (μ : (n : ℕ) → MeasureTheory.Measure (Ω n)) [∀ (n : ℕ), MeasureTheory.IsProbabilityMeasure (μ n)] {ι : ℕ → Type u_2} [inst_2 : (n : ℕ) → Fintype (ι n)] [inst_3 : (n : ℕ) → DecidableEq (ι n)] (X : (n : ℕ) → ι n → Ω n → ℝ) (N : (n : ℕ) → ι n → Finset (ι n)), (∀ (n : ℕ) (i : ι n), Measurable (X n i)) → ∀ (B : ℕ → ℝ), (∀ (n : ℕ), 0 ≤ B n) → (∀ (n : ℕ) (i : ι n) (ω : Ω n), |X n i ω| ≤ B n) → (∀ (n : ℕ) (i : ι n), ∫ (ω : Ω n), X n i ω ∂μ n = 0) → (∀ (n : ℕ) (i : ι n), ProbabilityTheory.IndepFun (X n i) (fun ω => ∑ j ∈ Finset.univ \\ N n i, X n j ω) (μ n)) → (∀ (n : ℕ), ∫ (ω : Ω n), Causalean.SteinMethod.depSum (X n) ω ^ 2 ∂μ n = 1) → Filter.Tendsto (fun n => ProbabilityTheory.variance (fun ω => ∑ i, X n i ω * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω) (μ n)) Filter.atTop (nhds 0) → Filter.Tendsto (fun n => ∑ i, ∫ (ω : Ω n), |X n i ω| * Causalean.SteinMethod.nbhdSum (X n) (N n) i ω ^ 2 ∂μ n) Filter.atTop (nhds 0) → ∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → Filter.Tendsto (fun n => ∫ (ω : Ω n), h (Causalean.SteinMethod.depSum (X n) ω) ∂μ n) Filter.atTop (nhds (Causalean.SteinMethod.gExpect h))","l":115,"f":"Causalean/Mathlib/Probability/SteinMethod/CLT.lean"},{"n":"Causalean.SteinMethod.stein_local_dependence_bound","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.DependencyCLT","a":"Mathlib/Probability/SteinMethod","t":1,"d":"The local-dependence Stein bound. For jointly measurable summands X i uniformly bounded in absolute value by a nonnegative constant B, each mean zero and independent of the sum of the summands outside its neighborhood set N i, with the standardized sum having unit variance, and for a test function h that is bounded in absolute value by C, has derivative bounded in absolute value by L, and is differentiable, the deviation of the expected test-function value of the local-dependence sum from its standard-normal expectation is at most 2L·√(Var(∑ᵢ Xᵢ·nbhdSumᵢ)) + L·∑ᵢ E[|Xᵢ|·nbhdSumᵢ²].","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [inst_2 : Fintype ι] [inst_3 : DecidableEq ι] (X : ι → Ω → ℝ) (N : ι → Finset ι), (∀ (i : ι), Measurable (X i)) → ∀ {B : ℝ}, 0 ≤ B → (∀ (i : ι) (ω : Ω), |X i ω| ≤ B) → (∀ (i : ι), ∫ (ω : Ω), X i ω ∂μ = 0) → (∀ (i : ι), ProbabilityTheory.IndepFun (X i) (fun ω => ∑ j ∈ Finset.univ \\ N i, X j ω) μ) → ∫ (ω : Ω), Causalean.SteinMethod.depSum X ω ^ 2 ∂μ = 1 → ∀ (h : ℝ → ℝ) {C L : ℝ}, (∀ (x : ℝ), |h x| ≤ C) → (∀ (x : ℝ), |deriv h x| ≤ L) → Differentiable ℝ h → |∫ (ω : Ω), h (Causalean.SteinMethod.depSum X ω) ∂μ - Causalean.SteinMethod.gExpect h| ≤ 2 * L * √(ProbabilityTheory.variance (fun ω => ∑ i, X i ω * Causalean.SteinMethod.nbhdSum X N i ω) μ) + L * ∑ i, ∫ (ω : Ω), |X i ω| * Causalean.SteinMethod.nbhdSum X N i ω ^ 2 ∂μ","l":161,"f":"Causalean/Mathlib/Probability/SteinMethod/DependencyCLT.lean"},{"n":"Causalean.SteinMethod.x_mul_phi_integrable","k":"theorem","m":"Causalean.Mathlib.Probability.SteinMethod.Bounds","a":"Mathlib/Probability/SteinMethod","t":2,"d":"The map x ↦ x·φ(x) is integrable over the real line.","s":"MeasureTheory.Integrable (fun x => x * Causalean.SteinMethod.phi x) MeasureTheory.volume","l":132,"f":"Causalean/Mathlib/Probability/SteinMethod/Bounds.lean"},{"n":"Causalean.ValuesOn","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given an underlying collection of nodes, a finite node set, and a family of value spaces, one for each node, a value assignment over that finite node set gives each node in the set one value from its associated value space.","s":"{M : Type u_1} → Finset M → (M → Type u_2) → Type (max u_2 u_1)","l":30,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.aeEq_restrict_of_indicator_aeEq","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"Recover a μ.restrict s a.e.-equality from a global equality of s-indicators.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : Set Ω}, MeasurableSet s → ∀ {β : Type u_2} [inst : Zero β] {f g : Ω → β}, s.indicator f =ᵐ[μ] s.indicator g → f =ᵐ[μ.restrict s] g","l":57,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_eq_of_ae_eq_restrict_arm","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Single-arm a.e.-equality lift for σ-measurable functions. For a sub-σ-algebra m such that m is coarser than the ambient σ-algebra, if f and g agree on an m-measurable set, they agree almost everywhere on the restriction of the measure to a set E, and the conditional probability of E given m is positive almost everywhere — every m-measurable set overlapping E only on a null set is itself null, then f and g agree almost everywhere on the whole space.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω}, ∀ m ≤ mΩ, ∀ {μ : MeasureTheory.Measure Ω} {E : Set Ω} {β : Type u_2} {f g : Ω → β}, MeasurableSet {ω | f ω = g ω} → f =ᵐ[μ.restrict E] g → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ E) = 0 → μ s = 0) → f =ᵐ[μ] g","l":85,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_le_YofA_of_ae_le_Y","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Y(a) inherits Y's a.e. upper bound under latent exchangeability and consistency. For a sub-σ-algebra σ_UX coarser than the ambient σ-algebra, a measurable treatment map A, a measurable potential outcome Ya, and a treatment value a, if Ya is conditionally independent of A given σ_UX — latent exchangeability, the observed outcome Y agrees with Ya almost everywhere on the restriction of the measure to the event {A = a} — consistency, the conditional probability of {A = a} given σ_UX is positive almost everywhere — overlap, and Y is at most M almost everywhere, then Ya is at most M almost everywhere as well.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {σ_UX : MeasurableSpace Ω} (_hσUX : σ_UX ≤ mΩ) {α : Type u_2} [inst_2 : MeasurableSpace α] [MeasurableSingletonClass α] {A : Ω → α} {Y Ya : Ω → ℝ}, Measurable A → Measurable Ya → ∀ (a : α), ProbabilityTheory.CondIndepFun σ_UX _hσUX Ya A μ → Y =ᵐ[μ.restrict {ω | A ω = a}] Ya → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ {ω | A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : Ω) ∂μ, Y ω ≤ M) → ∀ᵐ (ω : Ω) ∂μ, Ya ω ≤ M","l":125,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_le_YofA_of_ae_le_Y_below","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":1,"d":"Y(a) inherits Y's a.e. lower bound under latent exchangeability and consistency. For a sub-σ-algebra σ_UX coarser than the ambient σ-algebra, a measurable treatment map A, a measurable potential outcome Ya, and a treatment value a, if Ya is conditionally independent of A given σ_UX — latent exchangeability, the observed outcome Y agrees with Ya almost everywhere on the restriction of the measure to the event {A = a} — consistency, the conditional probability of {A = a} given σ_UX is positive almost everywhere — overlap, and Y is at least M almost everywhere, then Ya is at least M almost everywhere as well.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {σ_UX : MeasurableSpace Ω} (_hσUX : σ_UX ≤ mΩ) {α : Type u_2} [inst_2 : MeasurableSpace α] [MeasurableSingletonClass α] {A : Ω → α} {Y Ya : Ω → ℝ}, Measurable A → Measurable Ya → ∀ (a : α), ProbabilityTheory.CondIndepFun σ_UX _hσUX Ya A μ → Y =ᵐ[μ.restrict {ω | A ω = a}] Ya → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ {ω | A ω = a}) = 0 → μ s = 0) → ∀ {M : ℝ}, (∀ᵐ (ω : Ω) ∂μ, M ≤ Y ω) → ∀ᵐ (ω : Ω) ∂μ, M ≤ Ya ω","l":221,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.ae_pos_condExp_indicator_of_le","k":"theorem","m":"Causalean.Mathlib.CondIndep.Integrability","a":"Mathlib/CondIndep","t":1,"d":"σ-projection of indicator-conditional positivity. For a sub-σ-algebra m₁ coarser than the ambient σ-algebra, a measurable event E, and an overlap condition — every m₁-measurable set that meets E only on a null set is itself null, then the conditional probability of E given m₁ is strictly positive almost everywhere.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m₁ : MeasurableSpace Ω}, m₁ ≤ mΩ → ∀ {E : Set Ω}, MeasurableSet E → (∀ (s : Set Ω), MeasurableSet s → μ (s ∩ E) = 0 → μ s = 0) → ∀ᵐ (ω : Ω) ∂μ, 0 < μ[E.indicator fun x => 1 | m₁] ω","l":36,"f":"Causalean/Mathlib/CondIndep/Integrability.lean"},{"n":"Causalean.comap_eq_bot_of_subsingleton","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The comap σ-algebra of a function into a Subsingleton codomain is the trivial σ-algebra ⊥. Every measurable set in the codomain is either empty or the full space, so every preimage is either ∅ or univ.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace β] [Subsingleton β] (f : α → β), MeasurableSpace.comap f inferInstance = ⊥","l":412,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.comap_valuesProjection_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"For value assignments over a finite node set, if a finite node set W is a subset of a larger finite node set I, then the σ-algebra pulled back, via the coordinate restriction to W, from the measurable space on value assignments over W is a sub-σ-algebra of the ambient measurable space on value assignments over I.","s":"∀ {M : Type u_1} {I W : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hW : W ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection hW) inferInstance ≤ inferInstance","l":62,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.comap_valuesProjection_mono","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The comap σ-algebra is monotone in the projected set: if A ⊆ B ⊆ I, then σ(π_A) ≤ σ(π_B) on ValuesOn I.","s":"∀ {M : Type u_1} {I A B : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hAB : A ⊆ B) (hBI : B ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection (LE.le.trans hAB hBI)) inferInstance ≤ MeasurableSpace.comap (Causalean.valuesProjection hBI) inferInstance","l":38,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.comap_valuesProjection_union_eq_sup","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The σ-algebra generated by the projection to A ∪ B is the supremum of the σ-algebras generated by the separate projections to A and to B.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hA hB)) inferInstance = MeasurableSpace.comap (Causalean.valuesProjection hA) inferInstance ⊔ MeasurableSpace.comap (Causalean.valuesProjection hB) inferInstance","l":95,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.condDistrib_comp_right_measurableEquiv","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":2,"d":"Push-forward invariance of condDistrib under a measurable equivalence of the conditioning variable.","s":"∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {γ : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] [inst_1 : Nonempty Ω] {mγ : MeasurableSpace γ} (μ : MeasureTheory.Measure γ) [inst_2 : MeasureTheory.IsFiniteMeasure μ] {Y : γ → Ω} {X : γ → α} (e : α ≃ᵐ β), Measurable Y → Measurable X → ∀ᵐ (x : α) ∂MeasureTheory.Measure.map X μ, (ProbabilityTheory.condDistrib Y (⇑e ∘ X) μ) (e x) = (ProbabilityTheory.condDistrib Y X μ) x","l":93,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.condDistrib_map_comp","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Reparameterization of condDistrib through a pushforward. For a measurable map φ from the sample space Ω to Ω', a measurable outcome map g, and a measurable conditioning map f, the conditional distribution of g given f, computed under the pushforward of μ by φ, agrees almost everywhere on the f-marginal with the conditional distribution of the pullbacks g ∘ φ given f ∘ φ, computed under μ directly.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} {𝒳 : Type u_3} {𝒴 : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] [inst_2 : MeasurableSpace 𝒳] [inst_3 : MeasurableSpace 𝒴] [inst_4 : StandardBorelSpace 𝒴] [inst_5 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_6 : MeasureTheory.IsFiniteMeasure μ] {φ : Ω → Ω'} {g : Ω' → 𝒴} {f : Ω' → 𝒳}, Measurable φ → Measurable g → Measurable f → ⇑(ProbabilityTheory.condDistrib g f (MeasureTheory.Measure.map φ μ)) =ᵐ[MeasureTheory.Measure.map f (MeasureTheory.Measure.map φ μ)] ⇑(ProbabilityTheory.condDistrib (g ∘ φ) (f ∘ φ) μ)","l":262,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.condExpInd_preimage_of_map_eq","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"With a measurable source-to-target map, its pushforward identity, a target conditioning σ-algebra contained in the ambient σ-algebra, and a measurable target event, the event's conditional probability commutes almost everywhere with pullback along the map.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] (r : Ω → Ω'), Measurable r → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → ∀ m ≤ mΩ', ∀ {t : Set Ω'}, MeasurableSet t → μ[(r ⁻¹' t).indicator fun x => 1 | MeasurableSpace.comap r m] =ᵐ[μ] μ'[t.indicator fun x => 1 | m] ∘ r","l":108,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condExp_comp_of_map_eq","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"With a measurable source-to-target map, its pushforward identity, a target conditioning σ-algebra contained in the ambient σ-algebra, and an integrable target outcome, conditional expectation commutes almost everywhere with pullback along the map.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} {E : Type u_3} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] (r : Ω → Ω'), Measurable r → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → ∀ m ≤ mΩ', ∀ {f : Ω' → E}, MeasureTheory.Integrable f μ' → μ[f ∘ r | MeasurableSpace.comap r m] =ᵐ[μ] μ'[f | m] ∘ r","l":45,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condExp_indicator_aeEq_zero","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"If an integrable function vanishes after restriction by an m-measurable indicator, so does its conditional expectation.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {m : MeasurableSpace Ω} {s : Set Ω}, MeasurableSet s → ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E}, MeasureTheory.Integrable f μ → s.indicator f =ᵐ[μ] 0 → s.indicator μ[f | m] =ᵐ[μ] 0","l":70,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.condExp_mul_of_condIndep","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Conditional-independence factorization of a product's conditional expectation. For a sub-σ-algebra m coarser than the ambient σ-algebra, measurable maps f and g that are conditionally independent given m, and measurable real-valued functions u and v such that u ∘ f, v ∘ g, and their pointwise product are all integrable, then the conditional expectation, given m, of the product (u ∘ f)·(v ∘ g) equals the product of the separate conditional expectations of u ∘ f and v ∘ g given m.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm f g μ → ∀ {u : α → ℝ} {v : β → ℝ}, Measurable u → Measurable v → MeasureTheory.Integrable (fun ω => u (f ω)) μ → MeasureTheory.Integrable (fun ω => v (g ω)) μ → MeasureTheory.Integrable (fun ω => u (f ω) * v (g ω)) μ → μ[fun ω => u (f ω) * v (g ω) | m] =ᵐ[μ] μ[fun ω => u (f ω) | m] * μ[fun ω => v (g ω) | m]","l":436,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condExp_sup_comap_eq_of_condIndep","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Drop-of-conditioning for conditional expectation under conditional independence.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {h : β → ℝ}, Measurable h → MeasureTheory.Integrable (fun ω => h (g ω)) μ → μ[fun ω => h (g ω) | m ⊔ MeasurableSpace.comap f inferInstance] =ᵐ[μ] μ[fun ω => h (g ω) | m]","l":406,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condExp_sup_comap_eq_of_condIndep_comap","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"If g is conditionally independent of f given m, conditioning a σ(g)-measurable integrable real function on m ⊔ σ(f) is the same as conditioning it on m, up to μ-a.e. equality.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {Y : Ω → ℝ}, Measurable Y → MeasureTheory.Integrable Y μ → μ[Y | m ⊔ MeasurableSpace.comap f inferInstance] =ᵐ[μ] μ[Y | m]","l":312,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_bot_of_indepFun","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Bridge: if μ is a probability measure on a standard Borel space Ω, then unconditional IndepFun f g μ lifts to CondIndepFun ⊥ bot_le f g μ — conditional independence given the trivial σ-algebra.","s":"∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] {β : Type u_2} {γ : Type u_3} [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] {f : Ω → β} {g : Ω → γ}, Measurable f → Measurable g → ∀ {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsProbabilityMeasure μ], ProbabilityTheory.IndepFun f g μ → ProbabilityTheory.CondIndepFun ⊥ bot_le f g μ","l":190,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_comp_aeEquiv_iff","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":1,"d":"Let the source and target be standard Borel spaces carrying finite measures. With measurable maps in both directions, their two pushforward identities, an almost-everywhere right-inverse identity, three target random variables, and their measurability, conditional independence of the first two variables given the third is equivalent to conditional independence of their pullbacks given the pulled-back third variable.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} {𝒳 : Type u_3} {𝒴 : Type u_4} {𝒵 : Type u_5} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] [mΩ' : MeasurableSpace Ω'] [inst_1 : StandardBorelSpace Ω'] [inst_2 : MeasurableSpace 𝒳] [inst_3 : MeasurableSpace 𝒴] [inst_4 : MeasurableSpace 𝒵] (r : Ω → Ω') (s : Ω' → Ω) (hr : Measurable r), Measurable s → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [inst_5 : MeasureTheory.IsFiniteMeasure μ] [inst_6 : MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → MeasureTheory.Measure.map s μ' = μ → r ∘ s =ᵐ[μ'] id → ∀ (X : Ω' → 𝒳) (Y : Ω' → 𝒴) (Z : Ω' → 𝒵), Measurable X → Measurable Y → ∀ (hZ : Measurable Z), ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Z ∘ r) inferInstance) (Measurable.comap_le (Measurable.comp hZ hr)) (X ∘ r) (Y ∘ r) μ ↔ ProbabilityTheory.CondIndepFun (MeasurableSpace.comap Z inferInstance) (Measurable.comap_le hZ) X Y μ'","l":278,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condIndepFun_contraction_of_prodMk","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Contraction for conditional independence via a product right side. For a sub-σ-algebra m coarser than the ambient σ-algebra and measurable maps X, Y, and W, if X is conditionally independent of Y given the enlarged σ-algebra m joined with the σ-algebra generated by W and X is conditionally independent of W given m, then X is conditionally independent of the pair (Y, W) given m.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : Ω → α} {Y : Ω → β} {W : Ω → γ}, Measurable X → Measurable Y → ∀ (hW : Measurable W), ProbabilityTheory.CondIndepFun (m ⊔ MeasurableSpace.comap W inferInstance) (sup_le hm (Measurable.comap_le hW)) X Y μ → ProbabilityTheory.CondIndepFun m hm X W μ → ProbabilityTheory.CondIndepFun m hm X (fun ω => (Y ω, W ω)) μ","l":799,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_of_indepFun_indep","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"If X and Y are independent, and the joint (X, Y) is independent of the conditioning variable Z, then X and Y are conditionally independent given the σ-algebra generated by Z.","s":"∀ {Ω : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSpace γ] [inst_4 : MeasurableSpace δ] {μ : MeasureTheory.Measure Ω} [inst_5 : MeasureTheory.IsProbabilityMeasure μ] {X : Ω → β} {Y : Ω → γ} {Z : Ω → δ}, Measurable X → Measurable Y → ∀ (_hZ : Measurable Z), ProbabilityTheory.IndepFun X Y μ → ProbabilityTheory.Indep (MeasurableSpace.comap (fun ω => (X ω, Y ω)) inferInstance) (MeasurableSpace.comap Z inferInstance) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap Z inferInstance) (Measurable.comap_le _hZ) X Y μ","l":458,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_pi_cond_residual_of_disjoint","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"For a finite index set, a family of standard Borel probability spaces indexed by it, and finite subsets S0, T0, U of the index set, if S0 and T0 are disjoint from each other and S0 is disjoint from U, then under the product probability measure, the pair consisting of the U-coordinates and the S0-coordinates is conditionally independent of the pair consisting of the U-coordinates and the T0-coordinates, given the σ-algebra generated by the U-coordinates.","s":"∀ {ι : Type u_1} [inst : Finite ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] [inst_2 : ∀ (i : ι), StandardBorelSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [inst_3 : ∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S0 T0 U : Finset ι}, Disjoint S0 T0 → Disjoint S0 U → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.finsetCoordProj U) inferInstance) (Measurable.comap_le (Causalean.measurable_finsetCoordProj U)) (fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj S0 x)) (fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj T0 x)) (MeasureTheory.Measure.pi μ)","l":222,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_pi_of_inter_subset","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Under a finite product probability measure, if the overlap S ∩ T is contained in the conditioning block U, then the coordinate tuple on S is conditionally independent of the coordinate tuple on T, given the coordinate tuple on U.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] [inst_1 : Finite ι] {Ω : ι → Type u_2} [inst_2 : (i : ι) → MeasurableSpace (Ω i)] [inst_3 : ∀ (i : ι), StandardBorelSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [inst_4 : ∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S T U : Finset ι}, S ∩ T ⊆ U → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.finsetCoordProj U) inferInstance) (Measurable.comap_le (Causalean.measurable_finsetCoordProj U)) (Causalean.finsetCoordProj S) (Causalean.finsetCoordProj T) (MeasureTheory.Measure.pi μ)","l":349,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.condIndepFun_prodMk_of_measurable_left","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Extension of CondIndepFun by an m-measurable function.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {W : Ω → α} {X : Ω → β} {Z : Ω → γ}, Measurable W → Measurable Z → Measurable X → ProbabilityTheory.CondIndepFun m hm W Z μ → ProbabilityTheory.CondIndepFun m hm (fun ω => (W ω, X ω)) Z μ","l":666,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndepFun_threeBlock_of_density_factors","k":"theorem","m":"Causalean.Mathlib.CondIndep.ThreeBlockDensity","a":"Mathlib/CondIndep","t":1,"d":"On a product of three standard Borel spaces, take three σ-finite coordinate reference measures and a measurable joint density with respect to their product whose induced measure is finite. Suppose there are first and second block factors, both measurable, such that almost everywhere the density equals the first factor evaluated at the first and third coordinates times the second factor evaluated at the second and third coordinates. Then, under the finite density measure, the first and second coordinate maps are conditionally independent given the third coordinate.","s":"∀ {Y : Type uY} {Z : Type uZ} {C : Type uC} [inst : MeasurableSpace Y] [inst_1 : MeasurableSpace Z] [inst_2 : MeasurableSpace C] [inst_3 : StandardBorelSpace Y] [inst_4 : StandardBorelSpace Z] [inst_5 : StandardBorelSpace C] (muY : MeasureTheory.Measure Y) (muZ : MeasureTheory.Measure Z) (muC : MeasureTheory.Measure C) [MeasureTheory.SigmaFinite muY] [MeasureTheory.SigmaFinite muZ] [MeasureTheory.SigmaFinite muC] {d : Y × Z × C → ENNReal}, Measurable d → ∀ [inst_9 : MeasureTheory.IsFiniteMeasure ((muY.prod (muZ.prod muC)).withDensity d)] (a : Y × C → ENNReal) (b : Z × C → ENNReal), Measurable a → Measurable b → (d =ᵐ[muY.prod (muZ.prod muC)] fun q => a (q.1, q.2.2) * b (q.2.1, q.2.2)) → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (fun q => q.2.2) inferInstance) (Measurable.comap_le (Measurable.comp measurable_snd measurable_snd)) (fun q => q.1) (fun q => q.2.1) ((muY.prod (muZ.prod muC)).withDensity d)","l":24,"f":"Causalean/Mathlib/CondIndep/ThreeBlockDensity.lean"},{"n":"Causalean.condIndepFun_weak_union_of_prodMk","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":1,"d":"Weak union for conditional independence. For a sub-σ-algebra m coarser than the ambient σ-algebra and measurable maps W, V, and A, if W is conditionally independent of the pair (V, A) given m, then W is conditionally independent of V given the enlarged σ-algebra m joined with the σ-algebra generated by A.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst_3 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {W : Ω → α} {V : Ω → β} {A : Ω → γ}, Measurable W → Measurable V → ∀ (hA : Measurable A), ProbabilityTheory.CondIndepFun m hm W (fun ω => (V ω, A ω)) μ → ProbabilityTheory.CondIndepFun (m ⊔ MeasurableSpace.comap A inferInstance) (sup_le hm (Measurable.comap_le hA)) W V μ","l":536,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.condIndep_comap_aeEquiv_iff","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"Let the source and target be standard Borel spaces carrying finite measures. With measurable maps in both directions, their two pushforward identities, an almost-everywhere right-inverse identity, and three target σ-algebras contained in the target ambient σ-algebra, target conditional independence is equivalent to conditional independence of the three pullback σ-algebras.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] [mΩ' : MeasurableSpace Ω'] [inst_1 : StandardBorelSpace Ω'] (r : Ω → Ω') (s : Ω' → Ω) (hr : Measurable r), Measurable s → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [inst_2 : MeasureTheory.IsFiniteMeasure μ] [inst_3 : MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → MeasureTheory.Measure.map s μ' = μ → r ∘ s =ᵐ[μ'] id → ∀ (mX mY mZ : MeasurableSpace Ω'), mX ≤ mΩ' → mY ≤ mΩ' → ∀ (hZ : mZ ≤ mΩ'), ProbabilityTheory.CondIndep (MeasurableSpace.comap r mZ) (MeasurableSpace.comap r mX) (MeasurableSpace.comap r mY) (LE.le.trans (MeasurableSpace.comap_mono hZ) (Measurable.comap_le hr)) μ ↔ ProbabilityTheory.CondIndep mZ mX mY hZ μ'","l":246,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condIndep_comap_of_map_eq","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"With a measurable source-to-target map, its pushforward identity, three target σ-algebras contained in the target ambient σ-algebra, conditional independence on the target pulls back to conditional independence on the source.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] [mΩ' : MeasurableSpace Ω'] [inst_1 : StandardBorelSpace Ω'] (r : Ω → Ω') (hr : Measurable r) {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [inst_2 : MeasureTheory.IsFiniteMeasure μ] [inst_3 : MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → ∀ (mX mY mZ : MeasurableSpace Ω'), mX ≤ mΩ' → mY ≤ mΩ' → ∀ (hZ : mZ ≤ mΩ'), ProbabilityTheory.CondIndep mZ mX mY hZ μ' → ProbabilityTheory.CondIndep (MeasurableSpace.comap r mZ) (MeasurableSpace.comap r mX) (MeasurableSpace.comap r mY) (LE.le.trans (MeasurableSpace.comap_mono hZ) (Measurable.comap_le hr)) μ","l":133,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condIndep_of_comap_aeRetraction","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"With measurable maps in both directions, their two pushforward identities, an almost-everywhere right inverse, and three target σ-algebras contained in the target ambient σ-algebra, conditional independence of the three pullback σ-algebras implies target conditional independence.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} [mΩ : MeasurableSpace Ω] [inst : StandardBorelSpace Ω] [mΩ' : MeasurableSpace Ω'] [inst_1 : StandardBorelSpace Ω'] (r : Ω → Ω') (s : Ω' → Ω) (hr : Measurable r), Measurable s → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [inst_2 : MeasureTheory.IsFiniteMeasure μ] [inst_3 : MeasureTheory.IsFiniteMeasure μ'], MeasureTheory.Measure.map r μ = μ' → MeasureTheory.Measure.map s μ' = μ → r ∘ s =ᵐ[μ'] id → ∀ (mX mY mZ : MeasurableSpace Ω'), mX ≤ mΩ' → mY ≤ mΩ' → ∀ (hZ : mZ ≤ mΩ'), ProbabilityTheory.CondIndep (MeasurableSpace.comap r mZ) (MeasurableSpace.comap r mX) (MeasurableSpace.comap r mY) (LE.le.trans (MeasurableSpace.comap_mono hZ) (Measurable.comap_le hr)) μ → ProbabilityTheory.CondIndep mZ mX mY hZ μ'","l":194,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.condIndep_valuesProjection_contraction","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":1,"d":"Contraction for coordinate projections. Let X, Y, W, Z be subsets of a common finite index set I, with X, Y, W, and Z all contained in I, and let μ be a finite measure on the resulting finite-product value space indexed by I. If X is conditionally independent of Y given the union of Z and W and X is conditionally independent of W given Z, then X is conditionally independent of the union of Y and W given Z, where independence is always of the corresponding coordinate projections under μ.","s":"∀ {M : Type uM} [inst : DecidableEq M] {I X Y W Z : Finset M} {Ω : M → Type uΩ} [inst_1 : (n : M) → MeasurableSpace (Ω n)] [inst_2 : StandardBorelSpace (Causalean.ValuesOn I Ω)] (hX : X ⊆ I) (hY : Y ⊆ I) (hW : W ⊆ I) (hZ : Z ⊆ I) {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω)} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (Finset.union_subset hZ hW)) inferInstance) (Causalean.comap_valuesProjection_le (Finset.union_subset hZ hW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hY) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection hW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZ) inferInstance) (Causalean.comap_valuesProjection_le hZ) (Causalean.valuesProjection hX) (Causalean.valuesProjection (Finset.union_subset hY hW)) μ","l":206,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.condIndep_valuesProjection_weak_union","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":1,"d":"Weak union for coordinate projections. Let X, Y, W, Z be subsets of a finite index set I, with X, the union of Y and W, and the union of Z and W all contained in I, and let μ be a finite measure on the finite-product value space indexed by I. If X is conditionally independent of the union of Y and W given Z, then X is conditionally independent of Y given the union of Z and W, where independence is always of the corresponding coordinate projections under μ.","s":"∀ {M : Type uM} [inst : DecidableEq M] {I X Y W Z : Finset M} {Ω : M → Type uΩ} [inst_1 : (n : M) → MeasurableSpace (Ω n)] [inst_2 : StandardBorelSpace (Causalean.ValuesOn I Ω)] (hX : X ⊆ I) (hYW : Y ∪ W ⊆ I) (hZW : Z ∪ W ⊆ I) {μ : MeasureTheory.Measure (Causalean.ValuesOn I Ω)} [inst_3 : MeasureTheory.IsFiniteMeasure μ], ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hZW)) inferInstance) (Causalean.comap_valuesProjection_le (LE.le.trans Finset.subset_union_left hZW)) (Causalean.valuesProjection hX) (Causalean.valuesProjection hYW) μ → ProbabilityTheory.CondIndepFun (MeasurableSpace.comap (Causalean.valuesProjection hZW) inferInstance) (Causalean.comap_valuesProjection_le hZW) (Causalean.valuesProjection hX) (Causalean.valuesProjection (LE.le.trans Finset.subset_union_left hYW)) μ","l":131,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.edgeDiff","k":"def","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex set with decidable equality and two directed acyclic graphs on that vertex set, the directed-edge difference is the finite set of all ordered pairs of vertices that form a directed edge in the first graph but not in the second graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → Finset (V × V)","l":56,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.edgeDiffCount","k":"def","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"For a finite vertex set with decidable equality and two directed acyclic graphs on that vertex set, the directed-edge difference count is the number of directed edges in the first graph that are absent from the second graph.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → Causalean.DAG V → Causalean.DAG V → ℕ","l":62,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.edgeDiffCount_flipEdge_lt","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"Flipping a covered, oppositely oriented edge strictly decreases the edge-difference count (it fixes exactly the pair (a,b) and changes nothing else).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V} {a b : V} (hcov : G₁.IsCoveredEdge a b), G₂.edge b a → Causalean.edgeDiffCount (Causalean.DAG.flipEdge hcov) G₂ < Causalean.edgeDiffCount G₁ G₂","l":182,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.eventuallyEq_of_comp_aeRetraction","k":"theorem","m":"Causalean.Mathlib.CondIndep.DomainTransport.AeRetraction","a":"Mathlib/CondIndep/DomainTransport","t":2,"d":"With a source-to-target map, a measurable return map, the return-map pushforward identity, an almost-everywhere right-inverse identity, and an equality after pullback, the target functions agree almost everywhere.","s":"∀ {Ω : Type u_1} {Ω' : Type u_2} {A : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] (r : Ω → Ω') (s : Ω' → Ω), Measurable s → ∀ {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'}, MeasureTheory.Measure.map s μ' = μ → r ∘ s =ᵐ[μ'] id → ∀ {f g : Ω' → A}, f ∘ r =ᵐ[μ] g ∘ r → f =ᵐ[μ'] g","l":23,"f":"Causalean/Mathlib/CondIndep/DomainTransport/AeRetraction.lean"},{"n":"Causalean.exists_covered_reversed_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"AMP Lemma 3.2 (existence). For DAGs G₁ and G₂ on the same vertex set, suppose G₁ and G₂ have the same skeleton (the same underlying undirected adjacency), the same immoralities (unshielded colliders), and there is a pair a₀, b₀ with an edge a₀ → b₀ in G₁ that appears reversed as b₀ → a₀ in G₂. Then there is a *covered* edge a → b in G₁ that is likewise reversed to b → a in G₂. The proof chooses a head of a reversed edge that is minimal in the first graph's topological order, then a tail into that head that is maximal among reversed tails; the skeleton and immorality hypotheses force this edge to have the same non-tail parents at both endpoints.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ {a₀ b₀ : V}, G₁.edge a₀ b₀ → G₂.edge b₀ a₀ → ∃ a b, G₁.edge a b ∧ G₂.edge b a ∧ G₁.IsCoveredEdge a b","l":83,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.exists_half_net_card_le_five_pow_finrank","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":1,"d":"For a finite-dimensional real normed space E, there exists a finite set of unit vectors, of size at most 5 raised to the dimension of E, that forms a half-net of the unit sphere: every unit vector lies within distance 1/2 of some point in the set.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E], ∃ N, (∀ v ∈ N, ‖v‖ = 1) ∧ (∀ (x : E), ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) ∧ N.card ≤ 5 ^ Module.finrank ℝ E","l":160,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.exists_internal_net_card_le","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":2,"d":"Every subset of the closed unit ball of a finite-dimensional real normed space admits an internal net at any prescribed accuracy v: finitely many points, all taken from the set itself, such that every point of the set is within distance v of one of them, and the number of net points is at most (1 + 2 / v) raised to the dimension of the space.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] (A : Set E), (∀ x ∈ A, ‖x‖ ≤ 1) → ∀ {v : ℝ}, 0 < v → ∃ N, (∀ y ∈ N, y ∈ A) ∧ (∀ x ∈ A, ∃ y ∈ N, ‖x - y‖ ≤ v) ∧ ↑N.card ≤ (1 + 2 / v) ^ Module.finrank ℝ E","l":29,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.exists_unit_ball_net_card_le","k":"theorem","m":"Causalean.Stat.Concentration.FiniteDimensionalNet","a":"Stat/Concentration","t":2,"d":"The closed unit ball of a finite-dimensional real normed space has, at every accuracy v, a finite net of at most (1 + 2 / v) raised to the dimension of the space many points.","s":"∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {v : ℝ}, 0 < v → ∃ N, (∀ (x : E), ‖x‖ ≤ 1 → ∃ y ∈ N, ‖x - y‖ ≤ v) ∧ ↑N.card ≤ (1 + 2 / v) ^ Module.finrank ℝ E","l":148,"f":"Causalean/Stat/Concentration/FiniteDimensionalNet.lean"},{"n":"Causalean.finsetCoordProj","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Given an index collection, a family of coordinate value spaces, and a finite coordinate block, the coordinate-block projection maps a full coordinate assignment to the assignment obtained by retaining exactly the coordinates in that block.","s":"{ι : Type u_1} → {Ω : ι → Type u_2} → (S : Finset ι) → ((i : ι) → Ω i) → (i : ↥S) → Ω ↑i","l":88,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjFromCondResidual","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Given an index collection whose equality is decidable, a family of coordinate value spaces, a target coordinate block and a conditioning block, the residual-reassembly map reconstructs an assignment on the target block from an assignment on the conditioning block and an assignment on those target coordinates outside that block.","s":"{ι : Type u_1} → [inst : DecidableEq ι] → {Ω : ι → Type u_2} → (S U : Finset ι) → ((i : ↥U) → Ω ↑i) × ((i : ↥(S \\ U)) → Ω ↑i) → (i : ↥S) → Ω ↑i","l":107,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjFromCondResidual_comp","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Reassembling π_U with the residual projection π_{S \\ U} gives π_S.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} (S U : Finset ι), (Causalean.finsetCoordProjFromCondResidual S U ∘ fun x => (Causalean.finsetCoordProj U x, Causalean.finsetCoordProj (S \\ U) x)) = Causalean.finsetCoordProj S","l":138,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjPairFromUnion","k":"def","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":1,"d":"Given an index collection whose equality is decidable, a family of coordinate value spaces, and finite coordinate blocks, the union-sub-block extraction map maps an assignment on their union to the pair of its restrictions to the first and second blocks.","s":"{ι : Type u_1} → [inst : DecidableEq ι] → {Ω : ι → Type u_2} → (A B : Finset ι) → ((i : ↥(A ∪ B)) → Ω ↑i) → ((i : ↥A) → Ω ↑i) × ((i : ↥B) → Ω ↑i)","l":149,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.finsetCoordProjPairFromUnion_comp","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Projecting to A ∪ B and then extracting the two sub-blocks gives the pair of direct projections to A and B.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} (A B : Finset ι), Causalean.finsetCoordProjPairFromUnion A B ∘ Causalean.finsetCoordProj (A ∪ B) = fun x => (Causalean.finsetCoordProj A x, Causalean.finsetCoordProj B x)","l":178,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.half_norm_lt_inner_of_mem_net","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":2,"d":"Every nonzero vector in a subspace has a point in a finite half-net whose inner product with that vector is at least half of its norm.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {N : Finset E} {z : E} (V : Submodule ℝ E), (∀ x ∈ V, ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) → z ∈ V → z ≠ 0 → ∃ v ∈ N, ‖z‖ / 2 ≤ inner ℝ v z","l":23,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.indepFun_of_map","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Pushforward bridge for IndepFun: if X ∘ φ ⟂ᵢ Y ∘ φ under ν, then X ⟂ᵢ Y under ν.map φ. Analogue of LocalMarkov.condIndepFun_of_map at the IndepFun level.","s":"∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] {ν : MeasureTheory.Measure α} {φ : α → β}, AEMeasurable φ ν → ∀ {X : β → γ}, Measurable X → ∀ {Y : β → δ}, Measurable Y → ProbabilityTheory.IndepFun (X ∘ φ) (Y ∘ φ) ν → ProbabilityTheory.IndepFun X Y (MeasureTheory.Measure.map φ ν)","l":435,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.indepFun_pi_of_disjoint","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"Coordinate-tuple projections at disjoint Finsets are IndepFun under Measure.pi of a family of probability measures. This is the binary aggregation of iIndepFun_pi via iIndepFun.indepFun_finset.","s":"∀ {ι : Type u_1} [inst : Finite ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (μ : (i : ι) → MeasureTheory.Measure (Ω i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] {S T : Finset ι}, Disjoint S T → ProbabilityTheory.IndepFun (fun x i => x ↑i) (fun x i => x ↑i) (MeasureTheory.Measure.pi μ)","l":71,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.indicator_aeEq_of_aeEq_restrict","k":"theorem","m":"Causalean.Mathlib.CondIndep.AELift","a":"Mathlib/CondIndep","t":2,"d":"Push an a.e.-equality under μ.restrict s to a global equality of s-indicators.","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : Set Ω}, MeasurableSet s → ∀ {β : Type u_2} [inst : Zero β] {f g : Ω → β}, f =ᵐ[μ.restrict s] g → s.indicator f =ᵐ[μ] s.indicator g","l":41,"f":"Causalean/Mathlib/CondIndep/AELift.lean"},{"n":"Causalean.initialSWIG","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a finite directed acyclic graph, the initial single-world intervention graph is its split-node graph with no intervention targets, so every original edge joins random copies and every fixed copy is isolated.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Causalean.DAG (Causalean.SWIGNode N)","l":343,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.initialSWIG_fixed_isolated","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G and any node n, the fixed copy of n has no parents in the initial SWIG of G (the SWIG with no intervention targets).","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (n : N), (Causalean.initialSWIG G).parents (Causalean.SWIGNode.fixed n) = ∅","l":357,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.initialSWIG_random_edge","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G and any nodes u, v, in the initial SWIG of G (the SWIG with no intervention targets), the random copies of u and v are joined by an edge exactly when u and v are joined by an edge in G.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (u v : N), (Causalean.initialSWIG G).edge (Causalean.SWIGNode.random u) (Causalean.SWIGNode.random v) ↔ G.edge u v","l":350,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instDecidableEqEdgeType","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"DecidableEq Causalean.EdgeType","l":65,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instDecidableEqMonotonicityKind","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"DecidableEq Causalean.MonotonicityKind","l":43,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instDecidableEqSWIGNode","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"","s":"{N : Type u_1} → [DecidableEq N] → DecidableEq (Causalean.SWIGNode N)","l":86,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instDecidableSameImmoralities","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"For a finite vertex population with decidable equality and two directed acyclic graphs on that population, the decision procedure for the same-immoralities condition determines whether the graphs have identical v-structures for every ordered triple of vertices.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G₁ G₂ : Causalean.DAG V) → Decidable (Causalean.SameImmoralities G₁ G₂)","l":68,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.instDecidableSameSkeleton","k":"instance","m":"Causalean.Graph.MarkovEquiv.Defs","a":"Graph/MarkovEquiv","t":2,"d":"For a finite vertex population with decidable equality and two directed acyclic graphs on that population, the decision procedure for the same-skeleton condition determines whether the graphs have identical undirected adjacencies for every pair of vertices.","s":"{V : Type u_1} → [inst : DecidableEq V] → [inst_1 : Fintype V] → (G₁ G₂ : Causalean.DAG V) → Decidable (Causalean.SameSkeleton G₁ G₂)","l":64,"f":"Causalean/Graph/MarkovEquiv/Defs.lean"},{"n":"Causalean.instMeasurableSpaceSwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"For a collection of base variables, a family of base-variable value spaces, each equipped with a σ-algebra, and any split node, the σ-algebra on that node's SWIG value space is the σ-algebra of the corresponding base-variable value space.","s":"{N : Type u_1} → (Ω : N → Type u_2) → [(n : N) → MeasurableSpace (Ω n)] → (sn : Causalean.SWIGNode N) → MeasurableSpace (Causalean.swigΩ Ω sn)","l":135,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instNonemptySwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"For a collection of base variables, a family of nonempty base-variable value spaces, and any split node, the nonemptiness guarantee for that node's SWIG value space is inherited from the corresponding base-variable value space.","s":"∀ {N : Type u_1} (Ω : N → Type u_2) [∀ (n : N), Nonempty (Ω n)] (sn : Causalean.SWIGNode N), Nonempty (Causalean.swigΩ Ω sn)","l":169,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instReprEdgeType","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"Repr Causalean.EdgeType","l":65,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instReprMonotonicityKind","k":"instance","m":"Causalean.SCM.Model.EdgeType","a":"SCM/Model","t":2,"d":"","s":"Repr Causalean.MonotonicityKind","l":43,"f":"Causalean/SCM/Model/EdgeType.lean"},{"n":"Causalean.instReprSWIGNode","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"","s":"{N : Type u_1} → [Repr N] → Repr (Causalean.SWIGNode N)","l":86,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.instStandardBorelSpaceSwigΩ","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"For a collection of base variables, a family of value spaces each equipped with a σ-algebra and forming a standard Borel space, and any split node, the standard Borel-space structure on that node's SWIG value space is inherited from the corresponding base-variable value space.","s":"∀ {N : Type u_1} (Ω : N → Type u_2) [inst : (n : N) → MeasurableSpace (Ω n)] [∀ (n : N), StandardBorelSpace (Ω n)] (sn : Causalean.SWIGNode N), StandardBorelSpace (Causalean.swigΩ Ω sn)","l":162,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.integrableOn_of_condExp_indicator_mul","k":"theorem","m":"Causalean.Mathlib.CondIndep.Integrability","a":"Mathlib/CondIndep","t":1,"d":"Integrability on a stratum from a conditional-expectation indicator identity. For a sub-σ-algebra m coarser than the ambient σ-algebra, measurable events B and C, and an m-measurable real-valued function g that is almost everywhere nonnegative, if the conditional expectation of the indicator of B given m, multiplied pointwise by g, equals almost everywhere the conditional expectation of the indicator of C given m, then g is integrable on B (moreover ∫_B g dμ ≤ μ C).","s":"∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace Ω}, m ≤ mΩ → ∀ {B C : Set Ω}, MeasurableSet B → MeasurableSet C → ∀ {g : Ω → ℝ}, Measurable g → 0 ≤ᵐ[μ] g → (fun ω => μ[B.indicator fun x => 1 | m] ω * g ω) =ᵐ[μ] μ[C.indicator fun x => 1 | m] → MeasureTheory.IntegrableOn g B μ","l":103,"f":"Causalean/Mathlib/CondIndep/Integrability.lean"},{"n":"Causalean.iotaMap","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"The link map on split nodes sends every fixed copy of a base variable to its random copy and leaves every random copy at that random copy.","s":"{N : Type u_1} → Causalean.SWIGNode N → Causalean.SWIGNode N","l":267,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.iotaMap_fixed","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"The link map sends the fixed copy of a base variable to its random copy.","s":"∀ {N : Type u_1} (n : N), Causalean.iotaMap (Causalean.SWIGNode.fixed n) = Causalean.SWIGNode.random n","l":279,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.map_compProd_prodMap_left_eq_compProd_comap","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Pushforward of compProd through a measurable equivalence on the first coordinate. For a measurable equivalence e between the first-coordinate spaces, an s-finite measure ν on the source first coordinate, and an s-finite kernel κ from that coordinate to a second space, pushing the composed-product measure of ν and κ forward through e on the first coordinate (identity on the second) equals the composed-product measure of the pushed-forward ν and κ transported back along e's inverse.","s":"∀ {β : Type u_1} {β' : Type u_2} {γ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace β'] [inst_2 : MeasurableSpace γ] (ν : MeasureTheory.Measure β) [MeasureTheory.SFinite ν] (e : β ≃ᵐ β') (κ : ProbabilityTheory.Kernel β γ) [ProbabilityTheory.IsSFiniteKernel κ], MeasureTheory.Measure.map (Prod.map (⇑e) id) (ν.compProd κ) = (MeasureTheory.Measure.map (⇑e) ν).compProd (κ.comap (⇑e.symm) (MeasurableEquiv.measurable e.symm))","l":58,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.markovEquiv_iff_sameSkeleton_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv","a":"Graph/MarkovEquiv","t":1,"d":"Verma–Pearl (1990). For two DAGs G₁, G₂ on the same vertex set, they are Markov equivalent — they declare exactly the same d-separations, hence impose the same conditional-independence constraints — if and only if they have the same skeleton and the same v-structures (immoralities).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] (G₁ G₂ : Causalean.DAG V), Causalean.MarkovEquiv G₁ G₂ ↔ Causalean.SameSkeleton G₁ G₂ ∧ Causalean.SameImmoralities G₁ G₂","l":51,"f":"Causalean/Graph/MarkovEquiv.lean"},{"n":"Causalean.markovEquiv_of_sameSkeleton_sameImmoralities","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":1,"d":"Verma–Pearl hard direction (covered-edge route). For DAGs G₁ and G₂ on the same vertex set, if G₁ and G₂ have the same skeleton and the same immoralities, then G₁ and G₂ are Markov equivalent: they license exactly the same d-separation statements — proven via AMP Lemma 3.2 (covered-edge reversals), independent of the moralization/ancestral kernel.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → Causalean.MarkovEquiv G₁ G₂","l":232,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.markovEquiv_of_same_edge","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"Edge-congruence for Markov equivalence. DAGs with the same directed-edge relation are Markov equivalent (d-separation depends only on the edge relation).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₁.edge u w ↔ G₂.edge u w) → Causalean.MarkovEquiv G₁ G₂","l":40,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.measurable_finsetCoordProj","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"finsetCoordProj is measurable.","s":"∀ {ι : Type u_1} {Ω : ι → Type u_2} [inst : (i : ι) → MeasurableSpace (Ω i)] (S : Finset ι), Measurable (Causalean.finsetCoordProj S)","l":97,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_finsetCoordProjFromCondResidual","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The residual reassembly map is measurable.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (S U : Finset ι), Measurable (Causalean.finsetCoordProjFromCondResidual S U)","l":123,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_finsetCoordProjPairFromUnion","k":"theorem","m":"Causalean.Mathlib.Indep","a":"Mathlib","t":2,"d":"The union sub-block extraction map is measurable.","s":"∀ {ι : Type u_1} [inst : DecidableEq ι] {Ω : ι → Type u_2} [inst_1 : (i : ι) → MeasurableSpace (Ω i)] (A B : Finset ι), Measurable (Causalean.finsetCoordProjPairFromUnion A B)","l":162,"f":"Causalean/Mathlib/Indep.lean"},{"n":"Causalean.measurable_valuesProjection","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"For value assignments over a finite node set, if a finite node set J is a subset of a larger finite node set I, then restricting a value assignment over I to its coordinates in J is a measurable map.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hJI : J ⊆ I), Measurable (Causalean.valuesProjection hJI)","l":50,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurable_valuesProjection_union_sup","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"The projection to A ∪ B is measurable for the supremum of the separate projection σ-algebras.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), Measurable (Causalean.valuesProjection (Finset.union_subset hA hB))","l":50,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.measurable_valuesUnionMk","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Combining assignments is jointly measurable in both input assignments.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M}, Measurable fun p => Causalean.valuesUnionMk p.1 p.2","l":179,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurable_valuesUnionMk_right","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Combining assignments is measurable in the second assignment with the first held fixed.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω), Measurable fun b => Causalean.valuesUnionMk a b","l":153,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measurePreserving_valuesEquivOfEq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given two propositionally equal finite node sets I and J and a family μ of measures, one per coordinate of I, the canonical measurable equivalence between value assignments over I and over J carries the product measure Measure.pi μ to the product measure built from μ re-indexed along J through the equality.","s":"∀ {M : Type u_1} {I J : Finset M} {Ω : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω n)] (h : I = J) (μ : (i : ↥I) → MeasureTheory.Measure (Ω ↑i)), MeasureTheory.MeasurePreserving (⇑(Causalean.valuesEquivOfEq h)) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.pi fun j => μ ⟨↑j, (fun h_1 => Eq.ndrec (motive := fun {J} => ∀ (j : ↥J), ↑j ∈ J → ↑j ∈ I) (fun j h => h) h j h_1) j.property⟩)","l":94,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.measure_eq_bind_marginal_condDistrib","k":"theorem","m":"Causalean.Mathlib.CondDistrib","a":"Mathlib","t":1,"d":"Measure-theoretic chain rule / disintegration (Mathlib gap). For a finite measure μ on a product space, μ equals the composition obtained by first drawing the second coordinate from its marginal distribution and then drawing the first coordinate from its regular conditional distribution given that second coordinate.","s":"∀ {β : Type u_1} {γ : Type u_2} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [inst : StandardBorelSpace β] [inst_1 : Nonempty β] (μ : MeasureTheory.Measure (β × γ)) [inst_2 : MeasureTheory.IsFiniteMeasure μ], μ = (MeasureTheory.Measure.map Prod.snd μ).bind fun c => MeasureTheory.Measure.map (fun b => (b, c)) ((ProbabilityTheory.condDistrib Prod.fst Prod.snd μ) c)","l":169,"f":"Causalean/Mathlib/CondDistrib.lean"},{"n":"Causalean.measure_norm_gt_le_five_pow_finrank","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":1,"d":"Dimension-only sub-Gaussian norm tail. Let V be a finite-dimensional subspace of E. If Z lies in V almost surely under P and every unit vector v in V has the scalar projection ⟪v, Z⟫ sub-Gaussian with parameter 1 under P, then for any nonnegative t, the probability that ‖Z‖ exceeds 2 * t is at most 5 raised to the dimension of V, times exp (-t ^ 2 / 2).","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] (Z : Ω → E) (V : Submodule ℝ E) [FiniteDimensional ℝ ↥V], (∀ᵐ (ω : Ω) ∂P, Z ω ∈ V) → (∀ v ∈ V, ‖v‖ = 1 → ProbabilityTheory.HasSubgaussianMGF (fun ω => inner ℝ v (Z ω)) 1 P) → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 2 * t < ‖Z ω‖} ≤ ↑(5 ^ Module.finrank ℝ ↥V) * Real.exp (-t ^ 2 / 2)","l":87,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.measure_norm_gt_le_of_half_net","k":"theorem","m":"Causalean.Stat.Concentration.SubGaussianNorm","a":"Stat/Concentration","t":2,"d":"If a finite half-net controls every scalar projection with unit sub-Gaussian parameter, then the norm has a Gaussian upper tail. The support condition is only required almost everywhere.","s":"∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] (Z : Ω → E) (V : Submodule ℝ E) (N : Finset E), (∀ x ∈ V, ‖x‖ = 1 → ∃ v ∈ N, ‖x - v‖ ≤ 1 / 2) → (∀ᵐ (ω : Ω) ∂P, Z ω ∈ V) → (∀ v ∈ N, ProbabilityTheory.HasSubgaussianMGF (fun ω => inner ℝ v (Z ω)) 1 P) → ∀ {t : ℝ}, 0 ≤ t → P.real {ω | 2 * t < ‖Z ω‖} ≤ ↑N.card * Real.exp (-t ^ 2 / 2)","l":52,"f":"Causalean/Stat/Concentration/SubGaussianNorm.lean"},{"n":"Causalean.moralAdj_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral adjacency is a skeleton + v-structure invariant. Two DAGs with the same skeleton and the same v-structures induce the same moral adjacency on any ground set: a shielded pair is moral-adjacent via the shared skeleton edge, and an unshielded married pair is exactly the apex of a shared immorality.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S : Finset V) (u v : V), G₁.MoralAdj S u v ↔ G₂.MoralAdj S u v","l":523,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.moralConn_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral connectivity agrees across DAGs with the same skeleton and v-structures, for a fixed ground set S. (The ancestral sets used by MoralSep still differ between the graphs; that reconciliation is moralSep_congr.)","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S Z : Finset V) (u v : V), G₁.MoralConn S Z u v ↔ G₂.MoralConn S Z u v","l":561,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.moralStep_congr","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Moralization","a":"Graph/MarkovEquiv","t":2,"d":"Moral steps agree across DAGs with the same skeleton and v-structures (same ground set, same conditioning set).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.SameSkeleton G₁ G₂ → Causalean.SameImmoralities G₁ G₂ → ∀ (S Z : Finset V) (u v : V), G₁.MoralStep S Z u v ↔ G₂.MoralStep S Z u v","l":553,"f":"Causalean/Graph/MarkovEquiv/Moralization.lean"},{"n":"Causalean.sameImmoralities_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence ⇒ same v-structures.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameImmoralities G₁ G₂","l":351,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.sameSkeleton_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":2,"d":"Markov equivalence ⇒ same skeleton.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameSkeleton G₁ G₂","l":334,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.sameSkeleton_sameImmoralities_of_markovEquiv","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Readoff","a":"Graph/MarkovEquiv","t":1,"d":"Easy direction of Verma–Pearl. For two DAGs G₁, G₂, if they are Markov equivalent then they share the same skeleton and the same v-structures (immoralities).","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, Causalean.MarkovEquiv G₁ G₂ → Causalean.SameSkeleton G₁ G₂ ∧ Causalean.SameImmoralities G₁ G₂","l":384,"f":"Causalean/Graph/MarkovEquiv/Readoff.lean"},{"n":"Causalean.same_edge_of_edgeDiff_empty","k":"theorem","m":"Causalean.Graph.MarkovEquiv.Decompose","a":"Graph/MarkovEquiv","t":2,"d":"With a one-way skeleton inclusion, an empty edge-difference forces equal edge relations.","s":"∀ {V : Type u_1} [inst : DecidableEq V] [inst_1 : Fintype V] {G₁ G₂ : Causalean.DAG V}, (∀ (u w : V), G₂.UAdj u w → G₁.UAdj u w) → Causalean.edgeDiff G₁ G₂ = ∅ → ∀ (u w : V), G₁.edge u w ↔ G₂.edge u w","l":67,"f":"Causalean/Graph/MarkovEquiv/Decompose.lean"},{"n":"Causalean.setIntegral_condExp_indep_indicator","k":"theorem","m":"Causalean.Mathlib.CondIndep.CondExp","a":"Mathlib/CondIndep","t":2,"d":"Under conditional independence of two variables given a σ-algebra, integrating the conditional expectation of a constant times an event indicator over the intersection of a conditioning event and an event determined by one variable equals integrating that indicator directly.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} {m mΩ : MeasurableSpace Ω} (hm : m ≤ mΩ) [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : StandardBorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_3 : MeasureTheory.IsFiniteMeasure μ] {f : Ω → α} {g : Ω → β}, Measurable f → Measurable g → ProbabilityTheory.CondIndepFun m hm g f μ → ∀ {S A F : Set Ω}, MeasurableSet S → MeasurableSet A → MeasurableSet F → ∀ (c : ℝ), ∫ (x : Ω) in A ∩ F, μ[S.indicator fun x => c | m] x ∂μ = ∫ (x : Ω) in A ∩ F, S.indicator (fun x => c) x ∂μ","l":183,"f":"Causalean/Mathlib/CondIndep/CondExp.lean"},{"n":"Causalean.setIntegral_eq_setIntegral_mul_of_likelihoodRatio_swap","k":"theorem","m":"Causalean.Mathlib.LikelihoodRatioSwap","a":"Mathlib","t":1,"d":"Arm-swap via likelihood ratio. For a sub-σ-algebra m of the ambient σ-algebra on Ω, a finite measure μ, and measurable sets s and t, suppose the product f · L is almost-everywhere strongly measurable relative to m, the integrand f itself is almost-everywhere strongly measurable relative to m, f restricted to the source set t is integrable, the product f · L restricted to the target set s is integrable, and the likelihood ratio L satisfies the change-of-measure relation: the conditional probability of s given m, times L, agrees μ-almost everywhere with the conditional probability of t given m. Then the integral of f over t equals the integral of f · L over s: ∫ₜ f dμ = ∫ₛ f · L dμ.","s":"∀ {Ω : Type u_1} {m mΩ : MeasurableSpace Ω}, m ≤ mΩ → ∀ {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (s t : Set Ω), MeasurableSet s → MeasurableSet t → ∀ {L f : Ω → ℝ}, MeasureTheory.AEStronglyMeasurable (fun ω => f ω * L ω) μ → MeasureTheory.AEStronglyMeasurable f μ → MeasureTheory.IntegrableOn f t μ → MeasureTheory.IntegrableOn (fun ω => f ω * L ω) s μ → (fun ω => μ[s.indicator fun x => 1 | m] ω * L ω) =ᵐ[μ] μ[t.indicator fun x => 1 | m] → ∫ (ω : Ω) in t, f ω ∂μ = ∫ (ω : Ω) in s, f ω * L ω ∂μ","l":22,"f":"Causalean/Mathlib/LikelihoodRatioSwap.lean"},{"n":"Causalean.swigDAG","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a finite directed acyclic graph and a set of intervention targets, the single-world intervention graph as a directed acyclic graph is obtained by replacing every outgoing edge of a targeted variable by an edge from its fixed copy while retaining all incoming edges to its random copy.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Finset N → Causalean.DAG (Causalean.SWIGNode N)","l":250,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigEdge","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a finite directed acyclic graph and a set of intervention targets, the single-world-intervention edge relation declares that a random copy points to a random copy precisely when the corresponding original edge starts outside the targets, that a fixed copy points to a random copy precisely when its base variable is targeted and has the corresponding original outgoing edge, and that every other ordered pair has no edge.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Finset N → Causalean.SWIGNode N → Causalean.SWIGNode N → Prop","l":181,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigEdge_decidable","k":"instance","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"For a finite collection of base variables with decidable equality, a directed acyclic graph on those variables, and a finite set of intervention targets, the decision procedure for the single-world-intervention edge relation determines, for every ordered pair of split nodes, whether the pair is joined by a SWIG edge.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → (G : Causalean.DAG N) → (targets : Finset N) → DecidableRel (Causalean.swigEdge G targets)","l":199,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigTopo","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a finite directed acyclic graph, the interleaved topological order of its split nodes assigns each random copy one plus twice its base variable's topological position and each fixed copy twice that position.","s":"{N : Type u_1} → [inst : DecidableEq N] → [inst_1 : Fintype N] → Causalean.DAG N → Causalean.SWIGNode N → ℕ","l":209,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigTopo_lt","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"Every SWIG edge points from a lower to a higher position in the interleaved topological order.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (u v : Causalean.SWIGNode N), Causalean.swigEdge G targets u v → Causalean.swigTopo G u < Causalean.swigTopo G v","l":223,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_fixed_are_roots","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G, any set of intervention targets, and any node n, the fixed copy of n has no parents in the single-world intervention graph built from G and targets.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (n : N), (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.fixed n) = ∅","l":286,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_random_root_of_root","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":2,"d":"If n is a root in G, then random n is a root in the SWIG.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (n : N), G.parents n = ∅ → (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.random n) = ∅","l":321,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swig_target_parents","k":"theorem","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For any base DAG G, any set of intervention targets, and any node d, the parents of the random copy of d in the single-world intervention graph are exactly the copies of d's original parents in G, each represented by its random version if it is not a target and by its fixed version if it is.","s":"∀ {N : Type u_1} [inst : DecidableEq N] [inst_1 : Fintype N] (G : Causalean.DAG N) (targets : Finset N) (d : N) (x : Causalean.SWIGNode N), x ∈ (Causalean.swigDAG G targets).parents (Causalean.SWIGNode.random d) ↔ ∃ p, G.edge p d ∧ x = Causalean.SWIGNode.random p ∧ p ∉ targets ∨ G.edge p d ∧ x = Causalean.SWIGNode.fixed p ∧ p ∈ targets","l":296,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.swigΩ","k":"def","m":"Causalean.Graph.SWIG","a":"Graph","t":1,"d":"For a family of value spaces indexed by base variables, the single-world-intervention value-space family assigns to each random copy its base variable's value space and to each fixed copy that same base variable's value space.","s":"{N : Type u_1} → (N → Type u_2) → Causalean.SWIGNode N → Type u_2","l":123,"f":"Causalean/Graph/SWIG.lean"},{"n":"Causalean.valuesEquivOfEq","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given an underlying collection of nodes, finite node sets, a family of measurable value spaces, one for each node, and an equality of the two node sets, the canonical measurable equivalence identifies assignments over the first set with assignments over the second set by retaining the corresponding coordinate values.","s":"{M : Type u_1} → {I J : Finset M} → {Ω : M → Type u_2} → [inst : (n : M) → MeasurableSpace (Ω n)] → I = J → Causalean.ValuesOn I Ω ≃ᵐ Causalean.ValuesOn J Ω","l":76,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesProjection","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given an underlying collection of nodes, finite node sets, a family of measurable value spaces, one for each node, and evidence that the second node set is contained in the first, the coordinate-restriction map sends each assignment on the first set to its values on the second set.","s":"{M : Type u_1} → {I J : Finset M} → {Ω : M → Type u_2} → [(n : M) → MeasurableSpace (Ω n)] → J ⊆ I → Causalean.ValuesOn I Ω → Causalean.ValuesOn J Ω","l":37,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesProjection_comp","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"Composition identity for valuesProjection: restricting to K ⊆ J ⊆ I in one step equals composing the two restrictions.","s":"∀ {M : Type u_1} {I J K : Finset M} {Ω' : M → Type u_2} [inst : (n : M) → MeasurableSpace (Ω' n)] (hKJ : K ⊆ J) (hJI : J ⊆ I), Causalean.valuesProjection (LE.le.trans hKJ hJI) = Causalean.valuesProjection hKJ ∘ Causalean.valuesProjection hJI","l":28,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"Causalean.valuesUnionMk","k":"def","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":1,"d":"Given an underlying collection of nodes whose equality is decidable, a family of measurable value spaces, one for each node, finite node sets, and value assignments on the first and second sets, the combined assignment on their union uses the first assignment at nodes it contains and otherwise uses the second assignment.","s":"{M : Type u_1} → [inst : DecidableEq M] → {Ω : M → Type u_2} → [(n : M) → MeasurableSpace (Ω n)] → {A B : Finset M} → Causalean.ValuesOn A Ω → Causalean.ValuesOn B Ω → Causalean.ValuesOn (A ∪ B) Ω","l":120,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_apply_left","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Projecting a union assignment to a coordinate from the first input returns that value.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω) {v : M} (hA : v ∈ A), Causalean.valuesUnionMk a b ⟨v, Finset.mem_union_left B hA⟩ = a ⟨v, hA⟩","l":135,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_apply_right","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.FinsetValues","a":"Mathlib/MeasureTheory","t":2,"d":"Projecting a union assignment outside the first input returns the second input's value.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {Ω : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω n)] {A B : Finset M} (a : Causalean.ValuesOn A Ω) (b : Causalean.ValuesOn B Ω) {v : M} (hv : v ∈ A ∪ B) (hA : v ∉ A), Causalean.valuesUnionMk a b ⟨v, hv⟩ = b ⟨v, Or.resolve_left (Finset.mem_union.mp hv) hA⟩","l":144,"f":"Causalean/Mathlib/MeasureTheory/FinsetValues.lean"},{"n":"Causalean.valuesUnionMk_projection_comp","k":"theorem","m":"Causalean.SCM.Do.ValuesProjectionCI","a":"SCM/Do","t":2,"d":"Projecting to A and B and then assembling the union block recovers the direct projection to A ∪ B.","s":"∀ {M : Type u_1} [inst : DecidableEq M] {I A B : Finset M} {Ω' : M → Type u_2} [inst_1 : (n : M) → MeasurableSpace (Ω' n)] (hA : A ⊆ I) (hB : B ⊆ I), (fun ξ => Causalean.valuesUnionMk (Causalean.valuesProjection hA ξ) (Causalean.valuesProjection hB ξ)) = Causalean.valuesProjection (Finset.union_subset hA hB)","l":116,"f":"Causalean/SCM/Do/ValuesProjectionCI.lean"},{"n":"LinearMap.IsPositive.apply_eigBasis","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Applying the operator to an eigenbasis vector scales that vector by its indexed eigenvalue.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), T (hT.eigBasis i) = hT.eigVals i • hT.eigBasis i","l":43,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.eigBasis","k":"def","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"An orthonormal eigenbasis of T, obtained from the symmetry of a positive operator.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → {T : E →ₗ[ℝ] E} → T.IsPositive → OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E","l":35,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.eigVals","k":"def","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"The real eigenvalues of T, indexed compatibly with eigBasis.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → {T : E →ₗ[ℝ] E} → T.IsPositive → Fin (Module.finrank ℝ E) → ℝ","l":39,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.eigVals_nonneg","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Eigenvalues of a positive operator are nonnegative.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), 0 ≤ hT.eigVals i","l":50,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt","k":"def","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"The positive square root of T: the linear operator that acts as √(eigenvalue) on each vector of the chosen orthonormal eigenbasis.","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → {T : E →ₗ[ℝ] E} → T.IsPositive → E →ₗ[ℝ] E","l":60,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM","k":"def","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"posSqrt packaged as a continuous linear map (finite-dimensional domain).","s":"{E : Type u_1} → [inst : NormedAddCommGroup E] → [inst_1 : InnerProductSpace ℝ E] → [FiniteDimensional ℝ E] → {T : E →ₗ[ℝ] E} → T.IsPositive → E →L[ℝ] E","l":110,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_adjoint","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Adjoint form of self-adjointness.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), ContinuousLinearMap.adjoint hT.posSqrtCLM = hT.posSqrtCLM","l":123,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_apply","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The continuous-linear-map packaging of the positive square root has the same pointwise action as the linear-map square root.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (x : E), hT.posSqrtCLM x = hT.posSqrt x","l":113,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_comp_self","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"Composing the continuous positive square root of T with itself recovers T, packaged as a continuous linear map.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrtCLM ∘SL hT.posSqrtCLM = LinearMap.toContinuousLinearMap T","l":128,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrtCLM_isSelfAdjoint","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The continuous square root is self-adjoint.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), IsSelfAdjoint hT.posSqrtCLM","l":118,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_apply_eigBasis","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"The positive square root sends each eigenbasis vector to the same vector scaled by the square root of its eigenvalue.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (i : Fin (Module.finrank ℝ E)), hT.posSqrt (hT.eigBasis i) = √(hT.eigVals i) • hT.eigBasis i","l":65,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_inner","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"Inner-product form of posSqrt: a symmetric weighted sum over the eigenbasis.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive) (x y : E), inner ℝ (hT.posSqrt x) y = ∑ i, √(hT.eigVals i) * inner ℝ (hT.eigBasis i) x * inner ℝ (hT.eigBasis i) y","l":78,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_isPositive","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"posSqrt is itself a positive operator.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt.IsPositive","l":101,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_isSymmetric","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":2,"d":"posSqrt is symmetric.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt.IsSymmetric","l":94,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"LinearMap.IsPositive.posSqrt_mul_self","k":"theorem","m":"Causalean.Mathlib.OperatorSqrt","a":"Mathlib","t":1,"d":"Composing the positive square root of T with itself recovers T.","s":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {T : E →ₗ[ℝ] E} (hT : T.IsPositive), hT.posSqrt ∘ₗ hT.posSqrt = T","l":71,"f":"Causalean/Mathlib/OperatorSqrt.lean"},{"n":"MeasureTheory.AnalyticSet.isCapacitable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"Choquet's capacitability theorem. On a Polish sample space, if a set s is analytic — the continuous image of a Polish space, or empty, then s is capacitable for every Choquet capacity c: the capacity of s equals the supremum, over the compact subsets of s, of their capacities.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [PolishSpace Ω] {c : MeasureTheory.ChoquetCapacity Ω} {s : Set Ω}, MeasureTheory.AnalyticSet s → c.IsCapacitable s","l":208,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.AnalyticSet.nullMeasurableSet","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UniversalMeasurability","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"On a Polish sample space equipped with its Borel σ-algebra, if a set s is analytic, then s is null-measurable for every finite Borel measure μ: it becomes measurable after completing μ, so it differs from an ordinary measurable event only on a μ-null set.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {s : Set Ω}, MeasureTheory.AnalyticSet s → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], MeasureTheory.NullMeasurableSet s μ","l":71,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UniversalMeasurability.lean"},{"n":"MeasureTheory.ChoquetCapacity","k":"structure","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"A Choquet capacity on a topological sample space bundles an extended-nonnegative-real-valued set function that is monotone under set inclusion, continuous from below along increasing sequences of sets, finite on every compact set, and right-continuous at compact sets: any strict upper bound on a compact set's value is already an upper bound on some open neighborhood of that compact set.","s":"(Ω : Type u_2) → [TopologicalSpace Ω] → Type u_2","l":36,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.IsCapacitable","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"For every topological sample space, Choquet capacity on that space, and subset of the sample space, the property of being capacitable holds exactly when the capacity of the subset equals the supremum of the capacities of all compact subsets contained in it.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → MeasureTheory.ChoquetCapacity Ω → Set Ω → Prop","l":77,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.IsCapacitable.nullMeasurableSet","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UniversalMeasurability","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A set capacitable for the outer measure of a finite measure is null-measurable for that measure.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [inst_2 : PolishSpace Ω] [inst_3 : BorelSpace Ω] {μ : MeasureTheory.Measure Ω} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {s : Set Ω}, μ.toChoquetCapacity.IsCapacitable s → MeasureTheory.NullMeasurableSet s μ","l":28,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UniversalMeasurability.lean"},{"n":"MeasureTheory.ChoquetCapacity.exists_isOpen_superset_lt","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A strict upper bound for the capacity of a compact set also bounds some open neighborhood.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {K : Set Ω}, IsCompact K → ∀ {a : ENNReal}, c.toFun K < a → ∃ U, K ⊆ U ∧ IsOpen U ∧ c.toFun U < a","l":72,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.isCompact_lt_top","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Capacity values on compact sets are finite.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {K : Set Ω}, IsCompact K → c.toFun K < ⊤","l":67,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ChoquetCapacity.mono","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Capacity values are monotone under inclusion.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] (c : MeasureTheory.ChoquetCapacity Ω) {s t : Set Ω}, s ⊆ t → c.toFun s ≤ c.toFun t","l":63,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.Measure.eq_of_rnDeriv_eq","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.EqOfRnDerivEq","a":"Mathlib/MeasureTheory","t":1,"d":"A measure is determined by its Radon–Nikodym derivative against a fixed reference. If the measure μ is absolutely continuous with respect to a σ-finite reference measure ρ, the measure ν is likewise absolutely continuous with respect to ρ, and their Radon–Nikodym densities dμ/dρ and dν/dρ agree ρ-almost everywhere, then μ and ν are equal as measures.","s":"∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν ρ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ρ], μ.AbsolutelyContinuous ρ → ν.AbsolutelyContinuous ρ → μ.rnDeriv ρ =ᵐ[ρ] ν.rnDeriv ρ → μ = ν","l":26,"f":"Causalean/Mathlib/MeasureTheory/EqOfRnDerivEq.lean"},{"n":"MeasureTheory.Measure.toChoquetCapacity","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"For every Polish topological sample space equipped with its Borel σ-algebra and finite measure on that space, the Choquet capacity induced by the measure assigns each subset the measure’s value on that subset.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → [PolishSpace Ω] → [inst_2 : MeasurableSpace Ω] → [BorelSpace Ω] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → MeasureTheory.ChoquetCapacity Ω","l":300,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.Measure.toChoquetCapacity_apply","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"The capacity induced by a finite measure assigns every event exactly that measure's outer mass.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : PolishSpace Ω] [inst_2 : MeasurableSpace Ω] [inst_3 : BorelSpace Ω] (μ : MeasureTheory.Measure Ω) [inst_4 : MeasureTheory.IsFiniteMeasure μ] (s : Set Ω), μ.toChoquetCapacity.toFun s = μ s","l":311,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.ProbabilityMeasure.map_mul_eq_map_prod_dirac","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"Multiplying a deterministic real value and a random draw has the same distribution as scaling that random draw by the deterministic value.","s":"∀ (c : ℝ) (ν : MeasureTheory.ProbabilityMeasure ℝ), ((MeasureTheory.diracProba c).prod ν).map (Measurable.aemeasurable (Measurable.fun_mul (Measurable.fst measurable_id') (Measurable.snd measurable_id'))) = ν.map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))","l":84,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.ProbabilityMeasure.tendsto_map_mul_of_tendsto","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":1,"d":"Measure-level deterministic-scalar Slutsky theorem for weak convergence. For a filter l along which a family of probability measures on the reals converges weakly to a limit measure ν and a family of real scalars converges to a limit a₀, then the pushforwards of the measures by scalar multiplication x ↦ aᵢ·x converge weakly to the pushforward of ν by x ↦ a₀·x.","s":"∀ {ι : Type u_3} {l : Filter ι} {νs : ι → MeasureTheory.ProbabilityMeasure ℝ} {ν : MeasureTheory.ProbabilityMeasure ℝ} {a : ι → ℝ} {a₀ : ℝ}, Filter.Tendsto νs l (nhds ν) → Filter.Tendsto a l (nhds a₀) → Filter.Tendsto (fun i => (νs i).map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))) l (nhds (ν.map (Measurable.aemeasurable (Measurable.mul measurable_const measurable_id))))","l":102,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.TendstoInDistribution.const_mul_of_tendsto_const","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":1,"d":"Deterministic-scalar Slutsky theorem for random variables. If a sequence of random variables X n converges in distribution to Z, all under the same probability measure μ and a sequence of deterministic real scalars a n converges to a limit a₀, then the scaled sequence a n · X n converges in distribution to a₀ · Z.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [l.IsCountablyGenerated] {X : ι → Ω → ℝ} {Z : Ω → ℝ} {a : ι → ℝ} {a₀ : ℝ}, MeasureTheory.TendstoInDistribution X l Z (fun x => μ) μ → Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInDistribution (fun n ω => a n * X n ω) l (fun ω => a₀ * Z ω) (fun x => μ) μ","l":62,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.UpperSemianalytic","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"An extended-nonnegative-valued function on a topological sample space is upper-semi-analytic exactly when, for every extended-nonnegative threshold $a$, the set of sample points at which its value is strictly greater than $a$ is analytic.","s":"{Ω : Type u_1} → [TopologicalSpace Ω] → (Ω → ENNReal) → Prop","l":126,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.lintegral_completion_eq_outerLIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"On a Polish sample space equipped with its Borel σ-algebra and a finite measure μ, if f is upper-semi-analytic — every strict superlevel set {ω | a < f ω} is analytic, then the lower Lebesgue integral of f against the completion of μ equals the outer integral of f with respect to μ, i.e. the infimum of the lower integrals of all measurable pointwise majorants of f.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion = MeasureTheory.outerLIntegral μ f","l":151,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.measurable_completion","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"An upper-semi-analytic extended-nonnegative loss is measurable once a finite Borel sampling measure is completed, so it is available to ordinary completed-measure integration.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], Measurable f","l":144,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.UpperSemianalytic.nullMeasurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"An upper-semi-analytic extended-nonnegative function is null-measurable for every finite Borel measure.","s":"∀ {Ω : Type u_1} [inst : TopologicalSpace Ω] [inst_1 : MeasurableSpace Ω] [PolishSpace Ω] [BorelSpace Ω] {f : Ω → ENNReal}, MeasureTheory.UpperSemianalytic f → ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], MeasureTheory.NullMeasurable f μ","l":134,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.compProd_eq_prod_withDensity_fiber","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.RnDerivCompProdSigmaFinite","a":"Mathlib/MeasureTheory","t":2,"d":"A composition product is the product reference measure weighted by the fibrewise Radon–Nikodym derivative whenever the fibres are almost everywhere dominated by the reference measure and that derivative is almost everywhere measurable.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) [MeasureTheory.SFinite μ] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsSFiniteKernel κ], (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (μ.prod ρ) → μ.compProd κ = (μ.prod ρ).withDensity fun p => (κ p.1).rnDeriv ρ p.2","l":47,"f":"Causalean/Mathlib/MeasureTheory/RnDerivCompProdSigmaFinite.lean"},{"n":"MeasureTheory.condExp_add'","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpLinearity","a":"Mathlib/MeasureTheory","t":1,"d":"Conditional expectation is additive: the conditional expectation of a sum of two integrable functions is the sum of their conditional expectations, almost everywhere.","s":"∀ {α : Type u_1} {E : Type u_2} {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (m : MeasurableSpace α), autoParam (MeasureTheory.Integrable f μ) MeasureTheory.condExp_add'._auto_1 → autoParam (MeasureTheory.Integrable g μ) MeasureTheory.condExp_add'._auto_3 → μ[f + g | m] =ᵐ[μ] μ[f | m] + μ[g | m]","l":35,"f":"Causalean/Mathlib/MeasureTheory/CondExpLinearity.lean"},{"n":"MeasureTheory.condExp_finsetSum'","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpLinearity","a":"Mathlib/MeasureTheory","t":2,"d":"Conditional expectation commutes with a finite sum: conditioning a finite sum of integrable functions is the same, almost everywhere, as summing their conditional expectations.","s":"∀ {α : Type u_1} {E : Type u_2} {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} {s : Finset ι} {F : ι → α → E} (m : MeasurableSpace α), autoParam (∀ i ∈ s, MeasureTheory.Integrable (F i) μ) MeasureTheory.condExp_finsetSum'._auto_1 → μ[∑ i ∈ s, F i | m] =ᵐ[μ] ∑ i ∈ s, μ[F i | m]","l":57,"f":"Causalean/Mathlib/MeasureTheory/CondExpLinearity.lean"},{"n":"MeasureTheory.condExp_setIndicator_condExp_of_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpIndicatorTower","a":"Mathlib/MeasureTheory","t":1,"d":"Conditioning a masked outcome equals conditioning its inner regression. Given a tower of σ-algebras m ≤ m' ≤ m0 on the sample space, with the trim of the measure μ to m' σ-finite, a set s measurable with respect to the finer σ-algebra m', and an integrable function f, the conditional expectation given m of the masked outcome 1_s · f agrees μ-almost everywhere with the conditional expectation given m of f — first replaced by its conditional expectation given m', then masked by s.","s":"∀ {Ω : Type u_1} {m m' m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E], m ≤ m' → ∀ (hm' : m' ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm')] {s : Set Ω}, MeasurableSet s → ∀ {f : Ω → E}, MeasureTheory.Integrable f μ → μ[s.indicator f | m] =ᵐ[μ] μ[s.indicator μ[f | m'] | m]","l":39,"f":"Causalean/Mathlib/MeasureTheory/CondExpIndicatorTower.lean"},{"n":"MeasureTheory.condExp_sub'","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.CondExpLinearity","a":"Mathlib/MeasureTheory","t":2,"d":"Conditional expectation respects differences: the conditional expectation of the difference of two integrable functions is the difference of their conditional expectations, almost everywhere.","s":"∀ {α : Type u_1} {E : Type u_2} {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (m : MeasurableSpace α), autoParam (MeasureTheory.Integrable f μ) MeasureTheory.condExp_sub'._auto_1 → autoParam (MeasureTheory.Integrable g μ) MeasureTheory.condExp_sub'._auto_3 → μ[f - g | m] =ᵐ[μ] μ[f | m] - μ[g | m]","l":47,"f":"Causalean/Mathlib/MeasureTheory/CondExpLinearity.lean"},{"n":"MeasureTheory.instCoeFunChoquetCapacityForallSetENNReal","k":"instance","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.Capacity","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"For every topological sample space, the coercion from a Choquet capacity on that space to an extended-nonnegative-valued set function is given by the capacity's underlying set function.","s":"{Ω : Type u_1} → [inst : TopologicalSpace Ω] → CoeFun (MeasureTheory.ChoquetCapacity Ω) fun x => Set Ω → ENNReal","l":57,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/Capacity.lean"},{"n":"MeasureTheory.lintegral_completion_eq_of_measurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A measurable extended-nonnegative function has the same integral before and after completion of the measure.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) {f : Ω → ENNReal}, Measurable f → ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion = ∫⁻ (ω : Ω), f ω ∂μ","l":99,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_lintegral_completion","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Completing a measure can only increase the lower integral of an arbitrary extended-nonnegative function.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (f : Ω → ENNReal), ∫⁻ (ω : Ω), f ω ∂μ ≤ ∫⁻ (ω : MeasureTheory.NullMeasurableSpace Ω μ), f ω ∂μ.completion","l":67,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_outerLIntegral","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"The lower Lebesgue integral is bounded by the outer integral.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (f : Ω → ENNReal), ∫⁻ (ω : Ω), f ω ∂μ ≤ MeasureTheory.outerLIntegral μ f","l":44,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.lintegral_le_outerLIntegral_of_measurable_le","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"A pointwise lower bound may be integrated before comparison with an outer integral, without measurability of either function.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : Ω → ENNReal}, f ≤ g → ∫⁻ (ω : Ω), f ω ∂μ ≤ MeasureTheory.outerLIntegral μ g","l":51,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral","k":"def","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":1,"d":"The outer integral of an extended-nonnegative-valued function on a measurable sample space, under a measure on that sample space, is the infimum of the lower Lebesgue integrals of all measurable functions that dominate the given function pointwise.","s":"{Ω : Type u_1} → [inst : MeasurableSpace Ω] → MeasureTheory.Measure Ω → (Ω → ENNReal) → ENNReal","l":31,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral_eq_lintegral_of_measurable","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Outer integration agrees with Lebesgue integration for measurable functions.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f : Ω → ENNReal}, Measurable f → MeasureTheory.outerLIntegral μ f = ∫⁻ (ω : Ω), f ω ∂μ","l":58,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.outerLIntegral_mono","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.AnalyticSetUniversalMeasurability.UpperSemianalytic","a":"Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability","t":2,"d":"Outer integration is monotone in its extended-nonnegative integrand.","s":"∀ {Ω : Type u_1} [TopologicalSpace Ω] [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {f g : Ω → ENNReal}, f ≤ g → MeasureTheory.outerLIntegral μ f ≤ MeasureTheory.outerLIntegral μ g","l":37,"f":"Causalean/Mathlib/MeasureTheory/AnalyticSetUniversalMeasurability/UpperSemianalytic.lean"},{"n":"MeasureTheory.restrict_range_eq_of_forall_image","k":"theorem","m":"Causalean.Mathlib.MeasurableEmbeddingExtras","a":"Mathlib","t":1,"d":"If F is a measurable embedding of one measurable space into another and two measures μ and ν on the codomain assign the same mass to the image F '' A of every measurable set A in the domain, then restricting μ and ν to the range of F yields identical measures.","s":"∀ {α : Type u_1} {β : Type u_2} {x : MeasurableSpace α} {x_1 : MeasurableSpace β} {F : α → β}, MeasurableEmbedding F → ∀ (μ ν : MeasureTheory.Measure β), (∀ (A : Set α), MeasurableSet A → μ (F '' A) = ν (F '' A)) → μ.restrict (Set.range F) = ν.restrict (Set.range F)","l":24,"f":"Causalean/Mathlib/MeasurableEmbeddingExtras.lean"},{"n":"MeasureTheory.rnDeriv_compProd_prod_sigmaFinite","k":"theorem","m":"Causalean.Mathlib.MeasureTheory.RnDerivCompProdSigmaFinite","a":"Mathlib/MeasureTheory","t":1,"d":"σ-finite-reference Radon–Nikodym derivative of a composition-product. Fix an s-finite measure μ, a σ-finite base reference ν, a σ-finite fibre reference ρ, and an s-finite kernel κ from the base space to the fibre space, together with a candidate base density f. If μ is absolutely continuous with respect to ν, μ-almost every fibre κ a is absolutely continuous with respect to ρ, the raw per-slice fibre derivative (a, b) ↦ (dκ_a/dρ)(b) is almost-everywhere measurable with respect to the product reference ν × ρ, and f agrees ν-almost everywhere with the Radon–Nikodym derivative dμ/dν, then the Radon–Nikodym derivative of the composition-product μ ⊗ₘ κ with respect to the product reference ν × ρ agrees, ν × ρ-almost everywhere, with the pointwise product (a, b) ↦ f a · (dκ_a/dρ)(b) of the base density and the fibre derivative.","s":"∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] (μ ν : MeasureTheory.Measure α) (ρ : MeasureTheory.Measure β) (κ : ProbabilityTheory.Kernel α β) (f : α → ENNReal) [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ρ] [ProbabilityTheory.IsSFiniteKernel κ], μ.AbsolutelyContinuous ν → (∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous ρ) → AEMeasurable (fun p => (κ p.1).rnDeriv ρ p.2) (ν.prod ρ) → μ.rnDeriv ν =ᵐ[ν] f → (μ.compProd κ).rnDeriv (ν.prod ρ) =ᵐ[ν.prod ρ] fun p => f p.1 * (κ p.1).rnDeriv ρ p.2","l":70,"f":"Causalean/Mathlib/MeasureTheory/RnDerivCompProdSigmaFinite.lean"},{"n":"MeasureTheory.tendstoInMeasure_const_of_tendsto","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"Alias for tendstoInMeasure_const_of_tendsto_real whose name reflects that the result holds in any seminormed additive commutative group.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} {E : Type u_3} [inst_1 : SeminormedAddCommGroup E] {a : ι → E} {a₀ : E}, Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInMeasure μ (fun n x => a n) l fun x => a₀","l":54,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"MeasureTheory.tendstoInMeasure_const_of_tendsto_real","k":"theorem","m":"Causalean.Mathlib.ConvergenceInDistribution","a":"Mathlib","t":2,"d":"A deterministic sequence in a seminormed additive commutative group that converges in the usual topological sense also converges in measure when regarded as a sequence of constant random variables.","s":"∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {l : Filter ι} {E : Type u_3} [inst_1 : SeminormedAddCommGroup E] {a : ι → E} {a₀ : E}, Filter.Tendsto a l (nhds a₀) → MeasureTheory.TendstoInMeasure μ (fun n x => a n) l fun x => a₀","l":33,"f":"Causalean/Mathlib/ConvergenceInDistribution.lean"},{"n":"ProbabilityTheory.IndepFun.integral_restrict_preimage_eq_mul","k":"theorem","m":"Causalean.Mathlib.IndepIntegral","a":"Mathlib","t":1,"d":"Drop of conditioning. For random elements f and g that are independent, with f almost-everywhere measurable and g almost-everywhere measurable, given a measurable set E whose preimage f ⁻¹' E is measurable, and a function h that is almost-everywhere strongly measurable with respect to the pushforward measure of g, then restricting the integral of h ∘ g to the event f ⁻¹' E equals the probability of that event times the unrestricted integral of h ∘ g.","s":"∀ {Ω : Type u_1} {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace α] [inst_2 : MeasurableSpace β] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β}, ProbabilityTheory.IndepFun f g μ → AEMeasurable f μ → AEMeasurable g μ → ∀ {E : Set α}, MeasurableSet E → MeasurableSet (f ⁻¹' E) → ∀ {h : β → ℝ}, MeasureTheory.AEStronglyMeasurable h (MeasureTheory.Measure.map g μ) → ∫ (ω : Ω) in f ⁻¹' E, h (g ω) ∂μ = (μ (f ⁻¹' E)).toReal * ∫ (ω : Ω), h (g ω) ∂μ","l":23,"f":"Causalean/Mathlib/IndepIntegral.lean"},{"n":"ProbabilityTheory.condDistrib_map_of_condDistrib_fst_eq","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"Witness-kernel form of a conditional distribution under conditional independence. For measurable maps X, Z, and C and a jointly measurable structural-response function h, if the conditional distribution of C given the pair (X,Z) agrees almost everywhere with the conditional distribution of C given Z alone — conditional independence of C from X given Z, then the conditional distribution of the outcome h(X,Z,C) given (X,Z) agrees almost everywhere with the pushforward, by the map h x z, of the conditional distribution of C given Z.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] [inst_7 : StandardBorelSpace 𝒴] [inst_8 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_9 : MeasureTheory.IsFiniteMeasure μ] {X : Ω → 𝒳} {Z : Ω → 𝒵} {C : Ω → 𝒞} {h : 𝒳 → 𝒵 → 𝒞 → 𝒴}, Measurable X → Measurable Z → Measurable C → (Measurable fun p => h p.1.1 p.1.2 p.2) → ((fun p => (ProbabilityTheory.condDistrib C (fun ω => (X ω, Z ω)) μ) p) =ᵐ[MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ] fun p => (ProbabilityTheory.condDistrib C Z μ) p.2) → (fun p => (ProbabilityTheory.condDistrib (fun ω => h (X ω) (Z ω) (C ω)) (fun ω => (X ω, Z ω)) μ) p) =ᵐ[MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ] fun p => MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib C Z μ) p.2)","l":112,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.condDistrib_map_of_funext","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"No-treatment witness-kernel form of a conditional distribution. For measurable maps Z and C and a jointly measurable response function H, the conditional distribution of the outcome H(Z,C) given Z agrees almost everywhere with the pushforward, by the map H z, of the conditional distribution of C given Z.","s":"∀ {Ω : Type u_1} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒵] [inst_2 : MeasurableSpace 𝒞] [inst_3 : StandardBorelSpace 𝒞] [inst_4 : Nonempty 𝒞] [inst_5 : MeasurableSpace 𝒴] [inst_6 : StandardBorelSpace 𝒴] [inst_7 : Nonempty 𝒴] (μ : MeasureTheory.Measure Ω) [inst_8 : MeasureTheory.IsFiniteMeasure μ] {Z : Ω → 𝒵} {C : Ω → 𝒞} {H : 𝒵 → 𝒞 → 𝒴}, Measurable Z → Measurable C → (Measurable fun p => H p.1 p.2) → (fun z => (ProbabilityTheory.condDistrib (fun ω => H (Z ω) (C ω)) Z μ) z) =ᵐ[MeasureTheory.Measure.map Z μ] fun z => MeasureTheory.Measure.map (H z) ((ProbabilityTheory.condDistrib C Z μ) z)","l":189,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.instIsMarkovKernel_witnessKernel","k":"instance","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":2,"d":"For every sample space, treatment space, covariate space, nonempty standard Borel residual-coordinate space, and outcome space, each equipped with the stated measurable structure, finite sampling measure, covariate map, residual-coordinate map, structural response map, and joint measurability of that response map, the witness kernel is a Markov kernel.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] (μ : MeasureTheory.Measure Ω) [inst_7 : MeasureTheory.IsFiniteMeasure μ] {Z : Ω → 𝒵} {C : Ω → 𝒞} {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} (hh : Measurable fun p => h p.1.1 p.1.2 p.2), ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.witnessKernel μ hh)","l":100,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.witnessKernel","k":"def","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":1,"d":"Given a sample space equipped with a σ-algebra, a treatment space equipped with a σ-algebra, a covariate space equipped with a σ-algebra, a nonempty standard Borel residual-coordinate space, and an outcome space equipped with a σ-algebra, let the sampling measure be finite, the covariate map and the residual-coordinate map map sample points to their respective spaces, and let the structural response map map a treatment, covariate, and residual coordinate to an outcome. If this structural response is jointly measurable, then the witness kernel assigns to each treatment--covariate pair the distribution of the structural response after drawing the residual coordinate from its conditional distribution given the covariate.","s":"{Ω : Type u_1} → {𝒳 : Type u_2} → {𝒵 : Type u_3} → {𝒞 : Type u_4} → {𝒴 : Type u_5} → [inst : MeasurableSpace Ω] → [inst_1 : MeasurableSpace 𝒳] → [inst_2 : MeasurableSpace 𝒵] → [inst_3 : MeasurableSpace 𝒞] → [StandardBorelSpace 𝒞] → [Nonempty 𝒞] → [inst_6 : MeasurableSpace 𝒴] → (μ : MeasureTheory.Measure Ω) → [MeasureTheory.IsFiniteMeasure μ] → {Z : Ω → 𝒵} → {C : Ω → 𝒞} → {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} → (Measurable fun p => h p.1.1 p.1.2 p.2) → ProbabilityTheory.Kernel (𝒳 × 𝒵) 𝒴","l":45,"f":"Causalean/Mathlib/CondDistribWitness.lean"},{"n":"ProbabilityTheory.witnessKernel_apply","k":"theorem","m":"Causalean.Mathlib.CondDistribWitness","a":"Mathlib","t":2,"d":"Evaluating the witness kernel at a conditioning pair gives the residual conditional law pushed through the corresponding structural slice.","s":"∀ {Ω : Type u_1} {𝒳 : Type u_2} {𝒵 : Type u_3} {𝒞 : Type u_4} {𝒴 : Type u_5} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace 𝒳] [inst_2 : MeasurableSpace 𝒵] [inst_3 : MeasurableSpace 𝒞] [inst_4 : StandardBorelSpace 𝒞] [inst_5 : Nonempty 𝒞] [inst_6 : MeasurableSpace 𝒴] (μ : MeasureTheory.Measure Ω) [inst_7 : MeasureTheory.IsFiniteMeasure μ] (Z : Ω → 𝒵) (C : Ω → 𝒞) {h : 𝒳 → 𝒵 → 𝒞 → 𝒴} (hh : Measurable fun p => h p.1.1 p.1.2 p.2) (p : 𝒳 × 𝒵), (ProbabilityTheory.witnessKernel μ hh) p = MeasureTheory.Measure.map (h p.1 p.2) ((ProbabilityTheory.condDistrib C Z μ) p.2)","l":92,"f":"Causalean/Mathlib/CondDistribWitness.lean"}]}